TITLE: Closed form solution to recurrence: $g(n)=(k-2)g(n-1)+(k-1)g(n-2)$
QUESTION [3 upvotes]: For the number of paths with exactly $n$ hops from one node to another in a $k$ node fully connected graph, we get the following recurrence:
$$g_n=(k-2)g_{n-1}+(k-1)g_{n-2}$$
With $g_1=1$, $g_0=0$ and $g_m=0 \;\; \forall \;\; m<0$.
Is there a way to get a closed form for this recurrence? For $k=3$, this yields the Jacobsthal numbers.

For future reference, this recurrence represents the number of paths from one node to another in a $k$ node fully connected undirected graph and has the solution:
$$g(n) = \frac{(k-1)^n-(-1)^n}{(k)}$$

My attempt:
Let's try to construct the generating function.
$$B(x)=\sum\limits_{n=0}^\infty g_n x^n$$
We get:
$$B(x) = \sum\limits_{n=0}^\infty ((k-2)g_{n-1}+(k-1)g_{n-2})x^n$$
$$=>B(x) = (k-2)x\sum\limits_{n=0}^\infty g_{n-1}x^{n-1}+(k-1)x^2\sum\limits_{n=0}^\infty g_{n-2}x^{n-2}$$
$$=(k-2)x B(x)+(k-1)x^2 B(x)$$
But this just makes $B(x)$ cancel out and we don't get an expression for it.

REPLY [1 votes]: Let $g(n)=t^n$, then
$$t^2-(k-2)t-(k-1) \implies t=k-1,-1$$
Hence
$$g(n)=A (k-1)^n + B (-1)^n$$
The Proof
Let $ax^2+bx+c=0$ has two roots $p,q$. then
$$ap^2+bp+c=0~~~(1), ~~aq^2+bq+c=0~~~(2)$$
Multiply (1) and (2) by $C_1 p^n$ and $C_2 q^n$ and add them
we get $$(a C_1 p^{n+2}+ C_2 q^{n+2}))+b (C_1 p^{n+1}+ C_2 q^{n+1})+ c(C_1 p^n + C_2 q^n)=0~~~~(3)$$
Define $$U_n=C_1 p^n+ C_2 q^n~~~~(4)$$
So (3) becomes
$$aU_{n+2}+bU_{n+1}+cU_n=0~~~~~~(5)$$
Hence the solution of (5) is given by (4).


TITLE: Existence of solution to a system of linear PDEs with boundary conditions
QUESTION [2 upvotes]: I am looking to find a solution, or even just prove the existence of one, to the following system of linear PDEs. They come up in a construction I am trying to work out in symplectic geometry. Here $(r_1, \theta_1, r_2, \theta_2)$ are the polar coordinates on $\mathbb{R}^4$, and $\beta$ and $\gamma$ are smooth functions in terms of those coordinates, that I would like to solve for:
$$2\gamma + r_2\frac{\partial \gamma}{\partial r_2} = 1$$
$$\frac{\partial \beta}{\partial \theta_2}(r_1^2-1) - r_2^2\frac{\partial \gamma}{\partial \theta_1} = 0$$
$$2\beta r_1 + \frac{\partial \beta}{\partial r_1}(r_1^2-1) = r_1$$,
and I need the conditions that $\beta = \gamma = 1$ when $r_1^2 \geq \delta > 0$, and $\beta = r_1$ when $r_1 = 0$ and $\theta_2 = 0, 0.5$. For certain reasons, I know that a solution (i.e. functions $\beta$ and $\gamma$) will not exist if I want $\beta = r_1$ whenever $r_1 = 0$, but I want to know if they might exist around two values of $\theta_2$.

REPLY [4 votes]: The general solution of your equations in a simply connected domain on which $r_2\not=0$ and $r_1\not=\pm1$ is
$$
\beta = \frac12 + \frac1{{(r_1}^2{-}1)}\,
\left(\frac{\partial a}{\partial\theta_1}+b(\theta_1,r_2)\right)
\quad\text{and}\quad
\gamma= \frac12 + \frac1{{r_2}^2}\,
\left(\frac{\partial a}{\partial\theta_2}+c(\theta_2,r_1)\right),
$$
where $a = a(\theta_1,\theta_2)$ is a function of $\theta_1$ and $\theta_2$ only, $b$ is a function of $\theta_1$ and $r_2$ only, and $c$ is a function of $\theta_2$ and $r_1$ only.
To see this, set $\bar\beta = ({r_1}^2{-}1)\bigl(\beta-\tfrac12\bigr)$ and $\bar\gamma = {r_2}^2\bigl(\gamma-\tfrac12\bigr)$ and note that the given equations imply the constant coefficient linear equations
$$
\frac{\partial\bar\beta}{\partial r_1}
= \frac{\partial\bar\gamma}{\partial r_2}
= \frac{\partial\bar\beta}{\partial\theta_2}-\frac{\partial\bar\gamma}{\partial\theta_1} = 0,
$$
which are easily solved.
I do not see how you can choose $a$, $b$, and $c$ so that your 'boundary conditions' are satisfied.  Because the general solution depends on three functions of two variables, one would expect to be able to specify (initial) conditions along a surface in the domain, but not along a hypersurface (which is what boundaries generally are).


TITLE: Prove that $\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+R_8(x)$ where $|R_8(x)|\leq \frac{x^8}{8!}$
QUESTION [3 upvotes]: Prove that
$$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+R_8(x)$$
where $|R_8(x)|\leq \frac{x^8}{8!}$

REPLY [0 votes]: Hint:   $f(x)=\sum_{i=0}^n\frac{f^ {(i)}(x)x^i}{i! }$
 $\text if$ $$|f^{n+1}(x)|\le M_{n+1}$$then $$|R(x)|\le \frac{M_{n+1}|x|^n}{(n+1)!}$$


TITLE: When does $x^n = \sqrt[n]x$ for odd $n$ and real $x$?
QUESTION [1 upvotes]: Lead-up to the problem
Given that $n \in \mathbb Z$ and $x \in \mathbb R$, the function $f(x) = x^n$ will have an inverse only for odd values of $n$.
We find the inverse in these cases to be $f^{-1}(x) = g(x) = \sqrt[n]x$.
Problem
Find the points of intersection between $f$ and its inverse $g$.
Thoughts
Essentially we have to solve the equation $x^n = \sqrt[n]x$ for odd $n$ and real $x$.
Doing any "real" calculations gives arbitrarily powered equations like $x^{n^2} - x = 0$ but beyond the trivial solutions $x=0, 1$ or $n = \pm 1$, it's hard to predict if it's even worth trying to find more.
Conclusion (?)
$x \in \{0, 1\} \ \vee \ n=\pm1$ must be the only solutions.
Does that look right?

REPLY [2 votes]: You can see that $|x^m| > |x|$ for all  $m >1$ and all $x$ satisfying $|x| >1$. So $x^{n^2} \not = x$ for all $x$ satsifying $|x| >1$. Thus, the aforementioned solutions $x \in \pm 1,0$ are precisely the set of solutions.
ETA: For whatever reason I had read $n$ must be a positive integer. The same line of reasoning holds for every odd $n \not \in \pm 1$ positive or negative, except $x$ cannot be $0$ for negative $n$. For $n=1,-1$, note that $x^n =x^{1/n}$ for all $x \not = 0$; for $n=-1$, and for all $x$ for $n=1$.


text:
\begin{document}
\maketitle

\begin{abstract}
We define the weight $\theta$-sheaf $\underline{\mathbb{R}}_{X,\theta}$ and reinterpret Morse-Novikov cohomology via sheaf theory. We establish a theorem of Leray-Hirsch type for  Morse-Novikov cohomology. Eventually, using them, we prove two blow-up formulas and give the isomorphisms explicitly on (possibly noncompact) complex manifolds.
\end{abstract}

\textbf{Keywords}: Morse-Novikov cohomology, weight $\theta$-sheaf, K\"{u}nneth formula, Leray-Hirsch theorem, blow-up formula, modification formula.

\textbf{2010 MSC}: 57R19, 55N30, 53C56.

\section{Introduction}
For a  $n$-dimensional smooth manifold $X$,  let $\mathcal{A}^p(X)$ be the space of smooth $p$-forms and $\theta$ a closed 1-form on $X$. Define $\textrm{d}_\theta:\mathcal{A}^p(X)\rightarrow\mathcal{A}^{p+1}(X)$ as $\textrm{d}_\theta\alpha=\textrm{d}\alpha+\theta\wedge\alpha$ for any $\alpha$ in $\mathcal{A}^p(X)$.
Obviously, $\textrm{d}_\theta\circ \textrm{d}_\theta=0$, so $(\mathcal{A}^\bullet(X), \textrm{d}_\theta)$ is a complex. Denote its cohomolgy by $H^*_\theta(X)$, which is called \emph{Morse-Novikov cohomology} (\cite{M,OV,YZ}), \emph{Lichnerowicz cohomology} (\cite{BK,LLMP}),  \emph{adapted cohomology} (\cite{DO,Vai}), or \emph{$\emph{d}_\theta$-cohomology} (\cite{Ba1,Ba2,HR}). In this article, we call it Morse-Novikov cohomology. Similarly, the complex $(\mathcal{A}_c^\bullet(X), \textrm{d}_\theta)$ of the spaces of smooth forms with compact support also defines cohomology groups, denoted by $H^*_{\theta,c}(X)$, which are called \emph{Morse-Novikov cohomology with compact support}.

This cohomology was originally defined by Lichnerowicz (\cite{GL,L}) and Novikov (\cite{N}) in the context of Poisson geometry and Hamiltonian mechanics, respectively. It is well used to study the locally conformally K\"{a}hlerian (l.c.K.) and locally conformally symplectic (l.c.s.) structures (\cite{Ba1,Ba2,BK,DO,HR,LLMP,Vai}). For a Morse's function $f$ and $\lambda\in\mathbb{R}$, $\textrm{d}_{\lambda \textrm{d}f}$ is the Witten deformation, which is used to prove  strong Morse inequalities by E. Witten \cite{W}.

Unfortunately, Morse-Novikov cohomology is much more difficult to calculate than de Rham cohomology, since it depends on $\theta$. In \cite{Vai}, Vaisman studied the classical operators of forms on l.c.K. manifolds and proved the finiteness of the dimension of  Lichnerowicz cohomology for compact manfiolds. The Mayer-Vietoris sequence and Pincar\'{e} duality theorem were generalized on Morse-Novikov cohomology by Haller, S. and Rybicki, T. \cite{HR}. Millionschikov, D. V. found the Morse-Novikov cohomology concides with cohomology of the corresponding Lie algebra  for the solvmanifolds (more precisely, refer to \cite{M}). Le\'{o}n, M., L\'{o}pez, B, Marrero J. C. and Padr\'{o}n, E.,  (\cite{LLMP}), proved that a compact Riemmannian manifold $X$ endowed with a parallel one-form $\theta$ has trivial  Morse-Novikov cohomology, which is proved in a different approach in \cite{OV}.  By Atiyah-Singer index theorem, Bande, G. and Kotschick, D. (\cite{BK}) found that the Euler characteristic of Morse-Novikov cohomology coincides with the usual Euler characteristic. Moreover, there are some results for specific examples, referring to \cite{Ba2,HR,M,O}.

In this article, we proved a theorem of Leray-Hirsch type on Morse-Novikov cohomology, which can be used to compute the cohomology of projectivization bundle $\mathbb{P}(E)$, seeing Lemma \ref{3}.
\begin{theorem}\label{1.1}
Let $\pi:E\rightarrow X$ be a smooth fiber bundle over a smooth manifold $X$ and let $\theta$, $\Omega$ be closed $1$-forms on $X$, $E$ respectively. Set $\tilde{\theta}=\pi^*\theta$.

$(1)$ Assume there exist classes $e_1,\dots,e_r$ of pure degrees in $ H_{\Omega}^*(E)$, such that, for every $x\in X$,  their restrictions $e_1|_{E_x},\dots,e_r|_{E_x}$ freely linearly generate $H_{\Omega|_{E_x}}^*(E_x)$. Then, $\pi^*(\bullet)\wedge\bullet$ gives an isomorphism of graded vector spaces
\begin{displaymath}
H_{\theta}^*(X)\otimes_{\mathbb{R}} \emph{span}_{\mathbb{R}}\{e_1, ..., e_r\} \tilde{\rightarrow} H_{\tilde{\theta}+\Omega}^*(E).
\end{displaymath}

$(2)$ Assume there exist classes $e_1,\dots,e_r$ of pure degrees in $ H_{\Omega,cv}^*(E)$, such that, for every $x\in X$,  their restrictions $e_1|_{E_x},\dots,e_r|_{E_x}$ freely linearly generate $H_{\Omega|_{E_x},c}^*(E_x)$. Then, $\pi^*(\bullet)\wedge\bullet$ gives isomorphisms of graded vector spaces
\begin{displaymath}
H_{\theta,c}^*(X)\otimes_{\mathbb{R}} \emph{span}_{\mathbb{R}}\{e_1, ..., e_r\} \tilde{\rightarrow} H_{\tilde{\theta}+\Omega,c}^*(E)
\end{displaymath}
and
\begin{displaymath}
H_\theta^*(X)\otimes_{\mathbb{R}} \emph{span}_{\mathbb{R}}\{e_1, ..., e_r\} \tilde{\rightarrow} H_{\tilde{\theta}+\Omega,cv}^*(E).
\end{displaymath}

$(3)$ Assume there exist classes $e_1,\dots,e_r$ of pure degrees in $ H_{\Omega,c}^*(E)$, such that, for every $x\in X$,  their restrictions $e_1|_{E_x},\dots,e_r|_{E_x}$ freely linearly generate $H_{\Omega|_{E_x},c}^*(E_x)$. Then, $\pi^*(\bullet)\wedge\bullet$ gives isomorphisms of graded vector spaces
\begin{displaymath}
H_{\theta,c}^*(X)\otimes_{\mathbb{R}} \emph{span}_{\mathbb{R}}\{e_1, ..., e_r\} \tilde{\rightarrow} H_{\tilde{\theta}+\Omega,c}^*(E).
\end{displaymath}
\end{theorem}

By Theorem \ref{1.1}, we can get the following lemma, which plays a key role in writing out the blow-up formulas explicitly.
\begin{lemma}\label{1.2}
Let $Y$ be an oriented connected submanifold of an oriented connected smooth manifold $X$ and $i:Y\rightarrow X$  the inclusion.  Assume $\theta$ is a closed $1$-form on $X$.

$(1)$ For $\sigma\in H_{\theta|_Y}^*(Y)$,
\begin{displaymath}
i^*i_*\sigma=[Y]|_Y\cup\sigma.
\end{displaymath}

$(2)$ For $\sigma\in H_{\theta|_Y,c}^*(Y)$,
\begin{displaymath}
i^*i_*\sigma=[Y]|_Y\cup\sigma.
\end{displaymath}
where $[Y]\in H_{dR}^r(X)$ is the fundamental class of $Y$ in $X$ and $r=\emph{codim}Y$.
\end{lemma}

In \cite{YZ}, Yang, X. D. and Zhao, G. S. proved a blow-up formula of Morse-Novikov cohomology for compact l.c.K. manifolds with some restriction on closed $1$-forms $\theta$, where they do not give the isomorphism. We generalize their theorem on general complex manifolds and closed $1$-forms, and moreover, give the isomorphisms explicitly by Theorem \ref{1.1} and Lemma \ref{1.2}.

Let $\pi:\widetilde{X}_Y\rightarrow X$ is the blow-up of a connected complex manifold $X$ along a connected complex submaifold $Y$ and $\theta$ a  closed $1$-form on $X$. We know $\pi|_E:E=\pi^{-1}(Y)\rightarrow Y$ is the projectivization $E=\mathbb{P}(N_{Y/X})$ of the normal bundle $N_{Y/X}$. Let $h=c_1(\mathcal{O}_E(-1))\in H_{\textrm{dR}}^2(E)$ be the first Chern class of the universal line bundle $\mathcal{O}_E(-1)$ on $E$.
\begin{main theorem}\label{1.3}
Assume $X$, $Y$, $\pi$, $\widetilde{X}_Y$, $E$, $h$ are defined as above. Let $i_E:E\rightarrow \widetilde{X}_Y$ be the inclusion, $\tilde{\theta}=\pi^*\theta$ and $r=\emph{codim}_{\mathbb{C}}Y$. Then, for any $k$,
\begin{displaymath}\label{b-u-m}
\pi^*+\sum_{i=0}^{r-2}(i_E)_*\circ (h^i\cup)\circ (\pi|_E)^*
\end{displaymath}
gives isomorphisms
\begin{displaymath}
H_{\theta}^k(X)\oplus \bigoplus_{i=0}^{r-2}H_{\theta|_Y}^{k-2-2i}(Y)\tilde{\rightarrow} H_{\tilde{\theta}}^k(\widetilde{X}_Y)
\end{displaymath}
and
\begin{displaymath}
H_{\theta,c}^k(X)\oplus \bigoplus_{i=0}^{r-2}H_{\theta|_Y,c}^{k-2-2i}(Y)\tilde{\rightarrow} H_{\tilde{\theta},c}^k(\widetilde{X}_Y).
\end{displaymath}
\end{main theorem}
There are some results of this type for other homologies and cohomologies, for instance, singular cohomology \cite{Vo}, Chow group \cite{Vo}, Lawson homology \cite{Hu} and Deligne cohomology \cite{Gri}.

The paper is organized as follows. In Sec. 2, We define the weight $\theta$-sheaf $\underline{\mathbb{R}}_{X,\theta}$ and reinterpret Morse-Novikov cohomology as  cohomology of  sheaf $\underline{\mathbb{R}}_{X,\theta}$. In particular, we prove that several operators on Morse-Novikov cohomology are compatible with that on cohomology of sheaves. In Sect. 3, we give two K\"{u}nneth formulas which are used to prove Theorem \ref{1.1} and give a new proof of Bande, G. and Kotschick, D.'s result on the Euler characteristic. In Sec. 4, we prove Lemma \ref{1.2} and Theorem \ref{1.3}. Moreover, we prove a modification formula and obtain two bimeromorphic invariants.

\section{Morse-Novikov cohomology via sheaf theory}
\subsection{Preliminaries}
Let $X$ be a  $n$-dimensional smooth manifold and  $\theta$ a closed $1$-form on $X$. Let $\mathcal{A}^p(X)$ be the space of smooth $p$-forms and define $\textrm{d}_\theta:\mathcal{A}^p(X)\rightarrow\mathcal{A}^{p+1}(X)$ as $\textrm{d}_\theta\alpha=\textrm{d}\alpha+\theta\wedge\alpha$ for $\alpha\in\mathcal{A}^p(X)$.
Obviously, $\textrm{d}_\theta\circ \textrm{d}_\theta=0$, so we have a complex
\begin{displaymath}
\xymatrix{
\cdots\ar[r] &\mathcal{A}^{p-1}(X)\ar[r]^{\quad\textrm{d}_\theta}&\mathcal{A}^{p}(X)\ar[r]^{\textrm{d}_\theta\quad}&\mathcal{A}^{p+1}(X)\cdots\ar[r]&\cdots
},
\end{displaymath}
whose cohomology $H^p_\theta(X)=H^p(\mathcal{A}^\bullet(X),\textrm{d}_\theta)$ is called  \emph{$p$-th Morse-Novikov cohomology}.

The cohomology $H_{\theta}^*(X)$ can be computed via the cohomology with the value in  a flat line bundle (weight line bundle), referring to \cite{N}, \cite{OV}, \cite{YZ}. As we know, a flat bundle of finite rank and a locally constant sheaf of $\mathbb{R}$-modules of finite rank  are equivalent in some senses. In this article, we study the Morse-Novikov cohomology by the language of locally constant sheaves, which is much more convenient.

Suppose $\mathcal{A}_X^k$ is the sheaf of germs of smooth $k$-forms and $\underline{\mathbb{R}}_{X}$ is the constant sheaf with coefficient $\mathbb{R}$ on $X$. Define $\textrm{d}_\theta:\mathcal{A}_X^k\rightarrow\mathcal{A}_X^{k+1}$ as follows:
\begin{displaymath}
\textrm{d}_\theta\alpha=\textrm{d}\alpha+\theta\wedge\alpha,
\end{displaymath}
for $\alpha\in\mathcal{A}_X^k$.
\begin{definition}
We call the kernel of $\emph{d}_\theta:\mathcal{A}_X^0\rightarrow\mathcal{A}_X^1$ a weight $\theta$-sheaf, denoted by $\underline{\mathbb{R}}_{X,\theta}$.
\end{definition}
Locally, $\theta=\textrm{d}u$ for a smooth function $u$, so $\textrm{d}_\theta=e^{-u}\circ \textrm{d}\circ e^u$ and $\underline{\mathbb{R}}_{X,\theta}=\mathbb{R}e^{-u}$. Hence, the weight $\theta$-sheaf $\underline{\mathbb{R}}_{X,\theta}$ is a locally constant sheaf of of $\mathbb{R}$-modules of rank 1. We have a resolution of soft sheaves of $\underline{\mathbb{R}}_{X,\theta}$
\begin{displaymath}
\xymatrix{
0\ar[r] &\underline{\mathbb{R}}_{X,\theta}\ar[r]^{i} &\mathcal{A}_X^0\ar[r]^{\textrm{d}_\theta} &\mathcal{A}_X^1\ar[r]^{\textrm{d}_\theta}&\cdots\ar[r]^{\textrm{d}_\theta}&\mathcal{A}_X^n\ar[r]&0
},
\end{displaymath}
where $i$ is the inclusion.

Let $\mathcal{A}_X^\bullet\rightarrow \mathcal{I}^\bullet$ be an injective resolution of complex $(\mathcal{A}_X^\bullet,\textrm{d}_\theta)$ of sheaves in the category of sheaves on $X$, then it induces a  morphism
\begin{displaymath}
H_\theta^*(X)= H^*(\mathcal{A}^\bullet(X),\textrm{d}_{\theta})\rightarrow H^*(\Gamma(X,\mathcal{I}^\bullet))=H^*(X,\underline{\mathbb{R}}_{X,\theta}),
\end{displaymath}
denoted by $\rho$. Since $\mathcal{A}_X^\bullet$ is a resolution of soft sheaves of $\underline{\mathbb{R}}_{X,\theta}$, $\rho$ is an isomorphism. Similarly, $\mathcal{A}_X^\bullet\rightarrow \mathcal{I}^\bullet$ induces an isomorphism $H_{\theta,c}^*(X)\rightarrow H_c^*(X,\underline{\mathbb{R}}_{X,\theta})$, also denoted by $\rho$. Hence, Morse-Novikov cohomology (resp. Morse-Novikov cohomology with compact support) can be viewed as cohomology (resp. cohomology with compact support) of the weight $\theta$-sheaf $\underline{\mathbb{R}}_{X,\theta}$ via $\rho$. For $\textrm{d}_\theta$-closed $\alpha\in\mathcal{A}^*(X)$, denote $[\alpha]_{\theta}$ (resp. $[\alpha]_{\theta,c}$) the class in $H_\theta^*(X)$ (resp. $H_{\theta,c}^*(X)$).

In addition, assume $X$ is oriented. Let $\mathcal{D}_X^{\prime k}$ be the sheaf of germs of $k$-currents. Similarly, define $\textrm{d}_\theta:\mathcal{D}_X^{\prime k}\rightarrow\mathcal{D}_X^{\prime k+1}$ as follows:
\begin{displaymath}
\textrm{d}_\theta T=\textrm{d}T+\theta\wedge T,
\end{displaymath}
for $T\in\mathcal{D}_X^{\prime k}$. We have $\textrm{d}_{\theta}T(\alpha)=(-1)^{k+1}T(\textrm{d}_{-\theta}\alpha)$, for $\alpha\in\mathcal{A}_c^{n-k-1}(X)$, and
\begin{displaymath}
\textrm{d}_{\theta_1+\theta_2}(\beta\wedge\gamma)=\textrm{d}_{\theta_1}\beta\wedge\gamma+(-1)^{deg\beta}\beta\wedge \textrm{d}_{\theta_2}\gamma,
\end{displaymath}

where $\beta$ and $\gamma$ are in $\mathcal{A}^*(X)$ or $\mathcal{D}^{\prime*}(X)$, but not both in $\mathcal{D}^{\prime*}(X)$.

Since, locally, $\textrm{d}_\theta=e^{-u}\circ \textrm{d}\circ e^u$ on $\mathcal{D}_X^{\prime*}$,  we have $\underline{\mathbb{R}}_{X,\theta}=\textrm{ker}(\textrm{d}_\theta:\mathcal{D}_X^{\prime 0}\rightarrow\mathcal{D}_X^{\prime 1})$. There is another  resolution of soft sheaves of $\underline{\mathbb{R}}_{X,\theta}$
\begin{displaymath}
\xymatrix{
0\ar[r] &\underline{\mathbb{R}}_{X,\theta}\ar[r]^{i} &\mathcal{D}_X^{\prime 0}\ar[r]^{\textrm{d}_\theta} &\mathcal{D}_X^{\prime 1}\ar[r]^{\textrm{d}_\theta}&\cdots\ar[r]^{\textrm{d}_\theta}&\mathcal{D}_X^{\prime n}\ar[r]&0
},
\end{displaymath}
where $i$ is the inclusion. Moreover, $\mathcal{A}_X^\bullet\hookrightarrow\mathcal{D}_X^{\prime\bullet}$ induces isomorphisms $H_\theta^*(X)\tilde{\rightarrow} H^*(\mathcal{D}^{\prime\bullet}(X),\textrm{d}_{\theta})$ and $H_{\theta,c}^*(X)\tilde{\rightarrow} H^*(\mathcal{D}_c^{\prime\bullet}(X),\textrm{d}_{\theta})$.
For $\textrm{d}_\theta$-closed $T\in\mathcal{D}^{\prime*}(X)$, denote $[T]_{\theta}$ (resp. $[T]_{\theta,c}$ ) the class in $H_\theta^*(X)$ (resp. $H_{\theta,c}^*(X)$).

Let $f:X\rightarrow Y$ be a smooth map between connected oriented smooth manifolds and $\theta$ a closed $1$-form on $Y$. Set $\tilde{\theta}=f^*\theta$ and $r=\textrm{dim}X-\textrm{dim}Y$. If a current $T$ satisfies that $f|_{\textrm{supp}T}:\textrm{supp}T\rightarrow Y$ is proper, then $\textrm{d}_\theta(f_*T)=(-1)^rf_*(\textrm{d}_{\tilde{\theta}}T)$.

\begin{lemma}\label{lem fun}
Let $X$ be a connected smooth manifold and $\theta$ a closed $1$-form on $X$.

$(1)$ $\underline{\mathbb{R}}_{X,\theta}$ is a constant sheaf   if and only if $\theta$ is exact. More precisely, if $\theta=\emph{d}u$ for $u\in \mathcal{A}^0(X)$, then $h\mapsto e^{u}\cdot h$ gives an isomorphism $\underline{\mathbb{R}}_{X,\theta}\tilde{\rightarrow}\underline{\mathbb{R}}_X$ of sheaves.

$(2)$ If $\mu$ is a closed $1$-form on $X$, then $\underline{\mathbb{R}}_{X,\theta}\otimes_{\underline{\mathbb{R}}_X}\underline{\mathbb{R}}_{X,\mu}=\underline{\mathbb{R}}_{X,\theta+\mu }$.

$(3)$ Suppose $f:Y\rightarrow X$ is a smooth map between smooth manifolds.  Then inverse image sheaf $f^{-1}\underline{\mathbb{R}}_{X,\theta}=\underline{\mathbb{R}}_{Y,f^*\theta}$.
\end{lemma}

\begin{proof}
$(1)$ If $\underline{\mathbb{R}}_{X,\theta}$ is a constant sheaf, then $H_\theta^0(X)=H^0(X,\underline{\mathbb{R}}_{X,\theta})=\mathbb{R}$. By \cite{HR}, Example 1.6, $\theta$ is exact. Inversely, if $\theta=\textrm{d}u$, $\underline{\mathbb{R}}_{X,\theta}=\underline{\mathbb{R}}e^{-u}$, which implies the conclusion.

$(2)$ Locally, $\theta=\textrm{d}u$ and $\mu=\textrm{d}v$. Then,  $\underline{\mathbb{R}}_{X,\theta}=\mathbb{R}e^{-u}$, $\underline{\mathbb{R}}_{X,\mu}=\mathbb{R}e^{-v}$ and $\underline{\mathbb{R}}_{X,\theta+\mu}=\mathbb{R}e^{-u-v}$, locally. So, the product of functions gives an isomorphism $\underline{\mathbb{R}}_{X,\theta}\otimes_{\underline{\mathbb{R}}_X}\underline{\mathbb{R}}_{X,\mu}\rightarrow\underline{\mathbb{R}}_{X,\theta+\mu }$.

$(3)$ Locally, $\theta=\textrm{d}u$, $\underline{\mathbb{R}}_{X,\theta}=\mathbb{R}e^{-u}$ and $\underline{\mathbb{R}}_{Y,f^*\theta}=\mathbb{R}e^{-f^*u}$. So the pull backs of functions give an isomorphism $f^{-1}\underline{\mathbb{R}}_{X,\theta}\tilde{\rightarrow} \underline{\mathbb{R}}_{Y,f^*\theta}$ .
\end{proof}

\begin{notation}
For any topological subspace $i:W\rightarrow X$, we simply write $H^*(W,i^{-1}\underline{\mathbb{R}}_{X,\theta})$ and $H_c^*(W,i^{-1}\underline{\mathbb{R}}_{X,\theta})$ as $H^*(W,\underline{\mathbb{R}}_{X,\theta})$ and $H_c^*(W,\underline{\mathbb{R}}_{X,\theta})$ respectively. If $W$ is open, we also write $H^*_{\theta|_W}(W)$ and $H^*_{\theta|_W,c}(W)$ as $H^*_{\theta}(W)$ and $H^*_{\theta,c}(W)$ respectively.

\end{notation}

\subsection{Pullback and pushout}
Let $f:X\rightarrow Y$ be a smooth map between oriented connected smooth manifolds and $\theta$ a closed $1$-form on $Y$. Set $\tilde{\theta}=f^*\theta$ and $r=\textrm{dim}X-\textrm{dim}Y$. We can define \emph{pullback} $f^*:H^*_{\theta}(Y)\rightarrow H^*_{\tilde{\theta}}(X)$, $[\alpha]_{\theta}\mapsto [f^*\alpha]_{\tilde{\theta}}$ and \emph{pushout} $f_*:H^*_{\tilde{\theta},c}(X)\rightarrow H^{*-r}_{\theta,c}(Y)$, $[T]_{\theta,c}\mapsto [f_*T]_{\tilde{\theta},c}$. Moreover, if $f$ is proper, we can also define $f^*:H^*_{\theta,c}(Y)\rightarrow H^*_{\tilde{\theta},c}(X)$ and $f_*:H^*_{\tilde{\theta}}(X)\rightarrow H^{*-r}_{\theta}(Y)$, in the same way. Actually, it is unnecessary to request the condition ``\emph{oriented}" in definitions of the two pullbacks $f^*$. By \cite{Br}, p. 63, 8.1,  via $\rho$, the pullbacks $f^*$ are compatible with them on cohomology of sheaves.

\begin{proposition}\label {V-B}
 Let $f:X\rightarrow Y$ be a  closed surjective smooth map between smooth manifolds and $\theta$ a closed $1$-form on $Y$. Assume that every fibre $f^{-1}(y)$ is connected and
$ H^i(f^{-1}(y),\mathbb{R})=0$ for $1\leq i\leq r$.
Then $f^*:H^p_{\theta}(Y)\rightarrow H^p_{\tilde{\theta}}(X)$ is an isomorphism for $0\leq p\leq r$, and a monomorphism for $p=r+1$. In addition, if $f$ is proper, they also hold for Morse-Novikov cohomology with compact support.
\end{proposition}
\begin{proof}
The first part holds by  Vietoris-Begle Mapping theorem (\cite{SS}).

By Lemma \ref{lem fun}, $\underline{\mathbb{R}}_{X,\tilde{\theta}}|_{f^{-1}(y)}=\underline{\mathbb{R}}_{f^{-1}(y)}$. For a proper map $f$,
\begin{displaymath}
\left(R^qf_*\underline{\mathbb{R}}_{X,\tilde{\theta}}\right)_y=H^q(f^{-1}(y),\underline{\mathbb{R}}_{f^{-1}(y)}),
\end{displaymath}
which imply
\begin{displaymath}
R^qf_*\underline{\mathbb{R}}_{X,\tilde{\theta}}=\left\{
 \begin{array}{ll}
\underline{\mathbb{R}}_{Y,\theta},&~q=0\\
 &\\
 0,&~1\leq q\leq r.
 \end{array}
 \right.
\end{displaymath}
Consider the Leray spectral sequence
\begin{displaymath}
E_2^{p,q}=H_c^p(Y, R^qf_*\underline{\mathbb{R}}_{X,\tilde{\theta}})\Rightarrow H^{p+q}=H_c^{p+q}(X,\underline{\mathbb{R}}_{X,\tilde{\theta}}).
\end{displaymath}
Then $E_2^{p,q}=0$ for $1\leq q\leq r$. By \cite{F}, Prop. 2.2.3, $E_2^{p,0}\cong H^p$ for $0\leq p\leq r$ and $E_2^{r+1,0}\rightarrow H^{r+1}$ is injective, where the morphisms are all induced by the pull back of sheaves. We prove the second part.
\end{proof}

Assume $j:U\rightarrow X$ is the inclusion of an open subset $U$ into a smooth manifold $X$. Denote by $j_*: \mathcal{A}_c^*(U)\rightarrow\mathcal{A}_c^*(X)$ the extension by zero, which induces $j_*: H_{\theta,c}^*(U)\rightarrow H_{\theta,c}^*(X)$. If $X$ is oriented, it coincides with the pushout $j_*$ defined by currents.
\begin{proposition}\label{cpb2}
$j_*$ defined above is compatible with $j_!$ on cohomology of sheaves, i.e., the  diagram
\begin{displaymath}
\xymatrix{
 H_{\theta,c}^*(U) \ar[d]^{j_*} \ar[r]^{\rho\quad}& H_c^*(U,\underline{\mathbb{R}}_{U,\theta})\ar[d]^{j_!}\\
 H_{\theta,c}^*(X)      \ar[r]^{\rho\quad}& H_c^*(X,\underline{\mathbb{R}}_{X,\theta})  }.
\end{displaymath}
is commutative.
\end{proposition}
\begin{proof}
Let $\mathcal{A}_X^\bullet\rightarrow \mathcal{I}^\bullet$ and $j^{-1}\mathcal{I}^\bullet\rightarrow \mathcal{J}^\bullet$ be injective resolutions of complexes $\mathcal{A}_X^\bullet$ and $j^{-1}\mathcal{I}^\bullet$ of sheaves, respectively. Then $\mathcal{I}^\bullet$ and $\mathcal{J}^\bullet$ are injective resolutions of $\underline{\mathbb{R}}_{X,\theta}$ and $\underline{\mathbb{R}}_{U,\theta}$, respectively. Since $j_!$ is an exact functor
, $j_!j^{-1}\mathcal{I}^\bullet\rightarrow j_!\mathcal{J}^\bullet$ is quasi-isomorphic. By \cite{I}, p. 41, 6.2, there exists a morphism $j_!\mathcal{J}^\bullet\rightarrow\mathcal{I}^\bullet$ of complexes, such that the diagram
\begin{displaymath}
\xymatrix{
 j_!\mathcal{A}_U^\bullet=j_!j^{-1}\mathcal{A}_X^\bullet\ar[d] \ar[r]& j_!j^{-1}\mathcal{I}^\bullet \ar[d]\ar[r]&j_!\mathcal{J}^\bullet\ar[dl]\\
 \mathcal{A}_X^\bullet \ar[r]& \mathcal{I}^\bullet}
\end{displaymath}
satisfies that the left square is commutative and the right triangle is commutative up to a homotopy. Through the cohomology of sections $\Gamma_c(X,\cdot)$ of complexes of sheaves, we get the proposition immediately.
\end{proof}

We will frequently use the following proposition in sheaf theory.
\begin{proposition}[\cite{I}, p. 186 (7.8)]\label{L}
Let $f:X\rightarrow Y$ be a proper continuous map between locally compact spaces and $\mathcal{F}$ a sheaf on $Y$. Assume that $B$ is a closed subset on $Y$, $A=f^{-1}(B)$, $V=Y-B$, $U=X-A$, and $i_A$, $i_B$, $j_U$, $j_V$ are the corresponding inclusions. There is a commutative diagram of long exact sequences
\begin{displaymath}
\xymatrix{
\cdots\ar[r]&H_{c}^k(V,\mathcal{F})\ar[d]^{(f|_U)^*} \ar[r]^{j_{V!}}& H_{c}^k(Y,\mathcal{F}) \ar[d]^{f^*} \ar[r]^{i_B^*}& H_c^k(B,\mathcal{F}) \ar[d]^{(f|_A)^*}\ar[r]& H_{c}^{k+1}(V,\mathcal{F})\ar[d]^{(f|_U)^*}\ar[r]&\cdots\\
 \cdots\ar[r]&H_{c}^k(U,f^{-1}\mathcal{F})       \ar[r]^{j_{U!}}& H_{c}^k(X,f^{-1}\mathcal{F})     \ar[r]^{i_A^*} & H_c^k(A,f^{-1}\mathcal{F})     \ar[r]& H_{c}^{k+1}(U,f^{-1}\mathcal{F})         \ar[r]&\cdots }.
\end{displaymath}
\end{proposition}
\subsection{Cup product}
Let $X$ be a smooth manifold and $\theta$, $\mu$ closed  $1$-forms on $X$. The wedge product $\alpha\wedge \beta$ defines a \emph{cup product}
\begin{displaymath}
\cup: H^p_{\theta}(X)\times H^q_{\mu}(X)\rightarrow H^{p+q}_{\theta+\mu}(X).
\end{displaymath}
Similarly, we can define cup products between $H^p_{\theta}(X)$ or $H^p_{\theta,c}(X)$ and $H^q_{\mu}(X)$ or $H^q_{\mu,c}(X)$. By \cite{Br}, p. 63, 8.1, via $\rho$, the cup products are compatible with them on cohomology of sheaves.

By \cite{Dim}, Cor. 3.3.12, we get Poincar\'{e} duality for Morse-Novikov cohomology immediately.
\begin{corollary}[\cite{HR}, Cor. 1.4]\label {PD}
 Let $X$ be an oriented smooth manifold of dimension $n$ and $\theta$ a closed $1$-form on $X$. Then, for any $p$,
\begin{displaymath}
PD:H_\theta^p(X)\rightarrow (H_{-\theta,c}^{n-p}(X))^*
\end{displaymath}
is an isomorphism, where $PD([\alpha]_\theta)([\beta]_{-\theta,c})=\int_X\alpha\wedge\beta$.
\end{corollary}

\begin{proposition}[Lefschetz' hyperplane theorem]
Let $X$ be a $n$-dimensional nonsingular complex  projective variety and $Y$ a smooth hyperplane section of $X$. Assume $\theta$ is a closed $1$-form on $X$ and $i:Y\rightarrow X$ is the inclusion. Then

$(1)$ $i^*:H_\theta^p(X)\rightarrow H_{\theta|_Y}^p(Y)$ is isomorphic for $p\leq n-2$ and injective for $p= n-1$.

$(2)$ $i_*: H_{\theta|_Y}^p(Y)\rightarrow H_\theta^{p+2}(X)$ is isomorphic for $p\geq n$ and surjective for $p= n-1$.
\end{proposition}
\begin{proof}
$X-Y$ is a nonsingular complex affine variety with dimension $n$, hence a Stein manifold. By \cite{Dim}, Thm. 2.5.11, $H_{\theta,c}^p(X-Y)=0$ for $p\leq n-1$. We get $(1)$ by \ref{cpb2} and \ref{L}. By Poincar\'{e} duality, we get $(2)$ immediately.
\end{proof}

Let $f:X\rightarrow Y$ be a proper smooth map between oriented connected smooth manifolds and $\theta$, $\mu$ closed $1$-forms on $Y$. Set $\tilde{\theta}=f^*\theta$ and $r=\textrm{dim}X-\textrm{dim}Y$. We have \emph{projection formula}
\begin{equation}\label{pro-f}
f_*(\sigma\cup f^*\tau)=f_*(\sigma)\cup \tau
\end{equation}
for $\sigma$ $\in$ $H_{\tilde{\theta}}^*(X)$ or $H_{\tilde{\theta},c}^*(X)$ and $\tau\in H_{\mu}^*(Y)$ or $H_{\mu,c}^*(Y)$. Immediately, it is obtained by $f_*(T\wedge f^*\beta)=f_*T\wedge \beta$, where $T\in\mathcal{D}^{\prime*}(X)$ and $\beta\in\mathcal{A}^*(Y)$. If $\sigma\in H_{\tilde{\theta},c}^*(X)$ and $\tau\in H_{\mu}^*(Y)$, we can remove the condition ``\emph{proper}" in projection formula $($\ref{pro-f}$)$.  By projection formula $($\ref{pro-f}$)$, we get
\begin{corollary}\label{i-s}
Let $f:X\rightarrow Y$ be a proper smooth map of oriented connected smooth manifolds with the same dimensions and $\emph{deg}f\neq 0$. If $\theta$ is a closed $1$-form on $Y$, then $f^*:H^*_{\theta}(Y)\rightarrow H^*_{\tilde{\theta}}(X)$ is injective and $f_*:H^*_{\tilde{\theta}}(X)\rightarrow H^{*}_{\theta}(Y)$ is surjective. They also hold for Morse-Novikov cohomology with compact support.
\end{corollary}

\section{K\"{u}nneth formula and Leray-Hirsch theorem}
\subsection{K\"{u}nneth formula}
Let $\theta$ and $\mu$ be closed $1$-forms on smooth manifolds $X$ and $Y$ respectively. Set $\omega=pr_1^*\theta+pr_2^*\mu$, where $pr_1$ and $pr_2$ are projections from $X\times Y$ onto $X$ and $Y$ respectively. Then $pr_1^*(\bullet)\wedge pr_2^*(\bullet)$ defines a product
\begin{displaymath}
\times: H^p_{\theta}(X)\times H^q_{\mu}(Y)\rightarrow H^{p+q}_{\omega}(X\times Y).
\end{displaymath}
It is called  \emph{cartesian product}. Similarly, we can define cartesian product between $H^p_{\theta,c}(X)$ and  $H^q_{\mu,c}(Y)$. The cartesian products defined here are compatible with them on cohomology of sheaves, i.e. $\rho(\sigma\times\tau)=\rho(\sigma)\times\rho(\tau)$, for any $\sigma\in H^p_{\theta}(X)$ and $\tau\in H^q_{\mu}(Y)$, or $\sigma\in H^p_{\theta,c}(X)$ and $\tau\in H^q_{\mu,c}(Y)$. Actually,
\begin{displaymath}
\begin{aligned}
\rho(\sigma)\times\rho(\tau)=&pr_1^*\rho(\sigma)\cup pr_2^*\rho(\tau)\\
=&\rho (pr_1^*\sigma)\cup \rho (pr_2^*\tau)\\
=&\rho \left(pr_1^*\sigma\cup pr_2^*\tau\right)\\
= &\rho(\sigma\times\tau).
\end{aligned}
\end{displaymath}

Consider the trivial bundle $\pi:\mathbb{R}^n\times F\rightarrow \mathbb{R}^n$ on $\mathbb{R}^n$, where $F$ is a smooth manifold. Suppose $\Theta$ is a closed $1$-form on $\mathbb{R}^n\times F$. Let $pr_2:\mathbb{R}^n\times F\rightarrow F$ be the second projection and $i:F\rightarrow \mathbb{R}^n\times F$ a inclusion which map $f$ to $(0,f)$. Let $g:\mathbb{R}\times\mathbb{R}^n\times F\rightarrow\mathbb{R}^n\times F$ map $(t, x, f)$ to $((1-t)x, f)$, which gives a smooth homotopy between $\textrm{id}_{\mathbb{R}^n\times F}$ and $i\circ pr_2$. If $t$ is the coordinate of the first component of $\mathbb{R}\times\mathbb{R}^n\times F$, define the \emph{contraction} operator $i(\partial/\partial t): \mathcal{A}^*(\mathbb{R}\times\mathbb{R}^n\times F)\rightarrow  \mathcal{A}^{*-1}(\mathbb{R}\times\mathbb{R}^n\times F)$ as
\begin{displaymath}
i(\partial/\partial t)(\Upsilon)(X_1,...,X_{p-1})=\Upsilon(\partial/\partial t,X_1,...,X_{p-1})
\end{displaymath}
for any $\Upsilon\in \mathcal{A}^p(\mathbb{R}\times\mathbb{R}^n\times F)$ and smooth vector fields $X_1$, ..., $X_{p-1}$ on $\mathbb{R}\times\mathbb{R}^n\times F$. Set $u_s=\int_0^s i(\partial/\partial t)(g^*\Theta)\textrm{d}t$. Then
\begin{equation}\label{htp1}
pr_2^*i^*\Theta-\Theta=\textrm{d}u_1.
\end{equation}
Define $K: \mathcal{A}^*(\mathbb{R}^n\times F)\rightarrow  \mathcal{A}^{*-1}(\mathbb{R}^n\times F)$ as
\begin{equation}\label{operator}
K(\alpha)=\int_0^1 e^{u_t}\cdot i(\partial/\partial t)(g^*\alpha) \textrm{d}t.
\end{equation}
Then,
\begin{equation}\label{htp2}
e^{u_1}pr_2^*i^*\alpha-\alpha=\textrm{d}_\Theta K(\alpha)+K(\textrm{d}_\Theta \alpha),
\end{equation}
for any $\alpha\in \mathcal{A}^*(\mathbb{R}^n\times F)$. The formulas $($\ref{htp1}$)$ and $($\ref{htp2}$)$ were proved in \cite{HR}, Lemma 1.1.
\begin{lemma}\label{homotopic}
Let $\theta$ be a closed $1$-form on $F$ and $\tilde{\theta}=pr_2^*\theta$. Then $pr_2^*:H_\theta^*(F)\rightarrow H_{\tilde{\theta}}^*(\mathbb{R}^n\times F)$ is an isomorphism and $i^*$ is the inverse isomorphism.
\end{lemma}
\begin{proof}
Let $u_s$ be defined as above. By formula $($\ref{htp1}$)$, $\textrm{d}u_1=0$, so $u_1$ is a constant. By formula $($\ref{htp2}$)$, $pr_2^*\circ i^*$ is an isomorphism. Obviously, $i^*\circ pr_2^*=1$. Hence, we get the lemma.
\end{proof}

Let $X$ be a $n$-dimensional smooth manifold. A covering $\mathfrak{U}=\{U_\alpha\}$ of $X$ is called \emph{good}, if all nonempty finite intersections $U_{\alpha_0,\ldots,\alpha_p}=U_{\alpha_0}\cap\ldots\cap U_{\alpha_p}$ are diffeomorphic $\mathbb{R}^n$. For a sheaf $\mathcal{F}$ and a covering  $\mathfrak{U}=\{U_\alpha\}_{\alpha\in I}$ of $X$, denote $C^p(\mathfrak{U},\mathcal{F})$ the group of $\check{C}ech$ $p$-cochains and $\delta$ the differentials of the complex $C^\bullet(\mathfrak{U},\mathcal{F})$.

\begin{theorem}[K\"{u}nneth formula I]\label{Kun1}
Let $X$ and $Y$ be smooth manifolds and let $\theta$  and $\mu$ be closed $1$-forms on $X$ and $Y$ respectively. Set $\omega=pr_1^*\theta+pr_2^*\mu$, where $pr_1$, $pr_2$ are projections from $X\times Y$ onto $X$, $Y$ respectively.
Then, the cartesian product gives an isomorphism of graded vector spaces
\begin{displaymath}
H_{\theta,c}^*(X)\otimes_{\mathbb{R}}H_{\mu,c}^*(Y)\rightarrow H_{\omega,c}^*(X\times Y).
\end{displaymath}

Moreover, if $H_{\theta}^*(X)=\bigoplus_{p\geq0}H_{\theta}^p(X)$ or $H_{\mu}^*(Y)=\bigoplus_{p\geq0}H_{\mu}^p(Y)$ has finite dimension,  the cartesian product also gives an isomorphism of graded vector spaces
\begin{displaymath}
H_{\theta}^*(X)\otimes_{\mathbb{R}}H_{\mu}^*(Y)\rightarrow H_{\omega}^*(X\times Y).
\end{displaymath}
\end{theorem}

\begin{proof}
By \cite{Br}, II, 15.2, we get the first part.

Moreover, we assume $H_{\mu}^*(Y)$ has finite dimension. Suppose $\textrm{dim}X=n$ and $\mathfrak{U}=\{U_\alpha\}$ is a good covering of $X$. Define two double complexes
\begin{displaymath}
K^{p,q}=\bigoplus_{r+s=q}C^p(\mathfrak{U},\mathcal{A}_X^r)\otimes_{\mathbb{R}} H_{\mu}^s(Y),
\end{displaymath}
\begin{displaymath}
d'=\delta\otimes \textrm{id}, d''=\sum_{r+s=q}\textmd{d}_{\theta}\otimes \textmd{id}_{H_\mu^s(Y)},
\end{displaymath}
and
\begin{displaymath}
L^{p,q}=C^p(pr_1^{-1}\mathfrak{U},\mathcal{A}_{X\times Y}^q),
\end{displaymath}
\begin{displaymath}
d'=\delta, d''=\textmd{d}_{\omega}.
\end{displaymath}

Choose a system of forms $\{\beta_i\}$  of pure degrees in $A^*(Y)$, such that $\{[\beta_i]_{\mu}\}$ is a basis of $H_\mu^*(Y)$. Define $f:K^{*,*}\rightarrow L^{*,*}$ as
\begin{displaymath}
\{\eta_{\alpha_0,\ldots,\alpha_p}\}\otimes[\beta_i]_{\mu}\mapsto \{pr_1^*\eta_{\alpha_0,\ldots,\alpha_p}\wedge pr_2^*\beta_i\},
\end{displaymath}
which gives a morphism of double complexes. $\theta$ is exact on $U_{\alpha_0,...,\alpha_p}\cong\mathbb{R}^n$, hence $H_{\theta}^r(U_{\alpha_0,...,\alpha_p})\cong H^r(U_{\alpha_0,...,\alpha_p})$ is $\mathbb{R}$ for $r=0$ and zero otherwise. We get
\begin{displaymath}
\begin{aligned}
H_{d''}^q(K^{p,*})

= &\left(\prod_{\alpha_0,\ldots,\alpha_p}H_{\theta}^0\left(U_{\alpha_0,\ldots,\alpha_p}\right)\right)\otimes_{\mathbb{R}}H_{\mu}^q(Y)\\
\cong &\prod_{\alpha_0,\ldots,\alpha_p}H_{\mu}^q(Y),
\end{aligned}
\end{displaymath}
where the isomorphism uses the finiteness of dimension of $H_{\mu}^*(Y)$, and
\begin{displaymath}
\begin{aligned}
H_{d''}^q(L^{p,*})= &\prod_{\alpha_0,\ldots,\alpha_p}H_{\omega}^q(U_{\alpha_0,\ldots,\alpha_p}\times Y)\\
\cong &\prod_{\alpha_0,\ldots,\alpha_p}H_{pr_2^*\mu}^q(U_{\alpha_0,\ldots,\alpha_p}\times Y),
\end{aligned}
\end{displaymath}
where the isomorphism uses the fact that $pr_1^*\theta$ is exact on $U_{\alpha_0,\ldots,\alpha_p}\times Y$ and Lemma \ref{lem fun}. For every $p, q$, the morphism $H_{d''}^q(K^{p,*})\rightarrow H_{d''}^q(L^{p,*})$ induced by $f$ is exactly
\begin{displaymath}
\prod pr_2^*:\prod_{\alpha_0,\ldots,\alpha_p}H_{\mu}^q(Y)\rightarrow \prod_{\alpha_0,\ldots,\alpha_p}H_{pr_2^*\mu}^q(U_{\alpha_0,\ldots,\alpha_p}\times Y),
\end{displaymath}
which is an isomorphism by Lemma \ref{homotopic}. Hence, for any $p$, $f$ induces an isomorphism $H^p(f):H_D^p(K^*)\rightarrow H_D^p(L^*)$, where $K^*$ and $L^*$ are the simple complexes associated to double complexes $K^{*,*}$ and $L^{*,*}$,  respectively, and $D$ are the differentials of $K^*$ and $L^*$.

Consider the spectral sequence $E_2^{p,q}=H_{d''}^qH_{d'}^p(K^{*,*})\Rightarrow H_D^{p+q}(K^*)$. Since $\mathcal{A}_X^r$ are soft sheaf for all $r$, by Leray Theorem, we have
\begin{displaymath}
E_2^{p,q}=\left\{
 \begin{array}{ll}
 \bigoplus_{r+s=q}H_{\theta}^r(X)\otimes_{\mathbb{R}}H_{\mu}^s(Y),&~p=0\\
 &\\
 0,&~otherwise.
 \end{array}
 \right.
\end{displaymath}
So $H_D^k(K^*)=\bigoplus_{p+q=k}H_{\theta}^p(X)\otimes_{\mathbb{R}} H_{\mu}^q(Y)$. Similarly, $H_D^k(L^*)=H_{\omega}^k(X\times Y)$. It is eaisly to check that $H^k(f)$ is given by cartesian product.
\end{proof}

If $H_\theta^*(X)$ has finite dimension, define $b_p(X,\theta)=\textrm{dim}H_\theta^p(X)$ and $\chi(X,\theta)=\sum_p(-1)^pb_p(X,\theta)$, which are called \emph{betti number} and \emph{Euler-characteristic of Morse-Novikov cohomology}. Similarly, we can define $b_{c,p}(X,\theta)$ and $\chi_c(X,\theta)$ for Morse-Novikov cohomology with compact support. Let $X$ be a compact oriented smooth manifold and $f:X\rightarrow X$ a smooth map.  Assume $\theta$ is a closed $1$-form on $X$ satisfying $f^*\theta=\theta$. Then $f$ induces a linear map $f^*:H_\theta^*(X)\rightarrow H_\theta^*(X)$. Define
\begin{displaymath}
L(f,\theta)=\sum_{p\geq0}(-1)^p\textrm{tr}(f^*|_{H_\theta^p(X)}),
\end{displaymath}
where $\textrm{tr}(f^*|_{H_\theta^p(X)})$ is the trace of $f^*|_{H_\theta^p(X)}$.
We call it \emph{$\theta$-Lefschetz number} of $f$. In particular, $\chi(X,\theta)=L(\textrm{id}_X,\theta)$.
\begin{proposition}
$L(f,\theta)=L(f)$. In particular, $\chi(X,\theta)=\chi(X)$.
\end{proposition}
\begin{proof}
Suppose $\textrm{dim}X=n$. Let $\{[\alpha_i]_\theta\}$ be a basis of $H_\theta^*(X)$ and  $\{[\beta_j]_{-\theta}\}$ their dual basis in $H_{-\theta}^*(X)$ under Poincar\'{e} duality, i.e. $\int_X\alpha_i\wedge\beta_j=\delta_{ij}$, where $\alpha_i, \beta_j$ all have pure degrees. Suppose $pr_1$, $pr_2$ are two projections from $X\times X$ onto $X$.  Let $\Delta$ be the diagonal of $X\times X$ and $\Gamma_f$ the graph of $f$ in $X\times X$.  Let $i:\Delta\rightarrow X\times X$ and $i':\Gamma_f\rightarrow X\times X$ be inclusions and let $l:X\rightarrow\Delta$ be the diagonal map and $l':X\rightarrow \Gamma_f$ defined as $x\mapsto (x,f(x))$. We equip $\Delta$ and $\Gamma_f$ with suitable orientations such that $l$ and $l'$ are diffeomorphisms of preserving orientations. By Theorem \ref{Kun1}, the fundamental class $[\Gamma_f]\in H_{\textrm{dR}}^n(X\times X)$ can be written as $\sum_{i,j} c_{ij}pr_1^*[\alpha_i]_\theta\cup pr_2^*[\beta_j]_{-\theta}$ for some $c_{ij}\in\mathbb{R}$. Set $f^*[\alpha_i]_\theta=\sum_ja_{ij}[\alpha_j]_\theta$.

On the one hand,
\begin{displaymath}
\int_{\Gamma_f} i'^*(pr_1^*\beta_i\wedge pr_2^*\alpha_j)=\int_X l'^*i'^*pr_1^*\beta_i\wedge l'^*i'^*pr_2^*\alpha_j=\int_X \beta_i\wedge f^*\alpha_j
=(-1)^{\textrm{deg}\alpha_i\textrm{deg}\beta_i}a_{ji}.
\end{displaymath}

On the other hand,
\begin{displaymath}
\begin{aligned}
\int_{\Gamma_f} i'^*(pr_1^*\beta_i\wedge pr_2^*\alpha_j)=&\int_{X\times X}[\Gamma_f]\wedge pr_1^*\beta_i\wedge pr_2^*\alpha_j\\

= &\sum_{k,l} c_{kl}(-1)^{(\textrm{deg}\beta_i+\textrm{deg}\alpha_j)\textrm{deg}\beta_l}\cdot\int_{X\times X}pr_1^*(\alpha_k\wedge \beta_i)\wedge pr_2^*(\alpha_j\wedge \beta_l)\\

= &(-1)^{(\textrm{deg}\beta_i+\textrm{deg}\alpha_j)\textrm{deg}\beta_j}c_{ij}.
\end{aligned}
\end{displaymath}
So $c_{ij}=(-1)^{\textrm{deg}\alpha_i\textrm{deg}\beta_i+(\textrm{deg}\beta_i+\textrm{deg}\alpha_j)\textrm{deg}\beta_j}a_{ji}$. The intersection number
\begin{displaymath}
\begin{aligned}
\Gamma_f \cdot \Delta=&\int_{X\times X}[\Gamma_f]\cup[\Delta]\\
=&(-1)^n\sum_{i,j}c_{ij}\int_\Delta i^*pr_1^*\alpha_i\wedge i^*pr_2^*\beta_j\\
= &(-1)^n\sum_{i,j}c_{ij}\int_X \alpha_i\wedge \beta_j\\
=&\sum_i (-1)^{\textrm{deg}\alpha_i}a_{ii}\\
=&\sum_p (-1)^p \textrm{tr}(f^*|_{H_\theta^p(X)})\\
=&L(f,\theta).
\end{aligned}
\end{displaymath}
Hence $L(f,\theta)$ is independent of $\theta$.
\end{proof}
\begin{remark}
$(1)$ $\chi(X,\theta)=\chi(X)$ was first proved in \cite{BK} by Atiyah-Singer index theorem.

$(2)$ Let $Z$ be a closed oriented submanifold (without boundary) of an oriented smooth manifold $X$ of dimension $n$ and $\emph{codim}Z=r$. Assume $i:Z\rightarrow X$ is the inclusion. In \cite{Dem},
$\int_Zi^*\omega$, for any $\omega\in\mathcal{A}_c^{n-r}(X)$, defines a current, denoted by $[Z]$. It is closed, hence defines a class in $H_{\emph{dR}}^r(X)$, also denoted by $[Z]$. In \cite{BT},
\emph{the Poincar\'{e} dual $[\eta_Z]$} is defined as $\int_Zi^*\omega=\int_X\omega\wedge\eta_Z$ for any $\emph{d}$-closed $\omega\in\mathcal{A}_c^{n-r}(X)$. Then, $[\eta_Z]=(-1)^{r(n-r)}[Z]$ in $H_{\emph{dR}}^r(X)$. In this article, we use the notation $[Z]\in H_{\emph{dR}}^r(X)$ and call it \emph{the fundamental class} of $Z$.
\end{remark}

Before considering another version of K\"{u}nneth formula, we give a lemma, which is frequently used in follows .
\begin{lemma}\label{ind}
Denote $\mathcal{P}(X)$ a statement on a smooth manifold $X$. Assume $\mathcal{P}$ satisfies  conditions:

$(i)$ \textbf{\emph{Local condition}}\emph{:} There exists a basis $\mathcal{U}$ of topology of $X$, such that, for any finite $U_1,..., U_l\in \mathcal{U}$, $\mathcal{P}(U_1\cap...\cap U_l)$ holds.

$(ii)$ \textbf{\emph{Disjoint condition}}\emph{:} Let $\{U_\alpha|\alpha\in\Lambda\}$ be a collection of disjoint open subsets of $X$. If $\mathcal{P}(U_\alpha)$ hold for all $\alpha\in\Lambda$, $\mathcal{P}(\bigcup_{\alpha\in\Lambda}U_\alpha)$ holds.

$(iii)$ \textbf{\emph{Mayer-Vietoris condition}}\emph{:} For open subsets $U$, $V$ of $X$, if $\mathcal{P}(U)$, $\mathcal{P}(V)$ and $\mathcal{P}(U\cap V)$ hold, then $\mathcal{P}(U\cup V)$ holds.\\
Then $\mathcal{P}(X)$ holds.
\end{lemma}
\begin{proof}
We first prove:

$(\textbf{*})$ For open subsets  $U_1$,\ldots,$U_r$ of $X$, if $\mathcal{P}(U_{i_1}\cap\ldots \cap U_{i_k})$ holds for any $1\leq i_1<\ldots<i_k\leq r$, then $\mathcal{P}(\bigcup_{i=1}^r U_i)$ holds.

Evidently, for $r=1$, it is obviously. Suppose $(\textbf{*})$ holds for $r$. For $r+1$, set $U'_1=U_1,\ldots, U'_{r-1}=U_{r-1}, U'_r=U_r\cup U_{r+1}$. Then $\mathcal{P}(U'_{i_1}\cap\ldots \cap U'_{i_k})$ holds for any $1\leq i_1<\ldots<i_k\leq r-1$. Moreover, by Mayer-Vietoris condition, $\mathcal{P}(U'_{i_1}\cap\ldots \cap U'_{i_{k-1}}\cap U'_r)$ also holds for any $1\leq i_1<\ldots<i_{k-1}\leq r-1$, since $\mathcal{P}(U_{i_1}\cap\ldots \cap U_{i_{k-1}}\cap U_r)$, $\mathcal{P}(U_{i_1}\cap\ldots \cap U_{i_{k-1}}\cap U_{r+1})$ and  $\mathcal{P}(U_{i_1}\cap\ldots \cap U_{i_{k-1}}\cap U_r\cap U_{r+1})$  hold. By inductive hypothesis, $\mathcal{P}(\bigcup_{i=1}^{r+1} U_i)=\mathcal{P}(\bigcup_{i=1}^{r} U'_i)$ holds. We proved $(\textbf{*})$.

Let $\mathcal{U}_\mathfrak{f}$ be the collection of open sets  which is the finite union of open sets in  $\mathcal{U}$.

$(\textbf{**})$ $\mathcal{P}(V)$ holds for any finite intersection $V$ of open sets in $\mathcal{U}_\mathfrak{f}$.

Actually, $V=\bigcap_{i=1}^s U_i$, where $U_i=\bigcup_{j=1}^{r_i}U_{ij}$ and $U_{ij}\in\mathcal{U}$. Then $V=\bigcup_{J\in\Lambda}U_J$, where $\Lambda=\{J=(j_1,...,j_s)|1\leq j_1\leq r_1,\ldots,1\leq j_s\leq r_s\}$ and $U_J=U_{1j_1}\cap...\cap U_{sj_s}$. For any $J_1,\ldots,J_t\in\Lambda$, $\mathcal{P}(U_{J_1}\cap \ldots\cap U_{J_t})$ holds by Local condition. Hence $\mathcal{P}(V)=\mathcal{P}(\bigcup_{J\in\Lambda}U_J)$ holds by $(\textbf{*})$.

By \cite{GHV}, p. 16, Prop. II, $X=V_1\cup...\cup V_l$, where $V_i$ is a countable disjoint union of open sets in  $\mathcal{U}_\mathfrak{f}$. Obviously, for any $1\leq i_1<\ldots<i_k\leq l$, $V_{i_1}\cap \ldots \cap V_{i_k}$ is a disjoint union of the finite intersection of open sets in $\mathcal{U}_\mathfrak{f}$. By Disjoint condition and $(\textbf{**})$,  $\mathcal{P}(V_{i_1}\cap \ldots \cap V_{i_k})$ holds. So $\mathcal{P}(X)$ holds by $(\textbf{*})$.
\end{proof}

Let $\pi:E\rightarrow X$ be a smooth fiber bundle on a smooth manifold $X$. Set
\begin{displaymath}
cv=\{Z\subseteq E|\mbox{ Z is closed in }E\mbox{ and }\pi|_Z:Z\rightarrow X\mbox{ is proper}\},
\end{displaymath}
where $Z\in cv$ is called \emph{a compact vertical support}.  Evidently, $Z\in cv$, if and only if, $\pi^{-1}(K)\cap Z$ is compact for any compact subset $K\subseteq X$. By \cite{Br}, IV, 5.3 $(b)$ and 5.5, $cv$ is a paracompactifying family of supports on $E$. The sheaf $\mathcal{C}_E^\infty$ of germs of smooth functions on $E$ is $cv$-soft (\cite{Br}, II, 9.4), so is $\mathcal{A}_E^p$ for any $p$ (\cite{Br}, II, 9.16). By \cite{Br}, II, 9.11, $\mathcal{A}_E^p$ is $cv$-acyclic. Denote $\mathcal{A}_{cv}^*(E)=\Gamma_{cv}(E,\mathcal{A}_E^*)$. Obviously, $\mathcal{A}_{c}^*(E)\subseteq\mathcal{A}_{cv}^*(E)\subseteq\mathcal{A}^*(E)$. If $X$ is compact, $\mathcal{A}_{c}^*(E)=\mathcal{A}_{cv}^*(E)$ and if the fibre of $E$ is compact, $\mathcal{A}_{cv}^*(E)=\mathcal{A}^*(E)$.

Let $(\mathcal{A}_E^\bullet,\textrm{d}_\Theta)\rightarrow \mathcal{I}^\bullet$ be an injective resolution of the complex of sheaves in the category of sheaves on $X$. By \cite{Br}, II, 4.1, it induces an isomorphism
\begin{displaymath}
H_{\Theta,cv}^*(E):= H^*(\mathcal{A}_{cv}^\bullet(E),\textrm{d}_{\Theta})\rightarrow H^*(\Gamma_{cv}(E,\mathcal{I}^\bullet))=H_{cv}^*(E,\underline{\mathbb{R}}_{E,\Theta}).
\end{displaymath}
We call $H_{\Theta,cv}^*(E)$ \emph{compact vertical Morse-Novikov cohomology} of $E$.
For $\textrm{d}_\Theta$-closed $\alpha\in\mathcal{A}^{*}(E)$, denote by $[\alpha]_{\Theta,cv}$ the class in $H_{\Theta,cv}^*(X)$.

Let $E_x$ be the fiber of $E$ over $x\in X$. For $\omega\in \mathcal{A}_{cv}^*(E)$, $\textrm{supp}(\omega|_{E_x})\subseteq E_x\cap \textrm{supp}\omega$ is compact, so $\omega|_{E_x}\in\mathcal{A}_c^*(E_x)$. For a closed $1$-form $\Theta$ on $E$, the restriction gives a morphism $\mathcal{A}_{cv}^*(E)\rightarrow\mathcal{A}_c^*(E_x)$, which induces $H_{\Theta,cv}^*(E)\rightarrow H_{\Theta|_{E_x},c}^*(E_x)$.

If $E$ is an oriented manifold, $\mathcal{A}_E^\bullet\hookrightarrow\mathcal{D}_E^{\prime\bullet}$ induces an isomorphism $H_{\Theta,cv}^*(E)\tilde{\rightarrow} H^*(\mathcal{D}_{cv}^{\prime\bullet}(E),\textrm{d}_{\Theta})$, where $\mathcal{D}_{cv}^{\prime*}(E)=\Gamma_{cv}(E,\mathcal{D}_E^{\prime*})$.
For $\textrm{d}_\Theta$-closed $T\in\mathcal{D}_{cv}^{\prime*}(E)$, denote by $[T]_{\Theta,cv}$  the class in  $H_{\Theta,cv}^*(E)$.

Moreover, assume $X$ and $E$ are both oriented manifolds. Let $i:X\rightarrow E$ be the inclusion and $r=\textrm{dim}E-\textrm{dim}X$. For $T\in\mathcal{D}^{\prime*}(X)$, $i_*T\in \mathcal{D}_{cv}^{\prime*+r}(E)$. So $i_*$ induce a morphism $i_*:H_{\Theta|_X}^*(X)\rightarrow H_{\Theta,cv}^{*+r}(E)$.

\begin{lemma}\label{fib-cv1}
Let $X$ be a smooth manifold and let $\theta$, $\mu$ be closed $1$-forms on $X$, $\mathbb{R}^n$ respectively. Set $\omega=pr_1^*\theta+pr_2^*\mu$, where $pr_1$, $pr_2$ are  projections from $X\times \mathbb{R}^n$ onto $X$, $\mathbb{R}^n$  respectively. The cartesian product
gives an isomorphism of graded vector spaces
\begin{displaymath}
H_{\theta}^*(X)\otimes_{\mathbb{R}}H_{\mu,c}^*(\mathbb{R}^n)\rightarrow H_{\omega,cv}^*(X\times \mathbb{R}^n),
\end{displaymath}
where $pr_1:X\times \mathbb{R}^n\rightarrow X$ is viewed as a smooth fiber bundle.
\end{lemma}
\begin{proof}
For any open set $U$ in $X$, set
\begin{displaymath}
\Psi_U:H_{\theta}^*(U)\otimes_{\mathbb{R}}H_{\mu,c}^*(\mathbb{R}^n)\rightarrow H_{\omega,cv}^*(U\times \mathbb{R}^n).
\end{displaymath}
We need to prove that $\Psi_X$ is an isomorphism. Denote the statement by $\mathcal{P}(X)$.

Since $\mathbb{R}^n$ is contractible,  $\mu=\textrm{d}f$ for a smooth function $f$ on $\mathbb{R}^n$. If $\theta|_U=\textrm{d}g$ is an exact form on $U$, we have a commutative diagram
\begin{displaymath}
\xymatrix{
 H_{\theta}^*(U)\otimes_{\mathbb{R}}H_{\mu,c}^*(\mathbb{R}^n) \ar[d]^{e^g\otimes e^f} \ar[r]^{\quad\Psi_U}& H_{\omega,cv}^*(U\times\mathbb{R}^n) \ar[d]^{e^{pr_1^*f+pr_2^*g}}\\
 H^*(U)\otimes_{\mathbb{R}}H_{c}^*(\mathbb{R}^n) \ar[r]^{\quad\times}&  H_{cv}^*(U\times\mathbb{R}^n), }
\end{displaymath}
where the two vertical maps are isomorphic. Let $\phi$ be a generator of $H_c^n(\mathbb{R}^n)=\mathbb{R}$. By \cite{BT}, Prop. 6.18, $pr_2^*\phi$ is the Thom class of the vector bundle $U\times \mathbb{R}^n$ over $U$. By \cite{BT}, Thm. 6.17 and 6.17.1, the second row is an isomorphism. Through the commutative diagram, $\Psi_U$ is isomorphic, i.e. $\mathcal{P}(U)$ holds.

Let $\mathcal{U}$ be a basis of topology of $X$ satisfying that $\theta|_U$ is exact for any $U\in\mathcal{U}$. Then $\theta|_{U_1\cap...\cap U_l}$ is an exact form, for $U_1$,..., $U_l\in\mathcal{U}$. So $\mathcal{P}(U_1\cap...\cap U_l)$ holds, hence $\mathcal{P}$ satisfies Local condition in Lemma \ref{ind}.

Let $\{U_\alpha|\alpha\in\Lambda\}$ be a collection of disjoint open subsets of $X$ and $\Psi_{U_\alpha}$  isomorphisms for all $\alpha\in\Lambda$. Clearly, $\Psi_{\bigcup_{\alpha\in\Lambda}U_\alpha}$ is just the direct product
\begin{displaymath}
\prod_\alpha\Psi_{U_\alpha}:\prod_\alpha \left(H_{\theta}^*(U_\alpha)\otimes_{\mathbb{R}}H_{\mu,c}^*(\mathbb{R}^n)\right)\rightarrow \prod_\alpha H_{\omega,cv}^*(U_\alpha\times \mathbb{R}^n),
\end{displaymath}
which implies that $\mathcal{P}(\bigcup_{\alpha\in\Lambda}U_\alpha)$ holds. So $\mathcal{P}$ satisfies Disjoint condition in Lemma \ref{ind}.

For open subsets $U$ and $V$ in $X$, we have a commutative diagram of long exact sequences
\begin{displaymath}
\tiny{
\xymatrix{
   \left(H_{\theta}^*(U\cap V)\otimes_{\mathbb{R}}H_{\mu,c}^*(\mathbb{R}^n)\right)^{p-1}\ar[d]^{\Psi_{U\cap V}} \ar[r] &\left(H_{\theta}^*(U\cup V)\otimes_{\mathbb{R}}H_{\mu,c}^*(\mathbb{R}^n)\right)^p\ar[d]^{\Psi_{U\cup V}}\ar[r]&\left(H_{\theta}^*(U)\otimes_{\mathbb{R}}H_{\mu,c}^*(\mathbb{R}^n)\right)^p\oplus \left(H_{\theta}^*(V)\otimes_{\mathbb{R}}H_{\mu,c}^*(\mathbb{R}^n)\right)^p\ar[d]^{(\Psi_{U},\Psi_{V})}...\\
 H_{\omega,cv}^{p-1}((U\cap V)\times \mathbb{R}^n)   \ar[r] & H_{\omega,cv}^p((U\cup V)\times \mathbb{R}^n)\ar[r]&H_{\omega,cv}^p(U\times \mathbb{R}^n)\oplus H_{\omega,cv}^p(V\times \mathbb{R}^n)... }.}
\end{displaymath}
If $\Psi_{U}$,$\Psi_{V}$  and $\Psi_{U\cap V}$ are isomorphisms, then $\Psi_{U\cup V}$ is isomorphic by Five Lemma. So $\mathcal{P}$ satisfies Mayer-Vietoris condition in Lemma \ref{ind}.

By Lemma \ref{ind}, $\mathcal{P}(X)$ holds. We get the conclusion.
\end{proof}

Let $pr_1:X\times Y\rightarrow X$ be a trivial smooth fiber bundle and $\Theta$ a closed $1$-form on $X\times Y$. If $U$ and $V$  are open subsets in $Y$, there is an exact sequence of complexes
\begin{displaymath}
\begin{aligned}
\small{
\xymatrix{
0\ar[r] &\mathcal{A}_{cv}^\bullet(X\times(U\cap V))\ar[r]^{P\qquad} &\mathcal{A}_{cv}^\bullet(X\times U)\oplus \mathcal{A}_{cv}^\bullet(X\times V)\ar[r]^{\qquad Q} &\mathcal{A}_{cv}^\bullet(X\times (U\cup V))\ar[r]&0
},}
\end{aligned}
\end{displaymath}
where all the differentials are $\textrm{d}_\Theta$, $P(\alpha)=(\alpha, -\alpha)$, $Q(\beta, \gamma)=\beta+\gamma$ and  everything is  extended by zero. It can be proved as the exact sequence of complexes $\mathcal{A}_c^\bullet$, referring to \cite{BT}, Prop. 2.7.

Assume $Y=\bigsqcup_{\alpha\in I} Y_\alpha$ is a disjoint union of smooth manifolds. For  a form $\omega$ on $X\times Y$ and any compact subset $K\subseteq X$, $(K\times Y)\cap \textrm{supp}\omega$ is compact if and only if, $(K\times Y_\alpha)\cap \textrm{supp}\omega$ is nonempty for only finite $\alpha\in I$ and they are compact. So
\begin{displaymath}
\mathcal{A}_{cv}^*(X\times Y)=\bigoplus_{\alpha\in I}\mathcal{A}_{cv}^*(X\times Y_\alpha),
\end{displaymath}
where $X\times Y$ and $X\times Y_\alpha$ are viewed as smooth fiber bundles over $X$.

\begin{theorem}[K\"{u}nneth formula II]\label{Kun2}
Let $X$ and $Y$ be smooth manifolds and let $\theta$, $\mu$ be closed $1$-forms on $X$, $Y$ respectively. Set $\omega=pr_1^*\theta+pr_2^*\mu$, where $pr_1$, $pr_2$ are projections from $X\times Y$ onto $X$, $Y$ respectively. The cartesian product gives an isomorphism of graded vector spaces
\begin{displaymath}
H_{\theta}^*(X)\otimes_{\mathbb{R}}H_{\mu,c}^*(Y)\rightarrow H_{\omega,cv}^*(X\times Y),
\end{displaymath}
where $pr_1:X\times Y\rightarrow X$ is viewed as a smooth fiber bundle.
\end{theorem}
\begin{proof}
For any open set $U$ in $Y$, set
\begin{displaymath}
\Psi_U:H_{\theta}^*(X)\otimes_{\mathbb{R}}H_{\mu,c}^*(U)\rightarrow H_{\omega,cv}^*(X\times U).
\end{displaymath}
We want to prove $\Psi_Y$ is an isomorphism and the statement is called $\mathcal{P}(Y)$ here.

Let $\mathcal{U}$ be a good covering of $Y$ such that it also a basis of topology of $Y$. Then, for $U_1$,..., $U_l\in\mathcal{U}$, $U_1\cap...\cap U_l$ is diffeomorphic to $\mathbb{R}^n$, where $n=\textrm{dim}Y$. By Lemma \ref{fib-cv1}, $\mathcal{P}(U_1\cap...\cap U_l)$ holds, so $\mathcal{P}$ satisfies Local condition in Lemma \ref{ind}.

Let $\{U_\alpha|\alpha\in\Lambda\}$ be a collection of disjoint open subsets of $Y$ and $\Psi_{U_\alpha}$  isomorphisms for all $\alpha\in\Lambda$. Clearly, $\Psi_{\bigcup_{\alpha\in\Lambda}U_\alpha}$ is the direct sum
\begin{displaymath}
\bigoplus_\alpha\Psi_{U_\alpha}: \bigoplus_\alpha \left(H_{\theta}^*(X)\otimes_{\mathbb{R}}  H_{\mu,c}^*(U_\alpha)\right)\rightarrow \bigoplus_\alpha H_{\omega,cv}^*(X\times U_\alpha),
\end{displaymath}
which implies $\mathcal{P}(\bigcup_{\alpha\in\Lambda}U_\alpha)$ holds. So $\mathcal{P}$ satisfies Disjoint condition in Lemma \ref{ind}.

For open subsets $U$ and $V$ in $Y$, the diagram of exact sequences
\begin{displaymath}
\tiny{
\xymatrix{
   ...\left(H_{\theta}^*(X)\otimes_{\mathbb{R}}  H_{\mu,c}^*(U)\right)^p\oplus \left(H_{\theta}^*(X)\otimes_{\mathbb{R}}  H_{\mu,c}^*(V)\right)^p \ar[d]^{(\Psi_U,\Psi_V)} \ar[r] &\left(H_{\theta}^*(X)\otimes_{\mathbb{R}}  H_{\mu,c}^*(U\cup V)\right)^p\ar[d]^{\Psi_{U\cup V}}\ar[r]&\left(H_{\theta}^*(X)\otimes_{\mathbb{R}}  H_{\mu,c}^*(U\cap V)\right)^{p+1}\ar[d]^{\Psi_{U\cap V}}...\\
 ... H_{\omega,cv}^p(X\times U)\oplus H_{\omega,cv}^p(X\times V)     \ar[r] & H_{\omega,cv}^p(X\times (U\cup V))\ar[r]&H_{\omega,cv}^{p+1}(X\times (U\cap V))... }}
\end{displaymath}
is commutative. If $\Psi_{U}$,$\Psi_{V}$  and $\Psi_{U\cap V}$ are isomorphisms, then $\Psi_{U\cup V}$ is isomorphic by Five Lemma. So $\mathcal{P}$ satisfies Mayer-Vietoris condition in Lemma \ref{ind}.

By Lemma \ref{ind}, $\mathcal{P}(Y)$ holds. We complete the proof.
\end{proof}

\subsection{A theorem of Leray-Hirsch type}
\begin{lemma}\label{cv}
Let $F$ be a smooth manifold and $K: \mathcal{A}^*(\mathbb{R}^n\times F)\rightarrow  \mathcal{A}^{*-1}(\mathbb{R}^n\times F)$ defined as \emph{(\ref{operator})}. If $\alpha\in \mathcal{A}_{cv}^*(\mathbb{R}^n\times F)$, then $K(\alpha)\in \mathcal{A}_{cv}^{*-1}(\mathbb{R}^n\times F)$.
\end{lemma}
\begin{proof}
Let $pr_{23}:\mathbb{R}\times\mathbb{R}^n\times F\rightarrow \mathbb{R}^n\times F$ be the projection map and
\begin{displaymath}
C=pr_{23}\left(g^{-1}(\textrm{supp}\alpha)\cap ([0,1]\times\mathbb{R}^n\times F)\right).
\end{displaymath}
Obviously, $C$ is closed in $\mathbb{R}^n\times F$ and $g^{-1}(\textrm{supp}\alpha)\cap ([0,1]\times\mathbb{R}^n\times F)\subseteq [0,1]\times C$. By \cite{GHV}, p. 179, Remark, $\textrm{supp}\left(i(\partial/\partial t)g^*\alpha\right)\cap ([0,1]\times\mathbb{R}^n\times F)\subseteq [0,1]\times C$.

So $\textrm{supp}K(\alpha)\subseteq C$.

For any compact subset $L\subseteq \mathbb{R}^n$, let $h:[0,1]\times L\rightarrow\mathbb{R}^n$ map $(t, x)$ to $(1-t)x$. Then $h\times id_F$ is the restriction of $g$ on $[0,1]\times L\times F$. Choose a closed ball $B\supseteq L$ with finite volume, which contains original point as its center. Then
\begin{displaymath}
\begin{aligned}
\pi^{-1}(L)\cap \textrm{supp}K(\alpha)\subseteq& (L\times F)\cap pr_{23}\left(g^{-1}(\textrm{supp}\alpha)\cap ([0,1]\times\mathbb{R}^n\times F)\right)\\
=&pr_{23}\left(([0,1]\times L\times F)\cap g^{-1}(\textrm{supp}\alpha)\right)\\
\subseteq &pr_{23}\left((h\times id_F)^{-1}\left((B\times F)\cap\textrm{supp}\alpha\right)\right).
\end{aligned}
\end{displaymath}
Clearly, $h\times id_F$ is proper and $(B\times F)\cap\textrm{supp}\alpha$ is compact,  so $\pi^{-1}(L)\cap \textrm{supp}K(\alpha)$ is compact. We complete the proof.
\end{proof}

\begin{theorem}\label{L-H}
Let $\pi:E\rightarrow X$ be a smooth fiber bundle over a smooth manifold $X$ and let $\theta$, $\Omega$ be closed $1$-forms on $X$, $E$ respectively. Set $\tilde{\theta}=\pi^*\theta$.

$(1)$ Assume there exist classes $e_1,\dots,e_r$ of pure degrees in $ H_{\Omega}^*(E)$, such that, for every $x\in X$,  their restrictions $e_1|_{E_x},\dots,e_r|_{E_x}$ freely linearly generate $H_{\Omega|_{E_x}}^*(E_x)$. Then, $\pi^*(\bullet)\wedge\bullet$ gives an isomorphism of graded vector spaces
\begin{displaymath}
H_{\theta}^*(X)\otimes_{\mathbb{R}} \emph{span}_{\mathbb{R}}\{e_1, ..., e_r\} \tilde{\rightarrow} H_{\tilde{\theta}+\Omega}^*(E).
\end{displaymath}

$(2)$ Assume there exist classes $e_1,\dots,e_r$ of pure degrees in $ H_{\Omega,cv}^*(E)$, such that, for every $x\in X$,  their restrictions $e_1|_{E_x},\dots,e_r|_{E_x}$ freely linearly  generate $H_{\Omega|_{E_x},c}^*(E_x)$. Then, $\pi^*(\bullet)\wedge\bullet$ gives isomorphisms of graded vector spaces
\begin{displaymath}
H_{\theta,c}^*(X)\otimes_{\mathbb{R}} \emph{span}_{\mathbb{R}}\{e_1, ..., e_r\} \tilde{\rightarrow} H_{\tilde{\theta}+\Omega,c}^*(E)
\end{displaymath}
and
\begin{displaymath}
H_\theta^*(X)\otimes_{\mathbb{R}} \emph{span}_{\mathbb{R}}\{e_1, ..., e_r\} \tilde{\rightarrow} H_{\tilde{\theta}+\Omega,cv}^*(E).
\end{displaymath}

$(3)$ Assume there exist classes $e_1,\dots,e_r$ of pure degrees in $ H_{\Omega,c}^*(E)$, such that, for every $x\in X$,  their restrictions $e_1|_{E_x},\dots,e_r|_{E_x}$ freely linearly  generate $H_{\Omega|_{E_x},c}^*(E_x)$. Then, $\pi^*(\bullet)\wedge\bullet$ gives isomorphisms of graded vector spaces
\begin{displaymath}
H_{\theta,c}^*(X)\otimes_{\mathbb{R}} \emph{span}_{\mathbb{R}}\{e_1, ..., e_r\} \tilde{\rightarrow} H_{\tilde{\theta}+\Omega,c}^*(E).
\end{displaymath}
\end{theorem}
\begin{proof}
Let $F$ be the general fiber of $E$ and $\textrm{dim}X=n$.

$(1)$ For any open set $U\subseteq X$, set
\begin{displaymath}
\Psi_U=\pi^*(\bullet)\cup \bullet:H_{\theta}^*(U)\otimes_{\mathbb{R}} \textrm{span}_{\mathbb{R}}\{e_1, ..., e_r\} \tilde{\rightarrow} H_{\tilde{\theta}+\Omega}^*(E_U),
\end{displaymath}
where $E_U=\pi^{-1}(U)$. Our goal is to prove that $\Psi_X$ is an isomorphism, which is called the statement $\mathcal{P}(X)$.

For an open set $U$ of $X$, suppose that $U$ is diffeomorphic to $\mathbb{R}^n$ and $E_U$ is smooth trivial. Let $\varphi_U: U\times F\rightarrow E_U$ be a smooth trivialization and let $pr_1$, $pr_2$ be projections from $U\times F$ to $U$, $F$ respectively, which satisfy $\pi\circ \varphi_U=pr_1$. Fixed a point $o\in U$, set $j_o:F\rightarrow U\times F$ as $f\mapsto (o,f)$. Clearly, $pr_2\circ j_o=\textrm{id}_F$ and $i_o:=\varphi_U\circ j_o$ is the embedding $F\hookrightarrow E_U$ of the fiber $E_o$ over $o$ into $E_U$. Set $e_i^{\prime}=(\varphi_U)^*e_i$ in $H_{\varphi_U^*\Omega}^*(U\times F)$, $i=1,...,r$. Then $j_o^*e_i^{\prime}=i_o^*e_i$ for any $i$. $i_o^*e_1,...,i_o^*e_r$ are linearly independent, so are $e'_1,...,e'_r$. Then, mapping $e_i$ to $e'_i$ for $i=1,...,r$ give a natural linear isomorphism $\textrm{span}_{\mathbb{R}}\{e_1, ..., e_r\}\rightarrow\textrm{span}_{\mathbb{R}}\{e_1^{\prime}, ..., e_r^{\prime}\}$. Let $\{\beta_i\}_{i=1}^r$ be a system of forms of pure degrees in $\mathcal{A}^*(U\times F)$, such that $e'_i=[\beta_i]_{\varphi_U^*\Omega}$ for $1\leq i\leq r$. By formulas (\ref{htp1}) and (\ref{htp2}), we can choose a smooth function $u$ on $U\times F$ such that
\begin{displaymath}
pr_2^*i_0^*\Omega-\varphi_U^*\Omega=pr_2^*j_0^*\varphi_U^*\Omega-\varphi_U^*\Omega=\textrm{d}u
\end{displaymath}
and
\begin{displaymath}
\beta_i-e^{u}\cdot pr_2^*j_0^*\beta_i=\textrm{d}_{\varphi_U^*\Omega}\gamma_i,
\end{displaymath}
for some $\gamma_i\in \mathcal{A}^*(U\times F)$. Then
\begin{displaymath}
e^{-u}\cdot pr_1^*\alpha\wedge \beta_i=pr_1^*\alpha\wedge pr_2^*j_0^*\beta_i+(-1)^{\textrm{deg}\alpha}\textrm{d}_{pr_1^*\theta+pr_2^*i_0^*\Omega}(e^{-u}\cdot pr_1^*\alpha\wedge\gamma_i),
\end{displaymath}
for any $\textrm{d}_\theta$-closed form $\alpha\in\mathcal{A}^*(U)$. We obtain a commutative diagram
\begin{displaymath}
\xymatrix{
H_{\theta}^*(U)\otimes_{\mathbb{R}} \textrm{span}_{\mathbb{R}}\{e_1, ..., e_r\}\ar[d]^{\Psi_U} \ar[r]^{\cong}& H_{\theta}^*(U)\otimes_{\mathbb{R}} \textrm{span}_{\mathbb{R}}\{e'_1, ..., e'_r\}\ar[d]^{pr_1^*(\bullet)\cup \bullet} \ar[r]^{\quad \textrm{id}\otimes j_0^*}& H_{\theta}^*(U)\otimes_{\mathbb{R}} H_{i_0^*\Omega}^*(F) \ar[d]^{\times}\\
H_{\tilde{\theta}+\Omega}^*(E_U) \ar[r]^{\varphi_U^*}& H_{pr_1^*\theta+\varphi_U^*\Omega}^*(U\times F) \ar[r]^{e^{-u}}&  H_{pr_1^*\theta+pr_2^*i_0^*\Omega}^*(U\times F), }
\end{displaymath}
which implies $\Psi_U$ is isomorphic by K\"{u}nneth formual I. So $\mathcal{P}(U)$ holds.

Let $\mathcal{U}$ be a good covering of $X$ such that it is a basis of topology of $X$ and $E_U$ is smooth trivial for any $U\in\mathcal{U}$. For $U_1$,..., $U_l\in\mathcal{U}$, $U_1\cap...\cap U_l$ is diffeomorphic to $\mathbb{R}^n$ and $E_{U_1\cap...\cap U_l}$ is smooth trivial, so $\Psi_{U_1\cap...\cap U_l}$ is an isomorphism. Hence $\mathcal{P}$ satisfies Local condition in Lemma \ref{ind}.

Let $\{U_\alpha|\alpha\in\Lambda\}$ be a collection of disjoint open subsets of $X$ and $\Psi_{U_\alpha}$  isomorphisms for all $\alpha\in\Lambda$. Clearly, $\Psi_{\bigcup_{\alpha\in\Lambda}U_\alpha}$ is  the direct product
\begin{displaymath}
\prod_\alpha\Psi_{U_\alpha}:\prod_\alpha \left(H_{\theta}^*(U_\alpha)\otimes_{\mathbb{R}}\textrm{span}_{\mathbb{R}}\{e_1, ..., e_r\}\right)\rightarrow \prod_\alpha H_{\tilde{\theta}+\Omega}^*(E_{U_\alpha}).
\end{displaymath}
Then $\mathcal{P}(\bigcup_{\alpha\in\Lambda}U_\alpha)$ holds. So $\mathcal{P}$ satisfies Disjoint condition in Lemma \ref{ind}.

Put $L^*=\textrm{span}_{\mathbb{R}}\{e_1, ..., e_r\}$, which is a graded vector space. We have a commutative diagram of long exact sequences
\begin{displaymath}
\small{
\xymatrix{
   ... \left(H_{\theta}^*(U\cap V)\otimes_{\mathbb{R}}L^*\right)^{p-1}\ar[d]^{\Psi_{U\cap V}} \ar[r] &\left(H_{\theta}^*(U\cup V)\otimes_{\mathbb{R}}L^*\right)^p\ar[d]^{\Psi_{U\cup V}}\ar[r]&\left(H_{\theta}^*(U)\otimes_{\mathbb{R}}L^*\right)^p\oplus \left(H_{\theta}^*(V)\otimes_{\mathbb{R}}L^*\right)^p\ar[d]^{(\Psi_U,\Psi_V)}...\\
 ...   H_{\tilde{\theta}+\Omega}^{p-1}(E_{U\cap V})   \ar[r] & H_{\tilde{\theta}+\Omega}^p(E_{U\cup V})\ar[r]&H_{\tilde{\theta}+\Omega}^p(E_U)\oplus H_{\tilde{\theta}+\Omega}^p(E_V)... }.}
\end{displaymath}
If $\Psi_U$, $\Psi_V$ and $\Psi_{U\cap V}$ are isomorphisms, then $\Psi_{U\cup V}$ is an isomorphism by Five Lemma.  So $\mathcal{P}$  satisfies Mayer-Vietoris condition.

By Lemma \ref{ind}, we proved $(1)$.

$(2)$  For any open set $U\subseteq X$, set
\begin{displaymath}
\Psi_U=\pi^*(\bullet)\cup \bullet:H_{\theta,c}^*(U)\otimes_{\mathbb{R}} \textrm{span}_{\mathbb{R}}\{e_1, ..., e_r\} \tilde{\rightarrow} H_{\tilde{\theta}+\Omega,c}^*(E_U).
\end{displaymath}
We would like to prove that $\Psi_X$ is an isomorphism. Here, this statement is called  $\mathcal{P}(X)$.

 For an open set $U$ of $X$, suppose that $E_U$ is smooth trivial. Assume  $\varphi_U$, $pr_1$, $pr_2$, $o$, $j_o$ and $i_o$ are defined as them in the proof of $(1)$.  Set $e_i^{\prime}=(\varphi_U)^*e_i$ in $H_{\varphi_U^*\Omega,cv}^*(U\times F)$, $i=1,...,r$. Similar with the proof of $(1)$, $\textrm{span}_{\mathbb{R}}\{e_1, ..., e_r\}\tilde{\rightarrow}\textrm{span}_{\mathbb{R}}\{e_1^{\prime}, ..., e_r^{\prime}\}$. Let $\{\beta_i\}_{i=1}^r$ be a system of forms of pure degrees in $\mathcal{A}_{cv}^*(U\times F)$, such that $e'_i=[\beta_i]_{\varphi_U^*\Omega,cv}$ for $1\leq i\leq r$.  By formulas (\ref{htp1}), (\ref{htp2}) and Lemma \ref{cv},, we can choose a smooth function $u$ on $U\times F$ such that
\begin{displaymath}
pr_2^*i_0^*\Omega-\varphi_U^*\Omega=\textrm{d}u
\end{displaymath}
and
\begin{displaymath}
\beta_i-e^{u}\cdot pr_2^*j_0^*\beta_i=\textrm{d}_{\varphi_U^*\Omega}\gamma_i,
\end{displaymath}
for some $\gamma_i\in \mathcal{A}_{cv}^*(U\times F)$. Then
\begin{displaymath}
e^{-u}\cdot pr_1^*\alpha\wedge \beta_i=pr_1^*\alpha\wedge pr_2^*j_0^*\beta_i+(-1)^{\textrm{deg}\alpha}\textrm{d}_{pr_1^*\theta+pr_2^*i_0^*\Omega}(e^{-u}\cdot pr_1^*\alpha\wedge\gamma_i),
\end{displaymath}
for any $\textrm{d}_\theta$-closed form $\alpha\in\mathcal{A}_{c}^*(U)$. Since $pr_1^*\alpha\wedge\gamma_i$ has compact support, we get a commutative diagram
\begin{displaymath}
\xymatrix{
H_{\theta,c}^*(U)\otimes_{\mathbb{R}} \textrm{span}_{\mathbb{R}}\{e_1, ..., e_r\}\ar[d]^{\Psi_U} \ar[r]^{\cong}& H_{\theta,c}^*(U)\otimes_{\mathbb{R}} \textrm{span}_{\mathbb{R}}\{e'_1, ..., e'_r\}\ar[d]^{pr_1^*(\bullet)\cup \bullet} \ar[r]^{\quad \textrm{id}\otimes j_0^*}& H_{\theta,c}^*(U)\otimes_{\mathbb{R}} H_{i_0^*\Omega,c}^*(F) \ar[d]^{\times}\\
H_{\tilde{\theta}+\Omega,c}^*(E_U) \ar[r]^{\varphi_U^*}& H_{pr_1^*\theta+\varphi_U^*\Omega,c}^*(U\times F) \ar[r]^{e^{-u}}&  H_{pr_1^*\theta+pr_2^*i_0^*\Omega,c}^*(U\times F), }
\end{displaymath}
which implies $\Psi_U$ is isomorphic by K\"{u}nneth formual I. So $\mathcal{P}(U)$ holds.

Let $\mathcal{U}$ be a basis of topology of $X$, such that, $E_U$ is smooth trivial for any $U\in\mathcal{U}$. For $U_1$,..., $U_l\in\mathcal{U}$,  $E_{U_1\cap...\cap U_l}$ is smooth trivial, so $\Psi_{U_1\cap...\cap U_l}$ is an isomorphism. Hence $\mathcal{P}$ satisfies Local condition in Lemma \ref{ind}.

Let $\{U_\alpha|\alpha\in\Lambda\}$ be a collection of disjoint open subsets of $X$ and $\Psi_{U_\alpha}$  isomorphisms for all $\alpha\in\Lambda$. Obviously, $\Psi_{\bigcup_{\alpha\in\Lambda}U_\alpha}$ is just the direct sum
\begin{displaymath}
\bigoplus_\alpha\Psi_{U_\alpha}:\bigoplus_\alpha \left(H_{\theta,c}^*(U_\alpha)\otimes_{\mathbb{R}}\textrm{span}_{\mathbb{R}}\{e_1, ..., e_r\}\right)\rightarrow \bigoplus_\alpha H_{\tilde{\theta}+\Omega,c}^*(E_{U_\alpha}).
\end{displaymath}
It is an isomorphism. Hence $\mathcal{P}$ satisfies Disjoint condition.

Put $L^*=\textrm{span}_{\mathbb{R}}\{e_1, ..., e_r\}$, which is a graded vector space. There is a commutative diagram of long exact sequences
\begin{displaymath}
\small{
\xymatrix{
   ...\left(H_{\theta,c}^*(U)\otimes_{\mathbb{R}}L^*\right)^p\oplus \left(H_{\theta,c}^*(V)\otimes_{\mathbb{R}}L^*\right)^p \ar[d]^{(\Psi_U,\Psi_V)} \ar[r] &\left(H_{\theta,c}^*(U\cup V)\otimes_{\mathbb{R}}L^*\right)^p\ar[d]^{\Psi_{U\cup V}}\ar[r]&\left(H_{\theta,c}^*(U\cap V)\otimes_{\mathbb{R}}L^*\right)^{p+1}\ar[d]^{\Psi_{U\cap V}}...\\
 ... H_{\tilde{\theta}+\Omega,c}^p(E_U)\oplus H_{\tilde{\theta}+\Omega,c}^p(E_V)     \ar[r] & H_{\tilde{\theta}+\Omega,c}^p(E_{U\cup V})\ar[r]&H_{\tilde{\theta}+\Omega,c}^{p+1}(E_{U\cap V})... }.}
\end{displaymath}
If $\Psi_U$, $\Psi_V$ and $\Psi_{U\cap V}$ are isomorphisms, then $\Psi_{U\cup V}$ is an isomorphism by Five Lemma.  So $\mathcal{P}$  satisfies Mayer-Vietoris condition.

By Lemma \ref{ind}, we get the first part of $(2)$.

For the second part of $(2)$, if $U$ is diffeomorphic to $\mathbb{R}^n$ and $E_U$ is smooth trivial for an open set $U$ of $X$,  we have a commutative diagram
\begin{displaymath}
\xymatrix{
H_{\theta}^*(U)\otimes_{\mathbb{R}} \textrm{span}_{\mathbb{R}}\{e_1, ..., e_r\}\ar[d]^{\Psi_U} \ar[r]^{\cong}& H_{\theta}^*(U)\otimes_{\mathbb{R}} \textrm{span}_{\mathbb{R}}\{e'_1, ..., e'_r\}\ar[d]^{pr_1^*(\bullet)\cup \bullet} \ar[r]^{\quad \textrm{id}\otimes j_0^*}& H_{\theta}^*(U)\otimes_{\mathbb{R}} H_{i_0^*\Omega,c}^*(F) \ar[d]^{\times}\\
H_{\tilde{\theta}+\Omega,cv}^*(E_U) \ar[r]^{\varphi_U^*}& H_{pr_1^*\theta+\varphi_U^*\Omega,cv}^*(U\times F) \ar[r]^{e^{-u}}&  H_{pr_1^*\theta+pr_2^*i_0^*\Omega,cv}^*(U\times F), }
\end{displaymath}
where $\varphi_U$, $pr_1$, $pr_2$, $o$, $j_o$, $i_o$ and $e_i'$ are defined as them in the proof of the first part of $(2)$. From the commutative diagram, $\Psi_U$ is isomorphic by K\"{u}nneth formual II. The rest of the proof is the same with that of $(1)$, except that we use $H_{\tilde{\theta}+\Omega,cv}^*(E)$ instead of $H_{\tilde{\theta}+\Omega}^*(E)$.

$(3)$ Let $\hat{e}_1,\ldots,\hat{e}_r$ be the image of $e_1,\ldots,e_r$ under the natural map $H_{\Omega,c}^*(E)\rightarrow H_{\Omega,cv}^*(E)$. By the hypothesis,  it gives an isomorphism $\textrm{span}_{\mathbb{R}}\{e_1, ..., e_r\}\tilde{\rightarrow}\textrm{span}_{\mathbb{R}}\{\hat{e}_1, ..., \hat{e}_r\}$, which immediately implies $(3)$ by $(2)$.
\end{proof}
\begin{remark}
In \cite{Ha}, H. A. Haddou proved $(1)$ in above theorem  for the case that $X$ has a finite good covering and $\Omega=0$.
\end{remark}

Suppose $\pi:E\rightarrow X$ is an oriented smooth vector bundle of rank $r$ on a $($unnecessarily orientable$)$ smooth manifold $X$ and $\omega\in \mathcal{A}_{cv}^p(E)$. For a chart $U$ on $X$ satisfying $E|_U$ is trivial, let $(x_1,...,x_n;t_1,...,t_r)$ be the local coordinates of $E$ such that $\textrm{d}t_1\wedge...\wedge \textrm{d}t_r$ gives the orientation of $E$. If $\omega=\sum_{|I|+|J|=p} f_{I,J}(t,x)\textrm{d}t_I\wedge \textrm{d}x_J$ on $E|_U$, then
\begin{displaymath}
\sum_{|J|=p-r}\left(\int_{\mathbb{R}^r}f_{1...r,J}(t,x)\textrm{d}t_1\wedge...\wedge \textrm{d}t_r\right)\textrm{d}x_J
\end{displaymath}
define a $(p-r)$-form on $U$. For various charts of $X$, these forms give a global smooth form on $X$, denoted by $\pi_*\omega$.

\begin{remark}
$\pi_*\omega$ defined here coincides with that in \cite{BT} $($$\emph{cf. p.}$ $61$$)$, up to a sign $(-1)^{r(p-r)}$. Moreover, if $X$ is oriented,  $\pi_*\omega$ defined here  is exactly the pushout of $\omega$ as currents $($$\emph{cf. \cite{Dem}, p.}$ $18$, $2.16$$)$.
\end{remark}
By Thom isomorphism theorem,
$\pi_*:H_{cv}^*(E)\rightarrow H_{\textrm{dR}}^{*-r}(X)$ is an isomorphism. Let $\Phi\in\mathcal{A}_{cv}^r(E)$ satisfy $\pi_*[\Phi]_{cv}=1$ in $H_{\textrm{dR}}^0(X)$. $[\Phi]_{cv}\in H_{cv}^r(E)$ is called \emph{Thom class} of $E$. Evidently, $\pi_*\Phi=1$ in $\mathcal{A}^0(X)$.
\begin{corollary}[Thom Isomorphism]\label{is}
Let $\pi:E\rightarrow X$  be an oriented smooth vector bundle of rank $r$ on a smooth manifold $X$. Assume $\theta$ is a closed $1$-form on $X$. Then, $\Phi\wedge \pi^*(\bullet)$ gives isomorphisms
\begin{displaymath}
H_{\theta,c}^{*-r}(X)\tilde{\rightarrow} H_{\tilde{\theta},c}^*(E)
\end{displaymath}
and
\begin{displaymath}
H_{\theta}^{*-r}(X)\tilde{\rightarrow} H_{\tilde{\theta},cv}^*(E).
\end{displaymath}
which have the inverse isomorphism $\pi_*$. Moreover, if $X$ is oriented, they coincide with the pushout $i_*$.
\end{corollary}
\begin{proof}
By \cite{BT}, Prop. 6.18, the restriction $[\Phi]|_{E_x}$ is a generator of $H_{\textrm{dR},c}^*(E_x)$. By Theorem \ref{L-H} $(2)$, $\Phi\wedge \pi^*(\bullet)$ gives isomorphisms. For $\alpha\in\mathcal{A}^*(X)$, $\pi_*(\Phi\wedge\pi^*\alpha)=\alpha$, so $\pi_*$ is their inverse isomorphisms.

If $X$ is oriented, $i_*$ is defined well and $\pi_*i_*=\textrm{id}$. So $i_*=\pi_*^{-1}=\Phi\wedge\pi^*$.
\end{proof}
\section{Bimeromorphic formulas}
Let $X$ be a complex manifold, $\mathcal{A}_{X,\mathbb{C}}^k=\mathcal{A}_X^k\otimes_{\mathbb{R}}\mathbb{C}$ and $\mathcal{D}_{X,\mathbb{C}}^{'k}=\mathcal{D}_X^{'k}\otimes_{\mathbb{R}}\mathbb{C}$. For a complex closed $1$-form $\theta$, we can define $d_\theta$, $H_\theta^*(X)$ and $\underline{\mathbb{C}}_{X,\theta}$ as them in the real case. All the conclusions in this article also hold for the complex case. In follows, we still consider the real closed $1$-form $\theta$ and still call it closed $1$-form.
\subsection{Two blow-up formulas }
Yang and Zhao \cite{YZ} proved a formula of blow-up  for the compact locally conformal K\"{a}hler manifold with $\theta|_Y=0$, which generalized Voisin's result on compact K\"{a}hler manifolds, seeing \cite{Vo}. We will establish two formulas of blow-up for Morse-Novikov cohomology on  general complex manifolds. Moreover, we will write out the isomorphisms explicitly.
\begin{lemma}\label{1}
Let $\pi:E\rightarrow X$ be a smooth vector bundle of rank $r$ on a smooth manifold $X$ and $i:X\rightarrow E$ the inclusion of zero section.  Assume $\Theta$ is a closed $1$-form on $E$. Then, $i^*:H_{\Theta}^*(E)\tilde{\rightarrow} H_{\Theta|_X}^*(X)$ is an isomorphism. Moreover, if $X$ and $E$ are oriented, then $i_*: H_{\Theta|_X,c}^*(X)\tilde{\rightarrow}H_{\Theta,c}^{*+r}(E)$ and $i_*: H_{\Theta|_X}^*(X)\tilde{\rightarrow}H_{\Theta,cv}^{*+r}(E)$ are also isomorphisms.
\end{lemma}
\begin{proof}
Set $e_1=1\in H_{\textrm{dR}}^0(E)$. Then $e_1|_{E_x}=1$ generates $H_{\textrm{dR}}^*(E_x)$ for any $x\in X$. By Theorem \ref{L-H} $(1)$, $\pi^*:H_{\Theta|_X}^*(X)\rightarrow H_{\pi^*(\Theta|_X)}^*(E)$ is an isomorphism. Since $\pi i=\textrm{id}_X$,  $i^*:H_{\pi^*(\Theta|_X)}^*(E)\rightarrow H_{\Theta|_X}^*(X)$ is also an isomorphism. Since $i\pi$ and $\textrm{id}_E$ are smooth homotopic, $\Theta-\pi^*i^*\Theta=\textrm{d}u$ for some $u\in\mathcal{A}^0(E)$. Obviously, $\textrm{d}(i^*u)=0$, hence $i^*u$ is a constant. We can choose $u$ satisfying $i^*u=0$, which gives a commutative diagram
\begin{displaymath}
\xymatrix{
  H_{\Theta}^*(E) \ar[rr]^{e^u\cdot} \ar[dr]_{i^*}
                &  &    H_{\pi^*(
                \Theta|_X)}^*(E) \ar[dl]^{i^*}    \\
                & H_{\Theta|_X}^*(X) },
\end{displaymath}
which implies that $i^*:H_{\Theta}^p(E)\rightarrow H_{\Theta|_X}^p(X)$ is an isomorphism.

By the projection formula of currents, $e^u\cdot i_*\alpha=i_*(e^{i^*u}\cdot \alpha)=i_*\alpha$ for any $\alpha\in \mathcal{A}_c^*(X)$. So, there is a commutative diagram
\begin{displaymath}
\xymatrix@R=0.5cm{
                &     H_{\Theta,c}^{*+r}(E)     \ar[dd]^{e^u\cdot}     \\
 H_{\Theta|_X,c}^*(X) \ar[ur]^{i_*} \ar[dr]_{i_*}                 \\
                &        H_{\pi^*(
                \Theta|_X),c}^{*+r}(E)                  }.
\end{displaymath}
By Corollary \ref{is}, $i_*: H_{\Theta|_X,c}^*(X)\tilde{\rightarrow}H_{\Theta,c}^{*+r}(E)$ is an isomorphism. By the same way, we can prove that
$i_*: H_{\Theta|_X}^*(X)\tilde{\rightarrow}H_{\Theta,cv}^{*+r}(E)$ is also an isomorphism.
\end{proof}

The following lemma was proved in \cite{BT}, p. 67, but there, the proof seems not to be rigorous in using Stokes' theorem. Here we give a proof.
\begin{lemma}\label{2}
Let $E$ be an oriented vector bundle with rank $r$ on an oriented smooth manifold $X$ and $\Phi\in\mathcal{A}_{cv}^r(E)$ a representative of Thom class of $E$ defined in Sec. 3. Then $[\Phi]=[X]$ in $H_{dR}^r(E)$.
\end{lemma}

\begin{proof}
Assume $n=\textrm{dim} X$. Let $\pi:E\rightarrow X$ be the projection and $i:X\rightarrow E$ the inclusion of zero section. Since $i\circ \pi$ and $\textrm{id}_E$ are smooth homotopic, $\pi^*i^*=\textrm{id}$ on $H_{\textrm{dR}}^n(E)$. Then, for any closed form $\beta\in\mathcal{A}_c^n(E)$, $\beta=\pi^*i^*\beta+\textrm{d}\gamma$ for some $\gamma\in \mathcal{A}^{n-1}(E)$. $\Phi\wedge\beta$ and $\Phi\wedge\pi^*i^*\beta$ have compact supports, so does $\Phi\wedge \textrm{d}\gamma=(-1)^r\textrm{d}(\Phi\wedge\gamma)$. Let $\{U_\lambda\}$ be a locally finite open covering of $X$ such that every $\overline{U}_\lambda$ is compact and $\{\rho_\lambda\}$ a partition of unity subordinate to $\{U_\lambda\}$. Then $\sum_\lambda\pi^*\rho_\lambda=\pi^*(\sum_\lambda\rho_\lambda)=1$. Since $\pi^*\rho_\lambda\cdot\Phi\wedge\gamma$ has compact support, by Stokes' theorem, $\int_E\textrm{d}(\pi^*\rho_\lambda\cdot\Phi\wedge\gamma)=0$. So
\begin{displaymath}
\int_E\textrm{d}(\Phi\wedge\gamma)=\int_E\textrm{d}\left(\sum_\lambda\pi^*\rho_\lambda\cdot\Phi\wedge\gamma\right)=\sum_\lambda              \int_E\textrm{d}(\pi^*\rho_\lambda\cdot\Phi\wedge\gamma)=0.
\end{displaymath}
Therefore,
\begin{displaymath}
\int_E\Phi\wedge\beta=\int_E\Phi\wedge\pi^*i^*\beta+\int_E\Phi\wedge \textrm{d}\gamma=\int_X\pi_*\Phi\wedge i^*\beta=\int_Xi^*\beta,
\end{displaymath}
which implies $[\Phi]=[X]$ in $H_{\textrm{dR}}^r(E)$.
\end{proof}

Let $j:U\rightarrow X$ be the inclusion of an open subset $U$ into a $n$-dimensional oriented smooth manifold $X$. For a current $T\in \mathcal{D}^{'p}(X)$, the current $j^*T$ is defined as $\langle j^*T,\beta\rangle=\langle T,j_*\beta\rangle$ for any $\beta\in\mathcal{A}_c^{n-p}(U)$, where $\langle ,\rangle$ is the pair of the topological dual. Let $\theta$ be a closed $1$-form on $X$. Clearly, $\textrm{d}_{\theta|_U}(j^*T)=j^*(\textrm{d}_\theta T)$, so $j^*$ induces $H^p(\mathcal{D}^{'\bullet}(X),\textrm{d}_{\theta})\rightarrow H^p(\mathcal{D}^{'\bullet}(U),\textrm{d}_{\theta|_U})$, which is compatible with the pullback $j^*$ of Morse-Novikov cohomology,

since $\int_Uj^*\alpha\wedge\beta=\int_X\alpha\wedge j_*\beta$ for $\alpha\in\mathcal{A}^p(X)$ and $\beta\in\mathcal{A}_c^{n-p}(U)$.

\begin{lemma}\label{key}
Let $Y$ be an oriented connected submanifold of an oriented connected smooth manifold $X$ and $i:Y\rightarrow X$  the inclusion.  Assume $\theta$ is a closed $1$-form on $X$ and $[Y]$ is the fundamental class of $Y$ in $X$.

$(1)$ For $\sigma\in H_{\theta|_Y}^*(Y)$,
\begin{displaymath}
i^*i_*\sigma=[Y]|_Y\cup\sigma.
\end{displaymath}

$(2)$ For $\sigma\in H_{\theta|_Y,c}^*(Y)$,
\begin{displaymath}
i^*i_*\sigma=[Y]|_Y\cup\sigma.
\end{displaymath}
\end{lemma}

\begin{proof}
Set $r=\textrm{codim}Y$. Let $N$ be a tubular neighborhood of $Y$ in $X$ and denote $\tau$ the projection of the vector bundle $N$ over $Y$.  Let $l:Y\rightarrow N$ and $j:N\rightarrow X$ be inclusions. Denote $[Y]_N\in H_{\textrm{dR}}^r(N)$ the fundamental class of $Y$ in $N$.  We choose $u\in\mathcal{A}^0(N)$ satisfying $\theta|_N-\tau^*(\theta|_Y)=\textrm{d}u$ and $l^*u=0$. For any current $T$ on $Y$, $e^{-u}\cdot l_*T=l_*(e^{-l^*u}\cdot T)=l_*T$, by the projection formula of currents.

$(1)$ By the localization principle (\cite{BT}, Prop. 6.25), there is a representative $\eta_Y\in\mathcal{A}^r(X)$ of $[Y]$ such that $\textrm{supp}\eta_Y\subseteq N$. Then $\eta_Y|_N$ is a representative of $[Y]_N$. We have $l_*(1)=\eta_Y|_N+\textrm{d}S$ for some $S\in\mathcal{D}^{'r-1}(N)$. For $\sigma=[\alpha]_{\theta|_Y}\in H_{\theta|_Y}^*(Y)$, as a current on $N$,
\begin{displaymath}
\begin{aligned}
l_*\alpha=&e^{-u}\cdot l_*\alpha=e^{-u}\cdot l_*l^*\tau^*\alpha\\
=&e^{-u}\cdot l_*(1)\wedge\tau^*\alpha\\
=&e^{-u}\cdot \eta_Y|_N\wedge\tau^*\alpha+\textrm{d}_{\theta|_N}(e^{-u}\cdot S\wedge\tau^*\alpha).
\end{aligned}
\end{displaymath}
Since $\textrm{supp}(l_{*}\alpha)\subseteq Y$, $j|_{\textrm{supp}(l_{*}\alpha)}$ is proper, which implies that $j_*(l_{*}\alpha)$ is defined well.  Clearly, $j_*(l_{*}\alpha)=i_{*}\alpha$ and $j^*j_*(l_{*}\alpha)=l_{*}\alpha$. We have
\begin{displaymath}
\begin{aligned}
i^*i_*\sigma=&l^*j^*[i_*\alpha]_\theta=l^*[j^*j_*l_*\alpha]_{\theta|_N}\\
=&l^*[l_*\alpha]_{\theta|_N}=l^*[e^{-u}\cdot \eta_Y|_N\wedge\tau^*\alpha]_{\theta|_N}\\
=&[Y]|_Y\cup\sigma.
\end{aligned}
\end{displaymath}

$(2)$ Let $\Phi\in\mathcal{A}_{cv}^r(N)$ be a representative of Thom class of vector bundle $N$ satisfying $\tau_*\Phi=1$. For $\sigma=[\alpha]_{\theta|_Y,c}\in H_{\theta|_Y,c}^*(Y)$, by Lemma \ref{is},
\begin{displaymath}
\begin{aligned}
l_*\alpha=&e^{-u}\cdot l_*\alpha\\
=&e^{-u}\cdot\left(\Phi\wedge\tau^*\alpha+\textrm{d}_{\tau^*(\theta|_N)}S \right)\\
=&e^{-u}\cdot\Phi\wedge\tau^*\alpha+\textrm{d}_{\theta|_N}(e^{-u}S).
\end{aligned}
\end{displaymath}
for some  $S\in\mathcal{D}_c^{\prime p+r-1}(N)$. Therefore,
\begin{displaymath}
\begin{aligned}
i^*i_*\sigma=&i^*j_*[l_*\alpha]_{\theta|_N,c}=i^*j_*[e^{-u}\cdot\Phi\wedge\tau^*\alpha]_{\theta|_N,c}\\

=&[l^*j^*j_*(e^{-u}\Phi\wedge\tau^*\alpha)]_{\theta|_Y,c}\\
=&[\Phi]|_Y\cup\sigma,
\end{aligned}
\end{displaymath}
where we used that $j^*j_*=\textrm{id}$ on $\mathcal{A}_c^*(N)$. By Lemma \ref{2}, $[\Phi]=[Y]_N$ in $H_{\textrm{dR}}^r(N)$. Since $[Y]_N|_Y=[Y]|_Y$, $i^*i_*\sigma=[Y]|_Y\cup\sigma$.
\end{proof}

\begin{lemma}\label{3}
Let $\pi:\mathbb{P}(E)\rightarrow X$ be the projectivization of a complex vector bundle $E$ of rank $r$ on a complex manifold $X$ and $\theta$ a closed $1$-form on $X$. Assume $\tilde{\theta}=\pi^*\theta$ and $h=c_1(\mathcal{O}_{\mathbb{P}(E)}(-1))\in H_{\emph{dR}}^2({\mathbb{P}(E)})$ is the first Chern class of the universal line bundle $\mathcal{O}_{\mathbb{P}(E)}(-1)$ on ${\mathbb{P}(E)}$. Then $\pi^*(\bullet)\wedge\bullet$ gives isomorphisms of graded vector spaces
\begin{displaymath}
H_{\theta}^*(X)\otimes_{\mathbb{R}}\emph{span}_{\mathbb{R}}\{1,...,h^{r-1}\}\tilde{\rightarrow}H_{\tilde{\theta}}^*(\mathbb{P}(E))
\end{displaymath}
and
\begin{displaymath}
H_{\theta,c}^*(X)\otimes_{\mathbb{R}}\emph{span}_{\mathbb{R}}\{1,...,h^{r-1}\}\tilde{\rightarrow}H_{\tilde{\theta},c}^*(\mathbb{P}(E)).
\end{displaymath}
\end{lemma}
\begin{proof}
For every $x\in X$, $1$, $h$,\ldots, $h^{r-1}$ restricted to the fibre $\pi^{-1}(x)=\mathbb{P}(E_x)$ freely generate $H_{\textrm{dR}}^*(\mathbb{P}(E_x))$.  By Theorem \ref{L-H}, we proved the lemma.
\end{proof}

Let $\pi:\widetilde{X}_Y\rightarrow X$ is the blow-up of a connected complex manifold $X$ along a connected complex submanifold $Y$ and $\theta$ a closed $1$-form on $X$. We know the exceptional divisor $E=\pi^{-1}(Y)$ is naturally identified with the projectivization $E=\mathbb{P}(N_{Y/X})$ of the normal bundle $N_{Y/X}$. Let $h=c_1(\mathcal{O}_E(-1))\in H_{\textrm{dR}}^2(E)$ be the first Chern class of the universal line bundle $\mathcal{O}_E(-1)$ on $E$.
\begin{theorem}
Assume $X$, $Y$, $\pi$, $\widetilde{X}_Y$, $E$, $h$ are defined as above. Let $i_E:E\rightarrow \widetilde{X}_Y$ be the inclusion, $\tilde{\theta}=\pi^*\theta$ and $r=\emph{codim}_{\mathbb{C}}Y$. Then, for any $k$,
\begin{equation}\label{b-u-m}
\pi^*+\sum_{i=0}^{r-2}(i_E)_*\circ (h^i\cup)\circ (\pi|_E)^*
\end{equation}
gives two isomorphisms
\begin{displaymath}
H_{\theta}^k(X)\oplus \bigoplus_{i=0}^{r-2}H_{\theta|_Y}^{k-2-2i}(Y)\tilde{\rightarrow} H_{\tilde{\theta}}^k(\widetilde{X}_Y)
\end{displaymath}
and
\begin{displaymath}
H_{\theta,c}^k(X)\oplus \bigoplus_{i=0}^{r-2}H_{\theta|_Y,c}^{k-2-2i}(Y)\tilde{\rightarrow} H_{\tilde{\theta},c}^k(\widetilde{X}_Y).
\end{displaymath}

\end{theorem}

\begin{proof}
Set $U=X-Y$ and $\widetilde{U}=\widetilde{X}_Y-E$. Then $\pi|_{\widetilde{U}}:\widetilde{U}\rightarrow U$ is biholomorphic.

Choose a tubular neighbourhood $V$ of $Y$ in $X$ such that $\widetilde{V}=\pi^{-1}(V)$ is a tubular neighbourhood of $E$ in $\widetilde{X}_Y$. Set $W=U\cap V$ and $\widetilde{W}=\widetilde{U}\cap\widetilde{V}$. Then $\pi|_{\widetilde{W}}:\widetilde{W}\rightarrow W$ is biholomorphic.

We have a commutative diagram of long exact sequences
\begin{displaymath}
\small{\xymatrix{
    \cdots\ar[r]&H_{\theta}^{k-1}(W)\ar[d]^{\cong}\ar[r]&H_{\theta}^k(X) \ar[d]^{\pi^*} \ar[r]& H_{\theta}^k(U)\oplus H_{\theta}^k(V) \ar[d]^{(\pi|_{\widetilde{U}})^*\oplus(\pi|_{\widetilde{V}})^*}\ar[r]&  H_{\theta}^{k}(W)\ar[d]^{\cong}\ar[r]&H_{\theta}^{k+1}(X)\ar[d]^{\pi^*}\ar[r]&\cdots\\
 \cdots\ar[r]&H_{\tilde{\theta}}^{k-1}(\widetilde{W})     \ar[r] & H_{\tilde{\theta}}^k(\widetilde{X}_Y)\ar[r]&H_{\tilde{\theta}}^k(\widetilde{U})\oplus H_{\tilde{\theta}}^k(\widetilde{V})       \ar[r]& H_{\tilde{\theta}}^k(\widetilde{W})     \ar[r] & H_{\tilde{\theta}}^{k+1}(\widetilde{X}_Y)\ar[r]&\cdots     }.}
\end{displaymath}
By Corollary \ref{i-s}, $\pi^*$ is injective. By Snake Lemma (\cite{I}, p. 4), we have an isomorphism
\begin{equation}\label{*}
\textrm{coker}\pi^*\tilde{\rightarrow}\textrm{coker}\left((\pi|_{\widetilde{U}})^*\oplus(\pi|_{\widetilde{V}})^*\right)\cong\textrm{coker}(\pi|_{\widetilde{V}})^*.
\end{equation}

Let $i'_Y:Y\rightarrow V$ and $i'_E:E\rightarrow \widetilde{V}$ be the inclusions. By Lemma \ref{1}, $i_Y^{'*}: H_{\theta}^k(V)\rightarrow H_{\theta|Y}^k(Y)$ and  $i_E^{'*}:H_{\tilde{\theta}}^k(\widetilde{V})\rightarrow H_{\tilde{\theta}|_E}^k(E)$ are isomorphisms. Since $\pi|_{\widetilde{V}}\circ i'_E=i'_Y\circ\pi|_E$, $i_E^{'*}$ induces an isomorphism
\begin{equation}\label{**}
\textrm{coker}(\pi|_{\widetilde{V}})^*\tilde{\rightarrow}\textrm{coker}(\pi|_E)^*.
\end{equation}

Combining (\ref{*}) and (\ref{**}), we have a commuative diagram of short exact sequences
\begin{equation}\label{commutative1}
\xymatrix{
 0\ar[r]&H_\theta^k(X)\ar[d]^{i_Y^*} \ar[r]^{\pi^*}& H_{\tilde{\theta}}^k(\widetilde{X}_Y)\ar[d]^{i_E^*} \ar[r]& \textrm{coker}\pi^* \ar[d]^{\cong}\ar[r]& 0\\
 0\ar[r]&H_{\theta|_Y}^k(Y)       \ar[r]^{(\pi|_E)^*}& H_{\tilde{\theta}|_E}^k(E)   \ar[r]^{} &  \textrm{coker} (\pi|_E)^*    \ar[r]& 0 }.
\end{equation}

By Lemma \ref{3}, we have the isomorphism
\begin{equation}\label{pr-bun-1}
(\pi|_E)^*(\bullet)\cup\bullet: H_{\theta|_Y}^*(Y)\otimes_{\mathbb{R}}\textrm{span}_{\mathbb{R}}\{1,...,h^{r-1}\}\tilde{\rightarrow}  H_{\tilde{\theta}|_E}^*(E).
\end{equation}
Notice that $\mathcal{O}_E(-1)=\mathcal{O}_{\widetilde{X}_Y}(E)|_E$, so $h=[E]|_E$.

Suppose $\pi^*\alpha_k+\sum_{i=0}^{r-2}(i_E)_*\left(h^i\cup(\pi|_E)^*\beta_{k-2-2i}\right)=0$, where $\alpha_k\in H_{\theta}^k(X)$ and $\beta_{k-2-2i}\in H_{\theta|_Y}^{k-2-2i}(Y)$ for $i=0,1,...,r-2$. Pull back it by $i_E^*$, by Lemma \ref{key},
\begin{displaymath}
(\pi|_E)^*i_Y^*\alpha_k+\sum_{i=0}^{r-2}h^{i+1}\cup(\pi|_E)^*\beta_{k-2-2i}=0,
\end{displaymath}
where $i_Y:Y\rightarrow X$ is the inclusion. By the isomorphism (\ref{pr-bun-1}), $\beta_{k-2-2i}=0$ for every $i$. So $\pi^*\alpha_k=0$. By Corollary \ref{i-s}, $\alpha_k=0$. Hence the morphism $($\ref{b-u-m}$)$ is injective.

For any $\gamma\in H_{\tilde{\theta}}^k(\widetilde{X}_Y)$, by the isomorphism (\ref{pr-bun-1}), there exist $\beta_{k-2i}\in H_{\theta|_Y}^{k-2i}(Y)$ for $i=0,1,...,r-1$, such that $i_E^*\gamma=\sum_{i=0}^{r-1}h^i\cup(\pi|_E)^*\beta_{k-2i}$. By Lemma \ref{key},
\begin{displaymath}
i_E^*\left[\gamma-\sum_{i=0}^{r-2}(i_E)_*\left(h^i\cup(\pi|_E)^*\beta_{k-2-2i}\right)\right]=(\pi|_E)^*\beta_k,
\end{displaymath}
which is zero in $\textrm{coker}(\pi|_E)^*$. By commutative diagram (\ref{commutative1}),
\begin{displaymath}
\gamma-\sum_{i=0}^{r-2}(i_E)_*\left(h^i\cup(\pi|_E)^*\beta_{k-2-2i})\right)=\pi^*\alpha_k,
\end{displaymath}
for some $\alpha_k\in H_{\theta}^k(X)$. So the morphism $($\ref{b-u-m}$)$ is surjective.

We proved the first part.

By Proposition \ref{cpb2} and \ref{L}, we have the commutative diagram of long exact sequences
\begin{displaymath}
\xymatrix{
 \cdots\ar[r]&H_{\theta,c}^k(U)\ar[d]^{\cong} \ar[r]& H_{\theta,c}^k(X) \ar[d]^{\pi^*} \ar[r]& H_{\theta|_Y,c}^k(Y) \ar[d]^{(\pi\mid_E)^*}\ar[r]& H_{\theta,c}^{k+1}(U)\ar[d]^{\cong}\ar[r]&\cdots\\
 \cdots\ar[r]&H_{\tilde{\theta},c}^k(\widetilde{U})       \ar[r]& H_{\tilde{\theta},c}^k(\widetilde{X}_Y)     \ar[r]^{i_E^*} & H_{\tilde{\theta}|_E,c}^k(E)     \ar[r]& H_{\tilde{\theta},c}^{k+1}(\widetilde{U})         \ar[r]&\cdots }.
\end{displaymath}
By Corollary \ref{i-s}, $\pi^*$ is injective. By Snake Lemma, $i_E^*$ induces an isomorphism
\begin{displaymath}
\textrm{coker}\pi^*\tilde{\rightarrow}\textrm{coker}(\pi|_E)^*.
\end{displaymath}
Similarly with the proof of the first part, the morphism $($\ref{b-u-m}$)$ gives the second isomorphism.
\end{proof}

\subsection{A modification formula}
\begin{lemma}\label{top}
Let $X$ be a complex manifold and $\theta$ a closed $1$-form on $X$. Suppose $A\subseteq X$ is an analytic subset of dimension $r$ and denote by $A_{sm}$ the set of nonsingular points of $A$.

$(1)$ For $p>2r$, $H_c^p(A,\underline{\mathbb{R}}_{X,\theta})=0$.

$(2)$ If $A$ is irreducible, then
\begin{align*}
H_c^{2r}(A,\underline{\mathbb{R}}_{X,\theta})=\begin{cases}\mathbb{R},
\quad\text{if}\;\theta\; \text{is exact on}\; A_{sm}\\0,\quad
\text{otherwise}\;\end{cases}.
\end{align*}

$(3)$ Suppose $A$ is irreducible and $\pi:\widetilde{A}\rightarrow A$ is a desingularization.  Then
\begin{align*}
H_c^{2r}(A,\underline{\mathbb{R}}_{X,\theta})=\begin{cases}\mathbb{R},
\quad\text{if}\;(i\pi)^*\theta\; \text{is exact on}\; \widetilde{A}\\0,\quad
\text{otherwise}\;\end{cases},
\end{align*}
where $i:A\rightarrow X$ is the inclusion.

$(4)$ Let $\{A_\alpha\}$ be the collection of $r$-dimensional irreducible components of $A$. Then
\begin{displaymath}
H_c^{2r}(A,\underline{\mathbb{R}}_{X,\theta})=\bigoplus_{\alpha} H_c^{2r}(A_{\alpha},\underline{\mathbb{R}}_{X,\theta}).
\end{displaymath}
\end{lemma}

\begin{proof}
$(1)$ Assume $p>2r$. If $A$ is nonsingular, $H_c^p(A,\underline{\mathbb{R}}_{X,\theta})=H_{\theta|_A,c}^p(A)=0$. For the general case, let $A_s=A-A_{sm}$ be the set of singular points of $A$.  Set $A^0=A$, $A^i=(A^{i-1})_s$ for $i\geq 1$. There exists $i_0$, such that $A^{i_0+1}=\emptyset$ and $A^{i_0}\neq\emptyset$, i.e. $A^{i_0}$ is nonsingular. By Proposition \ref{L}, we have a long exact sequence for $(A^{i-1},A^i)$,
\begin{displaymath}
\tiny{
\xymatrix{
\cdots\ar[r] &H_c^p((A^{i-1})_{sm},\underline{\mathbb{R}}_{X,\theta})\ar[r] &H_c^p(A^{i-1},\underline{\mathbb{R}}_{X,\theta})\ar[r] &H_c^p(A^i,\underline{\mathbb{R}}_{X,\theta})\ar[r] &H_c^{p+1}((A^{i-1})_{sm},\underline{\mathbb{R}}_{X,\theta})\ar[r] &\cdots.
}}
\end{displaymath}
We have $H_c^p((A^{i-1})_{sm},\underline{\mathbb{R}}_{X,\theta})=H_c^{p+1}((A^{i-1})_{sm},\underline{\mathbb{R}}_{X,\theta})=0$, since $(A^{i-1})_{sm}$ is nonsingular.
Hence, for any $i$, $H_c^p(A^{i-1},\underline{\mathbb{R}}_{X,\theta})=H_c^p(A^i,\underline{\mathbb{R}}_{X,\theta})$, which imply
\begin{displaymath}
H_c^p(A,\underline{\mathbb{R}}_{X,\theta})=H_c^p(A_{i_0},\underline{\mathbb{R}}_{X,\theta})=0.
\end{displaymath}

$(2)$ By $(1)$, $H_c^{2r-1}(A_s,\underline{\mathbb{R}}_{X,\theta})=H_c^{2r}(A_s,\underline{\mathbb{R}}_{X,\theta})=0$. Similarly with the proof in $(1)$, for $(A,A_s)$, we have
\begin{displaymath}
H_c^{2r}(A,\underline{\mathbb{R}}_{X,\theta})=H_{\theta|_{A_{sm}},c}^{2r}(A_{sm}).
\end{displaymath}
We get $(2)$ by \cite{HR}, Example 1.6.

$(3)$ Since $\textrm{dim}_{\mathbb{C}}\pi^{-1}(A_s)<r$,
\begin{displaymath}
H_c^{2r-1}(\pi^{-1}(A_s),\underline{\mathbb{R}}_{\widetilde{A},(i\pi)^*\theta})=H_c^{2r}(\pi^{-1}(A_s),\underline{\mathbb{R}}_{\widetilde{A},(i\pi)^*\theta})=0.
\end{displaymath}
By Proposition \ref{L}, for $\pi:(\widetilde{A}, \pi^{-1}(A_s))\rightarrow (A, A_s)$, we have
\begin{displaymath}
H_c^{2r}(A,\underline{\mathbb{R}}_{X,\theta})=H_{(i\pi)^*\theta,c}^{2r}(\widetilde{A}).
\end{displaymath}
We get $(3)$ by \cite{HR}, Example 1.6.

$(4)$ Let $B=\bigcup_{\alpha}A_\alpha$ be the union of all $r$-dimensional components of $A$. Then $A-B$ is an analytic subset of the complex manifold $X-B$ satisfying $\textrm{dim}_{\mathbb{C}}(A-B)<r$. By $(1)$, $H_c^{2r}(A-B,\underline{\mathbb{R}}_{X,\theta})=0$. For $(A,B)$, we have
\begin{displaymath}
H_c^{2r}(A,\underline{\mathbb{R}}_{X,\theta})=H_c^{2r}(B,\underline{\mathbb{R}}_{X,\theta}).
\end{displaymath}
Set
\begin{displaymath}
C:=\bigcup_{\alpha\neq \beta}(A_\alpha\cap A_\beta),
\end{displaymath}
\begin{displaymath}
A'_\alpha:=A_\alpha-C\cap A_\alpha.
\end{displaymath}
Then $B-C=\bigcup_\alpha A'_\alpha$ and  $A'_\alpha\cap A'_\beta=\emptyset$ for $\alpha\neq\beta$. For $(B, C)$, we have
\begin{displaymath}
H_c^{2r}(B,\underline{\mathbb{R}}_{X,\theta})= H_c^{2r}(B-C,\underline{\mathbb{R}}_{X,\theta})= \bigoplus_{\alpha} H_c^{2r}(A'_\alpha,\underline{\mathbb{R}}_{X,\theta}).
\end{displaymath}
Similarly, $H_c^{2r}(A'_\alpha,\underline{\mathbb{R}}_{X,\theta})= H_c^{2r}(A_\alpha,\underline{\mathbb{R}}_{X,\theta})$. We complete the proof.
\end{proof}

A proper holomorphic map $\pi:X\rightarrow Y$ between connected complex manifolds is called a \emph{proper modification}, if there is a nowhere dense analytic subset $F\subset Y$, such that $\pi^{-1}(F)\subset X$ is nowhere dense and $\pi:X-f^{-1}(F)\rightarrow Y-F$ is biholomorphic. If $F$ is the minimal analytic subset satisfying above condition, then we call $E=\pi^{-1}(F)$ the \emph{exceptional set} of the proper modification $\pi$.
\begin{theorem}\label{mod}
Let $\pi:X\rightarrow Y$ be a proper modification of connected complex manifolds with the exceptional set $E$ and $\theta$ a closed $1$-form on $Y$.

$(1)$ There exists an exact sequence
\begin{displaymath}
\begin{aligned}
\xymatrix{
0\ar[r] &H_{\theta,c}^{2n-2}(Y)\ar[r]^{\pi^*} &H_{\tilde{\theta},c}^{2n-2}(X)\ar[r]^{} &\bigoplus_{\alpha\in I} H_c^{2n-2}(E_\alpha,\underline{\mathbb{R}}_{X,\tilde{\theta}})\ar[r]&0
},
\end{aligned}
\end{displaymath}
where $\{E_\alpha\}_{\alpha\in I}$ is the collection of all irreducible components of $E$.

$(2)$ $\pi^*$ gives isomorphisms $H_{\theta}^1(Y)\tilde{\rightarrow}H_{\tilde{\theta}}^1(X)$ and $H_{\theta,c}^{2n-1}(Y)\tilde{\rightarrow}H_{\tilde{\theta},c}^{2n-1}(X)$.
\end{theorem}
\begin{proof}
$(1)$ Set $F=\pi(E)$, $V=Y-F$ and $U=X-E$. By Proposition \ref{L}, there is a commutative diagram of long exact sequences
\begin{equation}\label{CL}
\small{\xymatrix{
\cdots\ar[r]&H_{\theta,c}^{2n-2}(V)\ar[d]^{\cong} \ar[r]& H_{\theta,c}^{2n-2}(Y) \ar[d]^{\pi^*} \ar[r]& H_c^{2n-2}(F,\underline{\mathbb{R}}_{Y,\theta}) \ar[d]^{\left(\pi\mid_E\right)^*}\ar[r]& H_{\theta,c}^{2n-1}(V)\ar[d]^{\cong}\ar[r]&\cdots\\
\cdots\ar[r]&H_{\tilde{\theta},c}^{2n-2}(U)       \ar[r]& H_{\tilde{\theta},c}^{2n-2}(X)     \ar[r] & H_c^{2n-2}(E,\underline{\mathbb{R}}_{X,\tilde{\theta}})     \ar[r]& H_{\tilde{\theta},c}^{2n-1}(U)       \ar[r]&\cdots }.}
\end{equation}
By \cite{GR2}, p. 215 or \cite{GR1}, $E$ has pure codimension $1$ and $\textrm{codim}_{\mathbb{C}}F\geq 2$. So $H_c^{2n-2}(F,\underline{\mathbb{R}}_{Y,\theta})=0$. By the injectivity of $\pi^*$ and Snake Lemma, $\textrm{coker}\pi^*=\textrm{coker}\left(\pi|_E\right)^*=H_c^{2n-2}(E,\underline{\mathbb{R}}_{X,\tilde{\theta}})$. By Lemma \ref{top} $(4)$, we proved $(1)$.

$(2)$ Through the commutative diagram $($\ref{CL}$)$, $\pi^*:H_{\theta,c}^{2n-1}(Y)\tilde{\rightarrow}H_{\tilde{\theta},c}^{2n-1}(X)$ is isomorphic. By this isomorphism for $-\theta$ instead of $\theta$ and by Poincar\'{e} duality, $\pi_*:H_{\tilde{\theta}}^1(X)\tilde{\rightarrow}H_{\theta}^1(Y)$ is isomorphic. By the projection formula, $\pi_*\pi^*=\textrm{id}$, so $\pi^*$ is also an isomorphism.
\end{proof}

We get two bimeromorphic invariants $b_1(X,\theta)$ and $b_{c,2n-1}(X,\theta)$ in following sense.
\begin{corollary}\label{bim1}
Let $f:X\dashrightarrow Y$ be a bimeromorphic map of complex manifolds and $\theta_X$, $\theta_Y$ closed $1$-forms on $X$, $Y$ respectively. Assume there exist nowhere dense analytic subsets $E\subseteq X$ and $F\subseteq Y$, such that $f:X-E\rightarrow Y-F$ is biholomorphic and $f^*(\theta_Y|_{Y-F})=\theta_X|_{X-E}$. Then, $H_{\theta_X}^1(X)\cong H_{\theta_Y}^1(Y)$ and $H_{\theta_X,c}^{2n-1}(X)\cong H_{\theta_Y,c}^{2n-1}(Y)$.
\end{corollary}
\begin{proof}
Choose a complex manifold $Z$ and  proper modifications $g:Z\rightarrow X$ and  $h:Z\rightarrow Y$, such that there exists a nowhere dense analytic subset $S\subseteq Z$ satisfying that, $E\subseteq g(S)$, $F\subseteq h(S)$, $g:Z-S\rightarrow X-g(S)$, $h:Z-S\rightarrow Y-h(S)$ are biholomorphic and $fg|_{Z-S}=h|_{Z-S}$. Obviously,
\begin{displaymath}
(g^*\theta_X-h^*\theta_Y)|_{Z-S}=g^*\left[\left(\theta_X|_{X-E}-f^*(\theta_Y|_{Y-F})\right)|_{X-g(S)}\right]=0.
\end{displaymath}
By the continuity, $g^*\theta_X=h^*\theta_Y$. Hence, we need only prove the corollary for the case that $f$ is a proper modification and $f^*\theta_Y=\theta_X$. It is just Theorem \ref{mod} $(2)$.
\end{proof}

\renewcommand{\refname}{\leftline{\textbf{References}}}

\makeatletter\renewcommand\@biblabel[1]{${#1}.$}\makeatother

config: arxiv


TITLE: Determine the stability of $(x,y)=(0,0)$
QUESTION [1 upvotes]: Determine the stability of $(x,y)=(0,0)$:
1/$$\bf{\begin{cases}
 & \mathrm{  } \dot{x}= -2x-y+2xy^2-3x^3\\
 & \mathrm{  } \dot{y}= \dfrac{x}{3}-y-x^2y-7y^3
\end{cases} \tag {1}}$$
2/
$$\bf{\begin{cases}
 & \mathrm{  } \dot{x}= x-xy^4\\
 & \mathrm{  } \dot{y}= y-x^2y^3
\end{cases} \tag {2}}$$
I'm trying to find a Liapunov function $V(x,y)=???$. But I have no solution! :( Anyone can post a few hints (like $V(x,y)=?$)
Any help will be appreciated.Thanks!

REPLY [0 votes]: Solution:
For system of equations: $$\bf{\begin{cases}
 & \mathrm{  } \dot{x}= x-xy^4\\
 & \mathrm{  } \dot{y}= y-x^2y^3
\end{cases} \tag {2}}$$
We use the Chetaev theorem to prove this.
Let the function $V(x,y)$ have the form $$V(x,y)=x^2-y^2$$
This function is positive definite in the subdomain $\mathcal{V}$ in which the inequality $|x| > |y|$ holds.
We calculate the derivative $\dfrac{dV}{dt}$ by virtue of the system and define its sign in the subdomain $\mathcal{V}$.
$$\dot{V}(x,y)=2(x^2-y^2)=2V(x,y)$$

It can be seen that the derivative $\dfrac{dV}{dt}$ is also positive definite in the subdomain $\mathcal{V}$ defined by the relation $|x| > |y|$.
Besides that, the function $V(x,y)$ is zero on the boundary of $\mathcal{V}$ including the point $(0,0)$.

So, all conditions of the Chetaev theorem are fulfilled. Therefore, the zero solution of the system is unstable.
============================================
Is that ok? Amzoti


TITLE: how to calculate $\sum_{x=1}^\infty xq^{2x}$?
QUESTION [2 upvotes]: Its been a long time since I last dealt with series's like this: $\sum_{x=1}^\infty xq^{2x}$, I tried to take to a form where I can calculate the derivative of other sum, but didn't work. I'd be happy if someone shows the way.
Note: $|q|<1$

REPLY [4 votes]: I thought it might be instructive to present an approach that relies on elementary tools and not on differentiation of a series.

Using $x=\sum_{y=1}^x (1)$, we can write
$$\begin{align}
\sum_{x=1}^\infty xq^{2x}&=\sum_{x=1}^\infty \sum_{y=1}^x q^{2x}\\\\
&=\sum_{y=1}^\infty \sum_{x=y}^\infty  q^{2x}\\\\
&=\sum_{y=1}^\infty \frac{q^{2y}}{1-q^2}\\\\
&=\frac{q^2}{(1-q^2)^2}
\end{align}$$

Tools Used:  Interchanging order of summation (Fubini) and Summing Geometric Series.