TITLE: Invariant subspace and minimal polynomial
QUESTION [3 upvotes]: Let $T$ be a linear map on a finite dimensional vector space $V$ over an arbitrary field. Show that the following 3 statements are equivalent.
(1) The minimal polynomial of $T$ is a product of distinct irreducibles.
(2) Each $T$-invariant subspace is admissible. (An invariant subspace $W$ is admissible if for any polynomial $f$ and $v \in V$ such that $f(T)v \in W$, then there exists $w \in W$ such that $f(T)w=f(T)v$.)
(3) For each $T$-invariant subspace, there exists a $T$-invariant complement.
The equivalence between (2) and (3) is a result in Hoffman and Kunze p.232. I am trying to show the equivalence between (1) and (3). However I don't know how to establish either direction. The problem is easier in complex case because the minimal polynomial is always a product of linear factors. So in the real case it requires just a little more work.  but in general field I don't know how to proceed in either direction, any hint/help is appreciated.

REPLY [2 votes]: Here's a succinct answer, assuming you know about primary decomposition. This allows reducing to the case where there is only one irreducible factor $P\in K[X]$ of the minimal polynomial (which therefore is a power $P^m$ of$~P$). The reduction is possible because the primary decomposition is canonical, and the primary decomposition of any $T$-invariant subspace$~W$ is obtained by intersecting $W$ with the primary components$~V_i$ of $V$; then a $T$-invariant subspace $W'$ is a complement of$~W$ if and only if $W'\cap V_i$ is a complement of $W\cap V_i$ in each$~V_i$, and so (3) holds in the whole space if and only if it holds in every $V_i$.
Now For the direction $(3)\implies(1)$ there is a simple argument proving the contrapositive. Assume the multiplicity$~m$ of$~P$ is larger than$~1$; we need to find a $T$-invariant subspace without $T$-invariant complement. I claim $W=\ker P[T]$ is such a subspace. By the multiplicity assumption $W\neq V$, so a hypothetical $T$-invariant complement $W'$ of $W$ will be of nonzero dimension. The endomorphism$~R$ of$~W'$ obtained by restricting $P[T]$ (which exists by $T$-invariance) is injective, as $\ker R=W'\cap \ker P[T]=W'\cap W=\{0\}$. But then $R^m$ is also injective, which contradicts that fact that it is the restriction of $P[T]^m=0$.
For the direction $(1)\implies(3)$ the question has been reduced to the question Irreducible minimal polynomial implies every invariant subspace has an invariant complement, for which see under the link.


text:
Let $F$ be a convex function over some convex domain $\cW$ equipped with norm $\|\cdot\|$. There is a $G$-bounded stochastic gradient oracle where on input $\bw$, it returns a random $\hat{\bg}$ such that $\Exp[\hat{\bg}]$ is the gradient of $F$ at $\bw$ and $\|\hat{\bg}\|^2\leq G^2$ almost surely for some constant $G>0$. The goal is to minimize $F$ over $\cW$ with the help of this stochastic gradient oracle. The stochastic gradient descent (SGD) algorithm maintains a vector $\bw_t$ at time $t$ as follows.

\vspace{0.5cm}
\begin{algorithm}[H]
	\caption{SGD for strongly convex function}\label{alg:SGD for strongly convex}
	    \textbf{Input:} Time parameter $T\in\N$ and step size parameters $\{\eta_t\}_{t\in\N}$.\\
	    \textbf{Initialize:} $\bw_0\in\cW$.\\
	    \For{$t=1,\dots,T$}{
	    Query the stochastic gradient oracle of $F$ at $\bw_{t-1}$ and get $\hat{\bg}_t$.\\
	    Let $\bw_t=\textsf{Proj}_{\cW}(\bw_{t-1}-\eta_t\hat{\bg}_t)$ where $\textsf{Proj}_{\cW}$ is an orthogonal projection operator for $\cW$.}
	    \textbf{Output:} $\bw_T$.
\end{algorithm}
\vspace{0.5cm}

We say $F$ is $\lambda$-strongly convex for some $\lambda>0$ if for all $\bw,\bw'\in\cW$ and any subgradient $\bg$ at $\bw$,
\[
F(\bw')\geq F(\bw)+\bg^\top(\bw'-\bw)+\frac{\lambda}{2}\|\bw'-\bw\|^2 \, .
\]
Due to the strong convexity, there exists a unique minimizer $\bw^*\in\cW$ of $F$. Thus, the potential function is defined as $X_t=\|\bw_t-\bw^*\|^2$. Note that due to the strong convexity again, we have
\[
G\|\bw - \bw^*\|\geq \|\bg\|\cdot\|\bw - \bw^*\|\geq \langle \bg, \bw - \bw^* \rangle\geq \frac{\lambda}{2}\|\bw - \bw^*\|^2
\]
where $\bg$ is the gradient of $\bw$. By dividing $\frac{2}{\lambda}\|\bw - \bw^*\|$ and square both side, we have for every $t\in\N$
\[
\ X_t\leq \frac{4G^2}{\lambda^2}
\]
almost surely. The goal is to prove the following theorem, which is a formal version of~\autoref{thm:sgd}.

\begin{theorem}[Convergence of SGD for strongly convex functions]\label{thm: sgd main appendix}
Let $F$ be a $\lambda$-strongly convex function over some convex domain. An SGD algorithm with gradient bounded by $G>0$ almost surely has the following convergence rate. For every $\delta\in(0,1)$,
\begin{itemize}
\item if $\eta_t=1/(\lambda t)$, we have the following last iterate convergence
\[
\Pr\left[ X_{T} > \frac{1000G^2\log\frac{1}{\delta}}{\lambda^2T} \right] < \delta
\]
for every $T\in\N$;
\item if $\eta_t=1/(\lambda t)$, we have the following strong uniform convergence
\[
\Pr\left[\exists  t\in\N,\ X_{t} > \frac{1000 G^2\left(\log\frac{1}{\delta}+2\log\log(t+1)\right)}{\lambda^2 t} \right] < \delta \, ;
\]
\end{itemize}
\end{theorem}

\paragraph{Section structure.} In the rest of this section, we provide the recursion analysis in~\autoref{app:sgd recursion}, the moment and concentration analysis in~\autoref{app:sgd moment}, and the improvement analysis in~\autoref{app:sgd improvement}.

\subsection{Recursion analysis}\label{app:sgd recursion}
Let us start with the linearization for $X_t$. For every $0\leq T_0<T_1$, we rewrite the dynamic as follows.
\begin{align*}
X_t&=\|\bw_t-\bw^*\|^2=\|\textsf{Proj}_{\cW}(\bw_{t-1}-\eta_t\hat{\bg}_t)-\bw^*\|^2\leq \|\bw_{t-1}-\eta_t\hat{\bg}_t-\bw^*\|^2\\
&=\|\bw_{t-1}-\bw^*\|^2-2\eta_t\hat{\bg}_t^\top(\bw_{t-1}-\bw^*)+\eta_t^2\|\hat{\bg}_t\|^2 \\
&=(1-2\eta_t\lambda)X_{t-1}+2\eta_t\left(\lambda X_{t-1}-\hat{\bg}_t^\top(\bw_{t-1}-\bw^*)\right)+\eta_t^2\|\hat{\bg}_t\|^2 \, .
\end{align*}

Let $M_{t}=\sum_{t'=T_0+1}^t\frac{2\eta_{t'}\left(\lambda X_{t'-1}-\hat{\bg}_{t'}^\top(\bw_{t'-1}-\bw^*)\right)+\eta_{t'}^2\|\hat{\bg}_{t'}\|^2}{\prod_{t''=T_0+1}^{t'}(1-2\eta_{t''}\lambda)}$ for all $T_0+1\leq t\leq T_1$. We can unfold the recursion and get the following.
\[
X_t = \prod_{t'=T_0+1}^t(1-2\eta_{t'}\lambda)\left(X_{T_0} + M_{t}\right) \, .
\]

\subsection{Moment and concentration analysis}\label{app:sgd moment}
Next, we calculate a moment profile for the recursion in~\autoref{app:sgd recursion}.

\begin{lemma}[Moment profile for SGD]\label{lem:sgd moment details}
Consider the setting in~\autoref{thm: sgd main appendix}. Let $\Lambda > 0$ and $\tau$ is the stopping time for the event $\lbrace X_t > \Lambda \rbrace$. For every $T_0+1\leq t\leq T$, the following functions $(B_{T_0},\mu_{T_0},\sigma^2_{T_0})$ form a moment profile for $\{X_t\}$, $\Lambda$, and $T_0$.
\begin{itemize}[leftmargin=*]
\item (Bounded difference) $|M_{t\wedge\tau}-M_{(t-1)\wedge\tau}|\leq B(t,\Lambda)=\eta_{t}\cdot\frac{12G/\lambda+\eta_{t}G^2}{\prod_{t'=T_0+1}^{t}(1-2\eta_{t'}\lambda)}$ almost surely.
\item (Conditional expectation) $\Exp\left[M_{t\wedge\tau}-M_{(t-1)\wedge\tau}\ |\ \cF_{t-1}\right]\leq\mu(t,\Lambda)=\eta_{t}^2\cdot\frac{G^2}{\prod_{t'=T_0+1}^{t}(1-2\eta_{t'}\lambda)}$.
\item (Conditional variance) $\Var\left[M_{t\wedge\tau}-M_{(t-1)\wedge\tau} \ |\ \cF_{t-1}\right]\leq\sigma^2(t,\Lambda)=\eta_{t}^2\cdot\frac{72G^2\Lambda+2\eta_{t}^2G^4}{\prod_{t'=T_0+1}^{t}(1-2\eta_{t'}\lambda)^2}$.
\end{itemize}
Specifically, if $\eta_t=1/(\lambda t)$ and $T_0 \geq100$, we can pick
\[
B_{T_0}(t,\Lambda) =\frac{20G^2t}{\lambda^2 T_0^2} \, ,\mu_{T_0}(t,\Lambda)=\frac{2G^2}{\lambda^2T_0^2} \, \text{, and }
\sigma_{T_0}^2(t,\Lambda) = \frac{G^2t^2}{\lambda^2T_0^4}\left(80\Lambda+\frac{3G^2}{\lambda^2t^2}\right)  \, .
\]
\end{lemma}
\begin{proof}
For bounded difference, we have
\begin{align}
|M_{t\wedge\tau}-M_{(t-1)\wedge\tau}| &= \left| \mathbf{1}_{\tau\geq t} \frac{2\eta_t\left(\lambda X_{t-1}-\hat{\bg}_t^\top(\bw_{t-1}-\bw^*)\right)+\eta_t^2\|\hat{\bg}_t\|^2}{\prod_{t'=T_0+1}^{t}(1-2\eta_{t'}\lambda)} \right| \, .\nonumber
\intertext{By Cauchy-Schwarz inequality and the fact that $\|\hat{\bg}_{t}\|\leq G$, we have}
&\leq \mathbf{1}_{\tau\geq t} \eta_{t}\cdot  \frac{2\lambda X_{t-1} + 2G\sqrt{X_{t-1}} + \eta_tG^2}{\prod_{t'=T_0+1}^{t}(1-2\eta_{t'}\lambda)} \, .\nonumber
\intertext{Because $X_{t}\leq \frac{4G^2}{\lambda^2}$, we have}
&\leq \eta_{t}\frac{12G^2/\lambda + \eta_tG^2}{\prod_{t'=T_0+1}^{t}(1-2\eta_{t'}\lambda)} \, . \label{eq: sgd bounded difference}
\end{align}

For the conditional expectation, notice that $\Exp[\hat{\bg}_{t}^\top(\bw_{t-1}-\bw^*)\ |\ \cF_{t-1}]\geq\lambda X_{t-1}$ due to strong convexity and $\|\hat{\bg}_{t}\|\leq G$ due to $G$-boundedness, we have
\[
\Exp\left[M_{t\wedge\tau}-M_{(t-1)\wedge\tau}\ |\ \cF_{t-1}\right]\leq\eta_{t'}^2\cdot\frac{G^2}{\prod_{t'=T_0+1}^{t}(1-2\eta_{t'}\lambda)} \, .
\]

For the conditional variance, we have by~\autoref{eq: sgd bounded difference} and the definition of $\tau$,
\[
\Var\left[M_{t\wedge\tau}-M_{(t-1)\wedge\tau} \ |\ \cF_{t-1}\right]\leq \Exp\left[\mathbf{1}_{\tau \geq t}\cdot(M_t-M_{t-1})^2\ |\ \cF_{t-1}\right]\leq \eta_{t}^2\cdot\frac{72G^2\Lambda+2\eta_{t}^2G^4}{\prod_{t'=T_0+1}^{t}(1-2\eta_{t'}\lambda)^2} \, .
\]

Notice that for $t>T_0\geq 100$
\begin{equation}\label{eq: sgd learning rate product}
\prod_{t' = T_0 + 1}^{t} (1 - 2\eta_{t'}\lambda) = \frac{T_0(T_0-1)}{t(t-1)}\geq \frac{99T_0^2}{100t^2} \, .
\end{equation}

Now plug the specific learning rate back, we obtain the bound in the lemma statement.
\end{proof}

\subsection{Improvement analysis}\label{app:sgd improvement}

\subsubsection{Last iterate convergence}

\begin{proof}[Proof of item 1 of~\autoref{thm: sgd main appendix}]

In the analysis for last iterate convergence, we consider a different stopping time $\tau$ defined for the event $\{X_t>\Lambda_t\}$ for some sequence of non-negative thresholds $\{\Lambda_t\}$ chosen properly later. Let $T_0=100$ and let $(B_{T_0},\mu_{T_0},\sigma^2_{T_0})$ be the moment profile obtained from~\autoref{lem:sgd moment details}. By concentration inequality (i.e.,~\autoref{lem:martingale concentration}), we have
\[
\Pr\left[\exists T_0+1\leq t\leq T_1,\ |M_{t\wedge\tau}|>\Delta\right]<\delta
\]
where
\[
\Delta=2\max\left\{\sqrt{\sum_{t=T_0+1}^{T_1}\sigma^2_{T_0}(t,\Lambda_t)\log\frac{1}{\delta}},\ \max_{T_0+1\leq t\leq T_1}B_{T_0}(t,\Lambda_t)\log\frac{1}{\delta}\right\}+\sum_{t=T_0+1}^{T_1}\mu_{T_0}(t,\Lambda_t) \, .
\]
We pick $\Lambda_t=\frac{T_1}{t^2}\Lambda$ where $\Lambda>0$ will be chosen later. The deviation $\Delta$ can hence be written out as follows.
\begin{align*}
\Delta&\leq2\max\left\{\sqrt{\sum_{t=T_0+1}^{T_1}\frac{G^2t^2(80T_1\Lambda/t^2+3G^2/(\lambda^2t^2))}{\lambda^2T_0^4}\log\frac{1}{\delta}},\ \max_{T_0+1\leq t\leq T_1}\frac{20G^2t}{\lambda^2T_0^2}\log\frac{1}{\delta}\right\}+\sum_{t=T_0+1}^{T_1}\frac{2G^2}{\lambda^2T_0^2}\\
&\leq2\max\left\{\frac{100G^2T_1\log\frac{1}{\delta}}{\lambda^2T_0^2}\sqrt{\frac{\lambda^2\Lambda}{125G^2\log\frac{1}{\delta}}+\frac{1}{10T_1\log\frac{1}{\delta}}},\ \frac{20G^2T_1\log\frac{1}{\delta}}{\lambda^2T_0^2}\right\}+\frac{2G^2T_1}{\lambda^2T_0^2} \, .
\intertext{Let $\Lambda=\frac{10000G^2\log\frac{1}{\delta}}{\lambda^2}$, we further have}
&\leq\frac{900G^2T_1\log\frac{1}{\delta}}{\lambda^2T_0^2} \, .
\end{align*}

Now, in order to \textit{pull out} the stopping time $\tau$, we have to check the pull-out condition in~\autoref{lem:pull out} as follows.
\[
\Pr[\tau\geq t+1\, |\, M_{t'}\leq\Delta,\ \forall t'\leq t]=1
\]
for every $T_0+1\leq t\leq T_1$. Observe that when $M_{t'}\leq\Delta$ for all $t'\leq t$, for every $T_0+1\leq t'\leq t$, by the recursion we have
\begin{align*}
X_{t'}&=\prod_{t''=T_0+1}^{t'}(1-2\eta_{t''}\lambda)(X_{T_0}+M_{t'})\\
&=\frac{T_0(T_0-1)}{t'(t'-1)}(X_{T_0}+M_{t'})\\
&\leq\frac{T_0^2}{t'^2}(X_{T_0}+\Delta)\\
&\leq\frac{T_0^2}{t'^2}\frac{2000G^2T_1\log\frac{1}{\delta}}{\lambda^2T_0^2}\\
&\leq\frac{T_1\Lambda}{10t'^2}=\frac{\Lambda_{t'}}{10}
\end{align*}
and hence $\tau>t'$ by the definition of $\tau$. Since the above holds for all $T_0+1\leq t'\leq t$, we have $\tau\geq t+1$ as desired. Finally, by the pull-out lemma (i.e.,~\autoref{lem:pull out}) we have
\[
\Pr\left[\exists T_0+1\leq t\leq T_1,\ |M_{t}|>\frac{900G^2T_1\log\frac{1}{\delta}}{\lambda^2T_0^2}\right]<\delta \, .
\]
Combine with the recursion and the fact that $X_{T_0}\leq\frac{4G^2}{\lambda^2}$ due to the $G$-boundedness of $F$, we have
\[
\Pr\left[X_{T_1}>\frac{1000G^2\log\frac{1}{\delta}}{\lambda^2T_1}\right]<\delta \, .
\]
This completes the proof for the last iterate convergence of SGD.
\end{proof}

\subsubsection{Strong uniform convergence}\label{app:sgd strong unif}
Recall that in~\autoref{sec:sgd} we used an interval analysis to achieve a tighter improvement analysis for the strong uniform guarantee. Here, we modularize the improvement analysis for a single interval into the lemma.

\begin{lemma}\label{lem:sgd concentration details}
Consider the setting in~\autoref{thm: sgd main appendix}. For every $100\leq T_0\in\N$, let  $(B_{T_0},\mu_{T_0},\sigma^2_{T_0})$ be the moment profile for $\{X_t\}$ and $T_0$ from~\autoref{lem:sgd moment details}. If $\eta_t=1/\lambda t$, then for every $A_0>A_1>0$, $\delta'>0$, $\Lambda>0$, and $100\leq T_0<T_1\in\N$, let
\[
\Delta=\frac{G^2T_1\log\frac{1}{\delta'}}{\lambda^2T_0^2} \left(\sqrt{\frac{70T_1\Lambda\lambda^2}{G^2
\log\frac{1}{\delta'}}} + 50\right) \, .
\]
Suppose that we have $A_0 + \Delta < \Lambda$ and $\frac{T_0(T_0-1)}{T_1(T_1-1)} \Lambda < A_1$. Then,
\[
\Pr\left[\exists{T_0 + 1\leq t\leq T_1}\, ,\ X_t > \frac{T_0(T_0-1)}{t(t-1)}\cdot\Lambda \ \middle| \ X_{T_0} < A_0\right] < \delta' \, .
\]
In particular, the above implies $\Pr[X_{T_1}>A_1\ |\ X_{T_0}\leq A_0]\leq\delta'$.
\end{lemma}
\begin{proof}
Given the moment profile for $\{X_t\}$ in~\autoref{lem:sgd moment details}, we apply~\autoref{prop:concentration} and get the following deviation using~\autoref{lem:martingale concentration}.
\begin{align*}
 &2\max\left\lbrace\sqrt{\sum_{t' = T_0 + 1}^{T_1}\sigma_{T_0}^2(t', \Lambda)\log\frac{1}{\delta'}}, 2\max_{T_0+1\leq t'\leq T_1}B_{T_0}(t',\Lambda)\log\frac{1}{\delta'}\right\rbrace + \sum_{t'=T_0+1}^{T_1}\mu_{T_0}(t', \Lambda)\\
 =\ &\max\left\{2\sqrt{\sum_{t' = T_0 + 1}^{T_1}\frac{G^2t'^2}{\lambda^2T_0^4}\left(80\Lambda+\frac{3G^2}{\lambda^2t'^2}\right)\log\frac{1}{\delta'}} , 40\frac{G^2T_1}{\lambda^2T_0^2}\log\frac{1}{\delta'}\right\} + \sum_{t'=T_0+1}^{T_1}\frac{2G^2}{\lambda^2 T_0^2}\\
 \leq\ & \sqrt{\frac{70G^2T_1^3\Lambda\log\frac{1}{\delta'}}{\lambda^2T_0^4}} + \frac{G^2\sqrt{12\log\frac{1}{\delta'}}}{\lambda^2T_0^2} +  40\frac{G^2T_1}{\lambda^2T_0^2}\log\frac{1}{\delta'} + \frac{2G^2T_1}{\lambda^2 T_0^2}\\
 \leq\ &\frac{G^2T_1\log\frac{1}{\delta'}}{\lambda^2T_0^2} \left(\sqrt{\frac{70T_1\Lambda\lambda^2}{G^2
\log\frac{1}{\delta'}}} + 50\right) = \Delta \, .
\end{align*}
Next, by~\autoref{prop:concentration} we get the desiring improvement inequalities.
\end{proof}

Now, we wrap up the proof for $O(\frac{G^2(\log\delta^{-1}+\log\log t)}{\lambda t^2})$ \textit{strong uniform} convergence bound.
\begin{proof}[Proof of item 2 in~\autoref{thm: sgd main appendix}]
Let $\ell=\infty$ and let
\[
t_0 = 0,\ t_1 = 100,\ t_i = 2t_{i-1},\ \forall i\geq2 \, .
\]
For $i\geq 1$, let
\[
\delta_i=\frac{\delta}{2i^2},\ a_0 = \frac{1000 G^2\log\frac{1}{\delta}}{\lambda^2 t_1},\ a_i = \frac{1000 G^2\log\frac{1}{\delta_i}}{\lambda^2 t_i}\text{, and } \Lambda_i = 2a_{i-1} \, .
\]
Now, for each $i\in\N$, we invoke~\autoref{lem:sgd concentration details} with $A_0=a_{i-1}$, $A_1=a_i$, $T_0=t_{i-1}$, $T_1=t_i$, and $\delta'=\delta_i$.
Let us verify the two conditions. First, we verify the pull out condition as follows.
\begin{align*}
a_{i-1} + \Delta_i
&=a_{i-1} + \frac{G^2t_i\log\frac{1}{\delta_i}}{\lambda^2t_{i-1}^2} \left(\sqrt{\frac{70t_{i-1}\Lambda_i\lambda^2}{G^2
\log\frac{1}{\delta_i}}} + 50\right)\\
&= a_{i-1} + \frac{2\sqrt{70\cdot 2000}\cdot a_{i-1}}{1000} + \frac{200a_{i-1}}{1000} < 2a_{i-1} = \Lambda_i \, .
\end{align*}
Next, we verify the improvement condition $\prod_{t=t_{i-1}+1}^{t_i}(1-2\eta_t\lambda)(a_{i-1}+\Delta_i) < a_{i}$ as follows.
\[
 \frac{t_{i-1}(t_{i-1} - 1)}{t_i(t_i - 1)} \cdot 2a_{i-1} < \frac{2a_{i-1}}{4} \leq a_i \, .
\]
As $a_i\geq \frac{1000G^2(\log(1/\delta)+\log\log (t_{i-1}+1))}{\lambda^2t_{i}}$ for every $t_{i-1}+1\leq t\leq t_i$, by~\autoref{lem:sgd concentration details}, this implies that
\[
\Pr\left[\exists t_{i-1} + 1\leq t\leq t_i \, ,\ X_{t} > \frac{1000 G^2\left(\log\frac{1}{\delta}+\log\log (t+1)\right)}{\lambda^2 t} \, \middle|\, X_{t_{i-1}}\leq a_{i-1} \right] < \delta_i \, .
\]
Also, since $\sum_{i=1}^\infty\delta_i\leq\delta$ and $X_{t}\leq \frac{4G^2}{\lambda^2}$ for all $t\leq t_1$ due to the $G$-boundedness of $F$, by union bound we have
\[
\Pr\left[\exists  t\in\N,\ X_{t} > \frac{1000 G^2\left(\log\frac{1}{\delta}+\log\log (t+1)\right)}{\lambda^2 t} \right] < \delta \, .
\]
This completes the proof for the strong uniform convergence of SGD.
\end{proof}

config: arxiv


text:
\begin{document}
\title[On homology classes of closed submanifolds in compact manifolds]{Realizing a homology class of a compact manifold by a homology class of an explicit closed submanifold--a new approach to Thom's works on homology classes of submanifolds--}
\author{Naoki Kitazawa}
\keywords{Differential topology. Submanifolds. Singularities of differentiable maps.}
\subjclass[2010]{Primary~57N15. Secondary~57R45}
\address{Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku Fukuoka 819-0395, Japan\\
 TEL (Office): +81-92-802-4402 \\
 FAX (Office): +81-92-802-4405}
\email{n-kitazawa@imi.kyushu-u.ac.jp}
\maketitle
\begin{abstract}
It is a classical important problem of differential topology by Thom; for a homology class of a compact manifold, can we realize this by a closed (smooth) submanifold with no boundary? This is true if the degree of the class is smaller or equal to the half of the dimension of the outer manifold under the condition that the coefficient ring is $\mathbb{Z}/2\mathbb{Z}$ and that the outer manifold is closed and smooth. If the degree of the class is smaller or equal to $6$ or equal to $k-2$ or $k-1$ under the condition that the coefficient ring is $\mathbb{Z}$ where $k$ is the dimension of the manifold and that the outer manifold is closed, orientable and smooth, then this is also true. As a specific study, for $4$-dimensional closed manifolds, the topologies of closed and connected surfaces realizing given 2nd homology classes have been actively studied, for example.

In the present paper, we consider the following similar problem; can we realize a homology class of a compact manifold by a homology class of an explicit closed manifold embedded in the (interior of the) given compact manifold? This problem is considered as a variant of previous problems.
We present an affirmative answer via important theory in the singularity theory of differentiable maps: lifting a given smooth map to an embedding or obtaining an embedding such that the composition of this with the canonical projection is the given map. Presenting this application of lifting smooth maps and related fundamental propositions is also a main purpose of the present paper.

\end{abstract}

\maketitle
\section{Introduction and fundamental notation and terminologies.}
\label{sec:1}

The following problem, essentially launched by Thom, is a classical important problem in differential topology. We discuss the problem and related problems here in the smooth category. Let $A$ be a module.

\begin{Prob}
For a homology class $c \in H_j(X;A)$ of a closed manifold $X$, can we realize this by a closed submanifold $Y$ with no boundary or can we represent $c$ as $i_{\ast}({\nu}_Y)=c$ for a generator ${\nu}_{Y} \in H_{\dim Y}(Y;A)$ of the module $H_{\dim Y}(Y;A)$ and the inclusion $i:Y \rightarrow X$?
\end{Prob}
This is true if the degree of the class is smaller or equal to the half of the dimension of the outer manifold under the condition that the
 coefficient $A$ is the ring $\mathbb{Z}/2\mathbb{Z}$. If the degree of the class is smaller or equal to 6 or equal to $\dim X-2$ or $\dim X-1$ under the conditions that the outer manifold $X$ is orientable and that the coefficient is the ring $\mathbb{Z}$, then this is also true. See \cite{thom} and see also \cite{suzuki} for related classical studies. Furthermore, \cite{bohrhankekotschick} is on classes we cannot represent in this way.
\begin{Prob}
For a 2nd homology class of a $4$-dimensional closed manifold, how about the orientability and the genus of a closed and connected surface realizing this?
\end{Prob}
This gives various, explicit, important and interesting problems. See \cite{kronheimer} for example. In low dimensional differential topology, these kinds of problems are actively studied via technique on low dimensional topology, gauge theory, and so on.
\subsection{Problems studied in this paper related to these problems by Thom}

In the present paper, related to the problems before, we consider the following problem.
\begin{MainProb}
For a homology class $c \in H_j(X;A)$ of a compact manifold $X$, can we realize the homology class by a homology class of a closed manifold $Y$ satisfying $\dim Y \geq j$ and $\partial Y=\emptyset$ embedded in the (interior of the) manifold or can we represent $c$ as $i_{\ast}(c^{\prime})=c$ for
 a class $c^{\prime} \in H_{j}(Y;A)$ of the module $H_{j}(Y;A)$ which may not be of degree $\dim Y$ and the inclusion $i:Y \rightarrow X$? Moreover, can we obtain $Y$ in a constructive way?
\end{MainProb}
Note that we consider manifolds of arbitrary dimensions in the present paper.
\subsection{Fold maps and Reeb spaces}
\subsubsection{Fold maps}
We introduce terminologies on differentiable maps. A {\it singular} point of a differentiable map $c:X \rightarrow Y$ is a point at which the rank of the differential of the map drops or the image of the differential is smaller than both the dimensions of $X$ and $Y$. A {\it singular value} of the map is a point realized as a value at a singular point of the map. The set $S(c)$ of all singular points is the {\it singular set} of the map. The {\it singular value set} is the image $c(S(c))$ of the singular set. The {\it regular value set} of the map is the complementary set $Y-c(S(c))$ of the singular value set and a {\it regular value} is a point in the regular value set.

Throughout the present paper, manifolds are assumed to be smooth or differentiable and of $C^{\infty}$ and so are maps between manifolds. A diffeomorphism on a manifold is always smooth and the {\it diffeomorphism group} of it is defined as the group of all diffeomorphisms on it.
\begin{Def}
\label{def:1}
Let $m \geq n \geq 1$ be integers. A smooth map from an $m$-dimensional smooth manifold with no boundary into an $n$-dimensional smooth manifold with no boundary is said to be a {\it fold} map if at each singular point $p$, the form is $(x_1, \cdots, x_m)\mapsto (x_1, \cdots, x_{n-1}, {\Sigma}_{k=n}^{m-i} {x_k}^2-{\Sigma}_{k=m-i+1}^{m} {x_k}^2)$ for some coordinates and an integer $0 \leq i(p) \leq \frac{m-n+1}{2}$.
\end{Def}
For a fold map in Definition \ref{def:1}, we have the following two properties.
\begin{enumerate}
\item For any singular point $p$, the $i(p)$ is unique {\rm (}$i(p)$ is called the {\it index} of $p${\rm )}.
\item The set consisting of all singular points of a fixed index of the map is a closed submanifold of dimension $n-1$ with no boundary of the domain and the restriction to the singular set is an immersion of codimension $1$.
\end{enumerate}
A fold map is a {\it special generic} map if $i(p)=0$ for all singular points. A Morse function with exactly two singular points on a homotopy sphere and canonical projections of unit spheres are simplest examples of special generic maps.
If $m=n$, then a fold map is always special generic.
\subsubsection{Reeb spaces}
The {\it Reeb space} of a continuous map is defined as the space of all connected components of preimages of the map.
\begin{Def}
\label{def:2}
 Let $X$ and $Y$ be topological spaces. For a continuous map $c:X \rightarrow Y$, we define a relation on $X$ as $p_1 {\sim}_c p_2$ if and only if $p_1$ and $p_2$ are in
 a same connected component of $c^{-1}(p)$ for some $p \in Y$. Thus ${\sim}_{c}$ is an equivalence relation on $X$ and we denote the quotient space $X/{\sim}_c$ by $W_c$ and call it the {\it Reeb space} of $c$.
\end{Def}

We denote the induced quotient map from $X$ into $W_c$ by $q_c$ and we can define $\bar{c}: W_c \rightarrow Y$ uniquely by $c=\bar{c} \circ q_c$. See also \cite{reeb} for example.
\begin{Prop}[\cite{shiota}]
\label{prop:1}
For a fold map, the Reeb space is a polyhedron whose dimension is equal to that of the target manifold.
\end{Prop}
For suitable classes of these maps, Reeb spaces inherit topological properties of the manifolds admitting the maps. See \cite{kitazawa}, \cite{kitazawa2}, \cite{kitazawa3} and \cite{saekisuzuoka} for example.
\subsection{Organization of the present paper}
This paper concerns Main Problem and also presents new answers to important problems on differentiable maps between manifolds.  The problems are on lifting of maps, which is a key ingredient in an explicit problem related to Main Pronlem.
The organization of the paper is as the following.

In the next section, we introduce a class of {\it standard-spherical} fold maps.
It is defined as a fold map such that indices of all singular points are $0$ or $1$, that preimages of regular values are points or disjoint unions of standard spheres: if the codimension of the map $f:M \rightarrow N$ is $n-m=-1$, then assume also that the domain is an orientable manifold. A {\it simple} fold map $f$ is a fold map such that ${q_f}{\mid}_{S(f)}$ is injective.
Special generic maps form a proper subclass of these classes.

In the third section, we present examples of simple standard-spherical fold maps. After that, we consider lifting these maps to embeddings: in other words we obtain embeddings such that the compositions with the canonical projections are the given maps. This is a fundamental, important and interesting studies in the theory of singularity of differentiable maps and this also gives strong tools in obtaining main results of the present paper. We also present a new answer for a kind of these problems as Theorem \ref{thm:1}: this is also an important ingredient in main results. The last section is devoted to main results or explicit answers to Main Problem via tools and theory in the third section.
\section{Special generic maps, standard-spherical fold maps and simple fold maps}
\label{sec:2}

Throughout this paper, let $m \geq n \geq 1$ be integers, $M$ be a closed and connected manifold of dimension $m$, $N$ be a manifold of dimension $n$ without boundary and $f:M \rightarrow N$ be a smooth map unless otherwise stated.
In addition, the structure groups of bundles such that the fibers are manifolds are assumed to be
 (subgroups of) diffeomorphism groups unless otherwise stated or these bundles are assumed to be so-called {\it smooth} bundles: {\it PL} bundles are discussed as exceptional cases. A {\it linear} bundle is a smooth bundle whose fiber is a ($k+1$)-dimensional (closed) unit disc or the $k$-dimensional unit sphere in ${\mathbb{R}}^{k+1}$ and whose structure group is a subgroup of the ($k+1$)-dimensional one $O(k+1)$ acting linearly in a canonical way. A {\it PL} bundle is a bundle whose fiber is a polyhedron and whose structure group is a group consisting of PL homeomorphisms of the fiber.
\begin{Def}
A fold map $f:M \rightarrow N$ is said to be {\it standard-spherical} if the following properties hold (we can easily know the definitions of a {\it crossing} and a {\it normal} crossing of a smooth immersion and we omit the definitions).
\begin{enumerate}
\item The restriction map $f {\mid}_{S(f)}$ is an immersion whose crossings are normal.
\item Indices of singular points are $0$ or $1$.
\item If $m-n=1$ and there exists a singular point of index $1$ of $f$, then $M$ is orientable.
\item Preimages of regular values are disjoint unions of points or standard spheres.
\end{enumerate}
\end{Def}
\begin{Ex}
A special generic map $f$ is a map of this class. Under the condition $m=n$ its Reeb space $W_f$ is regarded as a manifold diffeomorphic to $M$. Under the condition $m>n$ its Reeb space $W_f$ is an $n$-dimensional compact manifold we can smoothly immerse into $N$. $f$ is represented as the composition of $q_f$ with the immersion and furthermore, $q_f(S(f))=\partial W_f$ and ${q_f} {\mid}_{S(f)}$ is injective.
\end{Ex}
\begin{Def}
We say a fold map $f$ is {\it simple}
 if the restriction of $q_f$ to the singular set is injective.
\end{Def}
Special generic maps are simple. Let $m>n$.

Under the condition $m>n$, generally, for a simple fold map $f$, for each connected component $C$ of $q_f(S(f))$, its small regular neighborhood $N(C)$ is represented as a PL bundle over $C$ whose fiber is a closed interval or a $Y$-shaped $1$-dimensional polyhedron and the composition of the restriction of the map $q_f$ to the preimage ${q_f}^{-1}(N(C))$ of the total space $N(C)$ of the bundle with the canonical projection to $C$ is a smooth bundle.

Under the condition $m>n$, for a simple standard-spherical map $f$, for the complementary set of the union of the regular neighborhoods $N(C)$ for all connected components $C$ of $q_f(S(f))$, the restriction of $q_f$ to the preimage gives a smooth bundle whose fiber is $S^{m-n}$. The fiber
 of the bundle $N(C)$ is Y-shaped and the bundle ${q_f}^{-1}(N(C))$ is a smooth bundle whose fiber is diffeomorphic to a manifold obtained by removing the interior of a disjoint union of three standard closed discs of dimension $m-n$ smoothly and disjointly embedded into $S^{m-n}$ for each connected component $C$ consisting of singular points of index $1$. For this, see also \cite{saeki} for example. For each connected component $C$ consisting of singular points of index $0$, the fiber of the PL bundle is a closed interval and the smooth bundle is a linear bundle and this holds for general fold maps by \cite{saeki2}.

Under the condition $m=n$, for each connected component $C$ of the singular set, we can take a closed tubular neighborhood $N(C) \subset M$ and the canonical projection to $C$ gives a linear bundle whose fiber is a closed interval.

\begin{Rem}
In the case of a simple standard-spherical fold map $f$, we do not assume that the restriction map $f {\mid}_{S(f)}$ is an immersion whose crossings are normal. We may assume this but the assumption is not essential.
\end{Rem}

\section{S-trivial standard-spherical fold maps and lifting these maps to embeddings.}
\label{sec:3}
\begin{Def}
\label{def:5}
In section \ref{sec:2}, if in the case $m>n$ the PL bundle $N(C)$ over $C$ and the smooth bundle ${q_f}^{-1}(N(C))$ over $C$ are trivial for each $C$, then the simple fold map $f$ is said
 to be {\it S-trivial}. If in the case $m=n$, the linear bundle $N(C)$ over $C$ is trivial for each $C$, then the fold map $f$ is said to be {\it S-trivial}.
\end{Def}

We present several examples of S-trivial standard-spherical fold maps.

\begin{Ex}
\label{ex:2}
\begin{enumerate}
\label{ex:2.1}
\item A special generic map $f:M \rightarrow N$ between equidimensional manifolds such that the following properties hold.
\begin{enumerate}
\item $M$ is orientable.
\item $S(f)$ is orientable.
\end{enumerate}
For example, let $M$ be a homotopy sphere for example. \cite{eliashberg} implies that for any homotopy class of continuous maps between $S^m$ and $S^n$ where $m=n$, we can find a fold map satisfying the properties before.
\item
\label{ex:2.2}
Projections of bundles whose fibers are standard spheres: they are also special generic maps having no singular point.
\item
\label{ex:2.3}
\cite{costantinothurston}, \cite{ishikawakoda} and \cite{saeki3} present S-trivial standard-spherical fold maps. Through them, we can know that 3-dimensional closed and orientable manifolds of a class (the class of {\it graph manifolds}) admit such maps into surfaces. For such a map, we can regard the Reeb space $W_f$ as the {\it shadow} of the manifold. Roughly speaking, a {\it shadow} is a polyhedron at each point of which there exists a small regular neighborhood PL homeomorphic to a regular neighborhood of a point of a Reeb space of a simple fold map into the plane as before. Moreover, the space of all points the regular neighborhoods of which are $2$-dimensional discs containing the points in the interiors is a $2$-dimensional manifold and an integer called a {\it gleam} is assigned to each connected component of this. We will review the notion in Definition \ref{def:6}. See \cite{ishikawakoda} and see also \cite{turaev} and \cite{turaev2}. See also Remark \ref{rem:1}.
\item
\label{ex:2.4}
\cite{kitazawa5} and \cite{kitazawa7} present construction of families of infinitely many such maps and families of infinitely many closed and connected manifolds admitting them starting from fundamental examples. Fundamental examples are, special generic maps and examples in \cite{kitazawa}, \cite{kitazawa2} and \cite{kitazawa4} for example and we obtain new maps via surgery operations to the maps and the manifolds.
\end{enumerate}
\end{Ex}
We introduce {\it shadows}. We omit explanations on the notions of an {\it oriented} linear bundle and its ({\it relative}) {\it Euler class}.
\begin{Def}
\label{def:6}
Let $f$ be an S-trivial standard-spherical fold map on a 3-dimensional closed, connected and orientable manifold into an orientable surface. Orient the target surface. Define for the smooth bundle ${q_f}^{-1}(N(C))$ over $C$ a trivialization for each $C$.
$P$ denotes each connected component of the complementary set of the union of suitable small regular neighborhoods $N(C)$ of $C$ for all connected components $C$ of $q_f(S(f))$.
We can induce the orientation on $P$ from the target surface. We can define and let us define a trivialization for the trivial bundle ${q_f}^{-1}(P)$ over $P$ for each non-closed $P$, which is regarded as an oriented linear bundle which is trivial and whose fiber is diffeomorphic to $S^1$. We can define a relative Euler class $e_P \in H^2(P,\partial P;\mathbb{Z})$ for the oriented linear bundle over each $P$ respecting the trivializations of the bundles. The value of $e_P$ at the fundamental class of $P$, which is the generator of $H_2(P,\partial P;\mathbb{Z})$ respecting the orientation of $P$, is called the {\it gleam}. We can define a {\it shadow induced from} $f$ as the tuple of the Reeb space $W_f$, the orientation of the target surface, the family of all the trivializations of the trivial bundles, and the family of all gleams.
\end{Def}

\begin{Rem}
\label{rem:1}
In Example \ref{ex:2} (\ref{ex:2.3}) and Definition \ref{def:6}, a {\it shadow} is introduced and the definiton is in general a bit different. A {\it shadow} is defined as an object suitable for the content of the present paper here, and a {\it shadow} for the $2$-dimensional polyhedron of a wider class is in fact defined as a pair of the polyhedron and the family of integers or rational numbers of the form $\frac{k}{2}$ where $k$ is an integer assigned to all connected components of a $2$-dimensional manifold obtained as a subpolyhedron of the original polyhedron by removing the interior of a small regular neighborhood of a 1-dimensional subpolyhedron defined as the subset of all non-manifold points in the original 2-dimensional polyhedron. More precisely, a {\it shadow} is for a $3$-dimensional closed, connected and orientable manifold obtained in a natural way from a shadow introduced just before uniquely and there are in general shadows of more than one type for such a $3$-dimensional manifold. The assigned numbers are said to be {\it gleams}. Similarly, from a fold map $f$ such that the restriction map $f {\mid}_{S(f)}$ is an immersion whose crossings are normal on a $3$-dimensional closed, connected and orientable manifold into a surface, which always exists, and the Reeb space $W_f$, we can obtain a {\it shadow} similarly. Giving the rigorous definiton is left to readers or see the references.
\end{Rem}

The following proposition is a fundamental and key proposition in the present paper.
${\rm Emd}(X,Y)$ denotes the space of all the smooth embeddings of a manifold $X$ into a manifold $Y$ endowed with the so-called {\it $C^{\infty}$ Whitney topology} (see \cite{golubitskyguillemin}).
\begin{Fact}[\cite{budney}]
\label{fact:1}
${\rm Emd}(X,Y)$ is $\max\{0,\min \{2\dim
Y-3\dim X-4,\dim Y-dim X-2\}\}$-connected if $\dim Y \geq \dim X+2 \geq 3$ is assumed.
\end{Fact}
See also \cite{nishioka} for Fact \ref{fact:1}. The following is for example shown in Lemma 1 of \cite{kitazawa6}.
Hereafter, we call a manifold obtained by removing the interiors of three smoothly and disjointly embedded closed $k$-dimensional discs in a $k$-dimensional standard sphere a {\it pair of pants} of dimension $k$ for $k>1$.
\begin{Lem}
\label{lem:1}
For a Morse function $\bar{f}$ on a $k$-dimensional compact manifold satisfying $k \geq 1$ and either of the following properties, we can represent this as a composition of an embedding into ${\mathbb{R}}^{k+1}$ with a canonical projection.
\begin{enumerate}
\item A Morse function $\bar{f}$ is a function on a closed unit disc of dimension $k$ with exactly one singular point such that the singular point is in the interior, that the preimage of one of the two extrema and the boundary coincide, and that the preimage of the remaining extremum is the one-point set consisting of the singular point. In other word, it is a so-called {\it height function}.
\item $k>1$ and a Morse function $\bar{f}$ is a function on a pair of pants of dimension $k$ satisfying the following two.
\begin{enumerate}
\item It is a function with exactly one singular point and the singular point is in the interior.
\item The preimages of the two extrema are a connected component of the boundary and the disjoint union of the remaining two connected components of the boundary, respectively.
\end{enumerate}
\end{enumerate}
\end{Lem}
\begin{proof}[A sketch of the proof.]
The former case follows by the definition of a singular point of index $0$ of a fold map.
The latter case follows by the argument as the following.
\begin{enumerate}
\item Decompose the pair of pants of dimension $k$ into two $k$-dimensional compact submanifolds with corners along ($k-1$)-dimensional submanifolds: one is a disjoint union of two copies of the total space of a product bundle over a closed interval whose fiber is a standard closed disc of dimension $k-1$ and the other is diffeomorphic to the product of a closed interval and a standard closed disc of dimension $k-1$, which is so-called a $1$-handle of the Morse function.
\item On each connected component of the disjoin union of the total spaces of the product bundles, the original function is regarded as a projection of the bundle by virtue of (a relative version of) Ehresmann's fibration theorem (\cite{ehresmann}).
\end{enumerate}
\end{proof}
\begin{Prop}[\cite{nishioka} and an extension of the result there.]
\label{prop:2}
Let $n+k \geq \max\{\frac{3m+3}{2},m+n+1\}$.
For an S-trivial standard-spherical fold map $f:M \rightarrow N$, there exists an embedding $F$ such that
$f={\pi}_{N,k} \circ F$ where ${\pi}_{N,k}:N \times {\mathbb{R}}^k \rightarrow N$ is the canonical projection onto $N$.
\end{Prop}
\begin{proof}
For each connected component $C$ of $q_f(S(f))$, consider a small regular neighborhood $N(C)$ and the local smooth map represented as the composition of ${q_f} {\mid}_{{q_f}^{-1}(N(C))} $ with ${\bar{f}} {\mid}_{N(C)}$, we can consider an embedding so that the composition with the composition of the canonical projection $N(C) \times {\mathbb{R}}^k$ onto $N(C)$ with ${\bar{f}} {\mid}_{N(C)}$ is the original map: the assumptions that the map is S-trivial together with arguments related to so-called Thom's isotopy theorem and Lemma \ref{lem:1} and that $n+k \geq m+n+1$ enable us to do this.
We extend this over the complementary set of the union of $N(C)$ in $W_f$. The essential assumption is that the space ${\rm Emb}(S^{m-n},{\mathbb{R}}^k)$ is ($n-1$)-connected together with the relation $n+k \geq m+n+1$. We consider suitable cell decompositions of $W_f$ and the complementary set. On each $1$-cell in the complemantary set, the original map is regarded as the projection of a trivial smooth bundle and we can construct a smooth embedding as before. We can extend this as projections over $2$-cells. We can do this inductively over $k$-cells where $2 \leq k\leq n$. This completes the proof.
\end{proof}

We explain about the fact that ${\rm Emb}(S^{m-n},{\mathbb{R}}^k)$ is ($n-1$)-connected. In Fact \ref{fact:1}, the inequality $2k-3(m-n)-4 \geq 2(\frac{3m+3}{2}-n)-3(m-n)-4=n-1$ completes the proof.

Nishioka studied only cases of special generic maps on closed, connected and orientable manifolds into Euclidean spaces satisfying the relation $m>n$. For the methods here, see also \cite{saekitakase} for example. They are on a hot topic of the singularity theory of differentiable maps: lifting a smooth map to a smooth map of a suitable class into a higher dimensional space or finding a representation of the original map by a composition of a map of the suitable class with a canonical projection. Note also that in these problems, as the author know, the target manifolds of smooth maps are Euclidean spaces.

The following theorem presents another new explicit answer for a kind of these problems and important in a main result or Theorem \ref{thm:3} later.
\begin{Thm}
\label{thm:1}
Let $f:M \rightarrow N$ be an S-trivial standard-spherical fold map from a $3$-dimensional closed, connected and orientable manifold into a $2$-dimensional orientable manifold $N$ with no boundary such that for a shadow induced from the map {\rm (}, the Reeb space in Example \ref{ex:2} {(\rm }\ref{ex:2.3}{\rm )} or Remark \ref{rem:1}{\rm )}, all gleams are even. For the map and any integer $k \geq 3$, there exists an embedding $F:M \rightarrow N \times {\mathbb{R}}^k$ such that
$f={\pi}_{N,k} \circ F$ where ${\pi}_{N,k}:N \times {\mathbb{R}}^k \rightarrow N$ is the canonical projection onto $N$.

\end{Thm}
\begin{proof}
We prove for $k=3$ and we can prove for general $k \geq 3$.

Around each connected component $C$ of $q_f(S(f))$ and each connected component of $W_f-{\bigcup}_C N(C)$ where we abuse notation in the proof of Proposition \ref{prop:2}, we can construct smooth map into ${\mathbb{R}}^{2+k}$ similarly and we will construct.

First we investigate the case where each connected component $O$ of $W_f-{\bigcup}_C N(C)$ is not a closed
 surface.

 The restriction of $f$ to the preimage ${q_f}^{-1}(N(C))$ is, for suitable coordinates, represented as the composition of a product map of either of the following Morse functions, which are presented in Lemma \ref{lem:1}, and the identity map ${\rm id}_{C}$ with a suitable immersion into $N$. Note that the product map is, in other words, a trivial $1$-dimensional family of the Morse function.
\begin{enumerate}
\item A Morse function on a closed unit disc of dimension $2$ with exactly one singular point such that the singular point is in the interior, that the preimage of one of the two extrema and the boundary coincide and that the preimage of the remaining extremum is the one-point set consisting of the singular point.
\item A Morse function on a pair of pants of dimension $2$ with exactly one singular point such that the singular point is in the interior and that the preimages of the two extrema are a connected component of the boundary and the disjoint union of the remaining two connected components of the boundary.
\end{enumerate}
In the situation of the proof of Proposition \ref{prop:2}, they are represented as the compositions of embeddings into $I \times {\mathbb{R}}^3$ with the canonical projection to the first component where $I$ is a closed interval, regarded as a fiber of a suitable trivial bundle over $C$. We can consider functions obtained by considering the 1-dimensional higher versions of these two functions as presented in Lemma \ref{lem:1} as the second Morse functions. They are also represented as the compositions of embeddings into $I \times {\mathbb{R}}^3$ with the canonical projection to the first component where $I$ is a closed interval, regarded as a fiber of the trivial bundle over $C$, in the same situation. Furthermore, the original functions are obtained as restrictions of the corresponding functions on the $3$-dimensional closed unit discs or the pairs of pants of dimension $3$. Moreover, the boundary of the domain of each function on the surface is obtained as a closed submanifold with no boundary of the boundary of the domain of the Morse function on the $3$-dimensional manifold so that each connected component is an equator of each of the three $2$-dimensional standard spheres in the boundary of the $3$-dimensional manifold. Distinct circles are embedded as equators in distinct $2$-dimensional spheres in the boundary of the $3$-dimensional manifold.

On each connected component $O$ of $W_f-{\bigcup}_C N(C)$, the map $f$ is regarded as the composition of the projection of a trivial linear bundle whose fiber is a circle with a suitable immersion into $N$. We can represent the projection as the composition of an embedding into $O \times {\mathbb{R}}^3$ with the canonical projection to $O$. We can also construct a map regarded as the composition of the projection of a linear bundle whose fiber is a $2$-dimensional standard sphere with a suitable immersion into $N$ and we can represent this as the composition of the composition of an embedding into $O \times {\mathbb{R}}^3$ with the canonical projection to $O$ with the last immersion. Furthermore, the original projection is obtained as the restriction of the corresponding projection of the total space of the linear bundle whose fiber is the $2$-dimensional sphere by restricting each fiber to the equator of the sphere. In other words, the original bundle is regarded as a subbundle.

We can glue the local maps on $3$-dimensional and $4$-dimensional manifolds together to obtain global embeddings into $N \times {\mathbb{R}}^{3}$ and a smooth map into $N$. Note for example that we obtain the map on the $3$-dimensional closed manifold $M$ into $N$ as the given map $f$. Connected components of the boundaries where we glue the maps together are, regarded as total spaces of trivial linear bundles over circles whose fibers are $S^1$ or $S^2$ and the restrictions of the map $q_f$ to the spaces give the projections. Note that these fibers are in a preimage of the projection ${\pi}_{N,3}:N \times {\mathbb{R}}^3 \rightarrow N$ and $S^2$ is embedded as a so-called {\it unknot} in ${\mathbb{R}}^3$ in the smooth category. $S^1$ is, as explained, embedded in $S^2 \subset {\mathbb{R}}^3$ here as an equator. The assumption on the gleams implies that the way we glue the maps yields global maps: regard each copy of $S^1 \times S^2$ as a trivial linear bundle over $S^1$ equipped with a projection regarded as a canonical projection to the first component (where the trivialization is given). The bundle isomorphisms between two of these bundles is regarded as a product map of diffeomorphisms between the base spaces, regarded as $S^1 \times \{{\ast}_1\}$ and fibers, regarded as $\{{\ast}_2\} \times S^2$.

We explain this more precisely. We consider each pair of subbundles $S^1 \times S^1 \subset S^1 \times S^2$ whose fibers are equators to obtain the map $f$ on the $3$-dimensional manifold $M$. The assumption on the gleams implies that the bundle isomorphism is $(x,y) \mapsto r(x)(y)$ where $r:S^1 \rightarrow SO(2)$ is a smooth map and $r(x)$ acts on each fiber, diffeomorphic to $S^1$, linearly in a canonical way. The homotopy class of $r$ is $2a$ times a generator for an integer $a$. It is a fundamental argument on homotopy groups of classical Lie groups that the bundle isomorphism $(x,y) \mapsto r(x)(y)$ extend to an isomorphism $(x,y) \mapsto r_0(x)(y)$ between the two trivial linear bundles $S^1 \times S^2 \supset S^1 \times S^1$ over $S^1$ where $r_0:S^1 \rightarrow SO(3)$ is a null-homotopic smooth map and $r_0(x)$ acts on each fiber, diffeomorphic to $S^2$, linearly in a canonical way. This guarantees the argument of the global construction in the last paragraph.

Last we investigate the case where there exists no singular point of the map $f$ and this completes the proof. We can show the statement similarly by the assumption on the gleams or the discussion on the attachments of the local maps and spaces just before (there exists exactly one connected component a gleam is assigned to). Note also that this case accounts for the case which we cannot treat in the first case.

This completes the proof.
\end{proof}

Exmaple \ref{ex:3} presents infinitely many examples of maps and $3$-dimensional closed, connected and orientable manifolds to which we can apply Theorem \ref{thm:1}. We omit fundamental notions on general $k$-dimensional linear bundles
such that {\it oriented} linear bundles and the {\it Euler classes} of them, defined as $k$-th integral cohomology classes.

\begin{Ex}
\label{ex:3}
Projections of bundles whose fibers are circles over closed and orientable surfaces, which are also regarded as $2$-dimensional linear bundles such that the Euler classes are divisible by $2$ (if it is oriented), are simplest examples satisfying the assumption of Theorem \ref{thm:1}.
\end{Ex}

\section{Main theorems}
\begin{Def}
\label{def:7}
For a homology class of a compact manifold, if for the homology group of this, the coefficient is a commutative ring $R$ having the unique identity element $1 \neq 0 \in R$ and a homology class $c \neq 0$ satisfies the following properties, then it is said to be a {\it UFG}.
\begin{enumerate}
\item If $rc=0$ for $r \in R$, then $r=0$.
\item For any element $r \in R$ which is not a unit and any homology class $c^{\prime}$, $c$ is never represented as $rc^{\prime}$.
\end{enumerate}
\end{Def}
\begin{Def}
\label{def:8}
Let $R$ be a commutative ring having the unique identity element $1 \neq 0 \in R$.
For a UFG homology class $c \in H_j(X;R)$ of a compact manifold $X$, the class $c^{\ast} \in H^j(X;R)$ satisfying the following properties is said to be the {\it dual} $c$.
\begin{enumerate}
\item $c^{\ast}(c)=1$.
\item For any submodule $B$ of $H_j(X;R)$ such that the internal direct sum of the submodule generated by $c$ and $B$ is $H_j(X;R)$, $c^{\ast}(B)=0$.
\end{enumerate}
\end{Def}
\begin{Def}
\label{def:9}
For two closed, connected, oriented and equidimensional manifolds $X$ and $Y$, we denote by ${\nu}_X \in H_{\dim X}(X;\mathbb{Z})$ and ${\nu}_Y \in H_{\dim Y}(Y;\mathbb{Z})$ the {\it fundamental classes} of the manifolds or the generators compatible with the orientations: the groups are isomorphic to $\mathbb{Z}$. The {\it mapping degree} of a continuous map $c:X \rightarrow Y$ is the unique integer $d(c)$ satisfying ${\nu}_Y=d(c) c_{\ast}({\nu}_X)$.
For two closed, connected and equidimensional manifolds $X^{\prime}$ and $Y^{\prime}$, we denote by ${\nu}_{X^{\prime}} \in H_{\dim {X^{\prime}}}(X^{\prime};\mathbb{Z}/2\mathbb{Z})$ and
${\nu}_{Y^{\prime}} \in H_{\dim Y^{\prime}}(Y^{\prime};\mathbb{Z}/2\mathbb{Z})$ the {\it $\mathbb{Z}/2\mathbb{Z}$ fundamental classes} of the manifolds or the generators of the groups, isomorphic to $\mathbb{Z}/2\mathbb{Z}$.
The {\it $\mathbb{Z}/2\mathbb{Z}$ mapping degree} of a continuous map $c^{\prime}:X^{\prime} \rightarrow Y^{\prime}$ is the unique element $d(c)=0,1 \in \mathbb{Z}/2\mathbb{Z}$ satisfying ${\nu}_{Y^{\prime}}=d(c) c_{\ast}({\nu}_{X^{\prime}})$.
\end{Def}
\begin{Def}
\label{def:10}
Let $X$ be a compact manifold. Let $Y$ be a closed manifold satisfying $\dim Y<\dim X$. A class $c_X \in H_j(X;R)$ is {\it realized} by a class $c_Y \in H_j(Y;R)$ if for an embedding $i_{Y,X}:Y \rightarrow X$ whose image
is in ${\rm Int} X$, $c_X=i_{Y,X} {\ast}(c_Y)$ holds.

\end{Def}
\begin{Thm}
\label{thm:2}
Let $R:=\mathbb{Z}, \mathbb{Z}/l\mathbb{Z}$ for an integer $l >1$. Let $X$ be a compact manifold. Let a closed and connected manifold $M$ of dimension $m \geq n$ admit an S-trivial standard-spherical fold map $f:M \rightarrow N$ and the following properties hold.
\begin{enumerate}
\item If $R$ is not isomorphic to $\mathbb{Z}/2\mathbb{Z}$, then $M$ and $N$ are orientable and $N$ is oriented so that the fundamental class is ${\nu}_N \in H_n(N;R)$. We also
use ${\nu}_N \in H_n(N;R)$ for the $\mathbb{Z}/2\mathbb{Z}$ fundamental class where $R$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$.
\item There exists a preimage $F$ of a regular value for the map $f$. Let a class $c_F \in H_{m-n}(M;R)$ be represented as the sum of the fundamental classes of all connected components of $F$ oriented canonically respecting the orientation of $N$ and an orientation of $M$ if $R$ is not isomorphic to $\mathbb{Z}/2\mathbb{Z}$. If $R$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$, then let a class $c_F \in H_{m-n}(M;R)$ be represented
as the sum of the $\mathbb{Z}/2\mathbb{Z}$ fundamental classes of all connected components of $F$.
\item There exists an element $k \in R$ and a UFG $c_{F,0} \in H_{m-n}(M;R)$ satisfying $c_F=kc_{F_0}$ and $H_{m-n}(M;R)$ is the internal direct sum of the submodule generated by the one element set $\{c_{F_0}\}$ and a suitable submodule.
\item We can take an embedding $i_{N,X}:N \rightarrow X$ satisfying $i_{N,X}(N) \subset {\rm Int}  X$, $c={i_{N,X}}_{\ast}({\nu}_N)$ and that the normal bundle of the image is trivial.
\item $\dim X \geq \max\{\frac{3m+3}{2},m+n+1\}$.
\end{enumerate}
In this situation, $kc$ is realized by the class $k{\rm PD}_{R}({c_{F,0}}^{\ast})$.

\end{Thm}
\begin{proof}
In the situation of Proposition \ref{prop:2}, we consider a map $i_{N,X,0} \circ f:M \rightarrow i_{N,X}(N)$ where $i_{N,X,0}$ is the embedding defined by restricting the target space of $i_{N,X}$ to $i_{N,X}(N)$ and take the total space of the normal bundle of the image, which is a trivial linear bundle over the image, instead of "$N \times {\mathbb{R}}^k$" in the situation of Proposition \ref{prop:2}.

$F$ is the preimage of a regular value $p$ for the map $f$ and the preimage of the regular value $i_{N,X}(p)$ for $i_{N,X,0} \circ f:M \rightarrow i_{N,X}(N)$ is diffeomorphic to $F$ and $c_F=kc_{F_0} \in H_{m-n}(M;R)$ is
also represented as the sum of all fundamental classes of all connected component of the preimage where the image $i_{N,X}(N)$ is canonically oriented from $N$ and defining the fundamental classes before respecting an orientation of
$M$ (if $R$ is not isomorphic to $\mathbb{Z}/2\mathbb{Z}$). By virtue of the first three properties on $f$, we have ${(i_{N,X,0} \circ f)}_{\ast}({\rm PD}_{R}({c_{F}}^{\ast}))=k{i_{N,X,0}}_{\ast}({\nu}_N)$. Proposition \ref{prop:2} with the last property produces a smooth embedding into the total space of a normal bundle of $i_{N,X}(N)$ in $X$: by the fourth property the bundle is trivial. Furthermore, the embedding can be taken so that the composition with the projection onto $i_{N,X}(N)$ is $i_{N,X,0} \circ f$. $kc=k{i_{N,X}}_{\ast}({\nu}_N)={i_{N \subset X}}_{\ast}(k{i_{N,X,0}}_{\ast}({\nu}_N))={i_{N \subset X}}_{\ast}({(i_{N,X,0} \circ f)}_{\ast}({\rm PD}_{R}({c_{F}}^{\ast})))$ is represented by ${\rm PD}_{R}({c_{F}}^{\ast})$ where $i_{N \subset X}$ is the inclusion into $X$ by virtue of the relation before. We have the result.
\end{proof}
Note that the last condition on the dimensions is for lifting fold maps to smooth embeddings via Proposition \ref{prop:2} and Theorem \ref{thm:1} and in Theorem \ref{thm:2} we do not consider the case $(m,n,\dim X)=(3,2,5)$ or the case of Theorem \ref{thm:1}. However, in this case this argument works. Theorem \ref{thm:3} does not exclude the last case and it is a specific case of Theorem \ref{thm:2}. Theorem \ref{thm:4} is also another specific case.
\begin{Thm}
\label{thm:3}
Let $R:=\mathbb{Z},  \mathbb{Z}/l\mathbb{Z}$ for an integer $l >1$.Let $X$ be a compact and spin manifold and let $c \in H_{2}(X;R)$ be realized by the fundamental class of a closed, connected and orientable manifold $N_0$ of dimension $n=2$ for an orientation of $N_0$. Let $M$ be a closed and connected manifold of dimension $m \geq n=2$. Let $N$ be a closed, connected and orientable manifold $N$ of dimension $n=2$.
If $R$ is not isomorphic to $\mathbb{Z}/2\mathbb{Z}$, then $M$ is assumed to be orientable and $N$ and $N_0$ are oriented so that the fundamental classes are ${\nu}_N \in H_2(N;R)$ and ${\nu}_{N_0} \in H_2(N_0;R)$, respectively. In this case, we also assume that there exists a smooth map $c_N:N \rightarrow N_0$ of mapping degree $k_2$. We also use ${\nu}_N \in H_2(N;R)$ and ${\nu}_{N_0} \in H_2(N_0;R)$ for $\mathbb{Z}/2\mathbb{Z}$ fundamental classes where $R$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$ and in this case we also assume that there exists a smooth map $c_N:N \rightarrow N_0$ of $\mathbb{Z}/2\mathbb{Z}$ of mapping degree $k_2$. Suppose that an S-trivial standard-spherical fold map $f:M \rightarrow N$ exists. We also denote the element of $R$ by $k_2$ obtained by mapping $k_2$ into $R$ in the canonical way. Assume also he following properties.
\begin{enumerate}
\item There exists a preimage $F$ of a regular value for the map $f$. Let a class $c_F \in H_{m-n}(M;R)$ be represented as the sum of the fundamental classes of all connected components oriented canonically respecting the orientation of $N$ and an orientation of $M$ if $R$ is not isomorphic to $\mathbb{Z}/2\mathbb{Z}$. If $R$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$, then let a class $c_F \in H_{m-n}(M;R)$ be represented as the sum of the $\mathbb{Z}/2\mathbb{Z}$ fundamental classes of all connected components.
\item There exists an element $k \in R$ and a UFG $c_{F,0} \in H_{m-2}(M;R)$ satisfying $c_F=kc_{F_0}$ and $H_{m-2}(M;R)$ is the internal direct sum of the submodule generated by the one element set $\{c_{F_0}\}$ and a suitable submodule.
\item We can take an embedding $i_{N_0,X}:N_0 \rightarrow X$ satisfying $i_{N_0,X}(N_0) \subset {\rm Int} X$ and $c={i_{N_0,X}}_{\ast}({\nu}_{N_0})$, which is assumed in the beginning.
\item $\dim X \geq \max\{\frac{3m+3}{2},m+3\}$. If $f$ is an S-trivial map as in Theorem \ref{thm:1} with $(m,n)=(3,2)$, then $M$ is orientable and $\dim X \geq 5$.
\end{enumerate}
In this situation, $k k_2 c$ is realized by the class $k k_2 {\rm PD}_{R}({c_{F,0}}^{\ast})$.
\end{Thm}
\begin{proof}
First from the fourth property and other assumptions on the dimensions of the manifolds, $\dim X \geq 5$ follows.
The existence of $c_N:N \rightarrow N_0$ and $i_{N_0,X}:N_0 \rightarrow X$ in the third property yields the existence of a smooth embedding $i_{N,X}:N \rightarrow X$ of the $2$-dimensional manifold $N$ satisfying $i_{N,X}(N) \subset {\rm Int} X$ and $k_2 c={i_{N,X}}_{\ast}({\nu}_N)$. The assumptions that $X$ is spin and that $N$ and $N_0$ are closed, connected and orientable surfaces guarantee that the normal bundle of the image is trivial together with fundamental arguments on characteristic classes of linear bundles over manifolds related to smooth embeddings of smooth manifolds. In the situation of Theorem \ref{thm:2} let $n=2$ and we do not exclude the case $(m,n,\dim X)=(3,2,5)$. We can apply Theorem \ref{thm:2} together with Theorem \ref{thm:1} for $(m,n,\dim X)=(3,2,5)$.
\end{proof}
\begin{Thm}
\label{thm:4}
Let $R:=\mathbb{Z}, \mathbb{Z}/l\mathbb{Z}$ for an integer $l >1$.
Let $X$ be a compact and spin manifold and let $c \in H_{3}(X;R)$ be realized by the fundamental class of a closed, connected and orientable manifold $N_0$ of dimension $n=3$ for an orientation of $N_0$. Let $M$ be a closed and connected manifold of dimension $m \geq n=3$. Let $N$ be a closed, connected and orientable manifold $N$ of dimension $n=3$.
If $R$ is not isomorphic to $\mathbb{Z}/2\mathbb{Z}$, then $M$, $N$ and $N_0$ are orientable and $N$ and $N_0$ are oriented so that the fundamental classes are ${\nu}_N \in H_3(N;R)$ and ${\nu}_{N_0} \in H_3(N_0;R)$, respectively. In this case, we also assume that there exists a smooth map $c_N:N \rightarrow N_0$ of mapping degree $k_3$. We also use ${\nu}_N \in H_3(N;R)$ and ${\nu}_{N_0} \in H_3(N_0;R)$ for $\mathbb{Z}/2\mathbb{Z}$ fundamental classes where $R$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$ and in this case we also assume that there exists a smooth map $c_N:N \rightarrow N_0$ of $\mathbb{Z}/2\mathbb{Z}$ mapping degree $k_3$.
Suppose that an S-trivial standard-spherical fold map $f:M \rightarrow N$ exists. We also denote the element of $R$ by $k_3$ obtained by mapping $k_3$ into $R$ in the canonical way. Assume also the following properties.
\begin{enumerate}
\item There exists a preimage $F$ of a regular value for the map $f$. Let a class $c_F \in H_{m-3}(M;R)$ be represented as the sum of the fundamental classes of all connected components oriented canonically respecting the orientation of $N$ and an orientation of $M$ if $R$ is not isomorphic to $\mathbb{Z}/2\mathbb{Z}$. If $R$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$, then let a class $c_F \in H_{m-n}(M;R)$ be represented as the sum of the $\mathbb{Z}/2\mathbb{Z}$ fundamental classes of all connected components.
\item There exists an element $k \in R$ and a UFG $c_{F,0} \in H_{m-3}(M;R)$ satisfying $c_F=kc_{F_0}$ and $H_{m-3}(M;R)$ is the internal direct sum of the submodule generated by the one element set $\{c_{F_0}\}$ and a suitable submodule.

\item We can take an embedding $i_{N_0,X}:N_0 \rightarrow X$ satisfying $i_{N_0,X}(N_0) \subset {\rm Int} X$ and $c={i_{N_0,X}}_{\ast}({\nu}_{N_0})$, which is assumed in the beginning.
\item $\dim X \geq \max\{\frac{3m+3}{2},m+4\}$.
\end{enumerate}
In this situation, $k k_3 c$ is realized by the class $k k_3 {\rm PD}_{R}({c_{F,0}}^{\ast})$.
\end{Thm}
\begin{proof}
First from the assumption and other assumptions on the dimensions of the manifolds, $\dim X \geq 7=2 \times3+1$ follows. The existence of $c_N:N \rightarrow N_0$ and $i_{N_0,X}:N_0 \rightarrow X$ in the third property yields the existence of a smooth embedding $i_{N,X}:N \rightarrow X$ of the $3$-dimensional manifold $N$ satisfying
 $i_{N,X}(N) \subset {\rm Int} X$ and $k_3 c={i_{N,X}}_{\ast}({\nu}_N)$. The assumptions that $X$ is spin, that $N$ and $N_0$ are $3$-dimensional, closed, connected and orientable and that as a result $N$ and $N_0$ are spin guarantee that the normal bundle of the image is trivial together with fundamental arguments on characteristic classes of linear bundles over manifolds related to smooth embeddings of smooth manifolds.

In the situation of Theorem \ref{thm:2} let $n=3$.
We can apply Theorem \ref{thm:2} to complete the proof.
\end{proof}

Last we present important facts in constructing explicit cases explaining these theorems well.
\begin{enumerate}
\item For two closed, connected, oriented and equidimensional manifolds $X$ and $Y$ where $Y$ is a standard sphere of dimension larger than $0$ and an arbitrary integer $k$, there exists a smooth map $c:X \rightarrow Y$ whose mapping degree is $k$. This is important as an example for $c_N$ in Theorems \ref{thm:3} and \ref{thm:4}.
\item For two closed, connected and equidimensional manifolds $X$ and $Y$ where $Y$ is a standard sphere of dimension larger than $0$ and an arbitrary element $k \in \mathbb{Z}/2\mathbb{Z}$, there exists a smooth map $c:X \rightarrow Y$ whose $\mathbb{Z}/2\mathbb{Z}$ mapping degree is $k$. This is important as an example for $c_N$ in Theorems \ref{thm:3} and \ref{thm:4}.
\item The projections of smooth trivial bundles over closed and connected manifolds whose fibers are closed and connected manifolds present infinitely many examples for maps $f$ in Theorems \ref{thm:2}, \ref{thm:3} and \ref{thm:4}.
\item The projections of linear bundles over standard spheres which may not be trivial and whose fibers are unit spheres present infinitely many examples for maps $f$ in Theorems \ref{thm:2}, \ref{thm:3} and \ref{thm:4} for a suitable $R=\mathbb{Z}/l\mathbb{Z}$.
\end{enumerate}
In addition, Example \ref{ex:2} is also important.
\section{Acknowledgement}
\thanks{The author is a member of the project Grant-in-Aid for Scientific Research (S) (JP17H06128 Principal Investigator: Osamu Saeki)
"Innovative research of geometric topology and singularities of differentiable mappings"\\
(https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17H06128/)\\ and supported by this project.}


TITLE: Probability conservation in WKB tunneling
QUESTION [13 upvotes]: Suppose we have quantum mechanical plane waves of energy $E$ incident upon a one-dimensional potential barrier $V(x)$ with sloping sides.

One can compare the WKB solutions in the three relevant regions to asymptotics of the Airy function solutions to the linearized Schrödinger equation near the turning points to obtain matching formulae, relating $A, B$ (the coefficients of the right- and left-propagating waves, respectively, left of the barrier) to $F, G$ (the coefficients of the right- and left-propagating waves, respectively, right of the barrier). One can then construct the relevant scattering matrix $S$, defined by:
$$\left[\begin{array}{c}
B\\
F
\end{array}\right]=S\left[\begin{array}{c}
A\\
G
\end{array}\right] \, .$$
I have carried out the above computation and found that
$$S=\left[\begin{array}{cc}
i\left(\dfrac{1-4e^{2\gamma}}{1+4e^{2\gamma}}\right) & \dfrac{4e^{\gamma}}{1+4e^{2\gamma}}\\
\dfrac{4e^{\gamma}}{1+4e^{2\gamma}} & i\left(\dfrac{1-4e^{2\gamma}}{1+4e^{2\gamma}}\right)
\end{array}\right]$$
where
$$\gamma=\frac{1}{\hbar}\int_{x_{1}}^{x_{2}}\sqrt{2m\left[V\left(x\right)-E\right]}dx$$
measures the "strength" of the reflecting, classically-forbidden region.
This scattering matrix $S$ is in fact unitary, and hence this scattering process conserves the probability current across the barrier. But why should this be true?
I know that a solution of the Schrödinger equation will necessarily conserve probability current, but here we have both (1) assumed approximate WKB-type solutions and (2) carried out a complicated asymptotic matching procedure. It is not clear to me why, after this approximation and matching procedure, the resulting solution should still conserve probability current.
Said another way: is there an a priori reason one should expect the WKB-matched solutions to conserve probability current (in spite the fact that they are inexact), or is this simply a happy coincidence that falls out of the computation?

REPLY [3 votes]: I) Recall that the probability current 1D QM is
$$\tag{1} J(x)~:=~\frac{i\hbar}{2m}W(\psi,\psi^{\ast}) (x), $$
where
$$\tag{2}  W(\psi,\psi^{\ast}) (x)~:=~\psi(x) \psi^{\prime}(x)^{\ast}-  \psi^{\prime}(x)\psi(x)^{\ast}$$
is the Wronskian. Unitarity of the $S$-matrix is equivalent to the statement that
$$ \lim_{x\to -\infty} W(\psi,\psi^{\ast}) (x)~=~\lim_{x\to +\infty} W(\psi,\psi^{\ast}) (x), \tag{3}$$
cf. e.g. eq. (11) in my Phys.SE answer here. A sufficient condition for unitarity is if the Wronskian (2) does not depend on the $x$-position.
II) Firstly, there exists many semiclassical WKB approximations in the literature, e.g. to what order in $\hbar$ are we talking? If we consider some truncated WKB approximation scheme, there is no a priori reason why the unitarity condition (3) should be satisfied.
Secondly, the one-directional nature of the connection formulas often makes it impossible to determine the exponentially growing branch in a classically forbidden side from knowledge about the classically allowed side of a turning point, in particular in situations with many turning points, cf. e.g. Refs. 1-3.
OP is presumably interested in the case of a single potential barrier with two turning points. This case has been complete resolved in the literature using e.g. uniform approximation techniques, and the resulting WKB formula preserves unitarity, see e.g. Ref. 3. However, we shall not consider these methods here.
For the rest of this answer, we will consider the following WKB formula
$$ \psi(x)
~=~A(x)\sum_{\sigma=\pm 1} C_{\sigma}(x;x_0)~
\exp\left(\frac{i}{\hbar}\sigma S(x;x_0)\right)$$
$$ ~=~A(x)\sum_{\sigma=\pm 1} B_{\sigma}(x;x_0)~
\cos\left(\frac{1}{\hbar} S(x;x_0)+\sigma\frac{\pi}{4}\right),\tag{4}$$
written either in terms of an exponential function or a cosine function, where
$$A(x)~:=~\frac{1}{\sqrt{p(x)}} , \qquad
S(x;x_0)~:=~ \int_{x_0}^x\!dx^{\prime}~p(x^{\prime}), $$
$$ p(x)~:=~ \sqrt{2m(E-V(x))},\qquad  E,V(x)~\in~\mathbb{R}; \tag{5} $$
$x_0\in \mathbb{R}$ is fixed reference point; $\sigma\in\{\pm 1\}$ is a sign; and $B_{\sigma}(x;x_0),C_{\sigma}(x;x_0)\in\mathbb{C}$ are complex piecewise $x$-independent constants. As we shall see in Section VI, the piecewise constants $B_{\sigma}(x;x_0)$ and $C_{\sigma}(x;x_0)$ may jump at turning points because of the metaplectic correction/Maslov index, cf. e.g. this Phys.SE post and references therein. It is implicitly implied that the two square roots in formula (5) are appropriately chosen branches, not double-valued functions.
III) We have no a priori argument why the Wronskian of the WKB approximation (4) should be $x$-independent to all orders in $\hbar$. Nevertheless, it is interesting and instructive to analyze its form. The Wronskian of the WKB approximation (4) becomes
$$ W(\psi,\psi^{\ast})
~\stackrel{(2)+(4)}{=}~\sum_{\sigma=\pm 1}\sum_{\sigma^{\prime}=\pm 1}
\exp\left[\frac{i}{\hbar}\left(\sigma S -\sigma^{\prime}S^{\ast}\right)\right]$$
$$\left[|A|^2(C_{\sigma} C^{\prime\ast}_{\sigma^{\prime}}
- C^{\prime}_{\sigma} C^{\ast}_{\sigma^{\prime}})
+ C_{\sigma} C^{\ast}_{\sigma^{\prime}}\left\{ (AA^{\prime \ast} - A^{\prime}A^{\ast})
- \frac{i}{\hbar}|A|^2(\sigma p+\sigma^{\prime} p^{\ast}  ) \right\}\right] .\tag{6} $$
Firstly, one may argue (using arguments$^1$ similar to what is done below) that the term
$$AA^{\prime \ast} - A^{\prime}A^{\ast}~=~0\tag{7} $$
vanishes in eq. (6). Secondly, the term
$ C_{\sigma} C^{\prime\ast}_{\sigma^{\prime}}
- C^{\prime}_{\sigma} C^{\ast}_{\sigma^{\prime}}  $
is proportional to delta functions with supports at the turning points.
Hence, away from the turning points, the Wronskian (6) simplifies to
$$ W^{\neq}(\psi,\psi^{\ast})
~\stackrel{(6)}{=}~- \frac{i}{\hbar}|A|^2 \sum_{\sigma=\pm 1}\sum_{\sigma^{\prime}=\pm 1}
C_{\sigma} C^{\ast}_{\sigma^{\prime}}\left(\sigma p+\sigma^{\prime} p^{\ast}  \right)\exp\left[\frac{i}{\hbar}\left(\sigma S -\sigma^{\prime}S^{\ast}\right)\right] $$
$$~=~\frac{1}{\hbar}|A|^2 \sum_{\sigma=\pm 1}\sum_{\sigma^{\prime}=\pm 1}
B_{\sigma} B^{\ast}_{\sigma^{\prime}} \left[p\sin\left(\frac{1}{\hbar} S+\sigma\frac{\pi}{4}\right)\cos\left(\frac{1}{\hbar} S^{\ast}+\sigma^{\prime}\frac{\pi}{4}\right)\right. $$
$$\left. -p^{\ast}\cos\left(\frac{1}{\hbar} S+\sigma\frac{\pi}{4}\right)\sin\left(\frac{1}{\hbar} S^{\ast}+\sigma^{\prime}\frac{\pi}{4}\right)\right]. \tag{8} $$
IV) Classically allowed region $E>V(x)$. Then $p$ and $S$ are both real, and we must choose same signs $\sigma=\sigma^{\prime}$ in the exponential version of eq. (8) to get non-zero contributions. The exponential factor is then just $1$. The Wronskian (8) is $x$-independent
$$\tag{9}  W^>(\psi,\psi^{\ast})
~\stackrel{(8)}{=}~- \frac{2i}{\hbar}|A|^2p\sum_{\sigma=\pm 1}\sigma |C_{\sigma}|^2
~=~ \frac{1}{\hbar}|A|^2p\sum_{\sigma=\pm 1}\sigma B_{\sigma}B^{\ast}_{-\sigma}, $$
because $|A|^2p$ is $x$-independent, cf. eq. (5). The last expression of eq. (9) follows from the trigonometric version of eq. (8) with the help of a trigonometric addition formula.
V) Classically forbidden region $E<V(x)$. Then $p$ and $S$ are both imaginary, and we must choose opposite signs $\sigma=-\sigma^{\prime}$ in the exponential version of eq. (8) to get non-zero contributions. The exponential factor is then just $1$. The Wronskian (8) is $x$-independent
$$\tag{10}  W^<(\psi,\psi^{\ast})
~\stackrel{(8)}{=}~- \frac{2i}{\hbar}|A|^2p\sum_{\sigma=\pm 1}\sigma C_{\sigma}C^{\ast}_{-\sigma}
~=~ \frac{1}{\hbar}|A|^2p\sum_{\sigma=\pm 1}\sigma |B_{\sigma}|^2. $$
(The superscripts $>$ and $<$ refer to the classically allowed and forbidden regions, respectively.)
VI) A turning point $V(x_0)=E \Leftrightarrow p(x_0)=0$. Let us for notational simplicity assume $x_0=0$. Assume that in a neighborhood around $x_0=0$, the potential can be approximated with a linear$^2$ potential
$$\tag{11} V(x)-E~\propto ~x,$$
i.e. one side is classically allowed and the other side is classically forbidden. Then eq. (5) implies
$$\tag{12} p(x)~\propto ~x^{\frac{1}{2}}, \qquad
S(x)~\propto ~x^{\frac{3}{2}},\qquad A(x)~\propto ~x^{-\frac{1}{4}}. $$
The TISE of the linear potential (11) becomes the Airy equation. From the asymptotic expansion of the Airy functions, one derives the Airy connection formulas
$$\tag{13}
\exp\left(\sigma\frac{2}{3}|x|^{\frac{3}{2}}\right)
\quad\longleftrightarrow\quad\frac{3-\sigma}{2}\cos\left( \frac{2}{3}|x|^{\frac{3}{2}}+\sigma\frac{\pi}{4}\right) $$
between the positive and negative $x$-axis, where $\sigma\in\{\pm 1\}$. The Airy connection formulas (13) lead to the WKB connection formulas
$$\tag{14}  C^<_{+1}~=~e^{\frac{i\pi}{4}}B^>_{+1}, \qquad  2C^<_{-1}~=~e^{\frac{i\pi}{4}}B^>_{-1}, $$
for the complex coefficients $B_{\sigma},C_{\sigma}\in\mathbb{C}$, if we adapt the following sign conventions
$$\tag{15} p^<~=~ip^>\qquad A^<~=~e^{-\frac{i\pi}{4}}A^>$$
for the square roots in eq. (5). In matrix-form we have
$$\tag{16} C^<~=~\begin{pmatrix} 1 & i \cr \frac{i}{2}  &\frac{1}{2} \end{pmatrix} C^>\qquad\text{and}\qquad  C^>~=~\begin{pmatrix} \frac{1}{2} & -i \cr -\frac{i}{2}  &1 \end{pmatrix} C^<.$$
Note that the connection formulas (14) and (15) imply that the Wronskian
$$\tag{17} W^>(\psi,\psi^{\ast}) ~=~W^<(\psi,\psi^{\ast})  $$
of the WKB approximation (4) does not jump at the turning point, cf. eqs. (9) and (10).
The Airy connection formulas (13) may superficially look like double-directional connection formulas, but that's an illusion. Because of its asymptotic nature, it involves a (possibly illegitimate) implicit choice, which we have made for simplicity.
It is possibly to use more general complex methods involving the method of steepest descent, which leads to the principle of exponential dominance:
"A multiplier $C_+$, or $C_-$ can only change on a good path when crossing a Stokes line on which its exponential is subdominant."
One may show that this principle obeys unitarity, see e.g. Ref. 3 for details.
TL;DR: We have argued using various simplifying assumptions that the WKB formula (4) preserves unitarity.
--
References:

D. Griffiths, Intro to QM, Chapter 8.
A. Galindo & P. Pascual, QM2, Chapter 9.
M.V. Berry & K.E. Mount, Semiclassical approximations in wave mechanics, Rep. Prog. Phys 35 (1972) 315; Chapter 3 & 4.

$^1$ E.g. at a turning point $x_0$, the transformation $x\to -x$ induces a sign flip on the left-hand side of eq. (7). But minus zero is still zero!
$^2$ We leave it to the reader to analyze the cases of higher-order turning-points and the case of an infinite hard wall.


TITLE: Calculate the maximum and minimum of $f(x,y)=x+y$ at the unit square.
QUESTION [1 upvotes]: Let $[-1,1] \times [-1,1]$ the unit square, Im thinking in compute the extreme values at the frontier of the unit circle. So I describe each face of the unit
 square as:
$$F_{1}= \lbrace (1,y) | -1 \leq y \leq 1 \rbrace\\
  F_{2}= \lbrace (x,1) | -1 \leq x \leq 1 \rbrace\\
   F_{3}= \lbrace (-1,y) | -1 \leq y \leq 1 \rbrace\\
F_{4}= \lbrace (x,-1) | -1 \leq x \leq 1 \rbrace\\$$
So first question,how I compute the candidates for maximum and minimum values at these face from the unit circle?
Also Im looking in compute the stationary points of $f(x,y)=x+y$. But $\frac{\partial f}{ \partial x}=1$ and $\frac{\partial f}{ \partial y}=1$ which give non information.
I would aprecciate any help to solve this problem.

REPLY [2 votes]: If $(x,y)$ is in the boundary of the unit square, in particular you have $-1\leq x\leq 1$ and $-1\leq y\leq 1$. So $-2\leq x+y\leq 2$. So, $$-2\leq f(x,y)\leq 2$$
 And you actually have that $f(1,1)$ is maximum and $f(-1,-1)$ is minimum.


TITLE: Total Unimodularity in Integer QP
QUESTION [0 upvotes]: I have a Quadratic Programming problem with linear constraints. My objective is Quadratic-Convex, the constraint matrix is Totally Unimodular (similar to assignment or network flow problems), and the variables are Binary. I know that if an LP has Totally Unimodular constraint matrix $A$ and integral $b$ matrices, then the Optima will be integral.
Now, while solving my QP with a nonlinear solver (MINOS 5.51) with relaxed variable (ignoring the integrality), I get almost all the solutions as integral. Most of the time, I get all variables as integral. Only 2 in 20,000 variables or 2 in 84 thousands variables are non-integral.
My question is, is there any theory which talks about the implication of Total Unimodularity in QP or MIQP or convex optimization? Is there any explanation why my problem is getting an integral solution? Are there any articles or books which discuss this issue?

REPLY [0 votes]: What you're looking for is most likely an extension of the fundamental theorem of linear programming.
Such extensions give generalized sufficient conditions to when $\min_{x\in P} f(x)$, where $P$ is a polyhedron, has a global minimizer in one of its vertices.
For example, the fundamental theorem of linear programming also holds when $f(x)$ is a concave function.
This paper gives several possible extensions of the fundamental theorem of linear programming for when $f$ is concave, quasi-concave or differentiable and strictly quasi-convex.