Relative Morse inequalities for a cobordism

The critical set is finite by [F8]. The interior-supported bumps in [F3] separate its values while fixing a boundary collar: choose their supports away from that collar and their coefficients small enough to keep the function in $(0,1)$ on those supports. It suffices to prove the identity for this perturbation, again denoted $f$, since its critical points and indices are unchanged. Write its critical values as $c_1<\cdots <c_{\nu }$. Choose regular $0<t_0<\cdots <t_{\nu }<1$ with $t_0<c_1$, $c_i<t_i<c_{i+1}$ for $1\le i<\nu$, and $c_{\nu }<t_{\nu }$; if $\nu =0$ choose any $t_0\in (0,1)$. Put $W_i=W^{t_i}$. All these stages are compact $n$-manifolds with boundary.

Construct a downward gradient-like field for this $f$. Choose disjoint interior Morse charts by [F9] and smaller charts with closures inside them. On each Morse chart prescribe $X_p=(2u,-2v)$, so $df(X_p)=-4(\mathrm{\mid }u{\mathrm{\mid }}^2+\mathrm{\mid }v{\mathrm{\mid }}^2)$. Cover the complement of the smaller charts by regular coordinate neighborhoods avoiding still smaller critical neighborhoods. On each choose a smooth field $Y$ with $df(Y)=-1$: a nonzero coordinate derivative of $f$ can be inverted, also in boundary charts. Patch these fields and the $X_p$ by a partition of unity from [F8]. Near each critical point only its Morse-chart field contributes, giving the exact local model; elsewhere the derivative is a convex combination of negative numbers. At $M_0$ the resulting $X$ points outward, and at $M_1$ inward, since $f$ is constant on each face and its nonzero inward normal derivative has respectively positive and negative sign. Thus $X$ satisfies all adapted-field conditions except ambient completeness.

Append negative parameters to the fixed face collars to obtain the boundaryless extension $\widehat{W}$ of [F7]. Smoothness in boundary charts means that the coefficients of $X$ extend locally across the faces in signed collar charts. Compactness of the faces gives finitely many such extensions; together with $X$ on the interior, a partition of unity from [F8] patches them to a field $\tilde{X}$ on an open neighborhood $U$ of $W$ in $\widehat{W}$, agreeing with $X$ on $W$. Choose a relatively compact open neighborhood $V$ with $W\subseteq V\subseteq \bar{V}\subseteq U$. By [F10] take a bump $\rho$ equal to one near $W$ with support in $V$. Extend $\rho \tilde{X}$ by zero outside $U$. Its support lies in the compact set $\bar{V}$, so [F10] makes it complete, while its restriction to $W$ is $X$. Hence $(f,X)$ is adapted in the precise sense required by [F1]; trajectories in $W$ are followed only until a boundary exit. Empty faces need no extension, and if $\mathrm{\partial }W=\mathrm{\varnothing }$ take $\widehat{W}=W$.

The bottom and top bands are products even at the faces. On these compact regular bands normalize the field of step 3.1 to $Y:=X\mathrm{/}df(X)$, so $df(Y)=1$. Its ambient extension permits the local-flow theorem in [F8] across the faces. At $M_0$ the field $Y$ points inward and at $M_1$ outward. Along a trajectory $f({\mathrm{\Phi }}_s(x))=f(x)+s$; compactness permits continuation until the endpoint level. The inverse formula $y\mapsto ({\mathrm{\Phi }}_{a-f(y)}(y),f(y))$ then gives the product, as in [F8]. Choose $t_0$ sufficiently small and $t_{\nu }$ sufficiently close to $1$ when $\nu >0$. Thus $W_0\cong M_0\times [0,t_0]$ relative to $M_0$, and the top band is $f^{-1}(t_{\nu })\times [t_{\nu },1]$. If $\nu =0$, the same flow identifies the entire triad with $M_0\times [0,1]$; if a face is empty the corresponding regular band is empty. Hence $H_j(W_0,M_0;F)=0$ by [F6].

For each $1\le i\le \nu$ the closed band $f^{-1}([t_{i-1},t_i])$ lies in the interior of $W$ and contains exactly one nondegenerate critical point $p_i$ of index $k_i:=\mathrm{ind}(p_i)$. Apply [F1] to the adapted pair constructed in steps 2.1 and 3.1. With its lower-sublevel comparison up to homotopy of pairs, $W_i$ is obtained from $W_{i-1}$ by attaching one rounded $k_i$-handle, so [F2] gives $H_j(W_i,W_{i-1};F)=0(j\mathrm{\ne }{k}_i),H_{k_i}(W_i,W_{i-1};F)\cong F\mathrm{.}$

Induction on $i$: $H_j(W_i,M_0;F)$ is finite-dimensional for every $j$ and vanishes for $j<0$ and $j>n$. For $i=0$ it vanishes by step 4.1. For the step, apply [F5] to the exact sequence of the triple $(W_i,W_{i-1},M_0)$ from [F4] with $A_j=H_j(W_{i-1},M_0;F)$, $B_j=H_j(W_i,M_0;F)$, $C_j=H_j(W_i,W_{i-1};F)$: the hypotheses hold by the induction hypothesis and by step 4.2, and [F5] concludes that the $B_j$ are finite-dimensional.

The same application of [F5] gives, for each $i$, the polynomial identity $P_{i-1}^{\mathrm{r}\mathrm{e}\mathrm{l}}(t)+t^{k_i}=P_i^{\mathrm{r}\mathrm{e}\mathrm{l}}(t)+(1+t)Q_i(t),$ where $P_i^{\mathrm{r}\mathrm{e}\mathrm{l}}(t):={\sum }_j{\dim }_F{H}_j(W_i,M_0;F)t^j$, the middle term is the relative polynomial of the slab by step 4.2, and $Q_i\in \mathbb{Z}[t]$ has nonnegative coefficients.

Summing over $i=1,\dots ,\nu$ telescopes: $P_0^{\mathrm{r}\mathrm{e}\mathrm{l}}=0$ by step 5.1, so ${\sum }_{i=1}^{\nu }{t}^{k_i}=P_{\nu }^{\mathrm{r}\mathrm{e}\mathrm{l}}(t)+(1+t)Q(t),Q:={\sum }_{i=1}^{\nu }{Q}_i\in \mathbb{Z}[t],$ with $Q$ of nonnegative coefficients, and the left side is ${\sum }_k{m}_k^{\mathrm{r}\mathrm{e}\mathrm{l}}(f)t^k$ because the critical points $p_1,\dots ,p_{\nu }$ exhaust $\mathrm{Crit}(f)$ and $k_i=\mathrm{ind}(p_i)$.

Compress the top product of step 4.1 to its lower face and use the identity on $W_{\nu }$. This is a strong deformation retraction of $W$ onto $W_{\nu }$, fixing $M_0$. Alternatively the triple sequence [F4] and the vanishing of the product relative group [F6] show that $H_j(W_{\nu },M_0;F)\to H_j(W,M_0;F)$ is an isomorphism. Thus $P_{\nu }^{\mathrm{r}\mathrm{e}\mathrm{l}}=P_{W,M_0}$ and step 7.1 is the required identity. When $\nu =0$ the empty sum gives $P_{W,M_0}=Q=0$.

Uniqueness holds because $(1+t)R=0$ forces successively every coefficient of $R$ to be zero, and the coefficientwise and alternating partial-sum forms follow by comparing coefficients of $(1+t)Q$ exactly as in the absolute case; the relative Betti numbers $b_k(W,M_0;F)={\dim }_F{H}_k(W,M_0;F)$ are finite and eventually zero by step 5.1. Adaptedness is used through the interior-slab identification [F1]; a critical point on the boundary would not produce a handle stage and would break the count.