# Thursday January 23rd Recall from last time: $M$ is a closed smooth $n-1$ manifold and $\phi: S^{\lambda - 1} \cross D^{n-\lambda} \injects M$, and we used surgery to obtain $\chi(M, \phi)$ and a cobordism $W(M, \phi)$ from $M$ to $\chi(M, \phi)$. This yields a saddle $L_\lambda \subseteq \RR^n = \RR^{\lambda} \cross \RR^{n - \lambda}$. We construct the cobordism using \begin{align*} S^{\lambda - 1} \cross D^{n-\lambda} &\to \bd_L \\ (u, tv) &\mapsto (u\cosh t, v\sinh t) .\end{align*} This yields \begin{align*} M \setminus \theset{ \phi(S^{\lambda - 1}) \cross \theset{0} } \disjoint L_\lambda / \generators{ (u, tv)\cross c \sim (\vector x, \vector y) \suchthat \norm{x}^2 + \norm{y}^2 = c } \\ x, y\text{ are on a curve that starts from } (u\cosh t. v\sinh t) .\end{align*} ![Image](figures/2020-01-23-11:22.png)\ ![Image](figures/2020-01-23-11:23.png)\ > Todo: review! Suppose $W(; M_0, M_1)$ is an elementary cobordism and $f: W \to [-1, 1]$ is a Morse function with one critical point $p$, and $\xi$ a gradient-like vector field for $f$. The goal is to construct $\phi_L: S^{\lambda - 1} \cross D^{n-\lambda} \injects M_0$, the characteristic embedding. Let $\psi_x$ be the integral curve of $\xi$ such that $\psi_x(0) = x$, and define $W^s(p) = \theset{x\in W \suchthat \lim_{t\to\infty} \psi_x(t) = p}$ to be the stable manifold, and $W^u(p) = \theset{x\in W \suchthat \lim_{t\to -\infty} \psi_x(t)= p}$. Claim: $W^s(p), W^u(p)$ are diffeomorphic to disks of dimension $\lambda$ and $n-\lambda$ respectively. ![Image](figures/2020-01-23-11:33.png)\ Moreover, $\bd W^s(p) = W^s(p) \intersect M_0 \cong S^{\lambda - 1}$ (Milnor refers to this as the "left sphere" $S_L$ and $\bd W^u(p) = W^u(p) \intersect M_1 = S^{n - \lambda - 1}$ (the "right sphere" $S_R$). ### Proof ![Image](figures/2020-01-23-11:38.png)\ Choose an open $U \ni p$ and a coordinate chart $h: OD_{2\varepsilon} \to U$. Then $f\circ h(x, y) = c - \norm{x}^2 - \norm{y}^2$, which takes on a minimum value of $c - 4\varepsilon^2$ and a maximum of $c + 4\varepsilon^2$, and $\xi \circ h(x, y) = (-x, y)$. We thus obtain the inequalities \begin{align*} -1 < c - 4\varepsilon^2 < c + 4\varepsilon^2 < 1 \text{ if } 4\varepsilon^2 < c+1, 1-c .\end{align*} Now let $W_\varepsilon = f\inv[c-\varepsilon^2, c+\varepsilon^2]$, and we want a cobordism from $M_{-\eps} = f\inv(c - \eps^2)$ to $M_{+\eps} = f\inv(c + \eps^2)$. Then $$ M_{\mp} \intersect U = \theset{h(x, y) \suchthat c - \norm{x}^2 + \norm{y}^2 = c_{\mp} \eps^2 \iff -\norm{x}^2 + \norm{y}^2 = \mp \eps^2} .$$ ![Image](figures/2020-01-23-11:46.png)\ Then $W^2(p) \intersect U = \theset{h(x, 0) } \cong D^{\lambda}$, and $W^s(p) \intersect M_{-\eps} = \theset{h(x, 0) \suchthat \norm{x} = \eps}$. By flowing along the integral curves of $\xi$, we obtain a diffeomorphism $\Psi_{-\eps}: M_0 \to M_{-\eps}$. Then $W^s(p) = \theset{h(x,0) } \union \theset{\text{integral curves of } \xi \text{ passing through points in } W^s(p) \intersect M_{-\eps}}$. So $S_L = \Psi_{-\eps}\inv( W^s(p) \intersect M_{-\eps} )$, and we can define the embedding $\phi_L$ by the following composition: \begin{center} \begin{tikzcd} & S^{\lambda - 1} \cross D^{n-\lambda} \arrow[rrddd] \arrow[rrrr] & & & & M_0 \\ {L(u, tv)} \arrow[rdd] & & & & & \\ & & & & & \\ & {h(\eps u \cosh t, \eps v \sinh t)} & & M_{-\eps} \arrow[rruuu, "\Psi_{-\eps}\inv"] & & \end{tikzcd} \end{center} Similarly, one can show $W^u(p) \cong D^{n-\lambda}$ and define another embedding $\phi_R D^{\lambda} \cross S^{n-\lambda - 1} \injects M_1$. $\qed$ Theorem (?) : Let $(W; M_0, M_1)$ be an elementary cobordism with $(f, \xi, p)$ a Morse function with a gradient-like vector field and a critical point as above. Then there is a diffeomorphism of cobordisms \begin{align*} (W(M_0, \phi_L); M_0, \chi(M_0, \phi_L) ) \cong (W; M_0, M_1) ,\end{align*} where $\phi_L: S^{\lambda - 1} \cross D^{n-\lambda} \injects M_0$ is the characteristic embedding. Proof : Consider $f\inv[-1, c-\eps^2] \union W_\eps \union f\inv[c+\eps^2, 1]$, and note that $f\inv[-1, c-\eps^2], f\inv[c+\eps^2, 1]$ is a product (?). Then $(W; M_0, M_1) \cong (W_\eps; M_{-\eps}, M_\eps)$. We then have \begin{align*} ( W(M, \phi_L) ; M_0, \chi(M_0, \phi_L) ) \cong ( W(M_{-\eps}, \phi), M_{-\eps}, \chi(M_{-\eps}, \phi) ) .\end{align*} So we'll define a diffeomorphism from the RHS to the RHS of the former diffeomorphism. Define $f \to [c-\eps^2, c+ \eps^2]$ in the former and $f' \to [-1, 1]$ in the latter. Then take $f\circ k = c + \eps^2 f'$ to match up the domains. Take $(z, t) \in (M_{-\eps} \setminus \phi(\theset{S^{\lambda - 1} \cross \theset{0}}) ) \cross D^1$. Then $k(z, t)$ is the point on the integral curve which passes through $z$ with $f(k(z, t)) = c + \eps^2 t$. This map will take flow lines to flow lines. Now define \begin{align*} (x, y) \in L_\lambda \to h(\eps x, \eps y) .\end{align*} Exercise : Show that this is a well-defined diffeomorphism. Theorem (Cobordisms Retract Onto Surgery Components?) : For an elementary cobordism $(W; M_0, M_1)$ and $(f, \xi, p)$ as above, then $M_0 \union D_L$ (where $D_L \cong W^s(p)$) is a deformation retraction of $W$. Corollary : \hfill \begin{align*} H_i(W, M_0) = H_i(M_0 \union D_L, M_0) \equalsbecause{excision} H_i(D_L, S_L) \cong \begin{cases} \ZZ & i = \lambda \\ 0 & \text{otherwise} \end{cases} .\end{align*} Theorem (Reeb) : If a closed $n\dash$manifold $M$ has a Morse function with exactly 2 critical points, then $M$ is homeomorphic to $S^n$. Proof : If $m = \min_{x\in M} f(x) = f(A)$ and $M = \max_{x\in M} f(x) = f(B)$ where $A, B$ are critical points. Then points near $A$ are in the unstable manifold, so $\ind(A) = 0$, and points near $B$ are in the stable, so $\ind(B) = n$. ![Image](figures/2020-01-23-12:20.png)\ The middle piece of the cobordism is a product cobordism, and $M$ is called a twisted sphere. Exercise : Every twisted sphere is homeomorphic to $S^n$.