Talk:Homology braid II (Ex)

1) The diagram is composed of the exact sequences of the pair $(\widetilde W,\widetilde M)$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}(\widetilde W,\widetilde M)$, and of the triple $(\widetilde W,\widetilde M\cup\widetilde M',\widetilde M)$$(\widetilde W,\widetilde M\cup\widetilde M',\widetilde M)$, and of the analogous sequences with $\widetilde M$$\widetilde M$ and $\widetilde M'$$\widetilde M'$ swapped. Because $\widetilde M$$\widetilde M$ and $\widetilde M'$$\widetilde M'$ are disjoint, we have an isomorphism $H_k(\widetilde M\cup\widetilde M',\widetilde M')\cong H_k(\widetilde M)$$H_k(\widetilde M\cup\widetilde M',\widetilde M')\cong H_k(\widetilde M)$ and another one with $\widetilde M$$\widetilde M$ and $\widetilde M'$$\widetilde M'$ swapped. Both are implicit in the braid.

The squares that are half visible on the left and right hand side obviously commute. The upper triangles commute because of the following diagram,

$\displaystyle \xymatrix{ H_{i+1}(\widetilde W,\widetilde M) \ar[d] \ar[r]^{\partial} & H_i(\widetilde M) \ar[d]^{\cong} \ar[r] & H_i(\widetilde W) \ar[d] \\ H_{i+1}(\widetilde W,\widetilde M\cup\widetilde M') \ar[r]^{\partial} & H_i(\widetilde M\cup\widetilde M',\widetilde M') \ar[r] & H_i(\widetilde W,\widetilde M')}$

The lower triangles work similarly.

The central square only commutes up to sign: let~$a\in C_{i+1}(\widetilde w)$$a\in C_{i+1}(\widetilde w)$ be a relative cycle in~$W$$W$, so $\partial a=b+c\in C_i(\widetilde M)\oplus C_i(\widetilde M')$$\partial a=b+c\in C_i(\widetilde M)\oplus C_i(\widetilde M')$. Then the upper path maps $[a]$$[a]$ to $[b]$$[b]$, and the lower path maps $[a]$$[a]$ to $[c]=-[b+\partial a]=-[b]$$[c]=-[b+\partial a]=-[b]$. To repair this, put an additional minus sign on all boundary arrows landing in $H_\bullet(\widetilde M')$$H_\bullet(\widetilde M')$.

2) We have learned that $W$$W$ is homotopy equivalent to $M\cup_{g|}D^{k+1}$$M\cup_{g|}D^{k+1}$. Let $\tilde g|\colon S^k\to\widetilde M$$\tilde g|\colon S^k\to\widetilde M$ be a lift (ok if $k\ge2$$k\ge2$). If we pass to the universal cover of $M$$M$, we are attaching a $\pi$$\pi$-cell $\pi\times D^{k+1}$$\pi\times D^{k+1}$ by $(\gamma,t)\mapsto\gamma(\tilde g(t))$$(\gamma,t)\mapsto\gamma(\tilde g(t))$. Excision gives

$\displaystyle H_i(\widetilde W,\widetilde M)\cong H_i(\widetilde M\cup_{\pi\times S^k}(\pi\times D^{k+1}),\widetilde M) \cong \Zz \pi H_i(D^{k+1},S^k)\;.$ $$\displaystyle H_i(\widetilde W,\widetilde M)\cong H_i(\widetilde M\cup_{\pi\times S^k}(\pi\times D^{k+1}),\widetilde M) \cong \Zz \pi H_i(D^{k+1},S^k)\;.$$

Similarly, swapping $W$$W$ is homotopy equivalent to $M'\cup_{S^{n-k}}D^{n+1-k}$$M'\cup_{S^{n-k}}D^{n+1-k}$. This gives the second assertion.

3) By part 2, we get the following diagram.

$\displaystyle \xymatrix{ \Zz \pi H_{k+1}(D^{k+1},S^k) \ar[d]^{\cong} \ar[r]^{\cong} & \Zz \pi H_k(S^k) \ar[d]^{H_k\tilde g|}\\ H_{k+1}(\widetilde W,\widetilde M) \ar[r]^{\alpha} & H_k(\widetilde M).}$

This proves part (a).

To prove part (b), we decompose $\beta$$\beta$ using the above braid as

$\displaystyle \beta\colon H_k(\widetilde M) \longrightarrow H_k(\widetilde W) \longrightarrow H_k(\widetilde W,\widetilde M')\; .$ $$\displaystyle \beta\colon H_k(\widetilde M) \longrightarrow H_k(\widetilde W) \longrightarrow H_k(\widetilde W,\widetilde M')\; .$$

Now let $h$$h$ be a Morse function on $W$$W$ with $h|_M\equiv 0$$h|_M\equiv 0$ and $h|_{M'}\equiv 1$$h|_{M'}\equiv 1$ such that there is only one critical point of index $k$$k$. Fix a gradient vector field for $h$$h$ such that the stable and unstable disks are exactly the core disk of the $k$$k$-handle and its dual. We can multiply this vector field with a positive function such that the flow $X$$X$ moves points in $M$$M$ to $M'$$M'$ in time $1$$1$ except in a small neighbourhood of the image of $g|$$g|$. All this lifts to $\widetilde W$$\widetilde W$.

Now, let $y$$y$ be a smooth cycle in $\widetilde M$$\widetilde M$ that is transversal to $\gamma x$$\gamma x$, which is the image of $\gamma \circ \tilde g$$\gamma \circ \tilde g$, for all $\gamma\in\pi$$\gamma\in\pi$. The gradient flow moves $y$$y$ from $M$$M$ to $M'$$M'$ except in a small neighbourhood of the image of $\gamma\circ\tilde g$$\gamma\circ\tilde g$. Hence, $\alpha(y)$$\alpha(y)$ only depends on the intersections of $y$$y$ with the various~$\gamma x$$\gamma x$ for $\gamma\in\pi$$\gamma\in\pi$. This is the main step in the proof of (b).