Homology braid II (Ex)

1) Let $(W,M,M^\prime)$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}(W,M,M^\prime)$ be a cobordism of $CW$$CW$-complexes. Show that the commuting diagram

$\displaystyle \xymatrix{M^\prime \ar[dr] \ar[rr] && W/M \\ & W \ar[ur] & }$

induces a commutative diagram of cofibre sequences

$\displaystyle \xymatrix{ M^\prime \ar[dr] \ar@/^2pc/[rr] && W/M \ar[dr] \ar@/^2pc/[rr] && \Sigma M \\ & W \ar[dr] \ar[ur] && W/M\cup M^\prime \ar[dr] \ar[ur] && \\ M \ar[ur] \ar@/_2pc/[rr] && W/M^\prime \ar[ur] \ar@/_2pc/[rr] && \Sigma M^\prime \\ }$

and hence a commutative braid of long exact sequences in Homology

$\displaystyle \xymatrix{ H_*(M^\prime) \ar[dr] \ar@/^2pc/[rr] && H_*(W/M) \ar[dr] \ar@/^2pc/[rr] && H_{*-1}(M) \\ & H_*(W) \ar[dr] \ar[ur] && H_*(W/M\cup M^\prime) \ar[dr] \ar[ur] && \\ H_*(M) \ar[ur] \ar@/_2pc/[rr] && H_*(W/M^\prime) \ar[ur] \ar@/_2pc/[rr] && H_{*-1}(M^\prime) \\ }$

2) Now let $M$$M$ be a closed $n$$n$-dimensional manifold and $g:S^k\times D^{n-k}\hookrightarrow M$$g:S^k\times D^{n-k}\hookrightarrow M$ a framed embedding. Denote by $M'$$M'$ the effect of a surgery on $M$$M$ and by $W$$W$ the corresponding trace, i.e.

$\displaystyle M'=\overline{M\setminus g(S^n\times D^{n-k})}\cup D^{k+1}\times S^{n-k-1},$ $$\displaystyle M'=\overline{M\setminus g(S^n\times D^{n-k})}\cup D^{k+1}\times S^{n-k-1},$$

$\displaystyle W=M\times I\cup D^{k+1}\times D^{n-k}.$ $$\displaystyle W=M\times I\cup D^{k+1}\times D^{n-k}.$$

Denote by $\omega$$\omega$ the orientation character of $M$$M$, i.e. $\omega:\pi=\pi_1(M)\rightarrow \mathbb{Z}_2$$\omega:\pi=\pi_1(M)\rightarrow \mathbb{Z}_2$ and by $\widetilde{W},\widetilde{M},\widetilde{M}'$$\widetilde{W},\widetilde{M},\widetilde{M}'$ the corresponding universal covers. Write $H_i(\widetilde{M})$$H_i(\widetilde{M})$ etc. for the homology with $\mathbb{Z}[\pi]$$\mathbb{Z}[\pi]$-coefficients.

Show that there exists a commutative braid of exact sequences

$\displaystyle \def\curv{1.5pc}% Adjust the curvature of the curved arrows here \xymatrix@!R@!C@!0@R=2.5pc@C=4pc{% Adjust the spacing here H_{i+1}(\widetilde{W},\widetilde{M}) \ar[dr] \ar@/u\curv/[rr] && H_{i}(\widetilde{M}) \ar[dr] \ar@/u\curv/[rr] && H_{i}(\widetilde{W},\widetilde{M}') \\ & H_{i+1}(\widetilde{W},\widetilde{M}\cup\widetilde{M}') \ar[dr] \ar[ur] && H_{i}(\widetilde{W}) \ar[dr] \ar[ur] \\ H_{i+1}(\widetilde{W},\widetilde{M}') \ar[ur] \ar@/d\curv/[rr] && H_{i}(\widetilde{M}') \ar[ur] \ar@/d\curv/[rr] && H_{i}(\widetilde{W},\widetilde{M}) }$

3) Show that the relative homology modules are given by

$\displaystyle H_{i}(\widetilde{W},\widetilde{M})=\left\{\begin{array}{ll} \mathbb{Z}[\pi] & \textrm{if } i=k+1\\ 0 & \textrm{otherwise}\\ \end{array}\right.$ $$\displaystyle H_{i}(\widetilde{W},\widetilde{M})=\left\{\begin{array}{ll} \mathbb{Z}[\pi] & \textrm{if } i=k+1\\ 0 & \textrm{otherwise}\\ \end{array}\right.$$

$\displaystyle H_{i}(\widetilde{W},\widetilde{M}')=\left\{\begin{array}{ll} \mathbb{Z}[\pi] & \textrm{if } i=n-k\\ 0 & \textrm{otherwise}\\ \end{array}\right.$ $$\displaystyle H_{i}(\widetilde{W},\widetilde{M}')=\left\{\begin{array}{ll} \mathbb{Z}[\pi] & \textrm{if } i=n-k\\ 0 & \textrm{otherwise}\\ \end{array}\right.$$

4)Assume $n=2k$$n=2k$ and look at the top bit of the braid

$\displaystyle \def\curv{1.5pc}% Adjust the curvature of the curved arrows here \xymatrix@!R@!C@!0@R=2.5pc@C=4pc{% Adjust the spacing here H_{k+1}(\widetilde{W},\widetilde{M}) \ar@/u\curv/[rr]^{\alpha} && H_{k}(\widetilde{M}) \ar@/u\curv/[rr]^{\beta} && H_{k}(\widetilde{W},\widetilde{M}') }$

Let $x$$x$ be the Hurewicz image of $[g|]$$[g|]$ with $g|:S^{k}\times 0\hookrightarrow M$$g|:S^{k}\times 0\hookrightarrow M$ being the restriction of the framed embedding we do the surgery on.

a) Verify that $\alpha$$\alpha$ is (geometrically) given by sending the generator 1 to $x$$x$.

b) Verify that $\beta$$\beta$ is (geometrically) given by sending a class $y$$y$ to its (equivariant) homology intersection with $x$$x$, $\lambda(x,y)\in\mathbb{Z}[\pi]$$\lambda(x,y)\in\mathbb{Z}[\pi]$.