Homology braid II (Ex)

Let $M$$<wikitex>; Let $M$ be a closed $n$-dimensional manifold and $g:S^k\times D^{n-k}\hookrightarrow M$ a framed embedding. Denote by $M'$ the effect of a surgery on $M$ and by $W$ the corresponding trace, i.e. $$ M'=\overline{M\setminus g(S^n\times D^{n-k})}\cup D^{k+1}\times S^{n-k-1}, $$ $$ W=M\times I\cup D^{k+1}\times D^{n-k}. $$ Denote by $\omega$ the orientation character of $M$, i.e. $\omega:\pi=\pi_1(M)\rightarrow \mathbb{Z}_2$ and by $\widetilde{W},\widetilde{M},\widetilde{M}'$ the corresponding universal covers. Write $H_i(\widetilde{M})$ etc. for the homology with $\mathbb{Z}[\pi]$-coefficients. '''1)''' Show that there exists a commutative braid of exact sequences $$ \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}) } $$ '''2)''' Show that the relative homology modules are given by $$ H_{i}(\widetilde{W},\widetilde{M})=\left\{\begin{array}{ll} \mathbb{Z}[\pi] & \textrm{if } i=k+1\ 0 & \textrm{otherwise}\ \end{array}\right. $$ $$ H_{i}(\widetilde{W},\widetilde{M}')=\left\{\begin{array}{ll} \mathbb{Z}[\pi] & \textrm{if } i=n-k\ 0 & \textrm{otherwise}\ \end{array}\right. $$ '''3)'''Assume $n=2k$ and look at the top bit of the braid $$ \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}') } $$ Let $x$ be the Hurewicz image of $[g|]$ with $g|:S^{k}\times 0\hookrightarrow M$ being the restriction of the framed embedding we do the surgery on. '''a)''' Verify that $\alpha$ is (geometrically) given by sending the generator 1 to $x$. '''b)''' Verify that $\beta$ is (geometrically) given by sending a class $y$ to its (equivariant) homology intersection with $x$, $\lambda(x,y)\in\mathbb{Z}[\pi]$. </wikitex> == References == {{#RefList:}} [[Category:Exercises]]M$ be a closed $n$$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.

1) Show that there exists a commutative braid of exact sequences

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

2) Show that the relative homology modules are given by

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

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

3)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

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$x$ 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

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$x$. b) Verify that $\beta$$\beta$ is (geometrically) given by sending a class $y$$y$ to its (equivariant) homology intersection with

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$x$, $\lambda(x,y)\in\mathbb{Z}[\pi]$$\lambda(x,y)\in\mathbb{Z}[\pi]$.