Homology braid II (Ex)
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<wikitex>; | <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. | + | '''1)''' Let $(W,M,M^\prime)$ be a cobordism of $CW$-complexes. Show that the commuting diagram |
+ | $$ | ||
+ | \xymatrix{M^\prime \ar[dr] \ar[rr] && W/M \\ | ||
+ | & W \ar[ur] & | ||
+ | } | ||
+ | $$ | ||
+ | induces a commutative diagram of cofibre sequences | ||
+ | $$ | ||
+ | \xymatrix{ | ||
+ | M^\prime \ar[dr] \ar@/^2pc/[rr] && W/M \ar[dr] \ar@/^2pc/[rr] && \Sigma M \\ | ||
+ | & W \ar[dr] \ar[ur] && W/M\cup M^\prime \ar[dr] \ar[ur] && \\ | ||
+ | M \ar[ur] \ar@/_2pc/[rr] && W/M^\prime \ar[ur] \ar@/_2pc/[rr] && \Sigma M^\prime \\ | ||
+ | } | ||
+ | $$ | ||
+ | |||
+ | and hence a commutative braid of long exact sequences in Homology | ||
+ | |||
+ | $$ | ||
+ | \xymatrix{ | ||
+ | H_*(M^\prime) \ar[dr] \ar@/^2pc/[rr] && H_*(W/M) \ar[dr] \ar@/^2pc/[rr] && H_{*-1}(M) \\ | ||
+ | & H_*(W) \ar[dr] \ar[ur] && H_*(W/M\cup M^\prime) \ar[dr] \ar[ur] && \\ | ||
+ | H_*(M) \ar[ur] \ar@/_2pc/[rr] && H_*(W/M^\prime) \ar[ur] \ar@/_2pc/[rr] && H_{*-1}(M^\prime) \\ | ||
+ | } | ||
+ | $$ | ||
+ | |||
+ | |||
+ | '''2)''' Now let $M$ 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}, | M'=\overline{M\setminus g(S^n\times D^{n-k})}\cup D^{k+1}\times S^{n-k-1}, | ||
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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. | 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. | ||
− | + | 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 | \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}) \ar[dr] \ar@/u\curv/[rr] && H_{i}(\widetilde{M}) \ar[dr] \ar@/u\curv/[rr] && H_{i}(\widetilde{W},\widetilde{M}') \\ | ||
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} | } | ||
$$ | $$ | ||
− | + | ||
+ | '''3)''' 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\\ | H_{i}(\widetilde{W},\widetilde{M})=\left\{\begin{array}{ll} \mathbb{Z}[\pi] & \textrm{if } i=k+1\\ | ||
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\end{array}\right. | \end{array}\right. | ||
$$ | $$ | ||
− | + | '''4)'''Assume $n=2k$ and look at the top bit of the braid | |
$$ | $$ | ||
\def\curv{1.5pc}% Adjust the curvature of the curved arrows here | \def\curv{1.5pc}% Adjust the curvature of the curved arrows here | ||
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} | } | ||
$$ | $$ | ||
− | 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. | + | 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> | </wikitex> | ||
− | == References == | + | <!-- == References == |
− | {{#RefList:}} | + | {{#RefList:}} --> |
[[Category:Exercises]] | [[Category:Exercises]] | ||
+ | [[Category:Exercises with solution]] |
Latest revision as of 21:04, 25 August 2013
1) Let be a cobordism of -complexes. Show that the commuting diagram
induces a commutative diagram of cofibre sequences
and hence a commutative braid of long exact sequences in Homology
Tex syntax errorbe a closed -dimensional manifold and a framed embedding. Denote by the effect of a surgery on
Tex syntax errorand by the corresponding trace, i.e.
Denote by the orientation character of
Tex syntax error, i.e. and by the corresponding universal covers. Write etc. for the homology with -coefficients.
Show that there exists a commutative braid of exact sequences
3) Show that the relative homology modules are given by
4)Assume and look at the top bit of the braid
Let be the Hurewicz image of with being the restriction of the framed embedding we do the surgery on.
a) Verify that is (geometrically) given by sending the generator 1 to .
b) Verify that is (geometrically) given by sending a class to its (equivariant) homology intersection with , .