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Let $X$ be a closed, connected, orientable smooth n-manifold and let $Y\subset X$ be a smooth closed m- submanifold.

  1. Express homology of the complement $H_*(X\smallsetminus Y;\mathbb{Z})$ in terms of (co)homology of the pair $(X,Y)$.
  2. Compute $H_0(X\smallsetminus Y;\mathbb{Z})$ when $m=n-1$.
  3. Suppose $X$ has the integral homology of a sphere. Express $H_*(X\smallsetminus Y;\mathbb{Z})$ in tems of (co)homology of $Y$.
  4. Let $K\subset S^3$ be a knot, i.e., the image of an embedding $f:S^1\rightarrow S^3$. Compute $H_1(S^3\smallsetminus K;\mathbb{Z})$. How good is this invariant at distinguishing knots?

My attempts:

  1. I use MV for $A=N $ (tubular neighborhood of $Y$) and $B=X\smallsetminus Y$ to get: $H_3(X)\rightarrow H_2(A\cap B)\rightarrow H_2(N)\bigoplus H_2(X\smallsetminus Y)\rightarrow H_1(A\cap B)\rightarrow H_1(N)\bigoplus H_1(X\smallsetminus Y)\rightarrow H_1(X) \rightarrow \dots$ How can I describe $A\cap B$? Is $N\approx D^{m+1}\times Y$ ($D$ being a disc)? How does $H(X,Y)$ come into play?

  2. Is $H_0(X\smallsetminus Y;\mathbb{Z})$ simply due to its disconnectedness $\mathbb{Z}^2$?

  3. /

  4. By Aleksander duality $H_q(S^n\smallsetminus X)\approx H^{n-q-1}(X)$ is $H_1(S^3\smallsetminus K;\mathbb{Z})=H^{1}(S^1)=\mathbb{Z}$. I would prefer a comment on relevance of that knot invariant.

I thank you in advance for any clue.

Sumanta
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urbanog
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    Do you know Lefshectz duality (Poincare duality with boundary?) – Connor Malin Sep 08 '20 at 14:56
  • Yes, it says $H_k(X,\partial X)\approx H^{n-k}(X)$, but is $N\smallsetminus Y \approx \partial Y\times D$? – urbanog Sep 08 '20 at 15:29
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    Let $X'$ denote the manifold $X$ with the normal bundle of $Y$ removed. We have the pair $(X',S(N(Y))$, here $N(Y)$ denotes normal bundle and $S$ the sphere bundle of the vector bundle. Now by excision the inclusion $(X',S(N(Y)) \rightarrow (X,N(Y))$ is an isomorphism on homology. The inclusion $(X,Y) \rightarrow (X,N(Y))$ is a homotopy equivalence of pairs since the disk bundle is homotopy equivalent to the base space. Together with Lefschetz duality for $(X', S(N(Y)))$ this should be enough for the first part. – Connor Malin Sep 08 '20 at 16:56

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