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Wilking's connectivity theorem says, $X$ is a positively curved manifold and $Y$ is a totally geodesic submanifold of codimension $k$,then $Y$ is $n-2k+1$ connected in $X$.Then follow the theorem can we get that $X-Y$ has homology up to $2k-2$?

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Please type using TeX. –  J. GE Sep 11 '12 at 12:24
and separate sentences by space, else it is painful to read. –  Igor Belegradek Sep 11 '12 at 12:33

1 Answer 1

This is a consequence of Poincaré duality applied to the open $n$-manifold $X-Y$.

In particular, we have $H^i_c(X-Y)\cong H_{n-i}(X-Y)$ for all $i$, where $H^\ast_c$ denotes cohomology with compact supports. So it suffices to check that the compactly supported cohomology of $X-Y$ vanishes below degree $n-2k+2$.

There is a long exact sequence for compactly supported cohomology, which in this case looks like $$ \cdots \to H^i_c(X-Y)\to H^i_c(X)\to H^i_c(Y)\to H^{i+1}_c(X-Y)\to\cdots $$ and so the claim follows from the connectivity assumption on the inclusion map $Y\subseteq X$.

This all works with integral or mod 2 coefficients, depending on the orientability of the spaces involved.

Edit: The above argument works when $X$ and $Y$ are compact, which is the setting for Wilking's connectivity theorem mentioned by the OP (Acta Math. 191 (2003), no. 2, 259–297). In that case, compactly supported cohomology coincides with ordinary cohomology, and the map $H^i(X)\to H^i(Y)$ is an isomorphism for $i< n-2k+1$ and a monomorphism for $i=n-2k+1$ by naturality of the universal coefficient exact sequence and the five lemma.

It is false in general that a map of manifolds $f\colon\thinspace Y\to X$ which is $r$-connected induces an isomorphism $f^\ast\colon\thinspace H^i_c(X)\to H^i_c(Y)$ for $i< r$ and a monomorphism for $i=r$, because compactly supported cohomology is not homotopy invariant. Take $X=\mathbb{R}^n$ and $Y=\lbrace 0\rbrace$ for example.

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My algebraic topology is poor.Y is n−2k+1 connected in X,then homology H_k(Y) iso to H_k(X) when k<n-2k+1 and surjective when k=n-2k+1.So I think compact cohomology of X iso to Y when K>2k-1,surjective(?)when k=2k-1.Then by the long exact sequence,compact cohomology of X-Y =0 when k>=2k-2.So H_k(X-Y)=0 when k<=n-2k+2.Why I am wrong? –  jiangsaiyin Sep 12 '12 at 13:46
It looks like I am claiming that since the inclusion map $f\colon Y\to X$ is $n-2k+1$ connected, the map $f^\ast\colon H^i_c(X)\to H^i_c(Y)$ is iso for $i<n-2k+1$ and mono for $i=n-2k+1$. This is certainly true if $X$ and $Y$ are compact, by the naturality of the universal coefficient sequence and the five lemma. I think it should be true more generally, but may require an argument involving direct limits. –  Mark Grant Sep 12 '12 at 14:27
By the way, in your comment you seem to be composing the iso on homology with Poincaré duality to get an iso on cohomology with compact supports. But note that this introduces a degree shift, since $X$ and $Y$ have different dimensions. So you may have to use the UCT, or something similar. –  Mark Grant Sep 12 '12 at 15:25
Oh,different dimensions!Thank you for your answer.I wonder whether we can put forward a result nearer to the manifold case. –  jiangsaiyin Sep 13 '12 at 8:18
I'm not sure what you mean by "put forward a result nearer to the manifold case". As far as I understand it (from briefly skimming the review on MathSciNet(!)) Wilking's theorem is stated only for $X$ and $Y$ compact, and probably fails otherwise. So if by "manifold" you mean "possibly non-compact manifold" then there may not be any result to put forward. –  Mark Grant Sep 13 '12 at 9:37

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