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In the conference "on Manifolds with Non-negative Sectional Curvature" held in 2007, Problem 6 is:

Extend the Wilking Connectivity Theorem to Alexandrov spaces, i.e. if $X$ is a positively curved Alexandrov space and $Y$ is a totally geodesic subspace of codimension $k$, is it true that $X-Y$ has homology only up to dimension $2k-2$?

The Wilking connectivity theorem is:

If $X$ is a positively curved Riemannian manifold and $Y$ is a totally geodesic subspace of codimension $k$, then $Y\subset X$ is $n-2k+1$-connected and $X-Y$ are cells of dimension $\ge n-2k+2$.Then the homology groups are $H_0(X-Y)=\mathbb{Z}+\mathbb{Z}$, $H_i(X-Y)=0$ for $i\ge 0$.

I know the result does not holds for Alexandrov space (counterexample: the inclusion of $\mathbb{CP}^n$ into suspension of $\mathbb{CP}^n$). So the result has to be modified, but how did we get the conjecture for problem 6? Why $2k-2$?

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  • $\begingroup$ I've formatted the question using LaTeX, please check I haven't messed up the question. $\endgroup$
    – David Roberts ♦
    Commented Aug 29, 2012 at 7:40

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The $2k-2$ upper bound on the homology of $X-Y$ comes from Poincaré duality in the manifold case. See my answer to Wilking's connnectivity theorem.

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