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Poincaré duality for spaces with isolated singularities. (English) Zbl 1368.55005

This paper shows how one can, under reasonable hypotheses, assign to each pseudomanifold \(X\) with isolated singularities a rational Poincaré duality space \(\mathcal{DP}(X)\). If \(X\) is such a pseudomanifold then one can define \(X_{reg}\) to be the manifold obtained by excising a small neighbourhood of each singularity. Further \(\overline{X}\) is defined to be the result of coning off the boundaries of \(X_{reg}\). The paper constructs the space \(\mathcal{DP}(X)\), and a factorization \[ X_{reg}\overset{\phi}{\longrightarrow} \;\mathcal{DP}(X) \;\overset{\psi}{\longrightarrow} \;\overline{X} \] where (1) if dim \(X \;= \;2s\) the induced map in homology \(\phi_{r}: H_{r}(X_{reg}) \rightarrow H_{r}(\mathcal{DP}(X))\) is an isomorphism for \(2s-1 > r> s \) and an injection for \(r = s\) and \(\psi_{r} :H_{r}(\mathcal{DP}(X)) \rightarrow H_{r}(\overline{X})\) is an isomorphism for \(r >s\) and \(r=2s\). (2) if dim \( X = 2s+1 \), then the induced map in homology \(\phi_{r}: H_{r}(X_{reg}) \rightarrow H_{r}(\mathcal{DP}(X))\) is an isomorphism for \(2s >r > s + 1\) and an injection for \(r = s+1\) and \(\psi_{r} :H_{r}(\mathcal{DP}(X)) \rightarrow H_{r}(\overline{X})\) is an isomorphism for \(r< s\) and \(r = 2s+1\) and a surjection for \(r = s\). \(\mathcal{DP}(X)\) is a very good rational Poincaré approximation when dim \(X = 2s\); then \(\phi_{ s}\) is also an isomorphism and when dim \(X = 2s+1\), then \(\phi_{s+1}\) and \(\psi_{s}\) are also isomorphisms.
The methods are essentially those of using geometric constructions to obtain the desired space.

MSC:

55P62 Rational homotopy theory
57P10 Poincaré duality spaces
55N33 Intersection homology and cohomology in algebraic topology
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References:

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