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Homology manifold bordism
Authors:Heather Johnston   Andrew Ranicki
Affiliation:Department of Mathematics, University of Massachusetts, Amherst, Massachusetts 01003 ; Department of Mathematics and Statistics, University of Edinburgh, Edinburgh EH9 3JZ, Scotland, UK
Abstract:

The Bryant-Ferry-Mio-Weinberger surgery exact sequence for compact $ANR$ homology manifolds of dimension $geq 6$ is used to obtain transversality, splitting and bordism results for homology manifolds, generalizing previous work of Johnston.

First, we establish homology manifold transversality for submanifolds of dimension $geq 7$: if $f:M to P$ is a map from an $m$-dimensional homology manifold $M$ to a space $P$, and $Q subset P$ is a subspace with a topological $q$-block bundle neighborhood, and $m-q geq 7$, then $f$ is homology manifold $s$-cobordant to a map which is transverse to $Q$, with $f^{-1}(Q) subset M$ an $(m-q)$-dimensional homology submanifold.

Second, we obtain a codimension $q$ splitting obstruction $s_Q(f) in LS_{m-q}(Phi)$ in the Wall $LS$-group for a simple homotopy equivalence $f:M to P$ from an $m$-dimensional homology manifold $M$ to an $m$-dimensional Poincaré space $P$ with a codimension $q$ Poincaré subspace $Q subset P$ with a topological normal bundle, such that $s_Q(f)=0$ if (and for $m-q geq 7$ only if) $f$ splits at $Q$ up to homology manifold $s$-cobordism.

Third, we obtain the multiplicative structure of the homology manifold bordism groups $Omega^H_*congOmega^{TOP}_*[L_0(mathbb Z)]$.

Keywords:Homology manifolds   bordism   transversality   surgery
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