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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Homology manifold bordism


Authors: Heather Johnston and Andrew Ranicki
Journal: Trans. Amer. Math. Soc. 352 (2000), 5093-5137
MSC (2000): Primary 57P05; Secondary 19J25
Published electronically: March 16, 2000
MathSciNet review: 1778506
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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 $\Omega^H_*\cong\Omega^{TOP}_*[L_0(\mathbb Z)]$.


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Additional Information

Heather Johnston
Affiliation: Department of Mathematics, University of Massachusetts, Amherst, Massachusetts 01003
Email: johnston@math.umass.edu

Andrew Ranicki
Affiliation: Department of Mathematics and Statistics, University of Edinburgh, Edinburgh EH9 3JZ, Scotland, UK
Email: aar@maths.ed.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02630-1
PII: S 0002-9947(00)02630-1
Keywords: Homology manifolds, bordism, transversality, surgery
Received by editor(s): March 25, 1998
Published electronically: March 16, 2000
Additional Notes: This work was carried out in connection with the first named author’s EPSRC Visiting Fellowship in Edinburgh in August, 1997.
Article copyright: © Copyright 2000 American Mathematical Society