## Homology manifold bordism

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- by Heather Johnston and Andrew Ranicki
- Trans. Amer. Math. Soc.
**352**(2000), 5093-5137 - DOI: https://doi.org/10.1090/S0002-9947-00-02630-1
- Published electronically: March 16, 2000
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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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## Bibliographic 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
- MR Author ID: 144725
- Email: aar@maths.ed.ac.uk
- 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.
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**352**(2000), 5093-5137 - MSC (2000): Primary 57P05; Secondary 19J25
- DOI: https://doi.org/10.1090/S0002-9947-00-02630-1
- MathSciNet review: 1778506