Homology Manifold Bordism


The Bryant-Ferry-Mio-Weinberger surgery exact sequence for compact ANR homology manifolds of dimension≥ 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 ≥ 7: if f : M → P is a map from an m-dimensional homology manifold M to a space P , and Q ⊂ P is a subspace with a topological q-block bundle neighborhood, andm−q ≥ 7, then f is homology manifold s-cobordant to a map which is transverse to Q, with f−1(Q) ⊂M an (m− q)-dimensional homology submanifold. Second, we obtain a codimension q splitting obstruction sQ(f) ∈ LSm−q(Φ) in the Wall LS-group for a simple homotopy equivalence f : M → P from an m-dimensional homology manifold M to an m-dimensional Poincaré space P with a codimension q Poincaré subspace Q ⊂ P with a topological normal bundle, such that sQ(f) = 0 if (and for m− q ≥ 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 Ω∗ ∼= ΩTOP ∗ [L0(Z)].

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Cite this paper

@inproceedings{Johnston1999HomologyMB, title={Homology Manifold Bordism}, author={Heather Johnston and ANDREW RANICKI}, year={1999} }