Homology manifold bordism

Abstract

Abstract. The Bryant-Ferry-Mio-Weinberger surgery exact sequence for compact AN R 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, and m−q ≥ 7, then f is homology manifold s-cobordant to a map which is transverse to Q,Second, we obtain a codimension q splitting obstruction s Q (f ) ∈ LS m−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 s Q (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 H *

Problems on homology manifolds

“…The examples give maps approximately transverse to a point, but for which more geometric forms of transversality are obstructed. Transversality theories restricted to situations where indices must be integers have been developed by Johnston [30], Johnston-Ranicki [31] and Bryant-Mio [8], and a finiteness-obstruction case has been investigated by Bryant-Kirby [7]. The hope is that a more complete approximate transversality theory is possible.…”

Exotic homology manifolds (Oberwolfach 2003)

“…The examples give maps approximately transverse to a point, but for which more geometric forms of transversality are obstructed. Transversality theories restricted to situations where indices must be integers have been developed by Johnston [34], Johnston-Ranicki [35] and Bryant-Mio [9], and a finitenessobstruction case has been investigated by Bryant-Kirby [8]. The hope is that a more complete approximate transversality theory is possible.…”