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
\[ \Omega_*^H \cong \Omega_*^{TOP}[L_0(\mathbb{Z})] .\]
