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