Triangulating neighborhoods in topological manifolds

Abstract

Suppose X is a compact subset of the topological q-manifold Q, q ≥ 5, ∂Q = ▪ . It is shown that some neighborhood of X supports a piecewise linear structure provided the inclusion- induced homomorphism i ∗:H 4(Q;Z 2)→ H ̌ 4X;Z 2) is zero. Thus methods for studying embeddings of compacta in piecewise linear manifolds can often be applied without assuming piecewise linearity. As an example of such an application, it is pointed out that McMillan's criterion for cellularity in piecewise linear manifolds of dimension five or more also holds in topological manifolds. Examples are given to show that piecewise linear neighborhoods may fail to exist stably, even in the case when X is a piecewise linear manifold embedded as a locally flat submanifold of Q.