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The kind of approximation that one normally uses to prove something about topological manifolds is representing your manifold as an inverse limit of polyhedra (rather than smooth manifolds). … Whitney's embedding theorem applies only to smooth manifolds. …

some geometric topologists don't have a clue about regular neighborhoods, while others haven't heard of multijet transversality; but they all tend to be equally excited when it comes to Hilbert cube manifolds … Matveev, Algorithmic topology and classification of 3-manifolds 2D homotopy and combinatorial group theory Daverman-Venema, Embeddings in manifolds (about a third of the book is on PL embedding theory …

Some topologists, perhaps the majority, tend to think that smooth and topological manifolds are "present in nature" and are the genuine objects of study in geometric topology, while PL topology is a somewhat … ) and especially with Casson handles that occur in topological manifolds. …

Obviously not, because there exist non-shellable combinatorial spheres, but any combinatorial $n$-sphere is bistellar-equivalent to the boundary of the $(n+1)$-simplex. The observation you mentioned …

Let $N$ and $N'$ be regular neighborhoods of a subpolyhedron $P$ in a closed PL manifold $M$, and suppose that $t$ is a free PL involution on $M$ such that each of $\partial N$, $\partial N'$ is invar …

Akhmetiev that $S^6$ contains two smoothly embedded $5$-spheres invariant under the antipodal involution that are not equivariantly PL isotopic, and the reference is Lopez de Medrano's "Involutions on Manifolds …