On Siebenmann periodicity (Q1306208)

Elements of the structure set \(S(M)\) of a compact \(n\)-dimensional topological manifold \(M\) are equivalence classes of simple homotopy equivalences \(h:N\to M\), where \(h\) restricts to a homeomorphism on \(\partial N\). Two maps \(h_i:N_i\to M\) of topological manifolds \(N_i\) are equivalent if there is a homeomorphism \(\phi:N_1\to N_2\) such that \(h_2\cdot\phi\) is homotopic (rel \(\partial\)) to \(h_1\). The author gives a topological construction of the Cappell-Weinberger exact sequence \(0\to S(M)\to S(M\times D^4,\partial)\to Z\) using mapping cylinder neighborhood techniques. He uses his theory of Poincaré sheaves to obtain a relation between the Cappell-Weinberger map and the algebraic form of Siebenmann periodicity.