Classification of homotopy Wall's manifolds
If \(X\) is of a compact, connected, smooth, PL or topological manifold with or without boundary \(\partial X\), a homotopy smoothing (resp., homotopy PL or TOP triangulation) of \(X\) is a pair \((M,f)\), where \(M\) is a smooth, PL or topological manifold and \(f \colon (M, \partial M) \to (X, \partial X)\) is a simple homotopy equivalence of pairs, for which \(f| _{\partial M} \colon \partial M \to \partial X\) is a diffeomorphism (resp. a PL or TOP homeomorphism). Two homotopy smoothings (resp. PL or TOP triangulations) \((M,f)\) and \((M',f')\) are said to be equivalent if there is a diffeomorphism (resp. PL or TOP homeomorphism) \(h:(M, \partial M) \to (M', \partial M')\) such that the maps \(f'\circ h\) and \(f\) are homotopic relative to the boundary \(\partial M\). The set of equivalence classes of homotopy smoothings (resp. PL or TOP triangulations) of the manifold \(X\) is denoted by \(hS(X)\) (resp. \(hT_{\text{PL}}(X)\) or \(hT_{\text{TOP}}(X)\)). The aim of this paper is to determine completely the sets \(hT_{\text{PL}}(X)\) and \(hT_{\text{TOP}}(X)\), and partially the sets \(hS(X)\) for \(X=D^m \times Q(r,s)\), \(r,s >1\), where \(D^m\) is the disk of dimension \(m\geq 0\) and \(Q(r,s)\) is \textit{C. T. C. Wall}'s manifold defined in [Ann. Math. (2) 72, 292--311 (1960; Zbl 0097.38801)] as a mapping torus of a Dold manifold \(P(r,s)\), where \(P(r,s)\) can be seen as the total space of a fibre bundle over \(\mathbb RP^r\) with fibre \(\mathbb CP^s\). The method for determining the sets \(hS(X)\) (resp. \(hT_{\text{PL}}(X)\) or \(hT_{\text{TOP}}(X)\)) is the study of the Sullivan-Wall surgery exact sequence and, consequently, the determination of Wall's surgery obstruction groups \(L_n(\pi_1(X), \omega(X))\) and the sets of normal invariants of \(X\) \([X/\partial X, G/\text{CAT}]\), where \(\text{CAT}=\text{O,PL,TOP}\).
scientific article; zbMATH DE number 5083216
scientific article