Classification of homotopy real Milnor manifolds
For a compact connected smooth (resp. PL or topological) manifold \((X,\partial X)\), a pair \((M,\partial M)\) is called a homotopy smoothing (resp. homotopy PL or TOP triangulation) if \(M\) is a smooth (resp. PL or topological) manifold and \(f:(M,\partial M)\to (X,\partial X)\) is a simple homotopy equivalence such that \(f|_{\partial M}:\partial M\to \partial X\) is a diffeomorphism (resp. homotopy PL or TOP triangulation). Let \(hS(X)\) (resp. \(hT_{\text{PL}}(X)\) or \(hT_{\text{TOP}}(X)\)) denote the structure set of homotopy smoothings (resp. homotopy PL or TOP triangulations) consisting of all equivalence classes of \(X\) with base point \([(X,\text{id}_X)]\). In this paper the author determines the structure sets \(hS(X)\), \(hT_{\text{PL}}(X)\) and \(hT_{\text{TOP}}(X)\) completely for the case \(X=D^m\times \mathbb R H_{r,s}\) (\(r\geq s)\), where \(\mathbb R H_{r,s}\) denotes the real Milnor manifold given by \[ \mathbb R H_{r,s}=\{([z_0,\cdots ,z_r],[w_0,\cdots ,w_s])\in \mathbb R \text{P}^r\times \mathbb R \text{P}^s: \sum_{k=0}^sz_kw_k=0\}. \] For determining them, he studies the Sullivan-Wall surgery obstruction exact sequence and determines the normal invariant, surgery obstruction of a normal invariant and the action of Wall surgery obstruction group, \(\delta :L_{\dim X+1}(\pi_1(X),\omega (X))\to h{\text{CAT}}(X)\) \(({\text{CAT}}=S,T_{\text{PL}},T_{\text{TOP}})\).
scientific article; zbMATH DE number 2092796