Scanning for oriented configuration spaces (Q2354968)

The configuration space of a manifold \(M\) is defined as \(F_{k}(M) := M^{k} \setminus \Delta_{f}\) where \(\Delta_{f}\) is the fat diagonal. If \(C_{k}(M)\) resp. \(C_{k}^{+}(M)\) is the quotient of \(F_{k}(M)\) by the symmetric group \(\Sigma_{k}\) resp. by the alternating group \(A_{k}\) one obtains the the configurations spaces or ordered, unordered, and oriented collections of points in \(M\). This paper considers the homology of these spaces as \(k\) goes to infinity. More particularly the paper is concerned to show analogues of known results on the ordered configuration spaces for the oriented configuration spaces. In [Topology 14, 91--107 (1975; Zbl 0296.57001)] \textit{D. McDuff} showed that there is a scanning map \(s: C_{k}(M) \rightarrow \Gamma_{k}(M)\) where \(\Gamma_{k}(M)\) is the space of compactly supported degree \(k\) sections of the fibrewise one-point compatification of the tangent bundle of \(M\). For manifolds with boundary McDuff showed that the scanning maps \(C_{k}(M) \rightarrow \Gamma_{k}(M)\) induce an isomorphism \(H_{*}(C_{\infty}(M); {\mathbb Z} ) \rightarrow H_{*}(\Gamma_{\infty}(M); {\mathbb Z})\). The authors show that there are double coverings with a lifted scanning map \(s^{+} : C_{k}^{+}(M) \rightarrow \Gamma_{k}^{+}(M)\) which for manifolds with boundary induces an isomorphism on homology in the range \(* \leq (k-5)/3\) and a surjection for \(* \leq (k-2)/3\). The limit case as \(k\) goes to infinity follows.