Spaces of particles on manifolds and generalized Poincaré dualities. (Q2716947)

The author studies ``particle spaces'', identification spaces, and subspaces of k-fold products of \(SP^{\infty}(M)\), the union over all \(n\), of the spaces of \(n\) unordered points in a manifold \(M\). The natural structure of \(SP^{\infty}(M)\) as an abelian monoid is utilized. The spaces studied include for example, the classical configuration spaces of distinct points in \(M\), spaces of positive and negative particles [\textit{D. McDuff}, Topology 14, 91--107 (1975; Zbl 0296.57001)], and the divisor spaces of \textit{G. Segal} [Acta Math. 143, 39--72 (1979; Zbl 0427.55006)]. Classical theorems of Alexander, Lefshetz, Poincaré, and Spanier, Whitehead are recovered as well as recent theorems of \textit{M. A. Guest} [Acta Math. 174, 119--145 (1995; Zbl 0826.14035)] and McDuff [loc. cit.].NEWLINENEWLINEVarious equivalences are obtained between functors of the particle spaces and certain spaces of maps. The main theorem asserts the existence of a fibre bundle with a section over \(M\) with fibre a particle space, and a relative homology equivalence between a particle space and a space of sections of that bundle. A second theorem gives a condition under which the homology equivalence is a homotopy equivalence.