Particle Spaces on Manifolds and Generalized Poincar\'e Dualities

TL;DR

This paper introduces a new class of configuration spaces called particle spaces on manifolds, explores their relationships with mapping spaces via scanning maps, and applies these concepts to topology, geometry, and duality theorems, revealing new insights and results.

Contribution

It defines particle spaces on manifolds, establishes homology and homotopy equivalences with section spaces, and extends classical duality theorems using these new frameworks.

Findings

Particle spaces include classical and generalized configuration spaces.

Scanning maps relate particle spaces to section spaces as homology equivalences.

New results on configuration spaces, loop spaces, and duality theorems are obtained.

Abstract

It is quite an interesting phenomenon in Topology that configuration spaces on a manifold M are intrinsically related to certain mapping spaces from M. In this paper we interpret and greatly expand on this relationship. Building (mainly) on work of Segal, we introduce a new class of configuration spaces; the particle spaces, and these include the classical configuration spaces of distinct points, symmetric products, truncated products, divisor spaces, positive and negative particles of McDuff, as well as many others. We show that there are "scanning" maps that relate these spaces of particles on smooth manifolds to section spaces of appropriate bundles. These correspondences we prove are homology equivalences always and in some identifiable cases homotopy equivalences. Applications of this to both algebraic geometry and algebraic topology are given. We recover old as well as prove some new results on configuration spaces, bounded multiplicity polynomials and loop spaces. Our methods provide interesting insight into the stability phenomenon of holomorphic mapping spaces discovered in the early eighties and asserting that some spaces of rational maps give good topological approximations to second loop spaces. Finally, we rediscover a beautiful space level description of the classical duality theorems of Poincar\'e, Alexander and Lefshetz (first given by Gajer), and further extend it to include Spanier-Whitehead duality.