Spaces of particles on manifolds and generalized poincare dualities

Abstract

There are interesting results throughout the literature relating multi-configuration spaces to mapping spaces (cf. [B], [G], , [McD], [S1-2]). In this paper, we use a "local to global" scanning process based on a construction of Segal to unify and generalize these results.First of all by a configuration on a space X we mean a collection of unordered points on X (they can be distinct or not). A multi-configuration will then mean a tuple of configurations with (possibly) certain relations between them. Of course, more rigorous definitions are to follow.It is known by classical work of G. Segal [S1], that the space of configurations of distinct points in Euclidean space is equivalent in homology to an iterated loop space on a sphere. Later work of D. McDuff extended this result to an arbitrary smooth compact manifold (with boundary) where she showed that the space of configurations of distinct points there is equivalent in homology to a space of sections of an appropriate bundle. A bit more later, F. Cohen and C.F. Bodigheimer proved a similar result for spaces of configurations of distinct points with labels (see [B]).Both Segal and McDuff extended their ideas to spaces made out of pairs of configurations. While Segal worked with divisor spaces made out of pairs of configurations having no points in common on a punctured Riemann surface [S2], McDuff dealt with what she coined the space of positive and negative particles on a general smooth manifold. Both were able to identify these spaces with some function spaces. This paper extends and generalizes the work of Segal and McDuff in a great many directions. It also sets a context in which these types of results can be viewed and interpreted by relating them to more classical aspects of algebraic topology, as well as to some recent problems stemming from Gauge theory and dealing with the topology of holomorphic mapping spaces.A starting point for us has been to address the following question: which (multi-) configuration spaces can be used to model mapping spaces (and vice-versa). Such considerations have led us to * This is a revised version of a preprint that has been circulated in September 96† The author holds a postdoctoral fellowship with PIms (Pacific Institute for the mathematical sciences) 1 introduce a new and general class of multi-configurations, the particle spaces and these include symmetric and truncated products, divisor spaces, configurations of distinct points, spaces of positive and negative particles, and most other known examples in the literature.Our basic definition is: A particle space is a multi-configuration space with a partial monoid structure. If one defines the support of a multi-configuration to mean the locus of the points making up the multi-configuration, then the partial monoid structure will be "concatenation" defined on multiconfigurations having disjoint support.Given a manifold M , the most basic example of a particle space on M is the infinite symmetric product SP ∞ (M ) = SP n (M ) (and this is a monoid). Another standard part...

Configuration spaces and polyhedral products

“…Here C((D 1 , S 0 ) K ) is a configuration space that generalises classical labelled configuration spaces by allowing particles to collide under certain rules, and D i ((D 1 , S 0 ) K ) is a certain quotiented configuration space. We remark that there are various constructions of configuration spaces in the literature that are related to these [11,25,29,12,13]. The upshot of this one is that the summands ΣD i ((D 1 , S 0 ) K ) in the splitting of ΣΩ(D 2 , S 1 ) K are in a sense more combinatorial in their behaviour in comparison to polyhedral products, allowing us to obtain combinatorial statements about the homotopy type of (D 2 , S 1 ) K .…”

Factorization homology of topological manifolds

“…Our work is in essence a topological version of theirs, although the topological setting allows for arguments and conclusions ostensibly unavailable in the algebraic geometry. Secondly, it has an antecedent in the labeled configuration space models of mapping spaces dating to the 1970s; it is closest to the models of Salvatore [39] and Segal [43]; but see also [9,25,32,33,41]. Factorization homology thus lies at the broad nexus of Segal's ideas on conformal field theory [42] and his ideas on mapping spaces articulated in [41,43].…”

The Topology of the Space of Holomorphic Maps with Bounded Multiplicity

“…Because this was now well explained in several papers (cf. [5], [6], [7], [8], [10], [11]), we only sketch the rough idea.…”