Scanning for oriented configuration spaces

TL;DR

This paper extends homological stability results for oriented configuration spaces on open manifolds by identifying their limiting space and analyzing the scanning map, providing new insights into their topological structure.

Contribution

It identifies the homology limit of oriented configuration spaces as a specific double cover of a section space and proves the acyclicity of McDuff's scanning map in the limit.

Findings

Homological stability for oriented configuration spaces established.

Limiting space characterized as a double cover of a section space.

Scanning map shown to be acyclic in the limit.

Abstract

In [Pal13] (arXiv:1106.4540) the second author proved that the sequence of "oriented" configuration spaces on an open connected manifold exhibits homological stability as the number of particles goes to infinity. To complement that result we identify the corresponding limiting space, up to homology equivalence, as a certain explicit double cover of a section space. Along the way we also prove that the scanning map of McDuff in [McD75] for unordered configuration spaces is acyclic in the limit.