On the quasi polynomiality of extremal homology of configuration spaces

TL;DR

This paper precisely determines the degree of the top non-trivial quasi polynomial in the extremal homology of configuration spaces of manifolds, confirming the sharpness of previous upper bounds.

Contribution

It provides the exact degree of the extremal homology quasi polynomials, refining prior bounds and enhancing understanding of their algebraic structure.

Findings

The degree of the top non-trivial quasi polynomial is explicitly determined.

The upper bound for the degree of these quasi polynomials is shown to be sharp.

Results apply to all manifolds, confirming the generality of the bounds.

Abstract

Consider the unordered configuration spaces of manifolds. Knudsen, Miller and Tosteson proved that the extremal homology groups of configuration spaces of manifold are eventually quasi polynomials. In this paper, we give the precise degree of top non-trivial quasi polynomials. This shows that the upper bound of Knudsen, Miller and Tosteson for the degree of quasi polynomials is sharp for every manifold.