Infinite symmetric group and bordisms of pseudomanifolds

TL;DR

This paper explores the relationship between unitary representations of a product of infinite symmetric groups and functors from bordism categories of pseudomanifolds to Hilbert spaces, revealing a deep connection between algebraic and topological structures.

Contribution

It establishes a correspondence between unitary representations of a product of infinite symmetric groups and functors from bordism categories of pseudomanifolds to Hilbert spaces.

Findings

Unitary representations induce functors from bordism categories.

The category involves bordisms of pseudomanifolds with additional coloring.

Connections between algebraic representations and topological structures are demonstrated.

Abstract

We consider a category whose morphisms are bordisms of $n$-dimensional pseudomanifolds equipped with a certain additional structure (coloring). On the other hand, we consider the product $G$ of $(n+1)$ copies of infinite symmetric group. We show that unitary representations of $G$ produce functors from the category of $(n-1)$-dimensional bordisms to the category of Hilbert spaces and bounded linear operators.