Infinite symmetric group, pseudomanifolds, and combinatorial cobordism-like structures

Abstract

We consider a category [Formula: see text] whose morphisms are [Formula: see text]-dimensional pseudomanifolds equipped with certain additional structures (coloring and labeling of some cells), multiplication of morphisms is similar to a concatenation of cobordisms. On the other hand, we consider the product [Formula: see text] of [Formula: see text] copies of infinite symmetric group. We construct a correspondence between the sets of morphisms of [Formula: see text] and double coset spaces of [Formula: see text] with respect to certain subgroups. We show that unitary representations of [Formula: see text] produce functors from the category of [Formula: see text] to the category of Hilbert spaces and bounded linear operators.