Free topological inverse semigroups as infinite-dimensional manifolds

TL;DR

This paper characterizes when free topological inverse semigroups form infinite-dimensional manifolds, linking their manifold structure to properties of the underlying space and showing homeomorphism results for these semigroups.

Contribution

It provides a complete characterization of free topological inverse semigroups as $R^$-manifolds based on the properties of the generating space, including conditions for homeomorphism.

Findings

$F(X,K)$ is an $R^$-manifold iff $X$ has no isolated points and is a retract of an $R^$-manifold.

For retracts $X,Y$ of $R^$-manifolds with no isolated points, $F(X,K)$ and $F(Y,K)$ are homeomorphic.

Constructed examples of non-homeomorphic spaces with homeomorphic free topological inverse semigroups.

Abstract

Let $K$ be a complete quasivariety of topological inverse Clifford semigroups, containing all topological semilattices. It is shown that the free topological inverse semigroup $F(X,K)$ of $X$ in the class $K$ is an $R^\infty$-manifold if and only if $X$ has no isolated points and $F(X,K)$ is a retract of an $R^\infty$-manifold. We derive from this that for any retract $X$ of an $R^\infty$-manifold the free topological inverse semigroup $F(X,K)$ is an $R^\infty$-manifold if and only if the space $X$ has no isolated points. Also we show that for any homotopically equivalent retracts $X,Y$ of $R^\infty$-manifolds with no isolated points the free topological inverse semigroups $F(X,K)$ and $F(Y,K)$ are homeomorphic. This allows us to construct non-homeomorphic spaces whose free topological inverse semigroups are homeomorphic.