Möbius manifolds, monoids, and retracts of topological groups (Q2350937)

A Hausdorff space \(X\) is called a \textit{topological group retract} (GR-space, for short) if \(X\) is homeomorphic to a retract of some topological group \(G\). The aim of the paper is to present a general construction that leads in a natural manner to GR-spaces. The main tool for this construction is the notion of \textit{mapping cylinder}. If \(f: X\to Y\) is a continuous function between compact spaces, then the \textit{mapping cylinder of} \(f\), denoted by \({\mathrm MC}(f)\), is the quotient space \(\frac{\left({\mathbb I}\times X\right)\dot\cup Y}{R}\), where \({\mathbb I}\) is the interval \([0,1]\) of the real line, and \(R\) is the smallest closed equivalence relation identifying \((0,x)\in {\mathbb I}\times X\) and \(f(x)\in Y\). Möbius manifolds arise in this manner; more exactly, considering for \(n\in\{0,\,1,\dots\}\) the map \(f: {\mathbb S}^n\to{\mathbb R\mathbb P}^n\) that identifies antipodal points of the sphere \({\mathbb S}^n\), then \({\mathbb M}^{n+1}:={\mathrm M}{\mathrm C}(f)\) is called the \textit{\((n+1)\)-dimensional Möbius manifold}. The authors show that the Möbius manifold \({\mathbb M}^{n}\) is a GR-space if and only if \(n\in\{1,\,2,\,4,\,8\}\). (Note that \({\mathbb M}^1\) is naturally homeomorphic to the interval \([-1,1]\) and that \({\mathbb M}^2\) is the traditional Möbius band.) Another class of spaces studied in the paper is that of the so-called \((L)\)-\textit{semigroups}. A compact \(n\)-manifold \(X\) having a connected boundary \(B\) is called an \((L)\)-\textit{semigroup} if there is a monoid structure on \(X\) such that \(B\) is a subsemigroup of \(X\). It is shown that every commutative \(n\)-dimensional \((L)\)-semigroup is a GR-space and that, if \(n\leq 4\), then every \(n\)-dimensional \((L)\)-semigroup is a GR-space.