Every finite group is the group of self homotopy equivalences of an elliptic space

TL;DR

This paper proves that any finite group can be realized as the group of self-homotopy equivalences of infinitely many elliptic spaces, including rationalizations of certain compact manifolds.

Contribution

It demonstrates the universality of elliptic spaces in representing all finite groups as their self-homotopy equivalence groups.

Findings

Every finite group is realizable as a self-homotopy equivalence group.

Existence of elliptic spaces with prescribed self-homotopy groups.

Construction of such spaces as rationalizations of inflexible manifolds.

Abstract

In this paper we prove that every finite group $G$ can be realized as the group of self-homotopy equivalences of infinitely many elliptic spaces $X$. Moreover, $X$ can be chosen to be the rationalization of an inflexible compact simply connected manifold.