On The Group Of Self-homotopy Equivalences Of An Elliptic Space

TL;DR

This paper investigates the structure of the group of self-homotopy equivalences of simply connected rational elliptic spaces, establishing isomorphisms with certain Postnikov sections and skeletons, and analyzing conditions for finiteness and infiniteness.

Contribution

It provides new isomorphism results for the group of self-homotopy equivalences using Sullivan and Quillen models, and characterizes when this group is finite or infinite for elliptic spaces.

Findings

(X) d3 e9(X^{[n]})

(X) d3 e9(X^{m+1}) under certain conditions

(X) is infinite if X is 2-connected and () eq 0

Abstract

Let $X$ be a simply connected rational elliptic space of formal dimension $n$ and let $\E(X)$ denote the group of homotopy classes of self-equivalences of $X$. If $X^{[k]}$ denotes the $k^{\text{th}}$ Postikov section of $X$ and $X^{k}$ denotes its $k^{\text{th}}$ skeleton, then making use of the models of Sullivan and Quillen we prove that $\E(X)\cong\E(X^{[n]})$ and if $n>m=max\big\{k \,| \,\pi_{k}(X)\neq 0\big\}$ and $\E(X)$ is finite, then $\E(X)\cong\E(X^{m+1})$. Moreover, in case when $X$ is 2-connected, we show that if $\pi_{n}(X)\neq0$, then the group $\E(X)$ is infinite.