On the group of self-homotopy equivalences of a 2-connected and 6-dimensional CW-complex

TL;DR

This paper characterizes the group of self-homotopy equivalences of a specific 6-dimensional CW-complex, revealing its structure via Whitehead sequences and automorphisms, under certain homological conditions.

Contribution

It establishes an isomorphism between the quotient of the self-homotopy equivalence group and a group of automorphisms of the Whitehead exact sequence for the complex.

Findings

The group of self-homotopy equivalences modulo those inducing identity on homology is isomorphic to a group of automorphisms.

The structure of this quotient is explicitly described using Whitehead exact sequences.

Conditions on the homology of the CW-complex are crucial for the main result.

Abstract

Let $X$ be a \text{\rm{2}}-connected and \text{\rm{6}}-dimensional CW-complex $X$ such that $H_{3}(X)\otimes\Z_2=0$. This paper aims to describe the group $\E(X)$ of the self-homotopy equivalences of $X$ modulo its normal subgroup $\E_{*}(X)$ of the elements that induce the identity on the homology groups. Making use of the Whitehead exact sequence of $X$, denoted by WES$(X)$, we define the group $\Gamma\mathcal{S}(X)$ of $\Gamma$-automorphisms of WES$(X)$ and we prove that $\E(X)/\E_*(X)\cong \Gamma\mathcal{S}(X).$