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

Let $X$ be a $2$-connected and $6$-dimensional CW‑complex such that $H_3 (X) \otimes \mathbb{Z}_2 = 0$. This paper aims to describe the group $\mathcal{E}(X)$ of the self-homotopy equivalences of $X$ modulo its normal subgroup $\mathcal{E}_\ast (X)$ 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 $\Gamma S(X)$ of $\Gamma$-automorphisms of WES($X$) and we prove that $\mathcal{E}(X)/\mathcal{E}_\ast (X) \cong \Gamma \mathcal{S}(X)$.

Keywords

Whitehead’s exact sequence, $\Gamma$-automorphism, group of self-homotopy equivalences

2010 Mathematics Subject Classification

55P10, 55P15

Full Text (PDF format)

Received 17 October 2022

Received revised 10 March 2023

Accepted 28 March 2023

Published 20 March 2024