Homology, Homotopy and Applications

Volume 24 (2022)

Number 1

Realisability of the group of self-homotopy equivalences and local homotopy theory

Pages: 205 – 215

DOI: https://dx.doi.org/10.4310/HHA.2022.v24.n1.a11


Mahmoud Benkhalifa (Department of Mathematics, College of Sciences, University of Sharjah, United Arab Emirates)


We prove that any group $G$ occurs as the quotient $\mathcal{E}(X) / \mathcal{E}_\ast (X)$, where $\mathcal{E}(X)$ denotes the group of self-homotopy equivalence of a certain CW‑complex $X$ and $\mathcal{E}_\ast (X)$ denotes its subgroup of the elements inducing the identity on the homology groups.


Kahn’s realisability problem of groups, group of homotopy self-equivalences, Anick’s $R$-local homotopy theory

2010 Mathematics Subject Classification


Received 1 April 2021

Received revised 12 June 2021

Accepted 14 June 2021

Published 6 April 2022