Every group is the group of self homotopy equivalences of finite dimensional CW-complex
Mahmoud Benkhalifa
TL;DR
The paper solves Kahn's realisability problem for groups on finite-dimensional CW-complexes by encoding a given group $G$ as the automorphism group of a strongly connected digraph $\mathcal{G}$ (via de Groot's theorem) and translating this data into a chain of differential graded Lie algebras $\mathcal{L}_{}(\mathcal{G},s)$ using Anick's $\mathbb{Z}_{(p)}$-local homotopy theory. By progressively adding generators in increasing degrees to kill targeted cycles, the authors achieve $\mathcal{E}(\mathcal{L}_{}(\mathcal{G},27)) \cong \mathrm{aut}(\mathcal{G})$, and hence $\mathcal{E}(X) \cong G$ for a suitable $p$-local complex $X$, with $p>1114$. The construction yields $X$ as $116$-connected and $2341$-dimensional (when $G$ is finite) with a precise $\mathbb{Z}_{(p)}$-homology profile, thereby generalizing Costoya–Viruel's finite-group rational realizability to arbitrary groups on CW-complexes via Anick’s framework. This provides a concrete, algebraic-to-topological pipeline from graph automorphisms to self-homotopy equivalence groups of CW-complexes. The results have broad implications for the realizability of groups in homotopy theory and demonstrate the power of combining digraph automorphisms with $p$-local DGL methods.
Abstract
We prove that any group $G$ occurs as $\E(X)$, where $X$ is CW-complex of finite dimension and $\E(X)$ denotes its group of self-homotopy equivalence. Thus, we generalize a well know-theorem due to Costoya and Viruel \cite{CV} asserting that any finite group occurs as $\E(X)$, where $X$ is rational elliptic space.