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Self-Homotopy Equivalence Group of an Elliptic Space and Its Embedding in general Linear Groups

Mahmoud Benkhalifa

TL;DR

This work investigates when the self-homotopy equivalence group $\mathcal{E}(X)$ of a rational elliptic space $X$ is finite and how $X$'s rational homotopy data constrain its algebraic symmetries. Employing Sullivan minimal models and the Whitehead exact sequence, the authors derive split short exact sequences that yield a concrete semidirect-product description of $\mathcal{E}(\Lambda V^{\le n})$ in terms of indecomposables $V^{n}$ and lower-stage automorphisms, together with explicit maps $\Psi_n$, $\sigma_n$, $\Theta$, and $\Theta'$. They show that $\mathcal{E}(X)$ embeds into a product of vector-space homomorphisms $\mathcal{L}^{m_j}(X)$ with $\mathrm{GL}(p_j,\mathbb{Q})$ factors, and that finiteness forces a containment inside a finite product of GL-groups. For $F_0$-spaces, many $\mathcal{L}^{m_j}(X)$ terms vanish, yielding stronger structural constraints. These results connect the finiteness of $\mathcal{E}(X)$ to the rational homotopy invariants of $X$, providing practical tools to analyze symmetries of elliptic spaces and their embeddings into linear groups.

Abstract

For a rational elliptic space, this paper examines the relationship between its homotopy groups and its self-homotopy equivalence group. Moreover, we investigate how this group is embedded in general linear groups.

Self-Homotopy Equivalence Group of an Elliptic Space and Its Embedding in general Linear Groups

TL;DR

This work investigates when the self-homotopy equivalence group of a rational elliptic space is finite and how 's rational homotopy data constrain its algebraic symmetries. Employing Sullivan minimal models and the Whitehead exact sequence, the authors derive split short exact sequences that yield a concrete semidirect-product description of in terms of indecomposables and lower-stage automorphisms, together with explicit maps , , , and . They show that embeds into a product of vector-space homomorphisms with factors, and that finiteness forces a containment inside a finite product of GL-groups. For -spaces, many terms vanish, yielding stronger structural constraints. These results connect the finiteness of to the rational homotopy invariants of , providing practical tools to analyze symmetries of elliptic spaces and their embeddings into linear groups.

Abstract

For a rational elliptic space, this paper examines the relationship between its homotopy groups and its self-homotopy equivalence group. Moreover, we investigate how this group is embedded in general linear groups.

Paper Structure

This paper contains 7 sections, 10 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notion of homotopy for Sullivan algebras
  4. Whitehead exact sequence of a Sullivan algebra
  5. Elliptic spaces
  6. Short exact sequences associated with ${\mathcal{E}}(X)$ and ${\mathcal{E}}_{\#}(X)$.
  7. Topological applications

Key Result

Theorem 1

$($Theorem t3$)$. Let $X$ be an elliptic space, and let $\pi_{m_1}(X), \dots, \pi_{m_k}(X)$ denote the nontrivial homotopy groups of $X$, where $m_1 \leq \cdots \leq m_k$. For each $1 \leq j \leq k$, define Then, the group of self-homotopy equivalences $\mathcal{E}(X)$ is a subgroup of the group where $p_j = \mathrm{dim}\space \pi_{m_j}(X)$ for all $1 \leq j \leq k$.

Theorems & Definitions (23)

  • Theorem 1
  • Theorem 2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6
  • ...and 13 more