[Paper Review] Realizability of the group of rational self-homotopy equivalences

Mahmoud Benkhalifa|arXiv (Cornell University)|Oct 7, 2010

Homotopy and Cohomology in Algebraic Topology3 references14 citations

TL;DR

This paper proves that for every natural number $ n $, there exists a 1-connected rational CW-complex $ X_n $ whose group of rational self-homotopy equivalences is isomorphic to the direct sum of $ 2^{n+1} $ copies of $ \mathbb{Z}_2 $. Using Sullivan's rational homotopy theory, the authors construct minimal models of commutative cochain algebras and analyze automorphisms via cohomological obstructions and homotopy classes, ultimately realizing the desired group structure through explicit algebraic constraints on generators and their degrees.

ABSTRACT

For a 1-connected CW-complex $X$, let $\mathcal{E}(X)$ denote the group of homotopy classes of self-homotopy equivalences of $X$. The aim of this paper is to prove that, for every $n\in\Bbb N$, there exists a 1-connected rational CW-complex $X_{n}$ such that $\mathcal{E}(X_{n})\cong \underset{2^{n+1}\mathrm{. times}}{\underbrace{\Bbb Z_{2}\oplus... \Bbb \oplus \Bbb Z_{2}}}$.

Motivation & Objective

• To address the realizability problem in rational homotopy theory: for a given finite group $ G $, determine whether there exists a 1-connected rational CW-complex $ X $ such that $ \mathcal{E}(X) \cong G $.
• To specifically investigate whether elementary abelian 2-groups of the form $ \mathbb{Z}_2^{2^{n+1}} $ are rationally realizable for all $ n \in \mathbb{N} $.
• To construct explicit minimal models of 1-connected rational spaces whose self-equivalence groups match the target group structure.
• To extend prior results on realizability of $ \mathbb{Z}_2 $ and $ \mathbb{Z}_2 \oplus \mathbb{Z}_2 $ to higher-rank elementary abelian 2-groups using algebraic constraints on cochain morphisms.
• To establish a connection between the group structure of self-equivalences and cohomological data via the $ b^n $ maps and homotopy classes of automorphisms.

Proposed method

• Employ Sullivan's minimal model theory to translate the topological problem into the algebraic setting of 1-connected minimal commutative cochain algebras (mccas) over $ \mathbb{Q} $.
• Define the group $ \mathcal{E}(\Lambda V, \partial) $ of self-homotopy equivalences of a mcca as the group of homotopy classes of cochain automorphisms preserving the differential.
• Use the homotopy relation between cochain morphisms, defined via tensoring with $ (\Lambda(t, dt), d) $, to identify when two automorphisms are homotopic.
• Introduce the linear maps $ b^n: V^n \to H^{n+1}(\Lambda V^{\leq n-1}) $, defined by $ b^n(v) = [\partial(v)] $, to track obstruction data for extending automorphisms.
• Construct the group $ \mathcal{C}^{n+1} \subset \text{Aut}(V^{n+1}) \times \mathcal{E}(\Lambda V^{\leq n}, \partial) $ as the set of compatible pairs preserving the $ b^n $-structure.
• Use the surjective homomorphism $ \Phi^{n+1}: \mathcal{E}(\Lambda V^{\leq n+1}, \partial) \to \mathcal{C}^{n+1} $ to relate automorphism groups across filtration levels and track the group structure inductively.

Experimental results

Research questions

• RQ1 Can every finite group $ G $ be realized as the group of rational self-homotopy equivalences of some 1-connected rational CW-complex?
• RQ2 Are elementary abelian 2-groups of the form $ \mathbb{Z}_2^{2^{n+1}} $ rationally realizable for all $ n \in \mathbb{N} $?
• RQ3 What algebraic constraints on the minimal model of a rational space force its self-equivalence group to be isomorphic to $ \mathbb{Z}_2^{2^{n+1}} $?
• RQ4 How do the cohomological maps $ b^n $ and the compatibility conditions in $ \mathcal{C}^{n+1} $ control the structure of $ \mathcal{E}(\Lambda V, \partial) $?
• RQ5 What is the role of the choice of coefficients $ p_z, p_{y_i}, p_{x_k} $ in determining the number and order of self-equivalence classes?

Key findings

• For every $ n \in \mathbb{N} $, there exists a 1-connected rational CW-complex $ X_n $ such that $ \mathcal{E}(X_n) \cong \mathbb{Z}_2^{2^{n+1}} $, proving the rational realizability of these groups.
• The group $ \mathcal{E}(\Lambda V^{\leq 9.2^{n+2}-1}, \partial) $ is isomorphic to $ \mathbb{Z}_2^{2^{n+1}} $, constructed via a carefully designed minimal model with generators in specific degrees.
• The self-equivalence group arises from two cases: $ p_{n+1} = 1 $ and $ p_{n+1} = -1 $, each contributing $ 2^n $ homotopy classes of order 2.
• The total number of homotopy classes of self-equivalences is $ 2^n + 2^n = 2^{n+1} $, all of order dividing 2 (except the identity), confirming the group is elementary abelian.
• The coefficients $ p_z, p_{y_1}, p_{y_2}, p_{y_3}, p_w, p_k $ satisfy a system of equations that admit exactly $ 2^{n+1} $ solutions with values in $ \{ \pm 1 \} $, corresponding to the group elements.
• The construction relies on the fact that all cocycles in the relevant degrees are coboundaries, ensuring that any automorphism is homotopic to one that acts diagonally on generators, simplifying the classification.