[Paper Review] Right-angularity, flag complexes, asphericity. \ Criteria for asphericity: corrigenda for "Right-angularity, flag complexes, asphericity"

Michael W. Davis|arXiv (Cornell University)|Feb 23, 2011

Homotopy and Cohomology in Algebraic Topology12 references3 citations

TL;DR

This paper corrects and refines criteria for asphericity of polyhedral products associated with simplicial complexes and pairs of spaces. It establishes necessary and sufficient conditions—requiring each $A(i)$ to be aspherical, certain pairs to be nearly aspherical, $L^\prime$ to be a flag complex, and $L^\prime$ to decompose as a join—under a field-based acyclicity assumption, resolving gaps in earlier work on right-angled buildings and moment-angle complexes.

ABSTRACT

The "polyhedral product functor" produces a space from a simplicial complex L and a collection of pairs of spaces, {(A(i),B(i))}, where i ranges over the vertex set of L. We give necessary and sufficient conditions for the resulting space to be aspherical. There are two similar constructions, each of which starts with a space X and a collection of subspaces, {X_i} and then produces a new space. We give conditions for the results of these constructions to be aspherical. All three techniques can be used to produce examples of closed aspherical manifolds. Abstract for Corrigenda: This note concerns two refinements to the earlier work by the first author. First, when L is infinite, the definition of polyhedral product needs clarification. Second, the earlier paper omitted some subtle parts of the necessary and sufficient conditions for polyhedral products to be aspherical. Correct versions of these necessary and sufficient conditions are given in the present paper.

Motivation & Objective

• To correct and clarify the definition of polyhedral products when the underlying simplicial complex $L$ is infinite.
• To identify and fix missing or implicit assumptions in the original criteria for asphericity of polyhedral products.
• To provide a complete, necessary and sufficient condition for the asphericity of $\mathbf{A}^L$ under a field-based acyclicity assumption.
• To resolve cases where earlier results failed due to incorrect assumptions about fundamental group injectivity and acyclicity of $\widetilde{B}(i)$.
• To generalize the theory to infinite complexes and non-contractible base pairs, extending results on moment-angle complexes and right-angled buildings.

Proposed method

• Define the polyhedral product $\mathbf{A}^L$ as a subspace of $\prod_{i\in I} A(i)$ with conditions (I) finitely many non-basepoint coordinates and (II) support forming a simplex in $L$.
• Equip $\mathbf{A}^L$ with the colimit topology over finite subcomplexes when $L$ is infinite.
• Introduce the sets $I^{\prime\prime}$, $I_1$, $I_2$, and $I^\prime = I_2 \cup I^{\prime\prime}$, and define $L^\prime = \langle I^\prime \rangle$ to isolate critical components.
• Use the field-based acyclicity assumption $(*)$ to ensure clean homological behavior in the universal cover of the polyhedral product.
• Apply a homological criterion via the smash product of universal covers $\widetilde{B}(i)$, leveraging the formula from [1, Thm. 2.21] to detect nontrivial homology.
• Establish that $\widetilde{\mathbf{A}}^L$ is contractible by showing it is homotopy equivalent to $\widetilde{\mathbf{A}}^{L^\prime}$, which is shown to be contractible via join decomposition and acyclicity.

Experimental results

Research questions

• RQ1 What conditions are necessary and sufficient for the polyhedral product $\mathbf{A}^L$ to be aspherical when $L$ is infinite?
• RQ2 How do failures in fundamental group injectivity or acyclicity of $\widetilde{B}(i)$ affect the asphericity of $\mathbf{A}^L$?
• RQ3 In what way must the earlier criteria in [4, Theorem 2.22] be corrected when assumptions (a) and (b) are not satisfied?
• RQ4 How does the decomposition $L^\prime = \langle I_2 \rangle * L^{\prime\prime}$ relate to the asphericity of $\mathbf{A}^L$?
• RQ5 Under what conditions does the universal cover of $\mathbf{A}^L$ remain contractible despite non-contractible $B(i)$?

Key findings

• The corrected criterion for asphericity of $\mathbf{A}^L$ requires each $A(i)$ to be aspherical, and for each $i \in I^\prime$ not conelike in $L^\prime$, the pair $(A(i), B(i))$ to be aspherical.
• The complex $L^\prime$ must be a flag complex, and it must decompose as a join $\langle I_2 \rangle * L^{\prime\prime}$, where $I_2$ indexes vertices with non-acyclic $\widetilde{B}(i)$ over a field $\mathbb{F}$.
• The set $I^{\prime\prime}$, indexing vertices with $|E_i| \neq 1$, ensures that $L^{\prime\prime}$ captures the non-trivial fundamental group contributions.
• The universal cover $\widetilde{\mathbf{A}}^L$ is shown to be contractible by proving it is homotopy equivalent to $\widetilde{\mathbf{A}}^{L^\prime}$, which is contractible via the join structure and acyclicity.
• The homology of the universal cover detects non-triviality via the smash product of $\widetilde{B}(i)$'s; non-vanishing homology in degree $k$ implies the presence of an empty $k$-simplex, contradicting asphericity.
• The field-based assumption $(*)$ ensures that $I_1 \cup I_2$ partitions $I - I^{\prime\prime}$, which is essential for the homological argument and the final conclusion.