[Paper Review] Pontryagin algebras of some moment-angle-complexes

Yakov Veryovkin|arXiv (Cornell University)|Dec 1, 2015

Homotopy and Cohomology in Algebraic Topology4 references3 citations

TL;DR

This paper computes the Pontryagin algebra $ H_*(\Omega\mathcal{Z}_\mathcal{K}) $ of moment-angle complexes $ \mathcal{Z}_\mathcal{K} $ when $ \mathcal{K} $ is the boundary of a pentagon or hexagon, showing it is a one-relator algebra. Using iterated Whitehead products in the free tensor algebra on $ \mu_i $, the authors derive an explicit single relation among generators, providing a homotopy-theoretic proof of McGavran's result on the homotopy type of $ \mathcal{Z}_\mathcal{K} $ as a connected sum of sphere products.

ABSTRACT

We consider the problem of describing the Pontryagin algebra (loop homology) of moment-angle complexes and manifolds. The moment-angle complex Z_K is a cell complex built of products of polydiscs and tori parametrised by simplices in a finite simplicial complex K. It has a natural torus action and plays an important role in toric topology. In the case when K is a triangulation of a sphere, Z_K is a topological manifold, which has interesting geometric structures. Generators of the Pontryagin algebra H_*(ΩZ_K) when K is a flag complex have been described in the work of Grbic, Panov, Theriault and Wu. Describing relations is often a difficult problem, even when K has a few vertices. Here we describe these relations in the case when K is the boundary of a pentagon or a hexagon. In this case, it is known that Z_K is a connected sum of products of spheres with two spheres in each product. Therefore H_*(ΩZ_K) is a one-relator algebra and we describe this one relation explicitly, therefore giving a new homotopy-theoretical proof of McGavran's result. An interesting feature of our relation is that it includes iterated Whitehead products which vanish under the Hurewicz homomorphism. Therefore, the form of this relation cannot be deduced solely from the result of McGavran.

Motivation & Objective

• To describe the Pontryagin algebra $ H_*(\Omega\mathcal{Z}_\mathcal{K}) $ for moment-angle complexes when $ \mathcal{K} $ is the boundary of a pentagon or hexagon.
• To provide a homotopy-theoretic proof of McGavran's result that $ \mathcal{Z}_\mathcal{K} $ is a connected sum of products of spheres.
• To identify the single defining relation in the Pontryagin algebra, which involves iterated Whitehead products that vanish under the Hurewicz homomorphism.
• To demonstrate that the relation cannot be deduced from homological data alone, highlighting the role of higher homotopy operations.

Proposed method

• Use the isomorphism $ H_*(\Omega(\mathbb{C}P^\infty)^\mathcal{K}) \cong T\langle\mu_1,\dots,\mu_m\rangle / (\mu_i^2=0, \mu_i\mu_j + \mu_j\mu_i = 0 \text{ if } \{i,j\} \in \mathcal{K}) $ for flag complexes.
• Apply Theorem 2.4 to identify multiplicative generators of $ H_*(\Omega\mathcal{Z}_\mathcal{K}) $ as iterated commutators of the $ \mu_i $, with specific indexing conditions.
• Express all generators $ \alpha_i, \beta_j, \gamma_k, \delta_l $ as iterated commutators in the tensor algebra $ T\langle\mu_1,\dots,\mu_6\rangle $, and reduce to canonical form.
• Derive a system of linear equations by expanding all commutators and equating coefficients to zero, solving via computational algebra (using [WM]).
• Construct the key relation (4.1) as a sum of signed commutators of degree 6 elements, involving 9 $ \alpha_i $, 8 $ \beta_j $, 9 $ \gamma_k $, and 8 $ \delta_l $, with signs determined by $ \sigma_j $.
• Verify the relation by showing it induces a homotopy equivalence $ (S^3 \times S^5)^{\#9} \# (S^4 \times S^4)^{\#8} \to \mathcal{Z}_\mathcal{K} $, confirming the one-relator structure.

Experimental results

Research questions

• RQ1 What is the complete presentation of the Pontryagin algebra $ H_*(\Omega\mathcal{Z}_\mathcal{K}) $ when $ \mathcal{K} $ is the boundary of a pentagon or hexagon?
• RQ2 Can the homotopy type of $ \mathcal{Z}_\mathcal{K} $ as a connected sum of sphere products be recovered from the Pontryagin algebra structure?
• RQ3 What is the explicit form of the single defining relation in $ H_*(\Omega\mathcal{Z}_\mathcal{K}) $, and how does it involve iterated Whitehead products?
• RQ4 Why does the relation not follow from the Hurewicz image, and what does this imply about the role of higher homotopy operations?
• RQ5 How can the algebraic structure of the loop space homology be used to reconstruct the homotopy type of $ \mathcal{Z}_\mathcal{K} $?

Key findings

• The Pontryagin algebra $ H_*(\Omega\mathcal{Z}_\mathcal{K}) $ for $ \mathcal{K} = \partial\text{pentagon} $ or $ \partial\text{hexagon} $ is a one-relator algebra with a single defining relation among iterated commutators of the $ \mu_i $ generators.
• The defining relation is given by equation (4.1), a sum of 17 signed commutators of degree 6 elements, involving 9 $ \alpha_i $, 8 $ \beta_j $, 9 $ \gamma_k $, and 8 $ \delta_l $, with signs $ \sigma_j $ specified.
• The relation is equivalent to the long expression in equation (4.1) involving 20 nested commutators of the $ \mu_i $, which vanishes in the tensor algebra.
• The generators $ \alpha_i, \beta_j, \gamma_k, \delta_l $ are explicitly defined as iterated commutators of the $ \mu_i $, with $ \deg \alpha_i = 2 $, $ \deg \beta_j = \deg \delta_j = 3 $, $ \deg \gamma_k = 4 $.
• The relation (4.1) cannot be deduced from the Hurewicz image, as the involved iterated Whitehead products vanish under the Hurewicz homomorphism, indicating non-trivial higher homotopy structure.
• The relation induces a homotopy equivalence $ (S^3 \times S^5)^{\#9} \# (S^4 \times S^4)^{\#8} \to \mathcal{Z}_\mathcal{K} $, confirming the connected sum structure and providing a new homotopy-theoretic proof of McGavran's result.