Homotopy theory of looped polyhedral products

Abstract

This thesis studies the pointed loop space of spaces known as polyhedral products and gives loop space decompositions in various cases as a product of well-studied spaces. It is a research paper thesis which contains the following papers:

[1] L. Stanton, Loop space decompositions of moment-angle complexes associated to flag
complexes, Q. J. Math. 75 (2024), no. 2, 457–477,
[2] L. Stanton, Loop space decompositions of moment-angle complexes associated to two
dimensional simplicial complexes, (2024), to appear in Proceedings of the
Edinburgh Mathematical Society, https://arxiv.org/abs/2407.10781,
[3] L. Stanton and S. Theriault., Polyhedral products associated to pseudomanifolds, Int.
Math. Res. Not. 2025 (2025), rnaf164.

In [1], we show that the loop space of a moment-angle complex associated to the $k$-skeleton of a flag complex decomposes as a product of spheres and loops on spheres up to homotopy.

In [2], we show that the loop space of a moment-angle complex associated to a $2$-dimensional simplicial complex decomposes as a product of spheres, loops on spheres and well-studied torsion spaces up to homotopy.

In [3], we study the homotopy theory of polyhedral products associated to a combinatorial generalisation of manifolds known as a pseudomanifold. We use this to show that the loop space of a moment-angle manifold associated to a connected, orientable surface, or a triangulation of $S^3$ decomposes as a product of spheres and loops on spheres up to homotopy.

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Identifiers

ORCID for Lewis R. Stanton: orcid.org/0000-0003-4662-054X

ORCID for Stephen Theriault: orcid.org/0000-0002-7729-5527

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