Polyhedral products in abstract and motivic homotopy theory

TL;DR

This paper extends the concept of polyhedral products into an $ abla$-categorical framework, generalizes a key splitting theorem, and applies these results to compute motivic homotopy invariants such as $ abla^1$-homology and Euler characteristics.

Contribution

It introduces polyhedral products in an $ abla$-categorical setting, generalizes a known splitting theorem, and develops a motivic refinement of moment-angle complexes for homotopy computations.

Findings

Generalized splitting theorem for polyhedral products

Computed cellular $ abla^1$-homology and $ abla^1$-Euler characteristics

Established a motivic refinement of moment-angle complexes

Abstract

We introduce polyhedral products in an $\infty$-categorical setting. We generalize a splitting result by Bahri, Bendersky, Cohen, and Gitler that determines the stable homotopy type of the a polyhedral product. We also introduce a motivic refinement of moment-angle complexes and use the splitting result to compute cellular $\mathbb{A}^1$-homology, and $\mathbb{A}^1$-Euler characteristics.