Homeomorphic Model for the Polyhedral Smash Product of Disks and Spheres

TL;DR

This paper generalizes a known homeomorphic relationship between polyhedral smash products of disks and spheres, extending it from 2-dimensional cases to arbitrary dimensions and mixed sets, using a novel technique.

Contribution

It extends the homeomorphic model for polyhedral smash products from (D^2, S^1) to (D^{k+1}, S^k) for any k, and further to mixed disks and spheres of various dimensions.

Findings

Polyhedral smash product of (D^{k+1}, S^k) is homeomorphic to an iterated suspension of the geometric realization of K.

Generalization to sets of disks and spheres of different dimensions.

Unpublished work by David Stone is extended to broader cases.

Abstract

In this paper we present unpublished work by David Stone on polyhedral smash products. He proved that the polyhedral smash product of the CW-pair $(D^2, S^1)$ over a simplicial complex $K$ is homeomorphic to an iterated suspension of the geometric realization of $K$. Here we generalize his technique to the CW-pair $(D^{k+1}, S^{k})$, for an arbitrary $k$. We generalize the result further to a set of disks and spheres of different dimensions.