Combinatorial Invariants of Stratifiable Spaces II

TL;DR

This paper extends the concept of lego categories to stratifiable spaces, providing new methods for computing their Grothendieck classes and deriving classes of various quotient and configuration spaces.

Contribution

It introduces the notion of lego categories and demonstrates their application in simplifying the computation of Grothendieck classes for complex stratifiable spaces.

Findings

Derived classes of quotient spaces by stratified group actions

Computed classes of crystallographic quotients and polyhedral configuration spaces

Simplified calculations of spaces of 0-cycles

Abstract

In this follow-up to [16], we continue developing the notion of a lego category and its many applications to stratifiable spaces and the computation of their Grothendieck classes. We illustrate the effectiveness of this construction by giving very short derivations of the class of a quotient by the "stratified action" of a discrete group [1], the class of a crystallographic quotient, the class of both a polyhedral product and a polyhedral (or simplicial) configuration space [8], the class of a permutation product [19] and, foremost, the class of spaces of $0$-cycles [11].