On piecewise linear cell decompositions

TL;DR

This paper introduces PLCW complexes, a new class of cell decompositions for PL manifolds and polyhedra, and proves an Alexander-type theorem relating different PLCW decompositions through elementary moves.

Contribution

It defines PLCW complexes, a novel class of cell decompositions, and establishes an Alexander-type theorem for them, bridging triangulations and CW complexes.

Findings

Introduces PLCW complexes as a new class of cell decompositions.

Proves an Alexander-type theorem for PLCW complexes.

Shows PLCW complexes can be related by elementary moves.

Abstract

In this note, we introduce a class of cell decompositions of PL manifolds and polyhedra which are more general than triangulations yet not as general as CW complexes; we propose calling them PLCW complexes. The main result is an analog of Alexander's theorem: any two PLCW decompositions of the same polyhedron can be obtained from each other by a sequence of certain "elementary" moves. This definition is motivated by the needs of Topological Quantum Field Theory, especially extended theories as defined by Lurie.