Signature for piecewise continuous groups

TL;DR

This paper investigates the algebraic structure of a group of bijections on [0,1) that are piecewise continuous, showing the vanishing of a specific cohomology class and constructing a signature homomorphism to classify certain subgroups.

Contribution

It introduces a new signature homomorphism for the group of piecewise continuous bijections and uses it to analyze normal subgroups within this group.

Findings

Kapoudjian class of PC vanishes

Constructed a nonzero signature homomorphism to Z/2Z

Classified normal subgroups of certain subgroups of PC

Abstract

Let PC be the group of bijections from [0, 1[ to itself which are continuous outside a finite set. Let PC be its quotient by the subgroup of finitely supported permutations. We show that the Kapoudjian class of PC vanishes. That is, the quotient map PC $\rightarrow$ PC splits modulo the alternating subgroup of even permutations. This is shown by constructing a nonzero group homomorphism, called signature, from PC to Z 2Z. Then we use this signature to list normal subgroups of every subgroup G of PC which contains S fin and such that G, the projection of G in PC , is simple.