Continuous cohomology of topological quandles

TL;DR

This paper develops a continuous cohomology framework for topological quandles, compares it with algebraic theories, and introduces methods for computing cohomology groups, highlighting differences in second cohomology for specific examples.

Contribution

It introduces a continuous cohomology theory for topological quandles and provides new methods for computing their cohomology groups, especially for inverse limits.

Findings

Continuous cohomology differs from algebraic cohomology for certain quandles.

Extensions of topological quandles are characterized by continuous 2-cocycles.

A method for computing cohomology of inverse limits of quandles is presented.

Abstract

A continuous cohomology theory for topological quandles is introduced, and compared to the algebraic theories. Extensions of topological quandles are studied with respect to continuous 2-cocycles, and used to show the differences in second cohomology groups for specific topological quandles. A method of computing the cohomology groups of the inverse limit is applied to quandles.