Quasimorphisms of free products of racks and quandles

TL;DR

This paper demonstrates that the second bounded cohomology of free products of racks and quandles is infinite-dimensional, extending known results from group theory and providing new proofs using homogeneous quasimorphisms.

Contribution

It establishes the infinite-dimensionality of the second bounded cohomology for free racks and quandles, and offers alternative proofs via homogeneous quasimorphisms.

Findings

Second bounded cohomology of free racks and quandles is infinite-dimensional

Similarities between racks, quandles, and groups in cohomological properties

Alternative proof methods using homogeneous quasimorphisms

Abstract

We show that the second bounded cohomology of the free product of racks and quandles is infinite-dimensional as a real vector space. This is similar to the case of groups. As a corollary, we show that the second bounded cohomology of the free rack and the free quandle is infinite-dimensional. We also give another proof of this corollary using homogeneous group quasimorphisms.