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Quasimorphisms of free products of racks and quandles

Masamitsu Aoki

TL;DR

The paper proves that the second bounded cohomology $H_ ext{b}^2(X;\mathbb{R})$ is infinite-dimensional for the free product $X=\ast_{s\in S}X_s$ of finitely generated racks (and similarly for quandles). It constructs rack quasimorphisms on the free product from Rolli's group-quasimorphism framework, yielding an injective map from an infinite-dimensional space into $H_ ext{b}^2(X)$, and provides an alternate proof via homogeneous group quasimorphisms by exploiting the fact that the adjoint of the free rack is a free group. The results extend known phenomena for groups to racks and quandles, and give corollaries such as the infinite-dimensionality of $H_ ext{b}^2$ for the free rack and free quandle. The methods blend explicit free-product constructions with quasimorphism theory to establish nontrivial and large bounded cohomology classes for these algebraic structures.

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.

Quasimorphisms of free products of racks and quandles

TL;DR

The paper proves that the second bounded cohomology is infinite-dimensional for the free product of finitely generated racks (and similarly for quandles). It constructs rack quasimorphisms on the free product from Rolli's group-quasimorphism framework, yielding an injective map from an infinite-dimensional space into , and provides an alternate proof via homogeneous group quasimorphisms by exploiting the fact that the adjoint of the free rack is a free group. The results extend known phenomena for groups to racks and quandles, and give corollaries such as the infinite-dimensionality of for the free rack and free quandle. The methods blend explicit free-product constructions with quasimorphism theory to establish nontrivial and large bounded cohomology classes for these algebraic structures.

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.
Paper Structure (10 sections, 11 theorems, 35 equations)

This paper contains 10 sections, 11 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Racks and quandles
  4. Free products
  5. Bounded cohomology
  6. Rack quasimorphisms on the free product
  7. Group quasimorphisms
  8. Rack quasimorphisms
  9. Proof of Theorem \ref{['theorem:main']}
  10. Homogeneous group quasimorphisms and free racks

Key Result

Theorem 1.1

Let $X = \mathop{\hbox{$\ast$}}_{s \in S} X_s$ be the free product of finitely generated racks with $2 \leq \lvert S \rvert < \infty$. Then the second bounded cohomology $H_\mathrm{b}^2 (X, \mathbb{R})$ is infinite-dimensional as a real vector space.

Theorems & Definitions (26)

  • Theorem 1.1
  • Corollary 1.2
  • Example 2.1
  • Example 2.2
  • Example 2.3: Farinati2014-ztFenn1992-dd
  • Example 2.4: Farinati2014-ztJoyce1982-my
  • Example 2.5: Fenn1992-dd
  • Remark 2.6
  • Lemma 2.7
  • Example 2.8
  • ...and 16 more