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Generalized (co)homology of symmetric quandles over homogeneous Beck modules

Biswadeep Karmakar, Deepanshi Saraf, Mahender Singh

TL;DR

This work develops a Beck-module, category-theoretic framework for symmetric racks and quandles, introducing symmetric quandle modules and proving they classify Beck modules in the symmetric setting. It constructs a generalized (co)homology theory with homogeneous coefficient objects, showing it both recovers Kamada–Oshiro’s symmetric quandle (co)homology and extends to symmetric racks, with abelian extensions classified by $\mathcal{H}^2_{\mathrm{SR}}$ and $\mathcal{H}^2_{\mathrm{SQ}}$. A key achievement is the equivalence between module categories and abelian group objects in the corresponding slice categories, enabling a semi-direct product construction and a robust extension theory. The paper also connects the generalized cohomology to the cohomology of associated groups by establishing $H^2_{SR}((X,\rho_X),A) \cong H^1(G_{(X,\rho_X)}, \mathrm{Hom}(X,A))$ for trivial coefficients, highlighting deep links to group cohomology. Overall, the results provide a modular, coefficient-rich approach with potential to yield stronger invariants in low-dimensional topology and related areas.

Abstract

A quandle equipped with a good involution is referred to as symmetric. It is known that the cohomology of symmetric quandles gives rise to strong cocycle invariants for classical and surface links, even when they are not necessarily oriented. In this paper, we introduce the category of symmetric quandle modules and prove that these modules completely determine the Beck modules in the category of symmetric quandles. Consequently, this establishes suitable coefficient objects for constructing appropriate (co)homology theories. We develop an extension theory of modules over symmetric quandles and propose a generalized (co)homology theory for symmetric quandles with coefficients in a homogeneous Beck module, which also recovers the symmetric quandle (co)homology developed by Kamada and Oshiro [Trans. Amer. Math. Soc. (2010)]. Our constructions also apply to symmetric racks. We conclude by establishing an explicit isomorphism between the second cohomology of a symmetric quandle and the first cohomology of its associated group.

Generalized (co)homology of symmetric quandles over homogeneous Beck modules

TL;DR

This work develops a Beck-module, category-theoretic framework for symmetric racks and quandles, introducing symmetric quandle modules and proving they classify Beck modules in the symmetric setting. It constructs a generalized (co)homology theory with homogeneous coefficient objects, showing it both recovers Kamada–Oshiro’s symmetric quandle (co)homology and extends to symmetric racks, with abelian extensions classified by and . A key achievement is the equivalence between module categories and abelian group objects in the corresponding slice categories, enabling a semi-direct product construction and a robust extension theory. The paper also connects the generalized cohomology to the cohomology of associated groups by establishing for trivial coefficients, highlighting deep links to group cohomology. Overall, the results provide a modular, coefficient-rich approach with potential to yield stronger invariants in low-dimensional topology and related areas.

Abstract

A quandle equipped with a good involution is referred to as symmetric. It is known that the cohomology of symmetric quandles gives rise to strong cocycle invariants for classical and surface links, even when they are not necessarily oriented. In this paper, we introduce the category of symmetric quandle modules and prove that these modules completely determine the Beck modules in the category of symmetric quandles. Consequently, this establishes suitable coefficient objects for constructing appropriate (co)homology theories. We develop an extension theory of modules over symmetric quandles and propose a generalized (co)homology theory for symmetric quandles with coefficients in a homogeneous Beck module, which also recovers the symmetric quandle (co)homology developed by Kamada and Oshiro [Trans. Amer. Math. Soc. (2010)]. Our constructions also apply to symmetric racks. We conclude by establishing an explicit isomorphism between the second cohomology of a symmetric quandle and the first cohomology of its associated group.
Paper Structure (10 sections, 15 theorems, 105 equations)

This paper contains 10 sections, 15 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symmetric racks and symmetric quandles
  4. Trunks
  5. Modules over symmetric racks and symmetric quandles
  6. Category of modules over symmetric racks and symmetric quandles
  7. Abelian extensions of symmetric racks and symmetric quandles
  8. Generalized (co)homology for symmetric racks and symmetric quandles
  9. Relation between cohomology of symmetric racks and cohomology of associated groups
  10. Declaration

Key Result

Proposition 4.1

Let $(X,\rho_X)$ be a symmetric rack and $\mathscr{F}=(A,\phi,\psi,\eta)$ an $(X,\rho_X)$-module. Then the semi-direct product $({\mathscr{F} \rtimes X},\rho_{\mathscr{F} \rtimes X})$ is a symmetric rack. Furthermore, if $(X,\rho_X)$ is a symmetric quandle, then so is $({\mathscr{F} \rtimes X},\rho_

Theorems & Definitions (47)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 3.1
  • Definition 3.2
  • Example 3.3
  • Definition 3.4
  • Remark 3.5
  • ...and 37 more