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Automorphisms, cohomology and extensions of symmetric quandles

Biswadeep Karmakar, Deepanshi Saraf, Mahender Singh

TL;DR

The paper develops a generalized cohomology theory for symmetric racks and quandles with Beck-module coefficients and proves a Wells-type four-term exact sequence linking $Z^{1}_{SR}((X,\rho);A)$, automorphism groups of extensions, and the second symmetric rack cohomology $H^{2}_{SR}((X,\rho);A)$. It shows that the obstruction to lifting and extending automorphisms resides in $H^{2}_{SR}((X,\rho);A)$ and connects dynamical cocycles with ordinary 2-cocycles via a module structure $(A,\phi,\psi,\eta)$ together with a 2-cocycle $\sigma$. The work also relates group extensions to dynamical extensions, yielding decompositions like $SQ(E) \cong SQ(G) \times_{(\alpha,\beta)} SQ(A)$ and analogous core/conjugacy results. Overall, it provides a cohomological obstruction framework for automorphism lifting in symmetric rack/quandle theory and extends prior results on extensions and dynamical cocycles to a broader categorical setting.

Abstract

It is well-known that the cohomology of symmetric quandles generates robust cocycle invariants for unoriented classical and surface links. Expanding on the recently introduced module-theoretic generalized cohomology for symmetric quandles, we derive a four-term exact sequence that relates 1-cocycles, second cohomology, and a specific group of automorphisms associated with the extensions of symmetric quandles. This exact sequence shows that the obstruction to lifting and extending automorphisms is found in the second symmetric quandle cohomology. Additionally, some general aspects of dynamical cocycles and extensions are discussed.

Automorphisms, cohomology and extensions of symmetric quandles

TL;DR

The paper develops a generalized cohomology theory for symmetric racks and quandles with Beck-module coefficients and proves a Wells-type four-term exact sequence linking , automorphism groups of extensions, and the second symmetric rack cohomology . It shows that the obstruction to lifting and extending automorphisms resides in and connects dynamical cocycles with ordinary 2-cocycles via a module structure together with a 2-cocycle . The work also relates group extensions to dynamical extensions, yielding decompositions like and analogous core/conjugacy results. Overall, it provides a cohomological obstruction framework for automorphism lifting in symmetric rack/quandle theory and extends prior results on extensions and dynamical cocycles to a broader categorical setting.

Abstract

It is well-known that the cohomology of symmetric quandles generates robust cocycle invariants for unoriented classical and surface links. Expanding on the recently introduced module-theoretic generalized cohomology for symmetric quandles, we derive a four-term exact sequence that relates 1-cocycles, second cohomology, and a specific group of automorphisms associated with the extensions of symmetric quandles. This exact sequence shows that the obstruction to lifting and extending automorphisms is found in the second symmetric quandle cohomology. Additionally, some general aspects of dynamical cocycles and extensions are discussed.
Paper Structure (7 sections, 15 theorems, 61 equations)

This paper contains 7 sections, 15 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dynamical cocycles for symmetric quandles
  4. Dynamical cocycles and second cohomology
  5. From group extensions to dynamical extensions
  6. Wells type exact sequence for symmetric racks
  7. Declaration

Key Result

Proposition 3.1

Let $(X,\rho)$ be a symmetric rack and $S$ be a collection of sets $\{S_x \mid x \in X\}$. Let $\alpha:X \times X \rightarrow \cup_{x,y \in X}\mathrm{Map}(S_x \times S_y,S_{x*y})$ and $\beta:X \rightarrow \cup_{x \in X}\mathrm{Map} (S_x,S_{\rho(x)})$ be two maps such that $\alpha({x,y}) \in \mathrm{ and the good involution $\rho_{\alpha,\beta}:X \times S \rightarrow X \times S$ given by if and on

Theorems & Definitions (41)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • Proposition 3.1
  • proof
  • ...and 31 more