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Secondary cohomology operations and the loop space cohomology

Samson Saneblidze

TL;DR

The paper introduces secondary cohomology operations on $H^*(X; \mathbb{Z}_p)$ and shows how they form a Hopf algebra that controls loop space cohomology via $H^*(\Omega X; \mathbb{Z}_p)$. A mod $p$ filtered Hirsch model built from a minimal Hirsch resolution $RH$ and a perturbation $h$ yields a dg Hopf algebra model, whose bar-construction recovers the loop space cohomology and its coproduct. The secondary operations $\psi_{r,n}$ and the generators $\omega_{r,n}(x)$ are defined through the filtered Hirsch framework, enabling explicit power relations and kernel descriptions tied to Bockstein data. The authors apply the theory to compute the mod $p$ loop cohomology of the exceptional group $F_4$ at $p=2$ and $p=3$, illustrating a practical, constructive method for determining loop space cohomology and its Hopf-algebra structure in challenging cases.

Abstract

Motivated by the loop space cohomology we construct the secondary operations on the cohomology $H^*(X; \mathbb{Z}_p)$ to be a Hopf algebra for a simply connected space $X.$ The loop space cohomology ring $H^*(ΩX; \mathbb{Z}_p)$ is calculated in terms of generators and relations. This answers to A. Borel's decomposition of a Hopf algebra into a tensor product of the monogenic ones in which the heights of generators are determined by means of the action of the primary and secondary cohomology operations on $H^*(X;\mathbb{Z}_p).$ An application for calculating of the loop space cohomology of the exceptional group $F_4$ is given.

Secondary cohomology operations and the loop space cohomology

TL;DR

The paper introduces secondary cohomology operations on and shows how they form a Hopf algebra that controls loop space cohomology via . A mod filtered Hirsch model built from a minimal Hirsch resolution and a perturbation yields a dg Hopf algebra model, whose bar-construction recovers the loop space cohomology and its coproduct. The secondary operations and the generators are defined through the filtered Hirsch framework, enabling explicit power relations and kernel descriptions tied to Bockstein data. The authors apply the theory to compute the mod loop cohomology of the exceptional group at and , illustrating a practical, constructive method for determining loop space cohomology and its Hopf-algebra structure in challenging cases.

Abstract

Motivated by the loop space cohomology we construct the secondary operations on the cohomology to be a Hopf algebra for a simply connected space The loop space cohomology ring is calculated in terms of generators and relations. This answers to A. Borel's decomposition of a Hopf algebra into a tensor product of the monogenic ones in which the heights of generators are determined by means of the action of the primary and secondary cohomology operations on An application for calculating of the loop space cohomology of the exceptional group is given.
Paper Structure (13 sections, 6 theorems, 173 equations)

This paper contains 13 sections, 6 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. The Hirsch filtered models of a space
  3. A minimal complex to calculate $H^*(\Omega X;\mathbb{Z}_p)$
  4. The $(g,f)$-coderivation chain homotopy $\mathbf{b}$
  5. A minimal Hirsch resolution $RH$
  6. The Hirsch algebra structure on $RH$
  7. The description of multiplicative generators of $RH$
  8. The perturbation $h:RH\rightarrow RH$
  9. The perturbation $h$ on $\mathcal{V}_p$
  10. The Secondary cohomology operations $\psi_{r,n}$
  11. The generators $\omega_{r,n}(x)$
  12. The proofs of Theorems \ref{['premain']} and \ref{['sigmazero']}
  13. The mod $p$ loop cohomology ring of the exceptional group $F_4$

Key Result

Theorem 1

Let $X$ be a simply connected topological space such that the cohomology ring $H^*(X; \mathbb{Z}_p)$ is a Hopf algebra. Then $\Phi_0(X)\cup \Phi_n(X)$ is the set of multiplicative generators of $H^*(\Omega X;\mathbb{Z}_p),$$p\geq 2,$ and (i) For $k\geq 1$ and $\sigma x\in \Phi_0(X)$ unless $x\in \ma (ii) For $r,n,k\geq1$ and $\omega_{r,n}(x)\in \Phi_n(X):$

Theorems & Definitions (13)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Remark 2
  • Remark 3
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • ...and 3 more