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On the generalized brace product: relation to $H$-splitting of loop space fibrations \& the $J$-homomorphism

Somnath Basu, Aritra Bhowmick, Sandip Samanta

TL;DR

This work axiomatizes and analyzes the generalized James brace product for fibrations with a homotopy section, proving that its vanishing is exactly the obstruction to an $H$-split loop-space fibration $\Omega E \simeq \Omega B \times \Omega F$. It shows that, rationally, the generalized brace product collapses to the classical James brace product, enabling explicit calculations and rational decomposability results for many fibrations, including sphere bundles over spheres. A space-level map $J_s$ is constructed to realize the generalized brace product, and the authors connect the brace product to a generalized $J$-homomorphism, deriving relations that control the total space type and its rational homotopy type. The paper provides practical criteria and examples (notably free loop-space fibrations and sphere bundles) for when fibrations split, both integrally and rationally, and develops localization tools to study these obstructions in simpler (rational) settings. Collectively, the results yield a cohesive framework for understanding when fibrations admit $H$-space splittings, how the generalized brace product governs this, and how these phenomena interact with localization and generalized homotopy invariants like the $J$-homomorphism.

Abstract

Given a fibration $F \hookrightarrow E \rightarrow B$ with a homotopy section $s: B \rightarrow E$, James introduced a binary product $\left\{, \right\}_s: π_i B \times π_j F \rightarrow π_{i+j-1} F$, called the brace product, which was later generalized by Yoon. We show that the vanishing of this generalized brace product is the precise obstruction to the $H$-splitting of the loop space fibration, i.e., $ΩE \simeq ΩB \times ΩF$ as $H$-spaces. Using rational homotopy theory, we show that for rational spaces, the vanishing of the generalized brace product coincides with the vanishing of the classical James brace product, enabling us to perform the relevant computations. In addition, the notion of $J$-homomorphism is generalized and connected to the generalized brace product. Among the applications, we characterize the homotopy types of certain fibrations, including sphere bundles over spheres.

On the generalized brace product: relation to $H$-splitting of loop space fibrations \& the $J$-homomorphism

TL;DR

This work axiomatizes and analyzes the generalized James brace product for fibrations with a homotopy section, proving that its vanishing is exactly the obstruction to an -split loop-space fibration . It shows that, rationally, the generalized brace product collapses to the classical James brace product, enabling explicit calculations and rational decomposability results for many fibrations, including sphere bundles over spheres. A space-level map is constructed to realize the generalized brace product, and the authors connect the brace product to a generalized -homomorphism, deriving relations that control the total space type and its rational homotopy type. The paper provides practical criteria and examples (notably free loop-space fibrations and sphere bundles) for when fibrations split, both integrally and rationally, and develops localization tools to study these obstructions in simpler (rational) settings. Collectively, the results yield a cohesive framework for understanding when fibrations admit -space splittings, how the generalized brace product governs this, and how these phenomena interact with localization and generalized homotopy invariants like the -homomorphism.

Abstract

Given a fibration with a homotopy section , James introduced a binary product , called the brace product, which was later generalized by Yoon. We show that the vanishing of this generalized brace product is the precise obstruction to the -splitting of the loop space fibration, i.e., as -spaces. Using rational homotopy theory, we show that for rational spaces, the vanishing of the generalized brace product coincides with the vanishing of the classical James brace product, enabling us to perform the relevant computations. In addition, the notion of -homomorphism is generalized and connected to the generalized brace product. Among the applications, we characterize the homotopy types of certain fibrations, including sphere bundles over spheres.
Paper Structure (16 sections, 40 theorems, 127 equations)

This paper contains 16 sections, 40 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Generalized Brace Product
  4. Independence of base point and homotopy of sections
  5. Brace product of pullback and product fibrations
  6. Brace product in free loop space fibration
  7. Vanishing of the Brace Product
  8. H-splitting of the loop space of the total space
  9. The Samelson Product on the Loop Space of the Total Space
  10. Decomposability of a fibration over a suspension
  11. Fibrations with vanishing brace product
  12. Localization of Spaces and Brace Product
  13. Brace Product and the J-homomorphism
  14. Generalized J-Homomorphism
  15. Generalized Brace Product for a Fibration over a Suspension
  16. ...and 1 more sections

Key Result

Theorem A

Suppose $F\overset{i}{\hookrightarrow} E\overset{p}{\to} B$ is a fibration with a homotopy section $s:B\to E$. Then the James brace product $\{,\}_s$ vanishes for the given fibration if and only if the natural map $(\varphi_{s})_*:=(\Omega s\cdot\Omega i)_*:\pi_*(\Omega B\times\Omega F)\to \pi_*(\Om

Theorems & Definitions (98)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem F
  • Definition 2.1: Whitehead product
  • Definition 2.2: Samelson product
  • Definition 2.3: Brace product
  • Proposition 2.4
  • ...and 88 more