# nLab spherical fibration

### Context

#### Bundles

bundles

fiber bundles in physics

## Constructions

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

## Idea

A spherical fibration is a fiber bundle of spheres of some dimension (a sphere fiber bundle). Typically this is considered in homotopy theory where one considers fibrations whose fibers have the homotopy type of spheres; and this in turn is often considered in stable homotopy theory after stabilization (hence up to tensoring with trivial spherical fibrations) which makes spherical fibrations models for (∞,1)-module bundles for the sphere spectrum regarded as an E-∞ ring.

Every real vector bundle becomes a spherical fibration in the sense of homotopy theory upon removing its zero section and this construction induces a map from vector bundles and in fact from topological K-theory to spherical fibrations, called the J-homomorphism.

This is closely related to the Thom space/Thom spectrum construction for vector bundles.

## Definition

### In components

For $X$ (the homotopy type of) a topological space, a spherical fibration over it is a fibration $E \to X$ such that each fiber has the homotopy type of a sphere.

Given two spherical fibrations $E_1, E_2 \to X$, there is their fiberwise smash product $E_1 \wedge_X E_2 \to X$.

For $n \in \mathbb{N}$, write $\epsilon^n \colon X \times S^n \to X$ for the trivial sphere bundle of fiber dimension $n$. Two spherical fibrations $E_1, E_2 \to X$ are stably fiberwise equivalent if there exists $n_1, n_2 \in \mathbb{N}$ such that there is a map

$E_1 \wedge_X \epsilon^{n_1} \longrightarrow E_2 \wedge_X \epsilon^{n_2}$

over $X$ which is fiberwise a weak homotopy equivalence.

One consider the abelian group

$Sph(X) \in Ab$

to be the Grothendieck group of stable fiberwise equivalence classes of spherical fibrations, under fiberwise smash product.

(…)

## Properties

The Adams conjecture (a theorem) characterizes certain spherical fibrations in the image of the J-homomorphism as trivial.

### Gysin sequence

The long exact sequence in cohomology induced by a spherical fibration is called a Gysin sequence.

## References

An original reference is

Reviews include

• Howard Marcum, Duane Randall, The homotopy Thom class of a spherical fibration, Proceedings of the AMS, volume 80, number 2 (pdf)

• Per Holm, Jon Reed, section 7 of Structure theory of manifolds, Seminar notes 1971pdf

• Oliver Straser, Nena Röttgens, Spivak normal fibrations (pdf)

• S. Husseini, Spherical fibrations (pdf)

Last revised on October 23, 2017 at 04:59:12. See the history of this page for a list of all contributions to it.