nLab
spherical fibration

Contents

Context

Spheres

Bundles

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

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

Given two spherical fibrations E 1,E 2XE_1, E_2 \to X, there is their fiberwise smash product E 1 XE 2XE_1 \wedge_X E_2 \to X.

For nn \in \mathbb{N}, write ϵ n:X×S nX\epsilon^n \colon X \times S^n \to X for the trivial sphere bundle of fiber dimension nn. Two spherical fibrations E 1,E 2XE_1, E_2 \to X are stably fiberwise equivalent if there exists n 1,n 2n_1, n_2 \in \mathbb{N} such that there is a map

E 1 Xϵ n 1E 2 Xϵ n 2 E_1 \wedge_X \epsilon^{n_1} \longrightarrow E_2 \wedge_X \epsilon^{n_2}

over XX which is fiberwise a weak homotopy equivalence.

One consider the abelian group

Sph(X)Ab Sph(X) \in Ab

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

Classifying space

There is an associative H-space, G nG_n, of homotopy equivalences of the (n1)(n-1)-sphere with composition. Then BG nB G_n acts as the classifying space for spherical fibrations with spherical fibre S n1S^{n-1} (Stasheff 63).

There is an inclusion of the orthogonal group O(n)O(n) into G nG_n.

Suspension gives a map G nG n+1G_n \to G_{n+1} whose limit is denoted GG. Then BGB G classifies stable spherical fibrations.

As (,1)(\infty,1)-module bundles

(…)

Properties

Adams conjecture

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.

Rational homotopy type

See Sullivan model of a spherical fibration.

References

General

An original reference is

Treatment of the classifying space for spherical fibrations is in

  • James Stasheff, A classification theorem for fibre spaces, Topology Volume 2, Issue 3, October 1963, Pages 239-246.

Reviews include

  • Raoul Bott, Loring Tu, Chapter 11 of Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer 1982 (doi:10.1007/BFb0063500)

  • 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)

In rational homotopy theory

Discussion in rational homotopy theory (for more see at Sullivan model of a spherical fibration):

Last revised on December 1, 2019 at 14:11:45. See the history of this page for a list of all contributions to it.