nLab
spherical fibration

Contents

Context

Spheres

Bundles

bundles

(see also Chern-Weil theory, parameterized homotopy theory)

fiber bundles in physics

Context

Classes of bundles

Universal bundles

Presentations

Examples

Constructions

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 E→XE \to X such that each fiber has the homotopy type of a sphere.

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

For nβˆˆβ„•n \in \mathbb{N}, write Ο΅ n:XΓ—S nβ†’X\epsilon^n \colon X \times S^n \to X for the trivial sphere bundle of fiber dimension nn. Two spherical fibrations E 1,E 2β†’XE_1, E_2 \to X are stably fiberwise equivalent if there exists n 1,n 2βˆˆβ„•n_1, n_2 \in \mathbb{N} such that there is a map

E 1∧ Xϡ n 1⟢E 2∧ Xϡ 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 (nβˆ’1)(n-1)-sphere with composition. Then BG nB G_n acts as the classifying space for spherical fibrations with spherical fibre S nβˆ’1S^{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 n→G 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 RΓΆttgens, 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.