nLab
spherical fibration

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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

Stable Homotopy theory

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.

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.

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.