nLab spherical fibration

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.

Classifying space

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

There is an inclusion of the orthogonal group $O(n)$ into $G_n$.

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

As $(\infty,1)$-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.

Related concepts

• sphere spectrum

• Thom spectrum

• twisted cohomotopy

References

General

An original reference is

• Albrecht Dold, Richard Lashof, Principal quasifibrations and fibre homotopy equivalence of bundles, 1958 (pdf)

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):

• Yves Félix, Steve Halperin, Jean-Claude Thomas, p. 202 of: Rational Homotopy Theory, Graduate Texts in Mathematics 205 Springer (2000)

• Jesper Møller, Martin Raussen, Rational Homotopy of Spaces of Maps Into Spheres and Complex Projective Spaces, Transactions of the American Mathematical Society Vol. 292, No. 2 (Dec., 1985), pp. 721-732 (jstor:2000242)

• Ralph Cohen, Alexander Voronov, Notes on string topology (arXiv:math/0503625)

• Yves Félix, John Oprea, Daniel Tanré, Prop. 2.3 in Lie-model for Thom spaces of tangent bundles, Proc. Amer. Math. Soc. 144 (2016), 1829-1840 (pdf, doi:10.1090/proc/12829)