(see also Chern-Weil theory, parameterized homotopy theory)
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
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see also algebraic topology
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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.
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
over $X$ which is fiberwise a weak homotopy equivalence.
One consider the abelian group
to be the Grothendieck group of stable fiberwise equivalence classes of spherical fibrations, under fiberwise smash product.
(…)
The Adams conjecture (a theorem) characterizes certain spherical fibrations in the image of the J-homomorphism as trivial.
The long exact sequence in cohomology induced by a spherical fibration is called a Gysin sequence.
See Sullivan model of a spherical fibration.
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)
Discussion in rational homotopy theory (for more see at Sullivan model of a spherical fibration):
Yves Félix, Steve Halperin and J.C. Thomas, p. 202 of Rational Homotopy Theory, Graduate Texts in Mathematics, 205, Springer-Verlag, 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)
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