The Homotopy Thom Class of a Spherical Fibration
نویسندگان
چکیده
We investigate the following problems. Given a spherical fibration, does the Whitehead square of its homotopy Thom class vanish? If so, is the homotopy Thorn class a cyclic homotopy class?
منابع مشابه
Ring Spectra Which Are Thom Complexes
commutes up to homotopy where T is the map that exchanges factors. Let L be a space and let be a fibration over L classified by a mapf L BF (the classifying space of stable spherical fibrations). We can form the Thom spectrum T(f) of f as a suspension spectrum by letting (T(f)) be the Thorn complex of L n--YBFn where L is the n-skeleton of L. This makes T(f)-{(T(f))n) into a suspension spectrum...
متن کاملThe Adams Conjecture, after Edgar Brown
Let X be a finite CW-complex. Denote by Sph(X) the abelian group of stable fibre-homotopy classes of spherical fibrations on X. Let F (n) be the monoid of self-homotopy-equivalences of S and let F := colimn→∞ F (n). The classifying space BF represents Sph(X), i.e. Sph(X) ∼= [X;BF ]. For any vector bundle ξ on X, let J(ξ) ∈ Sph(X) be the stable fibre homotopy class of the spherical fibration ξ \...
متن کاملOn the cohomology of loop spaces for some Thom spaces
In this paper we identify conditions under which the cohomology H∗(ΩMξ; k) for the loop space ΩMξ of the Thom space Mξ of a spherical fibration ξ ↓ B can be a polynomial ring. We use the Eilenberg–Moore spectral sequence which has a particularly simple form when the Euler class e(ξ) ∈ H(B; k) vanishes, or equivalently when an orientation class for the Thom space has trivial square. As a consequ...
متن کاملThe Impact of Thom’s Cobordism Theory
At the 1958 International Congress of Mathematicians in Edinburgh, René Thom received one of two Field Medals for his development of cobordism. In his citation [11] Heinz Hopf described the definition of cobordism as one of those elementary and apparently trivial constructions which can hardly be expected to yield significant results. He compares this with Hurewicz’s definition of homotopy grou...
متن کاملThe spectra ko and ku are not Thom spectra: an approach using THH
We apply an announced result of Blumberg-Cohen-Schlichtkrull to reprove (under restricted hypotheses) a theorem of Mahowald: the connective real and complex K-theory spectra are not Thom spectra. The construction of various bordism theories as Thom spectra served as a motivating example for the development of highly structured ring spectra. Various other examples of Thom spectra followed; for i...
متن کامل