On the Stable Homotopy Type of Thom Complexes

Let α be a real vector bundle over a finite CW complex X and let T(α;X) be its associated Thorn complex. We propose to study the S-type (stable homotopy type) of Thorn complexes in the framework of the Atiyah-Adams J-Theory. Therefore we focus our attention on the group JR(X) which is defined to be the group of orthogonal sphere bundles over X modulo stable fiber homotopy equivalence.

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On the KOℤ/2-Euler class, II

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500027669 ◽

1991 ◽

Vol 117 (1-2) ◽

pp. 139-154 ◽

Cited By ~ 3

Author(s):

M. C. Crabb

Keyword(s):

Vector Bundle ◽

Finite Subset ◽

Homotopy Class ◽

Homotopy Group ◽

Euler Class ◽

Class Ii ◽

Stable Homotopy ◽

Stiefel Manifold ◽

Real Vector ◽

Characteristic Numbers

SynopsisLet ξ be an oriented n-dimensional real vector bundle over an oriented closed m-manifold X. An r-field on ξ defined outside a finite subset of X has an index in the homotopy group πm−l(Vn,r) of the Stiefel manifold of r-frames in ℝn. The principal theorems of this paper relate the d and e-invariants of an associated ℝ/2-equivariant stable homotopy class, in certain cases, to computable cohomology characteristic numbers. Results of this type were first obtained by Atiyah and Dupont [5].

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Stable summands of U(n)

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500026834 ◽

1997 ◽

Vol 127 (5) ◽

pp. 963-973 ◽

Cited By ~ 2

Author(s):

M. C. Crabb ◽

J. R. Hubbuck ◽

J. A. W. McCall

Keyword(s):

Homotopy Type ◽

Unitary Group ◽

Stable Homotopy ◽

Finite Complex ◽

Special Unitary Group ◽

Prime 2

SynopsisThe special unitary group SU(n) has the stable homotopy type of a wedge of n − 1 finite complexes. The ‘first’ of these complexes is ΣℂPn–1, which is well known to be indecomposable at the prime 2 whether n is finite or infinite. We show that the ‘second’ finite complex is again indecomposable at the prime 2 when n is finite, but splits into a wedge of two pieces when n is infinite.

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Rational Equivalence of Fibrations with Fibre G/K

Canadian Journal of Mathematics ◽

10.4153/cjm-1982-005-7 ◽

1982 ◽

Vol 34 (1) ◽

pp. 31-43 ◽

Cited By ~ 4

Author(s):

Stephen Halperin ◽

Jean Claude Thomas

Keyword(s):

Finite Type ◽

Homotopy Type ◽

Lie Group ◽

The Other ◽

Homotopy Groups ◽

Homotopy Equivalence ◽

Rational Homotopy ◽

Algebraic Invariants ◽

Associated Bundles ◽

Connected Lie Group

Let be two Serre fibrations with same base and fibre in which all the spaces have the homotopy type of simple CW complexes of finite type. We say they are rationally homotopically equivalent if there is a homotopy equivalence between the localizations at Q which covers the identity map of BQ.Such an equivalence implies, of course, an isomorphism of cohomology algebras (over Q) and of rational homotopy groups; on the other hand isomorphisms of these classical algebraic invariants are usually (by far) insufficient to establish the existence of a rational homotopy equivalence.Nonetheless, as we shall show in this note, for certain fibrations rational homotopy equivalence is in fact implied by the existence of an isomorphism of cohomology algebras. While these fibrations are rare inside the class of all fibrations, they do include principal bundles with structure groups a connected Lie group G as well as many associated bundles with fibre G/K.

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Stable homotopy type of the classifying space of virtually torsion-free groups

Topology ◽

10.1016/0040-9383(94)90005-1 ◽

1994 ◽

Vol 33 (4) ◽

pp. 721-728 ◽

Cited By ~ 3

Author(s):

Chun-Nip Lee

Keyword(s):

Homotopy Type ◽

Free Groups ◽

Stable Homotopy ◽

Classifying Space ◽

Torsion Free

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On the stable homotopy type of the classifying space of a compact lie group

Topology ◽

10.1016/0040-9383(93)90037-v ◽

1993 ◽

Vol 32 (1) ◽

pp. 55-59 ◽

Cited By ~ 3

Author(s):

Chun-nip Lee

Keyword(s):

Homotopy Type ◽

Lie Group ◽

Stable Homotopy ◽

Compact Lie Group ◽

Classifying Space

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The stable homotopy type of rank two 𝑝-groups

Contemporary Mathematics - Homotopy Theory and Its Applications ◽

10.1090/conm/188/02236 ◽

1995 ◽

pp. 93-103 ◽

Cited By ~ 1

Author(s):

Jill Dietz ◽

Stewart Priddy

Keyword(s):

Homotopy Type ◽

Stable Homotopy

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An n-dimensional Klein bottle

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2018.73 ◽

2019 ◽

Vol 149 (5) ◽

pp. 1207-1221

Author(s):

Donald M. Davis

Keyword(s):

Euclidean Space ◽

Homotopy Type ◽

Klein Bottle ◽

Stable Homotopy ◽

Cohomology Algebra ◽

Topological Complexity ◽

Integral Cohomology ◽

Planar Polygon

AbstractAn n-dimensional analogue of the Klein bottle arose in our study of topological complexity of planar polygon spaces. We determine its integral cohomology algebra and stable homotopy type, and give an explicit immersion and embedding in Euclidean space.

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