Analytic Evaluation of Certain Characteristic Classes(1)

1973 ◽  
Vol 16 (2) ◽  
pp. 219-223
Author(s):  
Clark D. Jeffries
Keyword(s):  
Vector Bundle ◽  
Orthonormal Frame ◽  
Alternative Proof ◽  
Characteristic Classes ◽  
Real Vector ◽  
Analytic Evaluation ◽  
Riemann Structure

We have an alternative proof of the following result of Kervaire [2]:Let V→M be a real vector bundle with fibre dimension n≥4k+l over a compact 4k-manifold. Suppose V restricted to M — {x} is trivial. Choose a Riemann structure for V and an orthonormal frame for V restricted to M —{x}. Thus the obstruction to extending the frame smoothly over M is an element λ in π4k+1SO(n))≅Z. Then up to sign the evaluation of the kth Pontrayagin class Pk on M is ak(2k— 1)!. λ, where ak is 1 or 2 depending upon whether k is even or odd.

On the Stable Homotopy Type of Thom Complexes

1973 ◽  
Vol 25 (6) ◽  
pp. 1285-1294 ◽  
Author(s):  
R. P. Held ◽  
D. Sjerve
Keyword(s):  
Vector Bundle ◽  
Homotopy Type ◽  
Stable Homotopy ◽  
Homotopy Equivalence ◽  
Real Vector ◽  
Fiber Homotopy

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.

A HOPF INDEX THEOREM FOR A REAL VECTOR BUNDLE

2002 ◽  
Vol 23 (04) ◽  
pp. 507-518 ◽  
Author(s):  
HUITAO FENG ◽  
ENLI GUO
Keyword(s):  
Vector Bundle ◽  
Index Theorem ◽  
Real Vector

A GENERALIZATION OF THE ATIYAH–DUPONT VECTOR FIELDS THEORY

2002 ◽  
Vol 04 (04) ◽  
pp. 777-796 ◽  
Author(s):  
ZIZHOU TANG ◽  
WEIPING ZHANG
Keyword(s):  
Vector Bundle ◽  
Cross Sections ◽  
Vector Fields ◽  
Complex Structure ◽  
Index Theory ◽  
Index Theorem ◽  
Obstruction Theory ◽  
Real Vector ◽  
Complete Answer ◽  
Linearly Independent

To generalize the Hopf index theorem and the Atiyah–Dupont vector fields theory, one is interested in the following problem: for a real vector bundle E over a closed manifold M with rank E = dim M, whether there exist two linearly independent cross sections of E? We provide, among others, a complete answer to this problem when both E and M are orientable. It extends the corresponding results for E = TM of Thomas, Atiyah, and Atiyah–Dupont. Moreover we prove a vanishing result of a certain mod 2 index when the bundle E admits a complex structure. This vanishing result implies many known famous results as consequences. Ideas and methods from obstruction theory, K-theory and index theory are used in getting our results.

On the KOℤ/2-Euler class, II

1991 ◽  
Vol 117 (1-2) ◽  
pp. 139-154 ◽  
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].

Sums of stably trivial vector bundles

1980 ◽  
Vol 87 (1) ◽  
pp. 97-107 ◽  
Author(s):  
Jacques Allard
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Vector Bundles ◽  
Real Vector

We say that a real vector bundle ξ over a finite C.W. complex X is stably trivial of type (n, k) or, simply, of type (n, k) if ξ ⊕ kε ≅ nε, where ε denotes a trivial line bundle. The following theorem is an immediate corollary (see (12)) of a theorem of T. Y. Lam ((9), theorem 2).

A REMARK ON THE STABLE REAL FORMS OF COMPLEX VECTOR BUNDLES OVER MANIFOLDS

2017 ◽  
Vol 96 (1) ◽  
pp. 69-76
Author(s):  
HUIJUN YANG
Keyword(s):  
Vector Bundle ◽  
Smooth Manifold ◽  
Vector Bundles ◽  
Characteristic Classes ◽  
Dimensional Manifold ◽  
Real Form ◽  
Complex Vector Bundle ◽  
Complex Vector ◽  
Simply Connected ◽  
Real Forms

Let$M$be an$n$-dimensional closed oriented smooth manifold with$n\equiv 4\;\text{mod}\;8$, and$\unicode[STIX]{x1D702}$be a complex vector bundle over$M$. We determine the final obstruction for$\unicode[STIX]{x1D702}$to admit a stable real form in terms of the characteristic classes of$M$and$\unicode[STIX]{x1D702}$. As an application, we obtain the criteria to determine which complex vector bundles over a simply connected four-dimensional manifold admit a stable real form.

scholarly journals Note on the characteristic rank of vector bundles

2014 ◽  
Vol 64 (6) ◽  
Author(s):  
Aniruddha Naolekar ◽  
Ajay Thakur
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Projective Spaces ◽  
Real Vector ◽  
Cw Complex ◽  
Moore Spaces

AbstractWe define the notion of characteristic rank, charrankX(ξ), of a real vector bundle ξ over a connected finite CW-complex X. This is a bundle-dependent version of the notion of characteristic rank introduced by Július Korbaš in 2010. We obtain bounds for the cup length of manifolds in terms of the characteristic rank of vector bundles generalizing a theorem of Korbaš and compute the characteristic rank of vector bundles over the Dold manifolds, the Moore spaces and the stunted projective spaces amongst others.

scholarly journals Thom isomorphism and push-forward map in twisted K-theory

2007 ◽  
Vol 1 (2) ◽  
pp. 357-393 ◽  
Author(s):  
Alan L. Carey ◽  
Bai-Ling Wang
Keyword(s):  
Vector Bundle ◽  
Real Vector ◽  
Anomaly Cancellation ◽  
Cancellation Condition ◽  
Canonical Element ◽  
Push Forward ◽  
Thom Isomorphism ◽  
K Theory

AbstractWe establish the Thom isomorphism in twisted K-theory for any real vector bundle and develop the push-forward map in twisted K-theory for any differentiable map f : X → Y (not necessarily K-oriented). We also obtain the wrong way functoriality property for the push-forward map in twisted K-theory. For D-branes satisfying Freed-Witten's anomaly cancellation condition in a manifold with a non-trivial B-field, we associate a canonical element in the twisted K-group to get the so-called D-brane charges.

scholarly journals An Integral Formula for the Chern from of a Hermitian Bundle

1971 ◽  
Vol 42 ◽  
pp. 135-172 ◽  
Author(s):  
Hideo Omoto
Keyword(s):  
Vector Bundle ◽  
Chern Class ◽  
Basic Characteristic ◽  
Grassmann Manifold ◽  
Integral Formula ◽  
Characteristic Classes ◽  
Characteristic Class ◽  
Chern Classes ◽  
Simplicial Decomposition ◽  
Hermitian Bundle

We shall consider a Hermitian n-vector bundle E over a complex manifold X. When X is compact (without boundary), S.S. Chern defined in his paper [3] the Chern classes (the basic characteristic classes of E) Ĉi(E), i = 1, · · ·, n, in terms of the basic forms Φi on the Grassmann manifold H(n, N) and the classifying map f of X into H(n, N). Moreover he proved ([3], [4]) that if Ek denotes the k-general Stiefel bundle associated with E, the (n — k + 1)-th Chern class Ĉn-k+1(E) coincides with the characteristic class C(Ek) of Ek defined as follows: Let K be a simplicial decomposition of X and K2(n-k)+1 the 2(n — k) + 1 — shelton of K.

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