On the KOℤ/2-Euler class, I

1991 ◽  
Vol 117 (1-2) ◽  
pp. 115-137 ◽  
Author(s):  
M. C. Crabb
Keyword(s):  
Vector Bundle ◽  
Euclidean Space ◽  
Cross Sections ◽  
Euler Class ◽  
Projective Spaces ◽  
Class I ◽  
Real Vector ◽  
Systematic Account ◽  
Linearly Independent

SynopsisThis paper presents a systematic account of ℤ/2-equivariant KO-theoretic methods in the study of r-fields (that is, r linearly independent cross-sections, r ≧ 1) on a real vector bundle. Applications of the theory to the problem of immersing complex and quaternionic projective spaces in Euclidean space are given in [13].

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].

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.

Euler classes of vector bundles over iterated suspensions of real projective spaces

2018 ◽  
Vol 68 (3) ◽  
pp. 677-684
Author(s):  
Aniruddha C. Naolekar ◽  
Ajay Singh Thakur
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Euler Class ◽  
Projective Spaces

Abstract We show that when k ≠ 2, 4, 8 the Euler class of any vector bundle over Σkℝℙm is zero if the rank of the bundle is not m + k, provided that m ≠ 3 when k = 6. If k = 2, 4, 8 we show that the Euler class of any vector bundle over Σkℝℙm is zero whenever the rank of the bundle is not kr + k, provided that m ≠ 6, 7 when k = 2, where r is the largest integer such that kr ≤ m.

Euler classes of vector bundles over manifolds

2021 ◽  
Vol 71 (1) ◽  
pp. 199-210
Author(s):  
Aniruddha C. Naolekar
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Euler Class ◽  
Homology Sphere ◽  
Rational Homology ◽  
Rational Homology Sphere ◽  
Orientable Manifolds

Abstract Let 𝓔 k denote the set of diffeomorphism classes of closed connected smooth k-manifolds X with the property that for any oriented vector bundle α over X, the Euler class e(α) = 0. We show that if X ∈ 𝓔2n+1 is orientable, then X is a rational homology sphere and π 1(X) is perfect. We also show that 𝓔8 = ∅ and derive additional cohomlogical restrictions on orientable manifolds in 𝓔 k .

WEIGHTED GRIDS IN COMPLEX JORDAN* TRIPLES

2010 ◽  
Vol 03 (01) ◽  
pp. 155-184
Author(s):  
L. L. STACHÓ
Keyword(s):  
Independent Sets ◽  
Vector Spaces ◽  
Complete List ◽  
Real Vector ◽  
Linearly Independent

Weighted grids are linearly independent sets {gw : w ∈ W} of signed tripotents in Jordan* triples indexed by figures W in real vector spaces such that {gugvgw} ∈ ℂgu-v+w (= 0 if u - v + w ∉ W). They arise naturally as systems of weight vectors of certain abelian families of Jordan* derivations. Based on Neher's grid theory, a classification of association free non-nil weighted grids is given. As a first step beyond the setting of classical grids, the complete list of complex weighted grids of pairwise associated signed tripotents indexed by ℤ2 is established.

scholarly journals An elementary proof of Minkowski's second inequality

1969 ◽  
Vol 10 (1-2) ◽  
pp. 177-181 ◽  
Author(s):  
I. Danicic
Keyword(s):  
Euclidean Space ◽  
Elementary Proof ◽  
Content Volume ◽  
Lattice Points ◽  
The Body ◽  
Dimensional Euclidean Space ◽  
Open Convex ◽  
Successive Minima ◽  
Linearly Independent ◽  
Independent Lattice

Let K be an open convex domain in n-dimensional Euclidean space, symmetric about the origin O, and of finite Jordan content (volume) V. With K are associated n positive constants λ1, λ2,…,λn, the ‘successive minima of K’ and n linearly independent lattice points (points with integer coordinates) P1, P2, …, Pn (not necessarily unique) such that all lattice points in the body λ,K are linearly dependent on P1, P2, …, Pj-1. The points P1,…, Pj lie in λK provided that λ > λj. For j = 1 this means that λ1K contains no lattice point other than the origin. Obviously

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.

scholarly journals Torus-equivariant vector bundles on projective spaces

1988 ◽  
Vol 111 ◽  
pp. 25-40 ◽  
Author(s):  
Tamafumi Kaneyama
Keyword(s):  
Vector Bundle ◽  
Toric Variety ◽  
Vector Bundles ◽  
Rational Point ◽  
Projective Spaces ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Algebraic Torus

For a free Z-module N of rank n, let T = TN be an n-dimensional algebraic torus over an algebraically closed field k defined by N. Let X = TN emb (Δ) be a smooth complete toric variety defined by a fan Δ (cf. [6]). Then T acts algebraically on X. A vector bundle E on X is said to be an equivariant vector bundle, if there exists an isomorphism ft: t*E → E for each k-rational point t in T, where t: X → X is the action of t. Equivariant vector bundles have T-linearizations (see Definition 1.2 and [2], [4]), hence we consider T-linearized vector bundles.

scholarly journals Chern Classes of Projective Modules

1963 ◽  
Vol 23 ◽  
pp. 121-152 ◽  
Author(s):  
Hideki Ozeki
Keyword(s):  
Vector Bundle ◽  
Cross Sections ◽  
Chern Class ◽  
Vector Bundles ◽  
Stein Manifold ◽  
Projective Modules ◽  
Singular Variety ◽  
Differentiable Functions ◽  
Holomorphic Vector Bundles ◽  
Differentiable Vector

In topology, one can define in several ways the Chern class of a vector bundle over a certain topological space (Chern [2], Hirzebruch [7], Milnor [9], Steenrod [15]). In algebraic geometry, Grothendieck has defined the Chern class of a vector bundle over a non-singular variety. Furthermore, in the case of differentiable vector bundles, one knows that the set of differentiable cross-sections to a bundle forms a finitely generated projective module over the ring of differentiable functions on the base manifold. This gives a one to one correspondence between the set of vector bundles and the set of f.g.-projective modules (Milnor [10]). Applying Grauert’s theorems (Grauert [5]), one can prove that the same statement holds for holomorphic vector bundles over a Stein manifold.

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