# On The Periodicity Theorem For Complex Vector Bundles

on the periodicity theorem for complex vector bundles

The periodicity theorem for the infinite unitary group can be interpreted as a state- ment about complex vector bundles. As such it describes the relation between vector bundles over X and X “ S 2, where X is a compact (1) space and S 2 is the 2-sphere.

On the periodicity theorem for complex vector bundles

Vector Bundle Complex Vector Complex Vector Bundle Periodicity Theorem These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Atiyah , Bott : On the periodicity theorem for complex ...

In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by Raoul Bott (1957, 1959), which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres.

Bott periodicity theorem - Encyclopedia of Mathematics

cap product de ned above) the periodicity theorem reduces to a calculation of an index of a speci c di erential operator. In the case of a complex vector bundle, this family is a family of Dolbeault operators while for real vector bundles it is a family of Dirac operators.

Vector Bundles & K-Theory Book - Cornell University

If E is a vector bundle aver X E m (the dimension cf the fibers may vary, on the components of X) we write IC(E) for the unit disc bundle of E (relative to sorne Riemann structure) and denote its boundary by lO(E). The pair (lD(E), lO(E)) as well as the quotient space lD(E)/13(E) will be denoted by XE.

algebraic topology - Is Atiyah's periodicity Theorem ...

Atiyah, M.F., Bott, R.: On the periodicity theorem for complex vector bundles. Acta Math. 112, 229–247 (1964) MathSciNet CrossRef zbMATH Google Scholar

Complex vector bundle - Wikipedia

The complex version ofKO(X)g, calledK(X)e, is constructed in the same way asKO(X)g but using vector bundles whose ﬁbers are vector spaces over Crather than R. The complex form of Bott Periodicity asserts simply thatK(Sen)is Zforneven and 0 fornodd, so the period is two rather than eight.

Bott Periodicity, Submanifolds, and Vector Bundles ...

Recall that given a complex vector bundle E of rank r on a smooth manifold M, and chosen a point x 0 ∈ M, if ∇, if ∇ is a flat connection on E, i.e, ∇ 2 = 0 (or, locally, dω + ω 2 = 0, with ω ∈ End E ⊗ Λ 2(M) − Λ 2(M) denoting the space of 2-forms over M − and ∇ = d + ω), then the Levi-Civita parallel transport T defined by ∇ gives rise to a representation T: π 1(M, x 0) → GL(r, ¢) through the expression

Chern class - Wikipedia

example, extrinsic vector bundles are used to model subatomic particles. Sections of vector bundles are generalized vector-valued functions. For example, sections of the tangent bundle TM Ñ M are vector ﬁelds on the manifold M. The set VectpXq of isomorphism classes of complex vector bundles on a topological space X is a homotopy invariant of X.

ALGEBRAIC TOPOLOGY: MATH 231BR NOTES

To check that the Chern character of any complex vector bundle on is integral, one thus reduces to analyzing the image of as above. This is convenient because the group is very simple. By the Bott periodicity theorem, it is generated by the th power of where is the Hopf bundle over , so that.

KR-theory - Wikipedia

theorem (1.4.2) and the periodicity theorem (1.5.4). The latter implies the existence of a global structure in the homotopy groups of many spaces called the chromatic ltration. This is the subject of Chapter 2, which begins with a review of some classical results about homotopy groups. The nilpotence theorem says that the complex bordism functor

Atiyah–Bott fixed-point theorem - Wikipedia

In these lectures I hope to trace the development of a subject which has grown up during the past ten years or so and which is now generally known as K-theory. As my starting point I have chosen the periodicity phenomenon in the homotopy of the classical groups which it was my good fortune to discover in 1957. This starting point is partly justified because these lectures traditionally deal ...

Vector Bundles and K-Theory - uni-freiburg.de

1. Formal structure of periodicity theorem For a compact space X we have the Grothendieck group K(X) of complex vector bundles on X [see for example (2)]. It is a commutative ring with identity. For locally compact X we introduce K with compact supports: R{X) = Ker{Z(X+) ^ K{+)} where X+ = X U {-(-} is the one-point oompactification of X. Alterna-

Comodules, sheaves, and the exact functor theorem

Chapter 3 The Hirzebruch-Riemann-Roch Theorem 3.1 Line Bundles, Vector Bundles, Divisors From now on, X will be a complex, irreducible, algebraic variety (not necessarily smooth). We have (I) X with the Zariski topology and O X = germs of algebraic functions. We will write X or X Zar. (II) X with the complex topology and O X = germs of algebraic functions. We will write XC for this.

Stable and Unitary Vector Bundles on a Compact Riemann Surface

(He and Gepner also recently showed that this works in the motivic setting too, though this other proof relies on the reader having already seen Bott periodicity for motivic complex K-theory.) Atiyah, Bott, and Shapiro in their seminal paper titled Clifford Modules produced an algebraic proof of the periodicity theorem. EDIT: Whoops x2! They ...

Analytic cycles and vector bundles on non-compact ...

An Introduction to Complex K-Theory May 23, 2010 Jesse Wolfson Abstract Complex K-Theory is an extraordinary cohomology theory de ned from the complex vector bundles on a space. This essay aims to provide a quick and accessible introduction to K-theory, including how to cal-culate with it, and some of its additional features such as characteristic

DIFFERENTIAL GEOMETRY OF COMPLEX VECTOR BUNDLES

Associated Fiber Bundles. 2. Classifying Vector Bundles. Pullback Bundles. Clutching Functions. The Universal Bundle. Cell Structures on Grassmannians. Appendix: Paracompactness Chapter 2. K-Theory. 1. The Functor K(X). Ring Structure. The Fundamental Product Theorem. 2. Bott Periodicity. Exact Sequences. Deducing Periodicity from the Product ...

Note on the Serre-Swan Theorem | Request PDF

account of the standard Bott periodicity theorem, from the point of view of C*-algebra theory, and with a view to our infinite dimensional generali-zation. In Section 3 we shall construct the C*-algebra SC(E) associated to an infinite dimensional Euclidean space and formulate our periodicity theorem for it.

Bott periodicity theorem | Project Gutenberg Self ...

and the Bott periodicity theorem in topological K-theory. This paper originates from the talk ... or BGL(n,C) is the classiﬁng space for real or complex vector bundles. In the following, we sketch the notions of classifying maps, classifying spaces and universal ... the name “universal bundle” is the following theorem.

BOTT PERIODICITY, SUBMANIFOLDS, AND VECTOR BUNDLES arXiv ...

The proof is, roughly speaking, an application of the local family index theorem for a perturbed twisted spin Dirac operator, a variational formula of the Bismut–Cheeger eta form without the kernel bundle assumption in the even dimensional fiber case, and some properties of the Cheeger–Chern–Simons class of complex flat vector bundle.

BOTT PERIODICITY, SUBMANIFOLDS, AND VECTOR BUNDLES

We provide a classification of type AII topological quantum systems in dimension d = 1, 2, 3, 4.Our analysis is based on the construction of a topological invariant, the FKMM-invariant, which completely classifies “Quaternionic" vector bundles (a.k.a. “symplectic" vector bundles) in dimension \({d\leqslant 3}\).This invariant takes value in a proper equivariant cohomology theory and, in ...

Introduction The Atiyah-Singer Index Theorem

In mathematics, Kuiper's theorem (after Nicolaas Kuiper) is a result on the topology of operators on an infinite-dimensional, complex Hilbert space H.It states that the space GL(H) of invertible bounded endomorphisms of H is such that all maps from any finite complex Y to GL(H) are homotopic to a constant, for the norm topology on operators.. A significant corollary, also referred to as Kuiper ...

(PDF) Local Index Theory and the Riemann-Roch-Grothendieck ...

I am trying to compute the number of classes of real vector bundles over spheres. I am reading Bott Tu (Differential Forms in Algebraic Topology) and have worked out using Bott Periodicity Theorem the following table (which I upload as a picture cause I was not able to make the code work here):

Automorphic vector bundles on connected Shimura varieties

These lecture notes are intended as an introduction to the methods of classification of holomorphic vector bundles over projective algebraic manifolds X. To be as concrete as possible we have mostly r

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