Topological K-theory is a generalized cohomology theory, for which $K_0(X)$ is the Grothendieck group of isomorphism classes of vector bundles over topological space $X$. See also (algebraic-k-theory).

learn more… | top users | synonyms

1
vote
1answer
28 views

Atiyah K theory

On page 3 of Atiyah's book on K theory (link here: http://www.cimat.mx/~luis/seminarios/Teoria-K/Atiyah_K_theory_Advanced.pdf) he states: "Since a vector bundle is locally trivial, any section of a ...
2
votes
0answers
69 views

Exercises in Topological K-Theory (Atiyah)

I'm currently working through Michael Atiyah's K-Theory. The main problem I'm finding with it is that it does not have any exercises. Does anyone have a good collection of exercises to go along with ...
2
votes
0answers
27 views

Holomorphic $K$-theory

$K$-theory is traditionally defined for arbitrary compact Hausdorff spaces. If instead we require the base to be a complex manifold and work with only holomorphic vector bundles, in what ways would ...
0
votes
1answer
35 views

$K$-theory for non-compact manifolds

It seems that usually (by which I mean, in every source I've looked at) people define the group $K^0(X)$ for $X$ compact Hausdorff. Sometimes they later extend this definition to all locally compact ...
0
votes
1answer
31 views

A smooth non-stably trivial smooth vector bundle

This may well be just a look-up, but do you have an example of a non-stably trivial smooth vector bundle? If it has a presentation as the vector bundle associated to the representation of some ...
1
vote
1answer
31 views

Complex bundles on $S^{2n+1}$

We know that the complex $K$ theory of spheres are 2 periodic. On the other hand every complex bundle on $S^{1}$ is trivial. So $K(S^{2n+1})=0$. So this is a motivation to ask: Is there a ...
2
votes
1answer
41 views

K-theory product trivial on $K^1$ due to stability?

Let $K^0$ be the topological complex vector bundle $K$-theory functor. On the one hand, we have that the product on $K^0(\Sigma X)$ vanishes by a Mayer-Vietoris argument. On the other hand, we define ...
2
votes
0answers
87 views

K-theory of $\mathbb{RP}^{\infty}$

what are the $K_0$ and $K_1$ group of $\mathbb{RP}^{\infty}$? Any reference would be good enough.
2
votes
0answers
46 views

Higher homotopy groups

Theorem 5.1 of this paper describes a map $K_n(R)\to \pi_{n+1}(SK(E(R),1))$, where $S$ denotes the suspension. My question: Do we have a map from $K_n(R)\to \pi_{n+1}(S^2K(E(R),1))$. Any reference is ...
3
votes
2answers
104 views

vector bundles on $X \times S^2$ built via clutching functions

Let $X$ be a compact Hausdorff space and $E \to X$ a (complex) vector bundle over $X$. We can build a vector bundle on $X \times S^2$ by clutching functions, i.e an automorphism $f:E \times S^1 \to E ...
1
vote
0answers
42 views

About the geometric interpretation of the reduced external product in K-theory

I'm trying to fill the details of the explanation of the interpretation of the reduced external product. I'll follow Hatcher second books, and I'll write part of his explanation in the gray boxes ...
3
votes
1answer
62 views

The long exact sequence in reduced K-theory. How to glue together the short ones?

I'm trying to filling the details about the construction of the long exact sequence in K-theory. I'm using the notation of Hatcher's book (pages 52-53). here is a related question, even thought it's ...
3
votes
1answer
103 views

Complex (topological) K-theory of $S \Sigma$ for a surface $\Sigma$.

In the following question here: What's the K-group of a surface? The $K$-theory of a compact orientable surface is computed. I was curious if it was also possible to compute the "higher" ...
2
votes
1answer
59 views

“Visual” interpretation of the Bott Periodicity for complex vector bundles

I've just read the Bott Periodicity proof (complex case) in Hatcher book about K-theory. At first it seems that using this theorem I could conclude that every complex vector bundles over $S^{2n+1}$ ...
0
votes
1answer
19 views

Linear clutching function properties

I'm trying to prove proposition $4.8$ of Husemöller (pay 148) Here it is the text: Where the only hints are: $[L^{n+1}(\xi),L^{n+1}(zp)] \approx [L^{n}(\xi),L^{n}(p)] \oplus [\xi,z]$ ...
2
votes
0answers
39 views

Filling the details of a construction via clutching function of a Vector Bundle

Let $(E,\pi,X)$ a complex vector bundle over X (which we assume to be Compact-Hausdorff) Let $$f \colon E \times S^1 \to E \times S^1$$ an automorphism of the product bundle $E \times S^1$. We define: ...
3
votes
1answer
20 views

About the generators of $\tilde{K}(S^2)$, the reduced K theory group

I've took the opportunity to join the community, because I didn't find a satisfying explanation to the following fact. Let $S^2$ the 2-sphere, let $H$ the tautological line bundle. Assume that $$ ...
2
votes
1answer
44 views

Working out the details of example 1.13 Hatcher: $\ E_{fg}\oplus n \approx E_f \oplus E_g$

I'm studying an example provided by Hatcher in his K-theory and Vector Bundle book. I'm referring to example 1.13 pag. 24 The first part is clear, $z^2$ is the Kronecker Product of the two ...
3
votes
1answer
39 views

Structures on vector bundles

I am reading the book K theory by Atiyah. In page number 32, he defines some additional structure on a vector bundle $V$. I have understood the definitions there. But there is a statement that says ...
1
vote
2answers
89 views

Elementary and purely Topological Proof of the non-triviality of Tautological Complex Line Bundle

I need some hints about the proof of the non triviality of the tautological complex line bundle, in a pure topological manner. Let $E$ be the t.c.l. bundle defined in this way $$ E= \{ (x,v) \in ...
0
votes
0answers
39 views

Hairy ball theorem, projections and L.I. vectors

I'm trying to understand this paper which proves that not every unimodular row is completable by invertible matrices: Why we have these implications: There are two linearly independent vectors at ...
5
votes
3answers
166 views

Relation between $K$-theories

I apologize in advance if this question is too vague/general. I am curious to know how all of the different $K$-theories are related (algebraic $K$-theory, topological $K$-theory, twisted $K$-theory, ...
3
votes
1answer
32 views

Necessity - Reasons of a passage in a proof of Hatcher vol. 2

Some words about the context of the proposition. Hatcher is defining operations on vector bundles over the same base space $B$. We are speaking about the Whitney sum here, and it is proving that if ...
2
votes
1answer
95 views

differential structure of $S^n$ and diffeomorphism map

Let $S^n$ be the unit sphere in $\mathbb{R}^{n+1}$, and $TS^n=\{(x,v): x\in S^n,v\in T_x S^n\}$. Show that $F: TS^n\times\mathbb{R}\to S^n\times\mathbb{R}^{n+1}$ given by ...
1
vote
1answer
100 views

$K$-theory exact sequence.

Let $Y$ be a closed subspace of a compact space $X$. Let $i:Y \to X$ the inclusion and $r:X \to Y$ a retraction ($r \circ i = Id_Y$). I have to prove that exists this short exact sequence $$ 0 \to ...
-1
votes
1answer
59 views

Are these equivalent definitions of the Grothendieck group construction?

I'm learning K-theory and I'm slightly confused. Given an abelian monoid $M$, we can construct the Grothendieck group $$ G(M) = (M \times M)/\sim$$ and I've seen two definitions for the equivalence ...
2
votes
0answers
40 views

Exterior algebra as a complex of vector bundles and an element in relative K-theory

There is a description of relative $K$-theory in terms of complexes of vector bundles. The following is an excerpt from Atiyah's book. Let $V$ be a complex vector space and consider the exterior ...
7
votes
1answer
97 views

Relationship between topological and Quillen's K-theory

Up until now, I've taken it for granted that the topological k-theory of a space $X$ is equal to the K-theory of vector bundles on $X$. $K_0$'s of both coincide (Serre-Swan) however, is it the case ...
3
votes
1answer
148 views

Prerequisite to start learning Topological K-Theory

Wanted to start learning K-theory to see if it is suitable for a undergraduate thesis (from an algebraic-topological view). For reference, I only know Hatcher's book (mostly because I've read his book ...
1
vote
1answer
70 views

Question on the map from algebraic K_0 to topological K_0

Let $X$ be a scheme over a field $F$ where $F$ is either the real or the complex numbers. If $q:Y\to X$ is an algebraic vector bundle over $X$, the $F$-points $q(F):Y(F)\to X(F)$ constitute a ...
3
votes
0answers
64 views

What is a $c_1$-map for Riemann-Roch theorems?

Atiyah and Hirzebruch define a $c_1$-map to spell out Riemann-Roch theorem for (compact and connected) smooth manifolds. The definition is following: a map $f:Y \to X$ is called a $c_1$-map if we are ...
3
votes
0answers
41 views

An isomorphism in the proof of Bott periodicity

In Efton Park's book on topological $K$-theory, to prove Bott periodicity, he considers the space $\mathcal{S}'X=(X\times S^{1})/(X\times\{1\})$ for a compact Hausdorff space $X$, and proves that ...
1
vote
1answer
61 views

Group completions and the induced homomorphisms

The group completion (aka Grothendieck group) of an abelian monoid $M$ is an abelian group $G(M)$ with a homomorphism $\iota:M \to G(M)$ of monoids satisfying the following universal property: for ...
2
votes
1answer
68 views

Property of elements in Grothendieck group

I'm reading Atiyah's K-Theory book and in the section where he introduces the Grothendieck group, he gives two constructions. One of them is as follows: Let $A$ be an abelian semigroup, let ...
3
votes
1answer
84 views

Yoneda's lemma and $K$-theory.

The K-theoretic form of Bott periodicity is that $\tilde{K}(X)\cong \tilde{K}(\Sigma^2 X)$, i.e., $\langle X, BU\times \mathbb{Z} \rangle \cong \langle \Sigma^2 X, BU\times \mathbb{Z} \rangle$. The ...
5
votes
1answer
109 views

$K$-theoretical interpretation of Bott periodicity.

We consider complex Bott periodicity $\pi_{i-1}(U) \simeq \pi_{i+1}(U)$. Why we can say that the $K$-theoretical interpretation of this assertion is $\tilde{K}(X) \cong \tilde{K}(\Sigma^2(X))$? Where ...
2
votes
1answer
312 views

Loop space suspension/adjunction

Let be $X,Y$ Hausdorff spaces. I denote with $\langle \cdot, \cdot \rangle$ the basepoint preserving homotopy classes of maps, with $\Sigma$ the reduced suspesion and with $\Omega$ the loop space. How ...
2
votes
0answers
68 views

Loop space and $K$-theory

How can I proove without using Yoneda's lemma that $$ \Omega^2(BU \times \mathbb{Z}) \cong BU \times \mathbb{Z} ?$$ In particular how can I define a cellular map $$ f: \Omega^2(BU \times \mathbb{Z}) ...
4
votes
0answers
74 views

Duality between K-theory and K-homology in the non-spin^c case.

Let M be a closed manifold. Then there is a cap product $K^\ast(M) \times K_\ast(M) \to K_\ast(M)$ between the K-theory of M and its K-homology. For a definition of it one could see my prior question ...
5
votes
2answers
163 views

Induced map on $K_{1}$-group

Let $A$ be a unital $C^{\ast}$-algebra. Any automorphism $\alpha$ of $A$ induces a map on $K_{1}(A)$ by $\alpha_{\ast}[u]=[\alpha(u)]$. Let the automorphism $\alpha$ be inner, does it follow that ...
4
votes
1answer
140 views

Index of twisted Dirac operator

I do not understand a step in the proof of the Lemma 11.4.1 in the book "Analytic K-Homology" by Higson, Roe. Let $S$ be a Dirac bundle over a closed manifold $M$ and $D$ the corresponding Dirac ...
1
vote
0answers
67 views

Chern Character Isomorphism for non-CW complexes

Atiyah and Hirzebruch showed in their paper "Vector bundles and homogeneous spaces" that $\mathrm{K}^\ast(X) \otimes \mathbb{Q} \cong \mathrm{H}^\ast(X; \mathbb{Q})$, where $\mathrm{H}^\ast$ denotes ...
2
votes
1answer
115 views

K-theory, $K_{0}$ of algebra of compact operators

I don't understand how to define the trace of a matrix with values in operators. This occurred in the following situation: Suppose that $H$ is an Hilbert space and $K$ is the algebra of compact ...
1
vote
0answers
23 views

CW-filtration in K-Theory

I'm just reading Atiyah's paper "Characters and cohomology of finite groups". In §2 he's defining a filtration on $K^*(X)$ by putting $K_p^*(X) = ker \{K^*(X) \rightarrow K^*{(X^{p-1}}) \}$, where $X$ ...
5
votes
0answers
115 views

How should we think of 'differences' of vector bundles?

Given a Hausdorff space $X$, the set of all vector bundles of finite dimensions $Vect(X)$ (up to isomorphism) with respect to the direct sum is a monoid. We can use the group completion construction ...
6
votes
1answer
171 views

How we do actually compute the topological index in Atiyah-Singer?

I am taking a lectured class in Atiyah-Singer this semester. While the class is moving on really slowly (we just covered how to use Atiyah-Singer to prove Gauss-Bonnet, and introducing ...
3
votes
1answer
110 views

long exact sequence in k theory

I am studying the basics of K-theory and given a CW pair $(X,A)$ I understand how to construct an long exact exact sequence $\cdots \rightarrow K(SX) \rightarrow K(SA) \rightarrow K(X/A) \rightarrow ...
2
votes
1answer
162 views

$K$-Theory of $BU(n)$

The following fact is asserted by wikipedia: The $K$-theory of $BU(n)$ is the numerical symmetric polynomials, i.e the subring of $\mathbb{Z}[x_1, \ldots, x_n]$ that is preserved under the action of ...
7
votes
0answers
123 views

K-theory for non-separable C*-algebras

Let $\kappa$ be an uncountable cardinal. What is the K-theory for the C*-algebras $\mathcal{K}(\ell_2(\kappa))$ and $\mathcal{B}(\ell_2(\kappa))$, of, respectively, compact and bounded operators on ...
3
votes
2answers
125 views

What's the K-group of a surface?

What's the K-group of a surface? I also want to know how to calculate such group and if there is a explicit characterization of the generators.