2
votes
0answers
28 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: ...
0
votes
0answers
46 views

Ring structure of $K(X)$ - definition of multiplication

Maybe it's a silly question, but I can't find a satisfactory answer to it. Hatcher defines the multiplication of two arbitrary elements in $K(X)$ as $$(E_1-E_1')(E_2-E_2') := E_1 \otimes E_2 - E_1 ...
1
vote
1answer
87 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 ...
3
votes
0answers
62 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
1answer
72 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
98 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
223 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
66 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}) ...
6
votes
1answer
139 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 ...
2
votes
1answer
87 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
156 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 ...
3
votes
2answers
106 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.
5
votes
2answers
112 views

How can I see that when $X$ is a trivial $G$-space, $K(X) \otimes R(G) \rightarrow K_G(X)$ is an isomorphism?

Here, $K$ is complex K-theory, $R(G)$ is the complex representation ring of $G$ (which -- for now, though it shouldn't matter -- is a finite group), and $K_G$ is $G$-equivariant complex K-theory. In ...
3
votes
1answer
429 views

8 periodicity: Clifford clock- Bott periodicity - KO-dimension in noncommutative geometries

Periodicity modulo 8 appears in the classification of real Clifford algebras $C\ell_{p,q}(\mathbb{R})$ (usualy refered to as the "Clifford Clock"), in real Bott periodicity and in the definition of a ...
7
votes
0answers
234 views

Weak Bott periodicity vs. strong Bott periodicity

Bruno Harris' proof (or I guess also Bott's original proof) of Bott periodicity (see here for instance) shows that there is a homotopy equivalence $h\colon\mathbb{Z}\times BU \rightarrow \Omega^2 ...
2
votes
1answer
155 views

Example of non-isomorphic vector bundles with the same element in $K$

Let $X$ be a paracompact and well behaved space. Topological K-Theory $K(X)$ of $X$ is group completing the monoid of isomorphism classes of vector bundles over $X$ with the Whitney sum. Two vector ...
7
votes
1answer
423 views

The Atiyah Hirzebruch Spectral Sequence

I am learning about the AHSS (for complex K-theory) by trying to compute the K-theory of some spaces. I have heard that the AHSS is functorial (maps of spaces induce maps of spectral sequences). Is ...
1
vote
1answer
129 views

Ring structure on K-theory of an even-dimensioned sphere

I am slightly confused by some statements in Hatcher's Vector Bundle book (page 60). To start with (and I am happy with) the natural ring homomorphism $$K(S^2) \simeq \mathbb{Z}[H]/(H-1)^2$$ I am ...
4
votes
1answer
131 views

Ring structure of K-theory of a wedge of spheres

I've just been using Bott Periodicity to calculate the K-theory of some simple spaces - spheres, torus, and wedge of spheres. The wedge of spheres is interesting. Given that $$\tilde{K}(X \vee Y) = ...
14
votes
2answers
603 views

How to understand the Todd class?

I am reading the article "K-Theory and Elliptic Operators"(http://arxiv.org/abs/math/0504555), which is about Atiyah-Singer index theorem. In page 14 the article discussed the Thom isomorphism: ...
13
votes
3answers
560 views

$K(\mathbb R P^n)$ from $K(\mathbb C P^k)$

EDIT: I found a brief discussion of this in Husemoller's Fibre Bundles, chapter 16 section 12. Here to compute $\tilde K(\mathbb R P^{2n+1})$ he says to consider the map $$ \mathbb R P^{2n+1} = ...
7
votes
1answer
164 views

Why Are These Two Morphisms the Same?

I am reading Max Karoubi's "K-Theory" and I think I'm overlooking some trivial fact. We have a vector bundle $E\rightarrow X$ and a morphism $p:E\rightarrow E$ with $p^2=p$. He is showing that $\ker ...
13
votes
1answer
692 views

Topological vs. Algebraic $K$-Theory

Suppose I can calculate the extraordinary cohomology encoded in topological $K$-groups of a topological space $X$ with CW structure. What information does this give me about $C^{*}$-algebras ...
9
votes
1answer
460 views

Why is stable equivalence necessary in topological K-theory?

The topological $K$-theory of a complex compact manifold $X$ is the commutative monoid $K(X)$ of isomorphism classes of complex vector bundles. Two classes $[E]$ and $[F]$ are equivalent in $K$-theory ...