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).
Questions tagged [topological-k-theory]
279 questions
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Asking an exact sequence of $K(X)$ modules
In page 76 of Atiyah's K-theory, he discussed the possibilities of extending Bott periodicity to the case when $\mathbb{S}^{2}\times X$ is replaced by a fibration over $X$ with structure group $U_{1}$. He quote an example as:
"..In particular let…
Kerry
- 2,286
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Proof of external product theorem using K-theory
I am using Hatcher's K-Theory book to work through the proof of the external product theorem:
$\mu:K(X) \otimes \mathbb{Z}[H]/(H-1)^2 \to K(X) \otimes K(S^2) \to K(X \times S^2)$ is an isomorphism
So far I have shown that $\mu$ is surjective. I am…
Juan S
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An example about trivial product of reduced K-theory
This is example 2.13 in Hatcher's K-theory notes. Suppose $X$ is a pointed compact Hausdorff space and $X=A\cup B$, where $A$ and $B$ are compact contractible subspaces of $X$ containing the basepoint. The book claims that
the product…
D. Huang
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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 $$ K(S^2) \approx \mathbb{Z}[x]/(x-1)^2$$ (thanks to the…
Luigi M
- 3,887
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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 there is a natural isomorphism $K^{0}(X)\rightarrow…
cyc
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Why $TN\cong \pi^{*}N\oplus \pi^{*}N$?
I think this question is quite trivial. We know that the classical tubular neighborhood theorem asserts the follows:
There exists an open neighborhood of $M$ in $A$ which is diffeomorphic to the total space of the normal bundle under a…
Kerry
- 2,286
2
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Complex topological K-theory - the book
I am reading the first chapter of the book "Complex topological K-theory" by Efton Park, which is in general very good. However, for some reasons, which I don't understand, when working with compact Hausdorff spaces, he assumes that the space has…
user446046
- 636
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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 $X$ is a finite CW-complex and the map is induced by…
tobiasm
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What is the real K-theory of the spheres?
In many sources one finds a computation of the complex topological K-theory of the spheres $S^n$, but the real theory $KO(S^n)$ is usually not computed. Maybe it can be read off some more advanced stuff, but I am unable to do so. I would like to see…
Nandor
- 418
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Visualizing the vector space structure of a vector bundle
In Bruce Blackadder's Book K-Theory for Operator Algebras, The very first definition in the book is as follows:
Definition : A vector bundle over X is a topological space $E$, a continuous map $p: E \rightarrow X $, and a finite dimensional vector…
Dominic Petti
- 498
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On clutching functions
I'm reading Hatcher's "Vector bundles and K-Theory" (version 2.2, November 2017). In chapter 1, section 1.2, he describes how to construct vector bundles with base space a sphere. I can follow his example (that I will state at the bottom for…
Schief
- 319
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Clarification on representing K-theory by vector bundles.
In the Vector bundles and K-theory text by Hatcher, on page 40 under Ring structures it says:
'For elements of $K(X)$ represented by vector bundles $E_1$ and $E_2$ their product in $K(X)$ will be represented by the bundle $E_1 \times E_2$, so for…
user500074
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How to prove a Kuenneth Theorem for K-theory
So I've been reading Atiyah's book titled $\textit{K-theory}$ and I'm stuck at Corollary 2.7.15 (pg 113) which is basically trying to prove a Kuenneth formula for K-theory. Ordinarily, I would try and be more specific about what I don't understand,…
Adam
- 155
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External product is not injective in real K theory
I am reading Vector Bundles and the Kunneth Formula in which Atiyah states (see second paragraph, page 1) that the external product $\mu : KO^*(X) \otimes KO^*(Y) \to KO^*(X \times Y)$ is not always injective. The counterexample he gives is $X = Y =…
James
- 989
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K theory and maps into unitary group - reference request
I am reading Vector Bundles and the Kunneth Formula and in the proof of lemma 2, Atiyah states $\widetilde K ^0(X) = [X,BU]$ and $K^1(X)=[X,U]$ without justification. Can someone give me a reference for this result?
Is this related to the…
James
- 989