Questions tagged [topological-k-theory]

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).

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59 views

Stably isomorphic complex vector bundles which are not isomorphic

Is there an example of complex vector bundles $E_1$ and $E_2$ which are not isomorphic but for which $E_1 \oplus \mathbb{C}^{n}$ isomorphic to $E_2 \oplus \mathbb{C}^{n}$ for some $n$? I'm a beginner ...
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Thom class in K-theory for real vector bundles of odd rank and pushforwards in K-theory with degree shift

If $E \to X$ is a complex vector bundle, its Thom class is defined using the exterior algebra of $E$, giving the Thom isomorphism $K^*(X) \cong \widetilde{K^*}(X^E)$. Atiyah-Bott-Shapiro use Clifford ...
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Relationship between Chern characters in the sense of Hatcher and in the sense of Roe

Let $\pi: E \to X$ be a vector bundle over some manifold $X=\mathcal M$. Definition 1: Hatcher defines Chern classes $$c_i(E)\in H^{2i}_\text{singular}(X)$$ as elements of the singular cohomology ring ...
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Different coefficient rings for the Chern classes and the Chern character

Following Hatcher, we define Chern classes $c_i$ taking vector bundles $E \to X$ to some equivalence class $c_i(E) \in H^{2i}(X; \mathbb Z)$ But when we define the Chern character, $$\operatorname{ch}(...
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Why is universal coefficient theorem for $C^*$-algebras natural in both variables?

This theorem 4.4 from The Kunneth Theorem and the Universal Coefficient Theorem for Kasparov’s Generalized K-functor, Jonathan Rosenberg and Claude Schochet. My question is, how does naturality of $\...
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Isomorphisms in Real $K$-theory

The Real $K$-theory introduced by Atiyah in the 60s is, in its most simple form, a functor from compact $C_2$ spaces to abelian groups $$KR:C_2Top_c\to Ab.$$ For $X,Y \in C_2Top_c$ an equivariant map $...
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What does degree mean here? Question about universal coefficient theorem and $KK$ theory.

Universal Coefficient Theorem 1.17. Let $A\in\mathcal N$. Then there is a short exact sequence $$ 0\to \text{Ext}(K_*(A),K_*(B))\stackrel{\delta}\to KK_*(A,B)\stackrel{\gamma}\to \text{Hom}(K_*(A),K_*(...
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What is the product in KO-theory in terms of concrete homogeneous spaces?

It is well-known that the spaces $KO_i$ of the $\Omega$-spectrum representing KO-theory a.k.a. real K-theory are $KO_0,KO_{-1}, \ldots, KO_{-7} = BO \times \mathbb{Z}, O, O/U, U/Sp, BSp \times \mathbb{...
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Spaces with only even dimensional cells has zero $K^{-1}$ group?

This is from D. Quillen's "On the Cohomology and K-Theory of the General Linear Groups Over a Finite Field". Here $BU= \bigcup _1^\infty X_m$ is the union of finite subcomplexes with only ...
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Is there a paper that explains Kasparov's KK theory in English?

I need to learn Kasparov's KK theory but the original paper is written in Russian. G.G. KASPAROV, The operator K-functor and extensions of C*-algebras, Izv. Akad. Nauk SSSR, Ser. Mat. 44 (1980), 571-...
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Milnor's construction

In the Milnor's construction, $$ \mathcal{J}(G):=\underrightarrow{\lim}G^{*(k+1)} $$ where $G$ is a topological group. I know that there is a natural freely (right) $G$-action on $\mathcal{J}(G)$ and ...
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Intuition behind 4-fold periodicity of $L$-theory

The quadratic and the symmetric L-groups are 4-fold periodic. What is the simple argument to obtain the intuition behind the 4-fold periodicity of $L$-theory? (For example, why not have the Bott ...
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K-Theory Equivalence Classes

Let $M$ be a finite dimensional compact manifold and $(Vect(M),\oplus)$ be the abelian monoid of complex vector bundles on $M$. I just read that it is possible to construct an equivalence relation on $...
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Calculate $K_0(\vee_{j=1}^{k}\mathbb{S}^{n})$ and $K_1(\vee_{j=1}^{k}\mathbb{S}^{n}).$

Let $\mathbb{S}^{n}\; \vee\cdots\;\vee\; \mathbb{S}^{n}=\vee_{j=1}^{k}\mathbb{S}^{n}$ the sum of the wedges of the $k$-spheres $\mathbb{S}^{n}:$ $\vee_{j=1}^{k}\mathbb{S}^{n}=\amalg_{j=1}^k\mathbb{S}^{...
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Generalizing the clutching construction to more contractible open sets

In computing the topological $K$-theory of $\mathbb{R}P^2$ I had an idea to mimic the clutching construction: Recall that this says that for the open cover $S^k = D^k_+\cup D^k_-$ a rank $n$ vector ...
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Alternate Reference for Topological K-theory

I have been trying to read Atiyah's Topological K-Theory. The book, in general, is quite painful to read as the proofs are in most cases too precise for the first reader to understand, and lacks to ...
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Definition of odd topological K-theory using circles

I wanted to check whether the following characterization of odd complex topological $K$-theory is correct. Let $X$ be a compact Hausdorff space. Then $K^{-1}(X)$ can be defined as $\tilde{K}^0((X\...
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K-theory proof of index theorem - some minor confusion

I am trying to understand the general approach to the $K$-theory proof of the Atiyah-Singer index theorem, using this https://arxiv.org/pdf/math/0504555.pdf paper. I ran into some confusion on page 29,...
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Example of a space whose complex K-theory is not easily computable from singular cohomology

I am looking for a counterexample to the formula $$ K^n(X) \cong \prod_{i\equiv n \mod 2} H^i(X) $$ where $K^*$ denotes complex topological $K$-theory, $H^*$ singular cohomology and $X$ a compact ...
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Topological index in Atiyah Singer

I'm a beginner at Atiyah-Singer index theorem and I've reviewed some results about theorem. Here's some questions. Ive seen the topological index is equal to $$\operatorname{ch}(D) \operatorname{Td}(X)...
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Help with the setup for Atiyah's proof of Bott Periodicity.

I'm trying to understand Atiyah's proof of Bott Periodicity from his little book on K-Theory - in particular his formulation in terms of $K(P(L \oplus 1))$ where $L$ is a line bundle on a space $X$. ...
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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 ...
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What is the meaning of $KO^{-1}(S^1)$?

I am interested in the KO-theory of the circle $S^1$. In particular $KO^{-1}(S^1)$. Using the suspension theorem and reduced $K$-theory I can easily show that \begin{equation} KO^{-1}(S^1) \simeq KO^{-...
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Equivalence of two definitions of $K^{-1}$ in complex $K$-theory

In complex $K$-theory, one way I have seen the group $K^{-1}(X)$ defined, for a compact Hausdorff space $X$, is $$K^{-1}(X):= K^0_c(X\times\mathbb{R}),$$ where the right-hand side refers to compactly ...
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A map from a $ G_1 $ - equivariant KK-theory of Kasparov, to a $ G_2 $ - equivariant KK-theory of Kasparov

Let $ G $ be a locally compact group. Let $ H $ and $ K $ be two normal subgroups of $ G $. In order to construct a map, $$ \psi \ : \ \ F(G/H,G/K) \to F(G/K,G/H) $$ where, $$ F(G/H,G/K) = KK^{G/H} ( ...
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1answer
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Why does this short exact sequence split?

I am currently reading K-Theory, Anderson and Atiyah, and trying to understand the index bundle (of families of Fredholm operators). Here is some context: let $X$ be a compact topological space, $F\...
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Is there a sense in which $\tilde{K}$ is an exact functor?

I'm going through Hatcher's K-Theory script for the first time and noticed that following theorem looked quite like a statement of the form “this functor is exact”: If $X$ is compact Hausdorff and $A\...
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Reference books on the Baum Connes conjecture

Do there exist readable reference books about Baum-Connes Conjecture for beginners ?
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Tangent bundle of the incidence variety and its Chern class

I am trying to learn about Thom polynomials and I often find arguments in the literature that just make no real sense to me. Maybe it is due to my lack of knowledge in $K$ - theory. I apologize for ...
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Equivalence of families indexes of Fredholm operators

Let $F=F(H,H)$ be the space of bounded Fredholm operators in a Hilbert space $H$ with topology inherited from the norm operator topology, and let $X$ be a compact topological space. For a continuous ...
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1answer
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$K_0(C(\mathbb{T}^{n})) \cong \mathbb{Z}^{2^{n-1}}$

I am new to this website and I have a question. I want to show that $K_0(C(\mathbb{T}^{n})) \cong \mathbb{Z}^{2^{n-1}}$ but first I want to show that $C(\mathbb{T}^{n}) \cong C(\mathbb{T} \rightarrow ...
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1answer
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Ideas for calculating $K_0(l_{\infty})$ and $K_1(l_{\infty})$.

Thank you for answering my question. I'm a bit new to K-theory. So I was wondering how can I calculate $K_0(l_{\infty})$ and $K_1(l_{\infty})$. I think if we have one, then by using bott periodicity ...
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1answer
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$K_1(A)$ is countable when A is separable C*-algebra

We know that when A is a separable C*-algebra then $K_0(A)$ is countable. How can I show that $K_1(A)$ is also countable?
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1answer
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$K_0(C_0(X, A))$ , when X is compact and contractible.

Let A be a $C^{*}$-algebra and $B = C_0(X, A)$ be the set of all continuous functions from a locally compact Hausdorff space $X$ to $A$, vanishing at infinity. Prove that $K_0(B) \cong K_0(A)$ and ...
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Topological K-theory and characteristic classes of module bundles

Let $R$ be a commutative ring and let $A$ be a commutative unital topological $R$-algebra. By means of replacing vector spaces with $A$-modules, one can define $A$-module bundle in analogy to vector ...
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1answer
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The generator of $K_0(C(\partial(]0,1[^2)))$ and $K_1(C(\partial(]0,1[^2)))$

Let $C=[0,1]^2 \subseteq \mathbb{C}$ and $\partial C$ the boundary of $C$. I'm looking for the $K_0(C(\partial C))$ and $K_1(C(\partial C))$ and its generator.
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Proving $K_1(\mathcal{T})=0$ (is trivial)

Let $\mathcal{T}$ the toeplitz algebra and we define the short exact sequence? where $C(\mathbb{T})=\{z\in \mathbb{C}/ |z|\leq 1\}$: $$ 0 \rightarrow \mathcal{K} \rightarrow \mathcal{T }\rightarrow C(...
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Is Atiyah's periodicity Theorem related to splitting principle?

I assumed all vector bundles are over complex number. Let $V$ be a vector bundle over $X$. Then $P(V)$ denotes the projectivization of fibers of $V-0$ as a projective space bundle over $X$. Denote $K(...
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Univesal Coefficient Theorem for $C^*$-algebras

The UCT theorem is shown in the sreenshot. I have a question : What is the definition of $Ext_{\Bbb Z}^1(K_{*}(A),K_{*}(B))$? Does it have a relationship with Tor functor?
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Interaction between reduced suspension and quotients

Let $SX$ denote the reduced suspension of $X$. If $A\subset X$, what is the relationship between $S(X/A)$ and $SX/SA$? In commutivity of suspension and quotient space, a comment says these spaces ...
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1answer
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Does $K$-theoretical Atiyah-Singer index formula hold for non-compact manifolds?

In famous The Index of Elliptic Operators: I Atiyah and Singer introduce two families of morphisms: $$\text{a-ind}^X,\, \text{t-ind}^X\colon K(TX)\to \mathbb Z$$ indexed by compact smooth manifolds $X$...
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1answer
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What are the K groups of $X_n$, the wedge sum of $n$ circles at a single point?

If $X_n$ is the bouquet of n circles, then what is $K_0(X_n)$ and $K_1(X_n)$? I am super confused on how to approach this problem, since I've only seen $K_0$, $K_1$ computed for algebras, but I don't ...
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Help with a problem in K-theory/C- algebras.

For this problem you may assume the fact that $K_1(C_0(D)) = 0$. Let $n > 1$, let $\omega = e^{\frac{2πi}{n}}$, and let $E_n$ be the space obtained from $D$ by identifying $z$ and $\omega z$ for ...
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What are the K groups of $X_n$?

If $X_n$ is the bouquet of n circles, then what is $K_0(X_n)$ and $K_1(X_n)$? I am super confused on how to approach this problem, since I've only seen $K_0$, $K_1$ computed for algebras, but I don't ...
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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 \...
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Geometric Resolutions of C* -Algebras

I am reading "K-theory FOR OPERATOR ALGEBRAS" Bruce Blackadar, Proposition 23.5.1. Let B be a separable $C^{*}$-algebra. Then there is a separable commutative $C^{∗}$-algebra F, whose spectrum ...
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Why do we need the local triviality condition when working with vector bundles?

I'm currently reading through Adams' paper on the image of the J homomorphism, and wanted to brush up on vector bundles and K-theory before tackling this paper. The definition of (real) vector bundle ...
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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 ...
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1answer
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How to get isomorphism $K(X)\cong \mathbb{Z}\times \underset{\to n}{\lim} {\rm Vect}_n(X)$

I'm reading Atiyah's K-theory, on page 44, the Lemma 2.1.1 claims that $$ K(X)\cong \mathbb{Z}\times \underset{\to n}{\lim} {\rm Vect}_n(X) $$ I'm confused about how to get this isomorphism. Please ...
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2answers
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Are these strengthenings of Serre-Swan and Gelfand-Naimark true?

Let $X$ be a compact Hausdorff space. The Serre-Swan theorem allows us to identify complex vector bundles with projective finitely generated modules over the ring $C(X;\mathbb{C})$ of complex-valued ...

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