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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48 views

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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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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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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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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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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Mayer-Vietoris sequence in topological K-theory

In topological K-theory, we define functors $K^{-n}$ on the category of compact Hausdorff spaces. With this theory we have the Mayer-Vietoris exact sequence: if $X = A \cup B$, we have an exact ...
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Functoriality of twisted K-theory

In order to avoid the XY problem I will first state what I want, then what I think is the solution and how that failed until now. I'm trying to use May-Sigurdsson's definition of Twisted $K$-theory, ...
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the isomorphism of $D(B(H))$,where ,$H$ is an infinite dimensional inseparable Hilbert space.

We define $D( B(H))=\cup_n P(M_n( B(H)))/\sim$, where $ P(M_n(B(H))$ is the set of projections in $M_n(B(H))$,$\sim$ is the equivalence relation as follows:suppose $p$ is a projection in $P(M_n(B(H)))$...
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a unitary does not in general lift to a unitary

Consider the restriction map $\pi:C(\mathbb{D})\rightarrow C(\mathbb{T})$, where $\Bbb D$ is the closed unit disk and $\Bbb T$ is the unit circle. Suppose $v\in C(\Bbb T)$ is a unitary such that $v(z)=...
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Complex line bundles with connection can be parametrized by unitary character of fundamental group

Recently I'm reading the book Geometric Quantization and Quantum Mechanics and it states in chapter $3$ that Assume $(X,\omega )$ is a symplectic manifold, $(L,\alpha )$ is a complex line bundle ...
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compute the abelian semigroup $V(A)$

$proj(A)$ is the set of algebraic equivalence classes of idempotents in $A$,we set $V(A)=proj(M_{\infty}(A))$. If $A$ is a $II_{1}$ factor,then $V(A)\cong \Bbb R_{+}\cup\{0\}.$If $A$ is a countably ...
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Theorem for the external product in K-theory and reduced version

In Hatcher's Vector Bundles and K-thery 1: http://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf a reduced version of the external product $\mu:K(X)\otimes K(Y)\to K(X\times Y)$ is used to prove the Bott ...
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K-theory functoriality understanding

Reading some K-theory I understand the construction of the abelian group $K(X)$ where $X$ is a compact Hausdorff space. However, I read in most of the references that $K$ is also a functor. Why? Which ...
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Some K-theory Examples

I have a dissertation on K-theory and I've been using some references like Hatcher's book and Karoubi's book. Googling a bit deeper for further examples I found that many sites use as reference the K-...
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Values of the $\mathbb{Z}/2$-spectrum $KR$ of $K$-theory with Reality

Atiyah's $K$-theory with Reality produces a $\mathbb{Z}/2$-spectrum $KR$. But I am stuck as I don't really know what the values of this equivariant spectrum should be on representations. Any ...
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Clarification of statement

I was reading Michael Atiyah's lectures on K-theory and I came across this paragraph which I couldn't understand. So we have a compact Hausdorff space $X$ and a vector bundle $\pi : E \rightarrow X $ ...
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Constructing abelian group from commutative monoid.

Studying K-Theory and the construction of the group $K(M)$ there are some details that still remain unclear for me (I'm reading Hatcher's book and Karoubi's book). First some background of the ...
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K-theory of classifying spaces

Can someone help me calculate the following groups in $ K $-theory 1) $ KU^0 (B\mathbb{S}^1) $ 2) $ KU^0 (\mathbb{RP}^\infty) $ where $ B \mathbb{S}^1$ is the classifying space of $ \mathbb{S}^1 $ ...
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What does it mean to say $K$-theory satisfy Mayer–Vietoris sequence?

I saw the statement "a $K$-theory is expected to satisfy Mayer–Vietoris property and Bott Periodicity" somewhere and I am trying to understand what it means. What does it mean to say a $K$-theory ...
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Characteristic class corresponding to $K_\mathbb{R}(S^1)$

In complex K theory, for any compact space X, Chern character $Ch$ induces a ring isomorphism $K_\mathbb{C}(X) \otimes \mathbb{Q} \rightarrow H^{even}(X;\mathbb{Q})$. For $X=S^2$, $K_\mathbb{C}(S^2) \...
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Proof for a classifying space for $K$ Theory.

The goal of my question is to understand a bijection between $K_0(A)$ to $[C_0(\Bbb R), M_2(M_\infty(A))]_*$ $$[C_0(\Bbb R), M_2(M_\infty(A))]_*$$ is the homotopy class of graded $*$-homomorphisms. $...
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What Cayley transformation does to a $*$-homomorphism

We let $A$ be a $C^*$ algebra. We consider a grading on $A=C_0(\Bbb R) $ by even and odd functions whilst a grading on $M:=M_2(M_\infty(A))$ by diagonal and off diagonal elements given by grading ...
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Computing $K$-theory elements in a $C^*$ algebra $A$

Let $A$ be a unital $C^*$ algebra. Let $p,q$ be projections in $M_n(A)$. Then $[p]-[q]$ defines an element in $K_0(A)$. Now consider the matrices, the projections, $$ \left[ \begin{pmatrix} 1-p &...
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$K$ theory of $C^*$ algebra is different to algebraic $K$ Theory?

Is the $K_0$ group for a $C^*$ algebras $A$ same as that for the $K_0$ group of ring $A$ from algebraic $K$ theory? We assume $A$ is unital (I am not sure if this matters), i.e. what is an example ...
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K Theory of $C^*$ algebras I, Higson's notes

Let $B$ be a $C^*$ algebra, and $A(B)$ denote the algebra of bounded continuous functions from $[1,\infty)$ to $B$. Let $I(B)$ be the ideal of functions which vanish at infinity. Let $Q(B)$ be the ...
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Question about Chern Character in Hatcher's book

I have a question about an argument from Allen Hatcher's script Vector Bundles and K-Theory in Cor. 4.4 (see page 110). Here the excerpt: We consider a vector bundle $E \to S^{2n}$, Then for Chern ...
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Loop Space of $BU \times \mathbb{Z}$

I have a question about an argument that occurred in the discussion about consequences of Bott periodicity in A Concise Course in Algebraic Topology by P. May on page 207. Here is the excerpt: ...
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$K$-Theory of operators I, Higson notes

I am having trouble understanding the following statement: 3.20 Proposition, pg44: Let $D$ be a symmetric, odd graded elliptic operator on a graded vector bundle $S$ over a compact manifold. The ...
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Understanding the map from $K_0(A)$ to homotopy class of maps,

In page 42, lectures on operator $K$-theory The writer defines a map from $$\Phi: K_0(A) \rightarrow [\mathcal S,A\otimes \mathcal K]$$ Notation: $\mathcal S:=C_0(\Bbb R)$ and $\mathcal K$ is a ...
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The complex topological $K$-theory spectrum is not an $H\mathbb{Z}$-module.

Why is it true that the complex topological $K$-theory spectrum is not an $H\mathbb{Z}$-module? I mean, why the nontriviality of the first $k$-invariant implies the claim above?
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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 ...
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Operator K-theory and Topological K-theory

I was trying to understand the relationship between these two K-theory's when you pick your C* algebra to be $C(X)$ for $X$ a compact Hausdorff space. For this you create a function between $P_\infty (...
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Role of Dirac operators in Index Theorems

I'm trying to approach the Atiyah-Singer Index Theorem by getting an overview of the area. One thing that confuses me a lot is that some treatments give (and hence prove) the theorem for Dirac ...
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Defining Bott Class by relative $K$-theory

I am really confused with this construction of Bott Class in Page 127, Example 8.4.12 If $V$ is a complex vector space of dimension $n$, we form the complex $$ 0 \rightarrow \wedge^0 V \...
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Relative $K$-theory, definition

On pg125 on proving that that the concordance classes quotiented by the acyclic ones froms a group, there is a lemma: Lemma 8.4.5 Let $(E,F,f)$ and $(E,F,g)$ be two $K$-cycles on $(X,Y)$. Assume ...
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KO theory v.s. ko theory

It looks that there are different types of topological K-theories, with similar names but they are totally different outputs for the same input. The first theory is called the KO theory. There are ...
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Is $M \oplus e^1 = e^2$ , i.e. trivial?

I am trying to prove formally that the Mobius bundle,M, over $S^1$ when summed with the trivial rank $1$ bundle $e_1$ isn't the trivial bundle. In other words $M\oplus e_1\neq e_2$. First we write ...
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two versions of the Euler class

I was wondering about cohomological and k-theoretical Euler class, or both versions of characteristic class in general. I mean, one knows that characteristic classes can measure, how twisted such a ...
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Short exact sequence of $K$ functor; applying Tietze extension

I have questions in page 52 of Hatcher's, for the proof of Proposition 2.9 If $X$ is a compact Hausdorff space and $A \subseteq X$ is a closed subspace, then the inclusion and quotient maps $$A \...
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Explaining the $K$ functor isomorphism $K(X) \cong \tilde{K}(X) \oplus \Bbb Z$

It seems unclear to me what the splitting means in page 40, of Hatcher's for a $K$ functor. There is a natural homomorphism, $\varphi:K(X)\rightarrow \tilde{K}(X)$, sending $[E-\epsilon^n] \...
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Proving continuity of homotopy of clutching functions

This is in page 45 of Hatcher's, that there is a linear homotopy between clutchig functions . I could not workout/justify the linear homotopy. Is there an easy way to see the continuity of homotopy? ...
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Operations on pullbacks of vector bundles.

I wanted to prove the following bundle isomorphisms, on page 20 of hatcher's $K$-theroy. That is $$f^*(E_1) \otimes f^*(E_2) \cong f^*(E_1 \otimes E_2)$$ $$f^*(E_1) \oplus f^*(E_2) \cong f^*(E_1 \...

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