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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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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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$K$ theory of vector bundle, $K(V)$, is a $K(X)$ module

This is on page 67, definition 5.6 when the author defines the Thom homomorphism: $V$ is a hermitian vector bundle over a compact space. $K(V)$ is a $K(X)$ module. How does $K(X)$ act on $K(V)$? ...
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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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Existence of Thom Class

In page 133, Theorem 8.5.5. (The Thom isomoprhism theorem) Let $\pi:V \rightarrow X$ be a complex vector bundle of rank $n$, over al locally comapct space $X$. Let $$ 0 \rightarrow \pi^* \...
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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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Clutching function for $S^n$: why a vector bundle?

On Page 22 of Hatcher's $K$ Theory we make the following construction For $n,k \in \mathbb N \setminus \{ 0\}$. Let $f:S^{k-1} \rightarrow GL_n(\mathbb R)$ be a continuous map. Define $$...
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the form of classes in $K(X)$ [Atiyah]

Anyone can explain me why taking two bundles $E,F$ we can conclude that any element in $K(X)$ is of the form $E-F$? It is mentioned in Atiyah on the bottom of the page $43$ (I can put screen if needed)...
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Existence of an element in $K^0$ group, Koszul complexes

I havee such a question on construction of the Koszul complex (further we are concerned about K-theoretical Euler class). I was thinking of introducing the Koszul complex, and the existing of elements ...
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Hatcher's, tensor product of vector bundles : topology explained

So this is a sumamry of the construction given in pg14 of Hatchers of the tensor of two vector bundles: Let $p_1:E _1 \rightarrow B$, and $p_2 : E_2 \rightarrow B$. We define $E_1 \otimes E_2$ to ...
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unit element in cohomology and k-theory

It is some well-known fact that there is the unit element in $K^{-2}(point)$ whereas there is no unit in $H^{-2}(point)$? Where it is come from? For details I add the link http://pages.uoregon.edu/...
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$X/A\wedge Y/B= (X\times Y)/(X\times B\cup A\times Y)$

In Hatcher's book on vector bundles and K-theory, page 55, in order to extend the external product to the relative form, he uses the following identification: $X/A\wedge Y/B= (X\times Y)/(X\times B\...
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Generators of $K_0(C(\mathbb{C}P^1))$

$\newcommand{\C}{\mathbb{C}}\newcommand{\R}{\mathbb{R}}$ I know that $K_0(C(\mathbb{C}P^1))\simeq K^0(\mathbb{C}P^1))\simeq K^0(S^2)\simeq \mathbb{Z}^2$ (we're talking about complex K-theory). This ...
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K-theory and usual (cohomoogical) Euler class

Can someone explain the difference (and hence the sense of studying) between the usual and K-theoretical Euler class? Most famous and for now the only difference is showed during computations of ...
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trying to understand the connecting homomorphism between K theory groups

the connecting homomorphism from $K_{1}(A/J)$ to $K_{0}(J)$ is defined by the composition $(j_{*})^{-1} k_{*}$ where $j$ is the inclusion of $J$ to the mapping cone $C_{\pi}$, which induce isomorphism ...
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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 ...
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An exercise in Atiyah's book K-Theory

This copy is from Atiyah's book K-Theory page 33, and it contains an exercise. I understand that real, quaternion bundle are orthogonal, symplectic bundle respectively. But I do not know how to figure ...
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property of vector bundles in thom spaces

Let $r$ be a positive integer such that $r([L]-1)=0$ in $\widetilde{KO}(\mathbb{R}P^{b-1})$, where $L$ is a line bundle. This implies that $rL\oplus \underline{s}\cong \underline{r}\oplus\underline{s}$...
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question about a notation in Atiyah's book K-Theory

Here is a copy of a theorem from Atiyah's book on page 29. My queation is that what is the meaning of the following notation? Or what is the exact definition of this set? I did not find the ...
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Why is $H^*(\mathbb{CP}^\infty, \mathbb{Z})$ not $\mathbb{Z}[[x]]$?

This is some confusion about standard facts. The cohomology ring of $\mathbb{CP}^\infty$ is a polynomial ring in one variable. However, it is also well-known that for generalized cohomology theories,...
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homotopy cofibers (Thom spaces)

I have a problem with understanding the construction which I marked as the red ractangle. The whole concept is showing the alternative definition of a Thom space. Construction up to the second diagram ...
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question about the Thom space

Let $E\rightarrow B$ be a bundle with an inner product. Using $D(E):=\left\{v\in E \ : \ u(v,v)\leq1\right\}$ and $S(E):=\left\{v\in E \ : \ u(v,v)=1\right\}$, we define a Thom space: $$\text{Th} \ E= ...
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Tietze extension theorem for vector bundles on paracompact spaces

In Atiyah's K-theory book, he gives a proof of the Tietze extension theorem to vector bundles. He gives the proof only for compact Hausdorff spaces, but I want to generalise it to paracompact ...
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How is $K(X\times S^2)$ a $K(X)$ module ?

I am reading K-theory by M.Atiyah and am having difficulty in understanding the proof of Bott periodicity. On page 72, he mentions that the homomorphism $\alpha : K(S^2 \times X) \to K(X)$ is a $K(X)$-...
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Definition of a (topologically) continuous functor

In Atiyah's K-theory page 6 (http://www.cimat.mx/~luis/seminarios/Teoria-K/Atiyah_K_theory_Advanced.pdf), he defines a covariant functor of one variable $T$ from finite dimensional vector spaces to ...
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Vector bundle over projective space

The following exercise is from Atiyah's book "K-theory", Example 2. Let $V$ be a vector space and let $\mathbb{P}(V)$ be its associated projective space. That is $\mathbb{P}(V)=\frac{V\backslash\{0\}}...
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Analytic Grothendieck Riemann Roch

I was wandering if is there an analytic version of the Grothendieck-Riemann-Roch theorem. If so, could you please tell me the references?
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What is a “formal” difference of vector bundles?

I saw this question: What does "formal" mean? But I don't think it answers my question. In Hatcher's book "Vector Bundles and K-Theory", he defines $K^0(X)$ to be the group of formal ...
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How does the class of the tangent bundle behave in the K-theory ring?

Seeing as the tangent bundle of a manifold is an important object in differential geometry, I was wondering if it has any specific behavior in the K-theory of a manifold. (Either real/complex, ...
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Confusion about relationship between operator $K$-theory and topological $K$-theory

I've often heard people say that $K$-theory of $C^*$-algebras generalises topological $K$-theory, for the reason that, say if $M$ is a compact manifold, we have $K_0(C(M)) = K^0(M)$, where $C(M)$ ...
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The semigroup of vector bundles by gluing of contractible patches

Assume $M_1$ and $M_2$ are two contractible compact manifolds with boundary and that $\partial M_1 = \partial M_2 = \Omega$. Fix a homeomorphism between the boundaries and consider $X = M_1 \sharp M_2$...
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2answers
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Very detailed book for topological K-theory

I have just learnt the first three chapters of Allen Hatcher's algebraic topology. And I would like to get exposed to some topological K-theory via self-study. But as a beginner, I am afraid of ...