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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K-theory of a classifying space (part two)

Continuing my previous question, given a compact, connected Lie group $G$, there is a sequence of maps $$R(G) \to \hat{R}(G) \overset\sim\to K^*(BG) \to \hat{H}^*(BG;\mathbb Q)$$ apparently first ...
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Injectivity of index map for $K_1(S^1)$

This example/problem is from Valette's notes on the Baum-Connes conjecture (p. 45). The exercise is to prove that the (trivially equivariant) $K$-homology group $K_1(S^1)$ is $\mathbb{Z}$. For this, ...
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Projections and projective modules of $C(X)$

This is a followup to this question I made yesterday (disclaimer: I'm new here and I'm not sure if asking a new but related question is the correct procedure). If $X$ is a connected, compact, ...
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How to construct the map relating topological and algebraic $K$ - theory ?.

Let $X$ be a complex projective variety, and let $ X^{an} $ denote the topological space of complex points of $ X $ equipped with the analytic topology. Then, any algebraic vector bundle $ E \to X $ ...
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How to compare the K-theory and singular cohomology of a classifying space

In their foundational paper "Vector bundles and homogeneous spaces," Atiyah and Hirzebruch show, among many other things, that for $G$ a compact, connected Lie group, the K-theory of the classifying ...
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Classification of vector bundles over the torus

In M. Rieffel's paper "The Cancellation Theorem for projective modules over irrational rotation $C^*$-algebras", he classifies finitely generated projective modules over the $C^*$-algebra ...
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Real topological K-theory of cyclic group

Letting $C_n$ be the cyclic group on $n$ elements we know through the use of the Atiyah-Segal completion theorem that $$ K^*(BG) = \pi_*(KU)[[t]]/((t+1)^n -1) $$ where $\pi_*(KU)=\mathbb{Z}[u^{\pm ...
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what does Whitney sum of vector bundles correspond to in the k-theory KO?

Let $X$ be a $CW$-complex and $\text{Vect}^n(X)$ the collection of $n$-dimensional real vector bundles over $X$. Let $$ \text{Vect}^*(X)=\bigoplus_{n=0}^\infty \text{Vect}^n(X) $$ with addition ...
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“Global” dimension for topological spaces // Geometric interpretation for global dimension of rings

The global dimension of a ring $R$ is the supremum of the projective dimensions of it's $R$-modules. $$\dim (R)=\sup \{\dim_\mathrm{proj}(M):M \in R\text{-mod} \}$$ I'd like to have some geometric ...
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Open questions in Topological K-Theory

I am interested in knowing about current research in the Topological K-Theory, especially its interactions with String Theory. About one and a half decade back, there were some papers by Physicists ...
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Homotopy fixed points of connective K-theory

Let $ku$ be the $p$-completion of the connective complex K-theory spectrum. The group $\Bbb{Z}_p^\times\cong \Delta \times \Bbb{Z}_p$ acts on $ku$, where $\Delta$ is the cyclic group of order $p-1$. ...
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Torsion element in the real topological K-group of S$^2$

As we know, the topological $K$-group $K_\mathbb{R}(S^2)=\mathbb{Z}\bigoplus\mathbb{Z}/2$. Which bundle represents the $\mathbb{Z}/2$ part? I know that the tangent bundle of S$^2$ is not trival by ...
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Tautological line bundle over rational projective space

Is the tautological line bundle over $\mathbb{Q}P^{n}$ a non trivial bundle? Here, $\mathbb{Q}P^{n}$ has the natural topology induced from the standard topology of $\mathbb{Q}$ as a subset of ...
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obstruction to lifting a projective bundle to a vector bundle

Why is the obstruction, to lifting a projective bundle to a (complex) vector bundle on a space $X$, given by an element $\alpha \in H^3(X, \mathbb{Z})$? Vector bundles are classified by $H^1(X, ...
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Proof that $K^*(BG)=K^*(BT)^W$.

I was wondering if anyone had a refrence for the fact that $K^*(BG)=K^*(BT)^W$ for $G$ a compact connected lie group, $T$ a maximal torus, and $W$ the associated Lie group. I was able to derive this ...
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Atiyah-Singer index theorem

I have seen the cohomological form of the index theorem usually stated in the following form: $$ \int_X \varphi^{-1}\left(\operatorname{ch}([\sigma(P)])\right).\operatorname{todd}(TX\otimes\mathbb C) ...
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Is paralellizability a topological invariant (Invariant under homemorphism)

This MO post is a motivation to ask: Is paralellizability a topological invariant (Invariant under homeomorphism)?
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How can one compare these two 4-manifolds

We would like to compare the following two real 4 dimensional manifolds: 1)$M$=The tangent bundle of $S^{2}$ 2)$N$= The total space of the canonical line bundle over $\mathbb{C}P^{1}\simeq S^{2}$ ...
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Definition of $K_1(A)$ for a Banach algebra $A$

When defining $K_1(A)$ for a Banach algebra $A$, one may consider $\bigcup_{n\in\mathbb{N}}\{x\in GL_n(A^+):x\equiv I_n\mod M_n(A)\}$ and take the quotient by the component containing the identity, or ...
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Does symplectic K-theory $KSp$ have products?

The real and unitary topological $K$-theories are cohomology theories defined by the $\Omega$-spectra $KO$ and $K$ respectively. These are multiplicative theories with products deriving from the ...
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A problem on isomorphic sections of locally trivial vector bundles

Let $E=(E,\pi,X)$ be a locally trivial vector bundle over a compact Hausdorff space $X$. Let $\Gamma(E)$ be the set of all sections in E. I am trying to prove that $E$ is ismorphic to the trivial ...
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$Z_2$ Equivariant K-theory of $S^1$

I am interested in the $\mathbf{Z}_2$ equivariant K-theory of $S^1$, but I cannot find any good references or methods to calculate it with the action I have in mind. The action on $S^1$ is an ...
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(basic?) isomorphism of topological K-theory and reduced K-theory

first I want to state which definitions I use for K-theory and reduced K-theory: Let $X$ be a compact topological Hausdorffspace and $V(X):=\{\text{Isomorphism classes of (complex) vector bundles ...
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Thom space for complex bundles

It is well known that for real vector bundle with Riemannian metric we can construct Thom space using associated disk and sphere bundles. Can we do it for complex bundle with hermitian metric ?
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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 ...
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Is the $K_1$-group of a compact subset of the plane free?

Let $X$ be a compact subset of the complex plane. Is it true that $K_1(C(X)) = K^1(X)$ is a free abelian group? My only basis for thinking this is that it is true for a finite wedge of circles.
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Uncertainties on the details of the Connor-Floyd isomorphism and Formal Group Laws

Let $\Omega^{\bullet}(-)$ be the complex cobordism cohomology. $\Omega^n(X) = \{ (M, f) | f: M \to X \}$ where cobordant maps are identified, $M$ and $X$ are smooth manifolds, and dim($M $) = $n$. ...
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K-theory formulation of the index theorem

The Atiyah-Singer index theorem is one of the most important results in twentieth century's mathematics. It states that for an elliptic differential operator $D$ on a smooth, oriented compact manifold ...
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Atiyah K theory

On page 3 of Atiyah's book on K theory (link here: http://www.cimat.mx/~luis/seminarios/Teoria-K/Atiyah_K_theory_Advanced.pdf) he states: "Since a vector bundle is locally trivial, any section of a ...
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Exercises in Topological K-Theory (Atiyah)

I'm currently working through Michael Atiyah's K-Theory. The main problem I'm finding with it is that it does not have any exercises. Does anyone have a good collection of exercises to go along with ...
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Holomorphic $K$-theory

$K$-theory is traditionally defined for arbitrary compact Hausdorff spaces. If instead we require the base to be a complex manifold and work with only holomorphic vector bundles, in what ways would ...
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$K$-theory for non-compact manifolds

It seems that usually (by which I mean, in every source I've looked at) people define the group $K^0(X)$ for $X$ compact Hausdorff. Sometimes they later extend this definition to all locally compact ...
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A smooth non-stably trivial smooth vector bundle

This may well be just a look-up, but do you have an example of a non-stably trivial smooth vector bundle? If it has a presentation as the vector bundle associated to the representation of some ...
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Complex bundles on $S^{2n+1}$

We know that the complex $K$ theory of spheres are 2 periodic. On the other hand every complex bundle on $S^{1}$ is trivial. So $K(S^{2n+1})=0$. So this is a motivation to ask: Is there a ...
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K-theory product trivial on $K^1$ due to stability?

Let $K^0$ be the topological complex vector bundle $K$-theory functor. On the one hand, we have that the product on $K^0(\Sigma X)$ vanishes by a Mayer-Vietoris argument. On the other hand, we define ...
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K-theory of $\mathbb{RP}^{\infty}$

what are the $K_0$ and $K_1$ group of $\mathbb{RP}^{\infty}$? Any reference would be good enough.
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Higher homotopy groups

Theorem 5.1 of this paper describes a map $K_n(R)\to \pi_{n+1}(SK(E(R),1))$, where $S$ denotes the suspension. My question: Do we have a map from $K_n(R)\to \pi_{n+1}(S^2K(E(R),1))$. Any reference is ...
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vector bundles on $X \times S^2$ built via clutching functions

Let $X$ be a compact Hausdorff space and $E \to X$ a (complex) vector bundle over $X$. We can build a vector bundle on $X \times S^2$ by clutching functions, i.e an automorphism $f:E \times S^1 \to E ...
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About the geometric interpretation of the reduced external product in K-theory

I'm trying to fill the details of the explanation of the interpretation of the reduced external product. I'll follow Hatcher second books, and I'll write part of his explanation in the gray boxes ...
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The long exact sequence in reduced K-theory. How to glue together the short ones?

I'm trying to filling the details about the construction of the long exact sequence in K-theory. I'm using the notation of Hatcher's book (pages 52-53). here is a related question, even thought it's ...
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Complex (topological) K-theory of $S \Sigma$ for a surface $\Sigma$.

In the following question here: What's the K-group of a surface? The $K$-theory of a compact orientable surface is computed. I was curious if it was also possible to compute the "higher" ...
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“Visual” interpretation of the Bott Periodicity for complex vector bundles

I've just read the Bott Periodicity proof (complex case) in Hatcher book about K-theory. At first it seems that using this theorem I could conclude that every complex vector bundles over $S^{2n+1}$ ...
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Linear clutching function properties

I'm trying to prove proposition $4.8$ of Husemöller (pay 148) Here it is the text: Where the only hints are: $[L^{n+1}(\xi),L^{n+1}(zp)] \approx [L^{n}(\xi),L^{n}(p)] \oplus [\xi,z]$ ...
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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: ...
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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 $$ ...
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Working out the details of example 1.13 Hatcher: $\ E_{fg}\oplus n \approx E_f \oplus E_g$

I'm studying an example provided by Hatcher in his K-theory and Vector Bundle book. I'm referring to example 1.13 pag. 24 The first part is clear, $z^2$ is the Kronecker Product of the two ...
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Structures on vector bundles

I am reading the book K theory by Atiyah. In page number 32, he defines some additional structure on a vector bundle $V$. I have understood the definitions there. But there is a statement that says ...
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Relation between $K$-theories

I apologize in advance if this question is too vague/general. I am curious to know how all of the different $K$-theories are related (algebraic $K$-theory, topological $K$-theory, twisted $K$-theory, ...
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Necessity - Reasons of a passage in a proof of Hatcher vol. 2

Some words about the context of the proposition. Hatcher is defining operations on vector bundles over the same base space $B$. We are speaking about the Whitney sum here, and it is proving that if ...
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differential structure of $S^n$ and diffeomorphism map

Let $S^n$ be the unit sphere in $\mathbb{R}^{n+1}$, and $TS^n=\{(x,v): x\in S^n,v\in T_x S^n\}$. Show that $F: TS^n\times\mathbb{R}\to S^n\times\mathbb{R}^{n+1}$ given by ...