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 [tag:algebraic-k-theory].

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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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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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Elementary and purely Topological Proof of the non-triviality of Tautological Complex Line Bundle

I need some hints about the proof of the non triviality of the tautological complex line bundle, in a pure topological manner. Let $E$ be the t.c.l. bundle defined in this way $$ E= \{ (x,v) \in ...
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Hairy ball theorem, projections and L.I. vectors

I'm trying to understand this paper which proves that not every unimodular row is completable by invertible matrices: Why we have these implications: There are two linearly independent vectors at ...
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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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$K$-theory exact sequence.

Let $Y$ be a closed subspace of a compact space $X$. Let $i:Y \to X$ the inclusion and $r:X \to Y$ a retraction ($r \circ i = Id_Y$). I have to prove that exists this short exact sequence $$ 0 \to ...
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Are these equivalent definitions of the Grothendieck group construction?

I'm learning K-theory and I'm slightly confused. Given an abelian monoid $M$, we can construct the Grothendieck group $$ G(M) = (M \times M)/\sim$$ and I've seen two definitions for the equivalence ...
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Exterior algebra as a complex of vector bundles and an element in relative K-theory

There is a description of relative $K$-theory in terms of complexes of vector bundles. The following is an excerpt from Atiyah's book. Let $V$ be a complex vector space and consider the exterior ...
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Relationship between topological and Quillen's K-theory

Up until now, I've taken it for granted that the topological k-theory of a space $X$ is equal to the K-theory of vector bundles on $X$. $K_0$'s of both coincide (Serre-Swan) however, is it the case ...
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Prerequisite to start learning Topological K-Theory

Wanted to start learning K-theory to see if it is suitable for a undergraduate thesis (from an algebraic-topological view). For reference, I only know Hatcher's book (mostly because I've read his book ...
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Question on the map from algebraic K_0 to topological K_0

Let $X$ be a scheme over a field $F$ where $F$ is either the real or the complex numbers. If $q:Y\to X$ is an algebraic vector bundle over $X$, the $F$-points $q(F):Y(F)\to X(F)$ constitute a ...
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What is a $c_1$-map for Riemann-Roch theorems?

Atiyah and Hirzebruch define a $c_1$-map to spell out Riemann-Roch theorem for (compact and connected) smooth manifolds. The definition is following: a map $f:Y \to X$ is called a $c_1$-map if we are ...
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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 ...
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Group completions and the induced homomorphisms

The group completion (aka Grothendieck group) of an abelian monoid $M$ is an abelian group $G(M)$ with a homomorphism $\iota:M \to G(M)$ of monoids satisfying the following universal property: for ...
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Property of elements in Grothendieck group

I'm reading Atiyah's K-Theory book and in the section where he introduces the Grothendieck group, he gives two constructions. One of them is as follows: Let $A$ be an abelian semigroup, let ...
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Yoneda's lemma and $K$-theory.

The K-theoretic form of Bott periodicity is that $\tilde{K}(X)\cong \tilde{K}(\Sigma^2 X)$, i.e., $\langle X, BU\times \mathbb{Z} \rangle \cong \langle \Sigma^2 X, BU\times \mathbb{Z} \rangle$. The ...
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$K$-theoretical interpretation of Bott periodicity.

We consider complex Bott periodicity $\pi_{i-1}(U) \simeq \pi_{i+1}(U)$. Why we can say that the $K$-theoretical interpretation of this assertion is $\tilde{K}(X) \cong \tilde{K}(\Sigma^2(X))$? Where ...
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Loop space suspension/adjunction

Let be $X,Y$ Hausdorff spaces. I denote with $\langle \cdot, \cdot \rangle$ the basepoint preserving homotopy classes of maps, with $\Sigma$ the reduced suspesion and with $\Omega$ the loop space. How ...
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Loop space and $K$-theory

How can I proove without using Yoneda's lemma that $$ \Omega^2(BU \times \mathbb{Z}) \cong BU \times \mathbb{Z} ?$$ In particular how can I define a cellular map $$ f: \Omega^2(BU \times \mathbb{Z}) ...
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Induced map on $K_{1}$-group

Let $A$ be a unital $C^{\ast}$-algebra. Any automorphism $\alpha$ of $A$ induces a map on $K_{1}(A)$ by $\alpha_{\ast}[u]=[\alpha(u)]$. Let the automorphism $\alpha$ be inner, does it follow that ...
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Index of twisted Dirac operator

I do not understand a step in the proof of the Lemma 11.4.1 in the book "Analytic K-Homology" by Higson, Roe. Let $S$ be a Dirac bundle over a closed manifold $M$ and $D$ the corresponding Dirac ...
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Chern Character Isomorphism for non-CW complexes

Atiyah and Hirzebruch showed in their paper "Vector bundles and homogeneous spaces" that $\mathrm{K}^\ast(X) \otimes \mathbb{Q} \cong \mathrm{H}^\ast(X; \mathbb{Q})$, where $\mathrm{H}^\ast$ denotes ...
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K-theory, $K_{0}$ of algebra of compact operators

I don't understand how to define the trace of a matrix with values in operators. This occurred in the following situation: Suppose that $H$ is an Hilbert space and $K$ is the algebra of compact ...
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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 $X$ ...
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How should we think of 'differences' of vector bundles?

Given a Hausdorff space $X$, the set of all vector bundles of finite dimensions $Vect(X)$ (up to isomorphism) with respect to the direct sum is a monoid. We can use the group completion construction ...
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How we do actually compute the topological index in Atiyah-Singer?

I am taking a lectured class in Atiyah-Singer this semester. While the class is moving on really slowly (we just covered how to use Atiyah-Singer to prove Gauss-Bonnet, and introducing ...
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long exact sequence in k theory

I am studying the basics of K-theory and given a CW pair $(X,A)$ I understand how to construct an long exact exact sequence $\cdots \rightarrow K(SX) \rightarrow K(SA) \rightarrow K(X/A) \rightarrow ...
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$K$-Theory of $BU(n)$

The following fact is asserted by wikipedia: The $K$-theory of $BU(n)$ is the numerical symmetric polynomials, i.e the subring of $\mathbb{Z}[x_1, \ldots, x_n]$ that is preserved under the action of ...
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K-theory for non-separable C*-algebras

Let $\kappa$ be an uncountable cardinal. What is the K-theory for the C*-algebras $\mathcal{K}(\ell_2(\kappa))$ and $\mathcal{B}(\ell_2(\kappa))$, of, respectively, compact and bounded operators on ...
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What's the K-group of a surface?

What's the K-group of a surface? I also want to know how to calculate such group and if there is a explicit characterization of the generators.
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How can I see that when $X$ is a trivial $G$-space, $K(X) \otimes R(G) \rightarrow K_G(X)$ is an isomorphism?

Here, $K$ is complex K-theory, $R(G)$ is the complex representation ring of $G$ (which -- for now, though it shouldn't matter -- is a finite group), and $K_G$ is $G$-equivariant complex K-theory. In ...
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8 periodicity: Clifford clock- Bott periodicity - KO-dimension in noncommutative geometries

Periodicity modulo 8 appears in the classification of real Clifford algebras $C\ell_{p,q}(\mathbb{R})$ (usualy refered to as the "Clifford Clock"), in real Bott periodicity and in the definition of a ...
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$K$-theory of smooth manifolds: continuous vs. smooth vector bundles

Suppose I have a smooth manifold $M$, and want to consider the $K$-theory $K^0(M)$. I could define this in the usual way (by taking the Grothendieck group of the monoid of equivalence classes of ...
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Weak Bott periodicity vs. strong Bott periodicity

Bruno Harris' proof (or I guess also Bott's original proof) of Bott periodicity (see here for instance) shows that there is a homotopy equivalence $h\colon\mathbb{Z}\times BU \rightarrow \Omega^2 ...
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Example of non-isomorphic vector bundles with the same element in $K$

Let $X$ be a paracompact and well behaved space. Topological K-Theory $K(X)$ of $X$ is group completing the monoid of isomorphism classes of vector bundles over $X$ with the Whitney sum. Two vector ...