Questions tagged [k-theory]

K-theory is the study of invariants of large matrices, in a suitable sense. It has many variations: (algebraic-k-theory), (topological-k-theory), or in the study of (operator-algebras).

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Defintion of a real algebraic space in Atiyah's K-theory and reality

In Atiyah's paper "K-theory and reality", p. 370, there is an example of a "real" algebraic space. Given the complex projective space $X=P(C^n)$, one considers the standard line-...
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Spin$^c$-structure necessary for K-orientation

In Atiyah, Bott, and Shapiro's Clifford Modules, Theorem 12.3, they prove that a Spin$(k)$-structure (resp. Spin$^c(2k)$)-structure gives a KO(K)-orientation on the associated vector bundle of rank $k$...
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Showing $K_0(F)=\mathbb{Z}$

In Morel's "$\mathbb{A}^1$-algebraic topology over a field", Milnor-Witt K-theory is defined to be the graded ring $K_*^{MW}(F)$ generated by the symbols, $[u]$ of degree $1$ and $\eta$ of ...
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Confused on Projective Bundle

I am working through Atiyah (https://www.maths.ed.ac.uk/~v1ranick/papers/atiyahk.pdf) and am confused on the projective bundle (pg. 44-45). Atiyah states that for a vector bundle $E$ over $X$, let $...
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The inverse of a map in the McKay Correspondence

So, I am trying to make my way through Gonzalez-Sprinberg and Verdier's paper on the McKay correspondence (here, in French), and I find myself very troubled by this one theorem that occurs very early ...
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Definition of vector bundle and K-theory

The definition of vector bundles seems to be split in the mathematical community: some sources insist that the rank of each fibre is the same, whereas some don't ask for this requirement. I was ...
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Husemoller Vector Bundles on $X \times S^2$

I'm currently studying Vector Bundles from Husemoller and I have two questions concerning Proposition $2.5$ (p.$142$) on Clutching Maps over $X \times S^2$. $1.$ Husemoller afirms that there is an ...
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Understanding the use of Hilbert modules in the definition of $K$-theory and $KK$-theory

Let $A$ be a unital $C^*$-algebra. Then elements of the $K$-group $K_0(A)$ are usually defined as equivalence classes of projections in matrix algebras over $A$. Such a projection, say $p\in M_n(A)$, ...
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Stably Equivalent and Isomorphism Class of Real Line Bundles

Let us denote by $\text{Vect}_{1}(X)$ the set of isomorphism classes of real line bundles of a compact nice manifold $X$. It is well known that the reduced $K$-group $\tilde{K}\mathcal{O}(X)$ can be ...
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Reference request: isomorphism in $K$-theory induced by inclusion of a full corner of a $C^*$-algebra

Let $A$ be a $C^*$-algebra $p$ a projection such that $ApA$ is dense in $A$. Let $B=pAp$. Then it is known that the inclusion $$i\colon B\hookrightarrow A$$ induces an isomorphism on operator $K$-...
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Identifying projections underlying projective modules over $C^*$-algebras

Let $A$ be a $C^*$-algebra and $B=pAp$ for some projection $p\in A$. Let $N=A^n q$ be a finitely generated projective (left) $A$-module, where $q$ is a projection in $M_n(A)$. Then there is a $B$-...
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Computing $K_1(l^\infty(\mathbb{N}))$

Let $l^\infty(\mathbb{N})$ be the $C^*$-algebra of bounded sequences $\mathbb{N}\to\mathbb{C}$. I would like to compute the $K_1(l^\infty(\mathbb{N}))$, which I suspect is the zero group. However, I'm ...
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Relative K-theory: constraint on dimensionality?

Following a lecture https://www.youtube.com/watch?v=hZHnUcy7rNc&list=PL3GPZrYvP8TjHuKUzrtUPiu_dwn8kYcqS&index=1&ab_channel=NDGeoTop on index and K-theory, the following definition is given ...
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K-Theory of $C(X)$ for $X$ not totally disconnected

My question is related to K-Theory of $C(X)$ for $X$ totally disconnected Question: Does anyone know an example of a compact Hausdorff space $X$ (that is not totally disconnected) where $K_0(C(X))$ is ...
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Question on Lemma 1.4.9 in Atiyah

The text can be found here: https://www.maths.ed.ac.uk/~v1ranick/papers/atiyahk.pdf I am having two struggles on the proof of Lemma 1.4.9. First, why is $C^+(X) \cap C^-(X) = X$? To me, it would ...
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$X$ is a connected compact metric space. Is every $C^*$-embedding $M_n(\mathbb C)\to M_n(C(X))$ unitarily equivalent to the trivial one?

$M_n=M_n(\mathbb C)$ is the $n\times n$ complex matrix algebra and $M_n(C(X))$ is the algebra of continuous functions from $X$ to $M_n(\mathbb C)$. There is a trivial homomorphism $M_n\to M_n(C(X))$ ...
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Do commutative $C^*$-algebras have torsion-free $K$-groups?

Assume $X$ is a metrix space. In the case that $X$ is an inverse limit of one-dimensional finite CW complexes, it is known that $K_0(C(X))$ and $K_1(C(X))$ are torsion free . So I tried some examples ...
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How do I show these two projections belong to different $K_0$ equivalence classes?

Denote by $S^2$ the sphere. By identifying $C(S^2)$ with functions on $D^2=\{z\in\mathbb C:|z|\leq 1\} $ which are constant on the boundary, one can define a projection in $M_2(C(S^2))$: $$f(z)=\left(\...
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Is real topological $K$-theory subsumed by complex topological $K$-theory?

The title pretty much captures my question. I understand that there are both real and complex $K$-theories of Hausdorff topological spaces depending on whether we look at real or complex bundles over ...
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What does countable weakly unperforated $\textbf{simple}$ ordered group mean here?

The second part of the main theorem states that, given any countable weakly unperforated simple ordered group $G_0$ with order unit $e$, any countable abelian group $G_1$, and any metrizable Choquet ...
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Simplifications in definition of KK groups ("Fredholm picture")

I want to understand the relation between the $KK$ groups (in the sense of Kasparov) and ordinary operator $K$-theory, i. e. I would like to prove something like this: For $B$ trivially graded $\sigma$...
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Finding the Picard Group of $\mathbb{C}[x,y,z]/(x^2+y-z)$. [closed]

I'm having problem comprehending the Picard group of a ring, and figured that perhaps it would be easier if I had a few examples of actual computations of Picard groups to look at. Thus, my question ...
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Milnor Squares and Milnor Patching: Examples?

In Weibel's book on K-theory, he introduces Milnor squares and Milnor patching as follows: I was wondering if someone might be able and willing to help me a little by constructing a nice friendly ...
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Lem. I.2.2 in Weibel's K-theory

Silly thing I cannot wrap my head around. Lem. I.2.2 in Weibel's K-book reads: If $R$ is a local ring, then every finitely generated projective $R$-module $P$ is free. In fact $P \cong R^p$, where $p ...
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References: Grothendieck Groups of Spaces and Varieties

Good people! I'm asking for some good references here! I need to find the Grothendieck groups for a couple of varieties, and I'm struggling with figuring out how to make this work... Would anyone of ...
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Why are stably free modules called stably free?

Being of the kind who easily get troubled by questions of semantics, I've been troubled for a while now about why "stably free" modules are called just "stably free". It seems to ...
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Relation between group of components and a torsion subgroup in a long exact sequence in cohomology

Consider the short exact sequence (call it "coefficient sequence'') $$ 0 \to {\bf{Z}} \to {\bf{R}} \to U(1) \to 0$$ Suppose $X$ is a topological space. The long-exact sequence in cohomology ...
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Isomorphism class of the zero module is not a set

I'm having a bit of trouble with this one question (3A.2) in Bruce A. Magurn's An Algebraic Introduction to K-Theory. It's about set theory, which has always managed to confuse me more than it should. ...
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Where do $k$-invariants actually live?

I'm looking at Brunner's version of Adam's computation showing $H^*(ku;\Bbb Z_2)=\mathcal A/\mathcal A(Q_0,Q_1)$. Here $ku$ is the connective cover of the usual complex $K$-theory spectrum $K=KU$. The ...
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Symmetric power of a virtual representation, and decomposition of tensor power

Given any representation of $G$ on $V$, one can define symmetric power and exterior power as usual. Then the symmetric algebra $S^\bullet (V)=\bigoplus_{k\geq 0}S^k(V)$ and exterior algebra $\Lambda^\...
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Why is $\mathcal H_B\oplus \mathcal H_B\simeq \mathcal H_B$? Question about Hilbert $C^*$-modules and Kasparov's $KK$-Theory.

I am reading THE OPERATOR K-FUNCTOR AND EXTENSIONS OF $C^*$-ALGEBRAS To cite this article: G G Kasparov 1981 Math. USSR Izv. 16 513 Definition 1.1. $G$ isa fixed compact group satisfying the second ...
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How is $\mathcal L(E)$ graded? Question about Hilbert $B$-module

I am reading G.G.KASPAROV's THE OPERATOR AT-FUNCTOR AND EXTENSIONS OF C*-ALGEBRAS Let $B$ be a C*-algebra and let $E$ be a $B$-right-module. Assume there is a $B$-valued inner product on $E$ which (...
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Lambda operations in K-theory

In Weibel's K-Book, before defining the lambda-operations on higher (Quillen) K-theory, he states that "Although many constructions of $\lambda$-operations have been proposed in more exotic ...
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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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Colimit of Chow groups on Zariski open subschemes.

Given a smooth affine scheme $X$, let's consider $\varinjlim\limits_{U\subset X}\tilde{K}_0(U)$, where $\tilde{K}_0(U)$ is the reduced zero-th $K$-group of the Zariski open subscheme $U$. Note that ...
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Atiyah-Hirzebruch spectral sequence for twisted K-theory for infinite-dimensional CW-complexes

I'm writing a thesis that runs an AHSS for twisted K-theory over an Eilenberg-MacLane space, namely an infinite-dimensional CW-complex. In Atiyahs works Twisted K-Theory and Twisted K-Theory and ...
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Analytic K-Homology (Higson, Roe) - Exercise 8.8.8

I am trying to understand the proof of Proposition 8.3.16 (***) in Higson and Roe's Analytic K-Homology book. They rely on Exercise 8.8.8 (which is the only argument I don't fully understand). It goes ...
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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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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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$\operatorname{trace}(e_1e_2)=\operatorname{trace}(f_1f_2)$ for certain idempotents of a $C^*$ algebra with trace

Indirectly inspired by this post we ask the following question: Let $A$ be a $C^*$ algebra which is equiped with a faithful positive normal trace. Assume that $e_1,e_2,f_1,f_2$ are idempotents in $...
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How to construct an $AF$-algebra $N$ with $K_0(N)=\mathbb Q$?

On the paper i'm reading, it says letting $N=\lim M_2\otimes ...\otimes M_{n!}$ will do. I wonder why it doesn't simply let $N=\lim M_2\otimes ...\otimes M_n$. Take arbitary $p\in N\otimes M_n$, there ...
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What is $K_1(B(H))$? (operator K-theory)

Let $H$ be a (separable) infinite-dimensional Hilbert space. Can someone give me a reference or a proof why the identity $$K_1(B(H)) = 0$$ is true? The more elementary the proof, the happier I am. ...
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identity matrices stably equivalent?

I'm reading the K-theory chapter in Murphy's C*-algebras and Operator Theory book. He defines stable equivalence to be $p \approx q$ if there exists positive integer $n$ such that $$ \begin{pmatrix}...
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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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4 votes
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K-Theory, how to start [duplicate]

This semester I'm studing courses about representation theory, ring theory, algebraic topology and homological algebra, and this huge mix between algebra and topology made my think about other ways to ...
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3 votes
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Prove that $v=vv^*v$ for a partial isometry $v$.

Let $A$ be a $C^*$-algebra and let $v$ be a partial isometry so that $v^*v$ is a projection. Show that $v=vv^*v$. The hint in the book recommended setting $z=(1-vv^*)v$ and then computing $z^*z$. I ...
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Path-lifting property.

What is the typical example $\pi \colon X \to B$ of a surjective morphism between topological (or $C^{\infty}$) manifolds, i.e. locally euclidean space via continuous bijections, where path lifting ...
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Resolution of coherent sheaves on abelian varieties.

If $A$ is a commutative unital ring and $E$ is a finite rank projective $A$-module there is a surjective $A$-linear map $\phi: A^n \rightarrow E$, with kernel $F:=ker(\phi)$ and $F\oplus E \cong A^n$ ...
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