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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Some questions about Cuntz’s proof of the $ K_{1} $-injectivity of purely infinite simple unital $ C^{*} $-algebras

I have some questions about Joachim Cuntz’s proof of the $ K_{1} $-injectivity of purely infinite simple unital $ C^{*} $-algebras, which is found in this paper. For this post, let us adopt the ...
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Analysing Exact Sequence

I have the following exact sequence $\mathbb{Z}\xrightarrow{f}\mathbb{Z} \xrightarrow{g} K_0(\mathcal{T})\xrightarrow{h}\mathbb{Z}\xrightarrow{0}0$. From here I want to conclude that ...
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A problem on isomorphic sections of locally trivial vector bundles

Edited: Let $E=(E,\pi,X)$ be a locally trivial vector bundle over a compact Housdorff space $X$. Let $\Gamma(E)$ be the set of all sections in E. I am trying to prove that $E$ is ismorphic to the ...
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K-theory,proper class,set,isomorphism types

Define $K$ as the free abelian group with generators $[A],[A'],[A'']$,the equivalence classes of isomorphism types, modulo $[A]=[A']+[A'']$ where $0\to A'\to A \to A''\to0$ is a s.e.s. of modules in ...
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K-Theory of $C(X)$ for $X$ totally disconnected

I am studying K-Theory for C*-algebras by the following book: Rordam, Larsen and Laustsen. I am having a problem with the the Exercise 3.4, which is: Let $X$ be any compact Housdorff space. In the ...
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What is the definition of polygonal loop?

How to prove "every loop can be uniformly approximated arbitrarily closed by a quotient of polynomial loops?" In Blackadar's book, the proof is completed with the help of "polygonal loops," but what ...
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Morita equivalence and KK-theory

Let $A,B,C$ be $C^\ast$-algebras. Suppose $B$ and $C$ to be strongly morita equivalent. Then $KK(A,B)\cong KK(A,C)$. Could someone provide a reference or proof of this fact? I guess the ...
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29 views

What is the Grothendieck group of finitely generated $R[G]$-modules?

Let $R$ be a ring with unity, $G$ a finite group and $R[G]$ the group ring. What is the definition of the Grothendieck group of finitely generated $R[G]$-modules? How is this connected to the ...
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Why is the K-theory of $X$ product of reduced $K$-theory and $\mathbb{Z}$

The reduced $K$-theory of $\tilde{K}(X)$ of the based space $X$ is the kernel of $d:K(X)\to \mathbb{Z}$, where $d$ is induced by $d:Vect(X)\to\mathbb{Z}$ that sends a vector bundle to the dimension of ...
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Computing the Grothendieck group of affine space.

For a Noetherian scheme $X$ the Grothendieck group $K(X)$ is defined as the free abelian group on coherent sheaves quotiented by the equivalence relation $\mathscr{F}=\mathscr{F}'+\mathscr{F}''$ for ...
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What is the quotient $\mathbb{T}^3/\mathbb{Z}_2$?

What is the quotient space $\mathbb{T}^3/\mathbb{Z}_2$, When $\mathbb{Z}_2$ acting on 3-dimensional torus $\mathbb{T}^3 := \mathbb{R}^3/\mathbb{Z}^3$ by sending $x$ to $-x$? Does anyone know how to ...
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58 views

Quillen's K-theory and ring homomorphisms

I am a beginner in algebraic K-theory and I want to make sure that I understand the following correctly: Let $f:A \to A'$ be an isomorphism of commutative rings. Denote by $\mathcal{P}(A)$ (resp. ...
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What is the 'Hom-description'?

I am trying to learn about the 'Hom-description' of the class group $Cl(A)$ of an $R_K$-order $A$ in $K[G]$ where $K$ is a number field with ring of integers $R_K$ and $G$ is a finite group. I've ...
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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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When does a smooth projective variety X have a free Grothendieck group

Let $X$ be a smooth projective variety (e.g. Grassmannians). Since $X$ is smooth, the groups $G_0(X):=K_0(CohX)$ and $K_0(X):=K_0(VectX)$, the Grothendieck groups of coherent sheaves of modules on $X$ ...
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A question on algebraic K-theory- show that $K_1(R)\cong R^\times$ if $R$ is a field.

If $R$ is a ring we write $K_1(R)$ for the abelian group $K_1({\rm category\; of\; finitely\; generated\; projective\;R-modules})$. Swan's 'Algebraic K-Theory' defines $K_1$ in terms of the usual ...
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Algebraic results using lower K-theory as a blackbox

There is an algebraic K-theory seminar at my school and we are struggling to find applications from areas other than topology. We'd like a nice statement like "If X then Y" whose proof makes ...
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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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72 views

$K$-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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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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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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Why is $K_{0}(C(\mathbb{D}))\rightarrow K_{0}(C(\mathbb{T}))$ injective?

There is the restriction map $\pi:C(\mathbb{D})\rightarrow C(\mathbb{T})$ where $\mathbb{D}$ is the closed unit disk and $\mathbb{T}$ is the unit circle. Why is $\pi_*:K_0(C(\mathbb{D}))\rightarrow ...
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Bott projection

This is part of an exercise problem (5.I) in Wegge-Olsen's book "K-theory and $C^*$-algebras". There he defines the Bott projection for $\mathbb{R}^2$ by $B:\mathbb{R}^2\rightarrow\mathbb{M}_2$, ...
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34 views

Finding a Monoid

I want to find the monoid $ colim_n \pi_0 (\mathcal{I} M_n(\mathbb{C})) $ which I know is isomorphic to $ \pi_0 (colim_n \mathcal{I} M_n(\mathbb{C})) $ where $ \mathcal{I}$ is the set of idempotents ...
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If there is already enough room to add all projections, does passing to matrices change anything?

Throughout, $A$ denotes a $*$-algebra. We always assume $A$ is representable in the sense that $A$ can be embedded into $B(H)$ for some Hilbert space $H$. The particular embedding is not important, ...
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K-theory computation for algebra of bounded continuous functions on $[0,\infty)$

I want to compute the K-theory of $C_{b}[0,\infty)$, the algebra of bounded, uniformly continuous functions on $[0,\infty)$, by considering the exact sequence $0\rightarrow C_{b,0}(\bigcup_{n\geq ...
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On an invertible element for equivariant K-theory

Fix a positive integer $m$. Let $G = \lbrace h\in\mathbb C | h^m = 1\rbrace$ and $(X,\pi)$ the standard representation of $G$. Namely $X = \mathbb C$ and $\pi:G \to GL(X)$ is defined by $\pi(h)v=h v$ ...
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Can anyone give an example of two stably equivalent projections that are not Murray Von Neumann equivalent?

Two projections $P$, $Q$ are MvN equivalent in $C^*$-algebra $A$ when there is an element $u\in A$ such that $uu^*=P$ and $u^*u=Q$, and two projections $P$, $Q$ are stably equivalent if $P\oplus ...
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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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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 ...
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Strong and weak equivalence of $C^*$-extensions by compacts

Let $A$ be a $C^*$-algebra. An extension of $A$ by the compact operators $K$ is an embedding $\epsilon$ of $A$ into the Calkin algebra $B(H)/K$. Two embeddings $\epsilon_1$ and $\epsilon_2$ are ...
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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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Proving the inclusion map induces isomorphism on $K$-theory

Let $M$ be a $C^\ast$-algebra, $A, B$ be closed, two-sided ideals of $M$ such that $A+B=M$. Define $T=\{f\in C([0, 1], M):f(0) \in A, f(1) \in B\}$. Why is that the inclusion map of $C([0, 1], A\cap ...
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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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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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Connection in the KK-Theory

I have some questions about the connection in the KK-Theory. 1)The definition is complicated, why? What is the motivation? 2)Does any relation bewteen the connection at here with the differential ...
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Duality between K-theory and K-homology in the non-spin^c case.

Let M be a closed manifold. Then there is a cap product $K^\ast(M) \times K_\ast(M) \to K_\ast(M)$ between the K-theory of M and its K-homology. For a definition of it one could see my prior question ...
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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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Cap product between K-Theory and K-Homology

In Exercise 9.8.9 of the book "Analytic K-Homology" by Higson and Roe one has to construct a cap product $K_p(A) \otimes K^q(A) \to K^{q-p}(A)$, if A is commutative. Is the commutativity ...
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234 views

Examples of Eilenberg-type Swindles

I am compiling a list of 'swindles' in the style of the Eilenberg-Mazur swindle. I've already got some swindles in K-theory, the Mazur Swindle and the proof of the Cantor–Bernstein–Schroeder theorem. ...
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What properties of a C*-algebra are reflected in their $K_0$ groups?

Title says it all. I've been looking at K-groups of a few C* algebras. My very rough understanding is that these groups reflect (in some appropriate sense) the algebraic structure that we get when ...
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555 views

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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Characteristic polynomial of the unique automorphism of the zero module

Is there any convention which makes sense of the characteristic polynomial of the unique automorphism of the zero module? This might seem like an odd question but it matters to me. The background ...
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Products in $C^*$-algebra $K$-theory

Let $A_1$ and $A_2$ be unital $C^*$-algebras. If $p_1 \in M_{n_1}(A_1)$ and $p_2 \in M_{n_2}(A_2)$ are projections then $p_1 \otimes p_2 \in M_{n_1 n_2}(A_1 \otimes A_2)$ is also a projection, ...
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Computing $K$ - theory groups of certain $C^{\ast}$- algebras

I am trying to show that the $K$-theory groups of the following $C^*$-algebra $A$ vanish: Let $\mathcal{H}$ be a separable Hilbert space. Now consider the subalgebra of ...
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Operator K-theory and the unitary group

Suppose that $A$ is a C*-algebra whose unitary group is contractible (e.g. $B(H)$ or more generally the stable multiplier algebra of any C*-algebra). It is clear from the definition that $K_1(A) = ...
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Effect of permuting coordinates on K-theory

Let $A$ be a C*-algebra and let $f: \mathbb{R}^n \to \mathbb{R}^n$ be the linear map which permutes the coordinates via a permutation $\sigma$. There is an induced map $K_0(C_0(\mathbb{R}^n) \otimes ...
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Applications of Elliott's theorem concerning the classification of AF-algebras

An AF-algebra is a $C^* $-algebra which is the inductive limit of an inductive sequence of finite-dimensional $C^*$-algebras. Elliott's theorem concerning the classification of AF-algebras says that ...