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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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Hopf invarient for $f:S^3 \to S^2$

Let $\alpha \in \Omega^{2}(S^2)$ be such that $\int_{S^2}\alpha=1$, and let $f:S^3 \to S^2$ be smooth map. i) Prove that there exists a l-form $\omega \in \Omega^{2}(S^3)$ such that $f^*\alpha=d\...
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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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Why we call monomial matrices?

Currently, I am reading Milnor’s book on algebraic K-theory, where he defined a monomial matrix over commutative ring with 1 to be a matrix of the form PD, where P is a permutation matrix and D is a ...
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Problems on exercise 7.G in the book “K-Theory and C*-Algebras”

I have a lot problems on exercise 7.G in the book K-Theory and C*-Algebras by Wegge-Olsen. $\newcommand{\C}{\mathbb{C}}$ $X\subset \mathbb{C}$? As I know the character space of $C^*(u_1,u_2)$ is ...
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$K_0(R)$ is generated by $[R]$?

Let $R$ be a unital associative ring. Let $F$ be the free abelian group on the set of all isomorphism classes $[P]$ of f.g. projective $R$-modules $P$. Let $K_0(R)$ be the quotient of $F$ modulo the ...
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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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Commuting Elements in Tensor Products of C*-Algebras

I am working on exercise 7.G in the book “K-Theory and C*-Algebras” by Wegge-Olsen. Let $A$ be some unital C*-algebra, $u$ a unitary in $M_n(A)$, and $u’$ a standard unitary (which is defined to be a ...
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Categories with zero Grothendieck group

I am interested in examples of triangulated categories that have zero Grothendieck group but are somehow still interesting, say have non-zero Hochschild homology. More example, for what rings $R$ is ...
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Clarifying how we can identify $SK_1(R)$ with set of path components

Let $R$ be a commutative Banach algebra. Let $E_n$ denote the group generated by $n\times n$ elementary matrices. $SK_1(R)$ denotes the kernel of the induced determinant map $K_1(R) \to R^{\times}$. ...
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Decomposition of $K_0$ with Whitehead group

Let $Wh_0(G) ={K_0(\mathbb Z[G])}/{\mathbb Z}$, where $\mathbb Z[G]$ is the group ring of $G$ over $\mathbb Z$. Weibel states that the augmentation map $f: \mathbb Z[G] \to \mathbb Z$ induces the ...
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1answer
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When $\mathbb Z$ is a direct summand of $K_0(R)$

Suppose that there is a ring homomorphism $R \to F$ where $F$ is a field. I'm trying to verify that $\mathbb Z$ is a direct summand of $K_0(R)$. We have an induced ring homomorphism $(K_0(R), \...
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Is there an example of a non-zero projection in a C$^{*}$-algebra that is infinite but not properly infinite?

For clarification: Given a projection $p$ in a C$^{*}$-algebra $A$, we say $p$ is infinite if there is a projection $q\in A$ satisfying $q\lneq p \sim q$; we say $p$ is properly infinite if there are ...
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Is the geometric realization of a pointed category contractible?

Given a pointed category $X$, is $|N(X)|$ contractible ($N(X)$ is the nerve of $X$)? Or are there counter-examples?
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Definition of algebraic K-theory space

Let $(C,wC)$ be a Waldhausen category. The algebraic K-theory space is the loop space of the classifying space of the simplicial pointed category $wS_*C$, i.e. of the topological realization of the ...
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1answer
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Degreewise cofibration in $S_nC$

Given a category $C$, we have $S_nC=Fun(Ar[n],C)$ and given $A,B\in Ob(S_n(C))$, (i.e. A,B are functors from $Ar[n]$ to $C$), then what does a morphism $f:A\to B$ in $S_nC$ mean by degreewise ...
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1answer
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Understanding Adams spectral sequence and Pontryagin-Thom isomorphism intuitively

The question is about understanding Adams spectral sequence intuitively and some of the meanings of its relations. In Adams spectral sequence, $$E_2^{s,t}=\text{Ext}_{\mathcal{A}}^{s,t}(H^*(MTG), \...
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1answer
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Atiyah's K Theory, pg 4

On page 4, of Atiyah's $K$ theroy he stated Suppose $V,W$ are real f.d. v.s, $E = X \times V$ and $F= X \times W$ are corresponding vector bundles. Then any homomoprhism $\varphi:E \rightarrow F$ ...
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How to calculate the $K_0$ and $K_1$ groups for $A$

Let $A=\{f\in C([0,1],M_n)\mid f(0)$ is scalar matrix $\}$. Then find the $K_0(A)$ and $K_1(A)$. I am trying to use the SES $J \rightarrow A \rightarrow A/J$ where $J$ can be taken as some closed ...
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is K theory functor continous with respect to inverse limit?

l know that K functor is continous with respect to direct limit, how about inverse limit ?does inverse limit exist in general in the caegory of topological space(or C star algebra). is there a ...
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Was Atiyah's proof of the odd order (Feit-Thompson) theorem false?

I read last year that Atiyah thought he had found a proof of the odd order theorem of only 12 pages, using $K$-theory, and that people were trying to figure out if it was correct or not. But I never ...
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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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Index map on the Hilbert A-module $A\otimes H$

I'm working on K-theory. Let H be an infinite dimensional separable Hilbert space and $A$ a $C^{\star}$-algebra. Let put $\mathcal{Q}(H):=\mathcal{B}(H)/\mathcal{K}(H)$ the Calkin algebra. I've ...
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2answers
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Projections in the Cuntz algebra which have the same $K_0$ class

Assume that $e,f$ are two projections in the Cuntz algebra $\mathcal{O}_n$ , which have the same $K_0$ class. Are they necessarily Murray von Neumann equivalent ? The following post is the ...
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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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$K_n(A)$ and $K_n(A/I)$ with $I^2=0$

For any commutative ring $A$, let $K_n(A)$ be the $n$-th algebraic $K$ group of $A$. Let $I$ be an ideal in $A$ such that $I^2=0$, there is a natural morphism $-\otimes A/I:K_n(A) \rightarrow K_n(A/I)$...
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Grothendieck group of $A$ and $A/I$ with $I^2=0$

For any commutative ring $A$, let $K_0(A)$ be the Grothendieck group of finitely generated projective modules, I want to study general $A$ by passing to the reduced case. Let $I$ be an ideal in $A$ ...
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1answer
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A continuous map arising from vector bundles

I can't see why something Atiyah says in page 27 of his book on K-theory is true. The context is the following. Let V be a complex vector space, and denote by $G_n(V)$ the set of all subspaces of $V$ ...
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higher K-theory: suspensions vs Clifford module bundles

Karoubi describes a model of K-theory built on triples: pairs of $C\ell(n+1)$ module bundles $E,F$ with isomorphisms $\alpha:E\rightarrow F$ of their underlying $C\ell(n)$ module bundles. The group of ...
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1answer
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question about one possible way of constructing Grothendieck group

There is a way to construct Grothendieck group from the given commutative monoid which can be found in many books with a chapter on K-theory( for example 'K-Theory' by Atiyah or 'Topology and Analysis'...
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K-theory and Clifford modules

I'm trying to wrap my head around the "Clifford modules" definition of K-theory. Let's just deal with K-theory of a point. One common definition of the $-n^\text{th}$ K-group is the quotient$$K^{-n}=M(...
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1answer
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Why is $U_n^+(A)/U_n^+(A)_0\simeq U_n(A^+)/U_n(A^+)_0$ for a unital C*-algebra $A$?

Was reading Wegge-Olsen's K-theory and C*-algebras and in chapter 4 they state that $U_n^+(A)/U_n^+(A)_0\simeq U_n(A^+)/U_n(A^+)_0$, to show that he says that $(a_{ij})+1_n$ is invertible (unitary) if ...
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1answer
55 views

Orthogonal complement of a vector bundle over paracompact base

In the book vector bundle and K-theory of Allan Hatcher, proposition $1.3$ state that If $p : E\to B$ is a vector bundle over a paracompact base $B$ and $E_{0} \subset E$ is a vector subbundle, ...
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Adams operations and an artificial grading on K-theory

In this article by Snaith (p. 575) appears the following comment: ... these transgressive elements [...] can be located by means of the Adams operations [...]. These operate (unstably) in both the ...
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1answer
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Traces on $K(H)$

Are there any traces on $K(H)$ for an infinite dimensional Hilbert space $H$? I think that the answer is no: viewing $K(H)$ as a direct limit of matrix algebras, if there was a trace we knew what it ...
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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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Milnor K-groups and classification of fields

For every field $F$ we can define $K_n^M(F)$ as the Milnor K-group of $F$ for each $n \in \Bbb N$ and form the Milnor K-ring $K^M(F)=\oplus_n K^M_n(F)$. My question is under which reasonable ...
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1answer
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Algebraic $K_2$ as “universal receptacle”?

In algebraic $K$-theory, $K_0$ and $K_1$ have nice descriptions in terms of the category of finitely generated projectives. $K_0$ is motivated as the "universal receptacle" for (additive) invariants ...
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1answer
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Are there examples of unital and nuclear $C^*$-algebras satisfying the UCT that are not groupoid algebras of an amenable etale groupoid?

Jean Louis Tu showed that the (maximal) groupoid $C^*$-algebra of a groupoid satisfying the Haagerup property (which includes all amenable groupoids) will satisfy the UCT. I am curious if there are ...
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1answer
61 views

Trivial K-theory implies trivial K-theory of hereditary corners?

Let $A$ be a $C^*$-algebra with trivial $K$-theory, that is $K_0(A)=K_1(A)=0$. Let $p$ be a projection in $A$. Does it follow that the hereditary corner $pAp$ has trivial $K$-theory? I think in the ...
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1answer
25 views

Opposite effective classes in a Grothendieck group

Let $\mathcal{A}$ be an Abelian category, and let $K_0(\mathcal{A})$ be its Grothendieck group. Is it possible to find two objects $A,B$ of $\mathcal{A}$ such that $[A]=-[B]$ in $K_0(\mathcal{A})$ ...
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An immediate result of fundamental theorem of algebraic $K$-theory.

The fundamental theorem of algebraic $K$-theory says that $K_1(R[t,t^{-1}])\cong K_1(R)\oplus K_0(R)\oplus NK_1(R)\oplus NK_1(R)$ On the page 153 of Rosenberg's book on algebraic $K$-theory, he said ...
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1answer
62 views

AF-algebras and K-theory

Suppose that $A$ and $B$ are AF-algebras and $\varphi, \psi \colon A \to B$ be $*$-homomorphisms with $K_0(\varphi) = K_0(\psi)$. Since $A$ is an AF-algebra, we may write $A = \bigcup_{n \in \mathbb{N}...
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1answer
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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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What is : $ \mu : RK_{*}^{ \Gamma } ( \underline{E \Gamma } ) \to K_{*} ( C_{r}^* ( \Gamma ) ) $?

Let $ \Gamma $ be a second countable locally compact group (for instance a countable discrete group). As it is said here : https://en.wikipedia.org/wiki/Baum%E2%80%93Connes_conjecture , one can ...
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1answer
52 views

On Elliot's classification of AF-algebras

I have two questions with a relation to Elliot's classification of Approximately Finite $C*$-algebras (from now on referred to as AF-algebras). Elliot's classical result yields that whenever $A$ and $...
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1answer
40 views

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
112 views

Understanding terminology of, fibers, clutchings and Hopf.

I have some questions regarding the terminology of fiber bundles as used in section 3 of this paper; http://www.sciencedirect.com/science/article/pii/S0723086907000151 The section starts off by ...
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1answer
306 views

Meaning of this exclamation mark?

In section 3 of the paper https://www.sciencedirect.com/science/article/pii/S0723086907000151 The author constructs a fiber bundle $(\rho_n)\zeta$ by taking the pullback of the diagram $S^8\...