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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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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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68 views

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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53 views

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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22 views

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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30 views

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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94 views

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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33 views

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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56 views

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
23 views

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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27 views

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
30 views

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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71 views

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
46 views

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
48 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
51 views

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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35 views

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
25 views

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
59 views

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
52 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
24 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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33 views

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
59 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
32 views

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
37 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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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
298 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\...
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1answer
30 views

Taking Grothendieck group of an already abelian group?

If $C$ is a commutative semigroup, then $K_0(C)$ is an abelian group associated to $C$, called the Groethendieck group of $C$. This satisfies the universal property; for each abelian group $A$, and ...
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The Exponential Map of the “Suspension-Cone Extension” of a C$^{*}$-algebra

Background. This is problem 12.4 in Rordam's C$^{*}$-algebra book. Let $A$ be a C$^{*}$-algebra. The suspension of $A$ is defined to be $SA:=\{f\in C([0,1]): f(0)=f(1)=0\}$. The cone of $A$ is ...
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Elliott-Natsume-Nest proof of Bott periodicity for $K$-theory of $C^\ast$-algebras

The following is an exercise in the book "$K$-theory and $C^\ast$-algebras" by Wegge-Olsen: Let $S$ denote the suspension functor. Suppose that we have natural transformations $\Phi^0$ from the ...
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30 views

Complex K-theory: Inducing natural transformations to $BU$

Consider topological complex $K$-theory, in this case for a connected finite CW-complex $X$ we have the following $K(X)=[X,BU]$. Naively I'd like to use the natural isomorphism $K(-)=[-,BU]$ to take ...
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1answer
88 views

Connected Unitaries in $K_{1}(C(\mathbb{T}^{3}))$

Let $\mathbb{T}=\{\xi:|\xi|=1\}$ and set $A:=C(\mathbb{T}^{3})$. Then, $A$ is a unital commutative C$^{*}$-algebra. For any C$^{*}$-algebra $B$, let $\mathcal{U}(B)$ denote the unitary elements in $B$...
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1answer
44 views

Fredholm operators, rank, and closure

Let $D$ be a densely defined symmetric unbounded operator on a Hilbert space $H$. Assume that the kernel and cokernel of $D$ are finite dimensional, and hence that we can assign a well-defined index ...
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1answer
53 views

Concerning the Connectedness of Certain Unitaries in the Stabilization $\mathcal{K}A$ of a C$^{*}$-algebra $A$

I am working through Rordam's book on $K$-Theory for C$^{*}$-algebras and I am stuck on one detail in exercise 8.17. Let $A$ be a unital C$^{*}$-algebra. In this book, the stabilization $\mathcal{K}A$...
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51 views

Linear algebra and K-theory

I am looking for a reference where I can learn both linear algebra, and K-theory. Does such a reference exist?
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1answer
27 views

What is the topology on the product bundle

If $X$ is a topological Space and $V$ is a vector Space then $X\times V$ forms a vector bundle over $X$, but I do not understand how do we topologise the space $X\times V$ so that it forms a vector ...
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2answers
163 views

How can I determine the Steenrod Square $Sq^2$ for complex projective space?

I am trying to learn about Steenrod Squares for algebraic varieties so that I can compute examples of complex topological K-theory using the Atiyah-Hirzebruch Spectral Sequence (AHSS). One of the key ...
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1answer
38 views

What is wrong in my computation of K groups?

I assume the sequence $0\to C_0(\mathbb{R})\to C(S^1)\to\mathbb{C}\to 0$ is an exact sequence of C*-algebras. Then I suppose we can use the Bott Periodicity, note $K_0(C_0(\mathbb{R}))=0, K_0(C(S^1))=\...
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1answer
88 views

Lemma 7.2 in Clifford Modules by M. A. Atiyah, et al.

I am trying to understand the proof of lemma 7.2 in the paper Clifford modules by Atiyah, Bott and Shapiro (Topology, Vol. 3, Supp. 1, 3-38). However I encountered a confusing statement: Right at the ...
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1answer
39 views

Ample Subspace of Sections

I am reading K-theory by Michael Atiyah, and I am confused with this definition. Let $E$ be a vector bundle over space $X$, and $\Gamma(E)$ be the vector space of all sections of $E$. We say a ...
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1answer
110 views

Atiyah–Jänich for $K_1$

Atiyah–Jänich's theorem says that $$ \left[X\to\mathcal{F}\left(\mathcal{H}\right)\right] = K_0\left(X\right) $$where $\mathcal{H}$ is any separable complex Hilbert space, $\mathcal{F}\left(\mathcal{H}...
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2answers
99 views

Detail in the proof of $\mathbb{R}^n$ is a division algebra only for $n=1,2,4,8$

One of the consequences of the Hopf invariant one problem is that $\mathbb{R}^n$ is a division algebra only for $n=1,2,4,8$. A division algebra structure $\odot$ on $\mathbb{R}^n$ need not play nicely ...
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31 views

In what way is $\sigma(L) \in K(T^*X)$ in the $K$-theoretic formulation of the Atiyah-Singer index theorem?

I am asking whether my interpretation of the $K$-theoretic description of the Atiyah-Singer index theorem is correct. First I state what I mean by "$K$-theoretic description". Let $X$ be a manifold ...
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1answer
45 views

Showing that $p\oplus e\sim_{0} q\oplus e$ whenever $e$ is a properly infinite full projection and $[p]_{0}=[q]_{0}$

I am having trouble with Exercise 4.9 (ii) from Rordam's book. Let $A$ be a unital C$^{*}$-algebra. I am trying to prove the following. Suppose $p\in \mathcal{P}_{n}(A)$ and $q\in\mathcal{P}_{m}(A)$...