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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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1answer
24 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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1answer
38 views

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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2answers
89 views

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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0answers
29 views

Motive of Pfisterforms spectral sequence

In this famous paper http://www.math.uni-bielefeld.de/~rost/data/motive.pdf Rost constructs the motive of a Pfister-Form/Pfister-Quadric. In the last proof on page 13 he writes: "By a spectral ...
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1answer
40 views

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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25 views

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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28 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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0answers
31 views

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, ...
5
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1answer
38 views

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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0answers
30 views

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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1answer
49 views

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 ...
4
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3answers
152 views

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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1answer
95 views

connected complete totally geodesic sub manifold of $S^n$

Let $M$ and $N$ be manifolds with Riemannian metrics $g$ and $h$ respectively. A diffeomorphism $F: M\to N$ is an isometry if \begin{equation*} h_{F(x)}(T_x F(u), T_x F(v))=g_x(u,v) ...
2
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1answer
91 views

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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0answers
34 views

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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1answer
94 views

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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1answer
109 views

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 ...
3
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1answer
136 views

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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1answer
60 views

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 ...
2
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1answer
63 views

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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0answers
58 views

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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2answers
159 views

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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1answer
135 views

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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0answers
77 views

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 ...
2
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1answer
190 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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0answers
52 views

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 ...
4
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1answer
458 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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2answers
80 views

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 ...
4
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0answers
77 views

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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1answer
133 views

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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1answer
111 views

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) = ...
5
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1answer
175 views

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 ...
5
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1answer
186 views

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 ...
3
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1answer
221 views

What's the map $BU \times \mathbb{Z} \to \prod K(\mathbb{Z},n)$ representing the total Chern class?

Recall that complex topological $K$-theory is representable on reasonable spaces by the space $BU \times \mathbb{Z}$ (where $BU$ is a colimit of various infinite Grassmannians), and that the total ...