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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3
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2answers
70 views

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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2answers
40 views

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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0answers
24 views

Bott Periodictity for Real and Complex Algebras

In Rordam's Introduction to K-Theory for $C^*$ Algebras they prove \begin{equation} K_{n+2}(A) \cong K_{n}(A) \end{equation} using isomorphism in terms of the suspensions of $A$. I have also heard ...
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46 views

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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1answer
28 views

Intuition behind 4-fold periodicity of $L$-theory

The quadratic and the symmetric L-groups are 4-fold periodic. What is the simple argument to obtain the intuition behind the 4-fold periodicity of $L$-theory? (For example, why not have the Bott ...
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1answer
76 views

Bott Periodicity of Class Group: Linkage to $K$-theory

I am currently reading Bott's The Stable Homotopy of the Classical Groups (1959), which was his original proof towards Bott periodicity. As a consequence of his proof, the stable homotopy group of ...
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1answer
55 views

K theory of the wedge of circles

I am interested in finding the $K_0, K_1$ groups of $C(S^1 \vee S^1)$. We know that $K_0 (C(S^1)) = K_1(C(S^1)) = \mathbb Z$, but is this directly helpful?
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1answer
28 views

$K_0$ of an inner automorphism

An exercise in a book I am studying says $K_0(\alpha) =id$ for every inner automorphism $\alpha$. I am not sure why this is true, I suppose it has to do with the fact that inner automorphisms look ...
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1answer
151 views

Does every vector bundle have a 'tensor inverse'?

For any vector bundle $E$ over a finite-dimensional CW complex, there is a vector bundle $E'$ such that $E\oplus E'$ is trivial. For a compact Hausdorff base, this is Proposition 1.4 of Hatcher's ...
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Cyclic homology of $C(S^1)$?

As an exercise for myself, I am trying to figure out the cyclic homology of the algebra of functions over $S^1$, i.e. $C(S^1, \mathbb{C})$. Since the cyclic homology is defined via the universal ...
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32 views

Restriction of $G$-equivariant sheaves to fixed locus

Let $G$ be a group acting on a Noetherian scheme $X$ over $\mathbb{C}$ and $\mathcal{E}$ be a $G$-equivariant coherent sheaf on $X$ where the support of $\mathcal{E}$ contains the fixed locus $X^G$. I ...
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1answer
46 views

Equivalence of triangulated categories of perfect complexes

I'm reading a paper by Paul Balmer where he supposes to have a nice (let's say noetherian) scheme $X$, with an open $U\subseteq X$, and proves the existence of an equivalence of triangulated ...
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1answer
55 views

K-Theory Equivalence Classes

Let $M$ be a finite dimensional compact manifold and $(Vect(M),\oplus)$ be the abelian monoid of complex vector bundles on $M$. I just read that it is possible to construct an equivalence relation on $...
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1answer
22 views

Proof of $\tilde{K}^*_G((G/H)_+\bigwedge X)\cong \tilde{K}^*_H(X)$

I have been reading the paper "A generalization of the Atiyah-Segal completion theorem" by Adams, Haeberly, Jackowski and May, where I found the Following claim: Let $G$ be finite group, $H\...
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1answer
39 views

Special case of Elliott's Theorem

Let $A$ and $B$ be unital $AF$-algebra. By Elliott's theorem we know that if there an order isomorphism $\psi: K_0(A) \rightarrow K_0(B)$ with $\psi([1_{A}]) = [1_{B}]$, then there exists an ...
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13 views

Calculate $K_0(\vee_{j=1}^{k}\mathbb{S}^{n})$ and $K_1(\vee_{j=1}^{k}\mathbb{S}^{n}).$

Let $\mathbb{S}^{n}\; \vee\cdots\;\vee\; \mathbb{S}^{n}=\vee_{j=1}^{k}\mathbb{S}^{n}$ the sum of the wedges of the $k$-spheres $\mathbb{S}^{n}:$ $\vee_{j=1}^{k}\mathbb{S}^{n}=\amalg_{j=1}^k\mathbb{S}^{...
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1answer
25 views

Why is $ \frac{1}{4\pi} \int \int_S dx \, dy\, \vec{n} \cdot ( \partial_x \vec{n} \times \partial_y \vec{n} ) = 1 $?

I am trying to understand this notion of "topological charge" from a paper in Physics Review Letters. They talk about skyrmions and give this integral (here $\vec{n}$ is a vector field on ...
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1answer
32 views

Is the KO dimension of commutative real spectral triple agree with it's dimension of the manifold?

Given a spectral triple with some conditions(such as the algebra is commutative), Connes's reconstruction theorem states a we can recover a Riemannian manifold with spin structure.(see here) Now I ...
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31 views

What is the usual defintion of $K_1$ group?

On the book I am reading, $K_1$ group of a unital $C^*$-algebra $A$ is defined to be $K_1(A)=\lim_{n\to\infty} GL(M_n(A))/GL(M_n(A))_0=\lim_{n\to\infty} U(M_n(A))/U(M_n(A))_0$, where $A_0$ is the path ...
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How to compute $K_0(S^1)$?

I am taking a course on K-theory and we have been asked to compute $K_0(S^1)$. Now I have a few conceptual problems: I see that $M_\infty(C(S_1))$ corresponds to "matrices of potentially infinite ...
2
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1answer
110 views

Bounding $\|e^{2\pi iq}-1\|$ for $q$ an almost idempotent

Let $A$ be a unital $C^*$-algebra, and let $q\in A$ such that $\|q^2-q\|<\varepsilon$, and $\|q\|<K$. Question: How small does $\varepsilon$ need to be (possibly in terms of $K$) to guarantee ...
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43 views

Do group actions on smooth algebraic varieties induce group actions on K-theory?

(Edited after discussion in comments): If a group $G$ acts on a smooth algebraic variety $V$, then the cohomology ring $H^*(V;\mathbb C)$ is a representation of $G$. If, furthermore, $V$ admits a ...
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3answers
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K Theory: Book Recommendations

Good people! So I've been hoping to get into K Theory for a while now, and the book that I have been trying to use (and failing) has been Charles Weibel's book by that very title. The book itself isn'...
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40 views

Elementary computations for hermitian and non-hermitian algebraic K-theory

As the title states, I am looking for some examples of non trivial albeit elementary calculations of algebraic hermitian and non-hermitian $K$-theory groups. Concretely, I would be interested in ...
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25 views

Definition of odd topological K-theory using circles

I wanted to check whether the following characterization of odd complex topological $K$-theory is correct. Let $X$ be a compact Hausdorff space. Then $K^{-1}(X)$ can be defined as $\tilde{K}^0((X\...
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2answers
86 views

Example of a space whose complex K-theory is not easily computable from singular cohomology

I am looking for a counterexample to the formula $$ K^n(X) \cong \prod_{i\equiv n \mod 2} H^i(X) $$ where $K^*$ denotes complex topological $K$-theory, $H^*$ singular cohomology and $X$ a compact ...
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0answers
155 views

On stable isomorphism of separable nuclear C$^*$-algebras and Morita equivalence

I'm a new one to the theory of C$^*$-algebras, and I'm really missing something significant. According to Blackadar,"Operator algebras", page 153 Brown-Green-Rieffel theorem - For $\sigma$-...
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0answers
123 views

Help with the setup for Atiyah's proof of Bott Periodicity.

I'm trying to understand Atiyah's proof of Bott Periodicity from his little book on K-Theory - in particular his formulation in terms of $K(P(L \oplus 1))$ where $L$ is a line bundle on a space $X$. ...
4
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1answer
78 views

What is the meaning of $KO^{-1}(S^1)$?

I am interested in the KO-theory of the circle $S^1$. In particular $KO^{-1}(S^1)$. Using the suspension theorem and reduced $K$-theory I can easily show that \begin{equation} KO^{-1}(S^1) \simeq KO^{-...
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0answers
76 views

Why is this inner product positive definite?

I’m currently working on some Hilbert module constructions and I have a problem proving that an inner product is positive definite. One can define on $ C_c(X) $, where $ X $ is a $ G $-compact proper $...
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0answers
60 views

About the K-theory of rational rotation algebras

The K-theory of the non-commutative torus or rotation algebra $A_\theta$ was studied in the early 80's by several mathematicians with a special focus on the irrational case. However I have not been ...
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1answer
57 views

Equivalence of two definitions of $K^{-1}$ in complex $K$-theory

In complex $K$-theory, one way I have seen the group $K^{-1}(X)$ defined, for a compact Hausdorff space $X$, is $$K^{-1}(X):= K^0_c(X\times\mathbb{R}),$$ where the right-hand side refers to compactly ...
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0answers
24 views

A map from a $ G_1 $ - equivariant KK-theory of Kasparov, to a $ G_2 $ - equivariant KK-theory of Kasparov

Let $ G $ be a locally compact group. Let $ H $ and $ K $ be two normal subgroups of $ G $. In order to construct a map, $$ \psi \ : \ \ F(G/H,G/K) \to F(G/K,G/H) $$ where, $$ F(G/H,G/K) = KK^{G/H} ( ...
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0answers
48 views

Is there a sense in which $\tilde{K}$ is an exact functor?

I'm going through Hatcher's K-Theory script for the first time and noticed that following theorem looked quite like a statement of the form “this functor is exact”: If $X$ is compact Hausdorff and $A\...
4
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1answer
92 views

Complex K-theory cohomology ring $K^*(T^2)$

I was considering the complex K-theory of $T^2$ and I found, using the split exact sequences associated with the pair $(T^2,S^1 \vee S^1)$ that $\widetilde{K}^0(T^2)\cong\mathbb{Z}$ and $\widetilde{K}^...
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1answer
53 views

Reference books on the Baum Connes conjecture

Do there exist readable reference books about Baum-Connes Conjecture for beginners ?
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1answer
87 views

Tangent bundle of the incidence variety and its Chern class

I am trying to learn about Thom polynomials and I often find arguments in the literature that just make no real sense to me. Maybe it is due to my lack of knowledge in $K$ - theory. I apologize for ...
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0answers
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Equivalence of families indexes of Fredholm operators

Let $F=F(H,H)$ be the space of bounded Fredholm operators in a Hilbert space $H$ with topology inherited from the norm operator topology, and let $X$ be a compact topological space. For a continuous ...
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0answers
28 views

Including space of embeddings into space of immersions and general position of handles

Let $E$ be the space of codimension zero embeddings of a manifold N (PL or Diff, whichever you prefer) into a manifold V that are homotopy equivalences. Let $I$ be the analogous space for immersions. ...
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92 views

Showing that for $f : S^{4n-1} \to S^{2n}$ that if the Hopf invariant of $f$ is $\pm 1$ then $n = 1, 2$ or $4$

So recall that if I have a map $f : S^{4n-1} \to S^{2n}$ the Hopf invariant $h(f)$ of $f$ is defined in the following way. Letting $X = S^{2n} \cup_f e^{4n}$ there exists a short exact sequence $$0 \...
3
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1answer
51 views

$K_0(C(\mathbb{T}^{n})) \cong \mathbb{Z}^{2^{n-1}}$

I am new to this website and I have a question. I want to show that $K_0(C(\mathbb{T}^{n})) \cong \mathbb{Z}^{2^{n-1}}$ but first I want to show that $C(\mathbb{T}^{n}) \cong C(\mathbb{T} \rightarrow ...
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1answer
38 views

Ideas for calculating $K_0(l_{\infty})$ and $K_1(l_{\infty})$.

Thank you for answering my question. I'm a bit new to K-theory. So I was wondering how can I calculate $K_0(l_{\infty})$ and $K_1(l_{\infty})$. I think if we have one, then by using bott periodicity ...
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1answer
52 views

$K_1(A)$ is countable when A is separable C*-algebra

We know that when A is a separable C*-algebra then $K_0(A)$ is countable. How can I show that $K_1(A)$ is also countable?
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1answer
42 views

$K_0(C_0(X, A))$ , when X is compact and contractible.

Let A be a $C^{*}$-algebra and $B = C_0(X, A)$ be the set of all continuous functions from a locally compact Hausdorff space $X$ to $A$, vanishing at infinity. Prove that $K_0(B) \cong K_0(A)$ and ...
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0answers
23 views

Is $St(n,R) \rightarrow St(n+1,R)$ injective?

For every (not necessarily commutative) ring R and every $n\geq 3$, one can define the n-th (unstable) Steinberg group St(n,R) as in https://ncatlab.org/nlab/show/Steinberg+group. Is the canonical map ...
5
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0answers
72 views

Chern-Weil theory in the cohomological Atiyah-Singer theorem

I am interested in the following piece of data appearing in the cohomological Atiyah-Singer theorem. My reference is "The index of elliptic operators. III" by Atiyah and Singer. Let $D:\...
2
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1answer
39 views

The generator of $K_0(C(\partial(]0,1[^2)))$ and $K_1(C(\partial(]0,1[^2)))$

Let $C=[0,1]^2 \subseteq \mathbb{C}$ and $\partial C$ the boundary of $C$. I'm looking for the $K_0(C(\partial C))$ and $K_1(C(\partial C))$ and its generator.
2
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1answer
38 views

Proving $K_1(\mathcal{T})=0$ (is trivial)

Let $\mathcal{T}$ the toeplitz algebra and we define the short exact sequence? where $C(\mathbb{T})=\{z\in \mathbb{C}/ |z|\leq 1\}$: $$ 0 \rightarrow \mathcal{K} \rightarrow \mathcal{T }\rightarrow C(...
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0answers
51 views

Identity element in the K-group

This is a question from page 552 of Atiyah and Singer's paper "The index of elliptic operators III". The line is "We suppose first that $X$ is compact so that $K(X)$ has an identity element." Even if ...
1
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1answer
70 views

$ C_0(\mathbb{R}^2)$, $C(\mathbb{D})$, $C(\mathbb{T})$ and the index map

Consider the short exact sequence $$0 \longrightarrow C_0(\mathbb{R}^2) \overset{\varphi}\longrightarrow C(\mathbb{D}) \overset{\psi}\longrightarrow C(\mathbb{T}) \longrightarrow 0$$ I need to show ...

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