Algebraic and topological K-theory, relations with topology, commutative algebra, and operator algebras

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58 views

Use of algebraic topology in geometry(differential and complex analytic) [on hold]

I'm curious in seeing connections of algebraic topology with other areas of mathematics. As far as I know, it's not really used in scheme-theoretic algebraic geometry and arithmetics(aside from ...
4
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0answers
37 views

K theory for pre $C^*$-algebras

In noncommutative geometry when one want to go to the differentiable level, one is forced to work with algebras which are no longer $C^*$. It is nice if we don't loose much information by the ...
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248 views
+500

Is this a model for $K$-theory of a triangulated category?

The recent question Complete the following sequence: point, triangle, octahedron, . . . in a dg-category reminded me of something I wanted to clarify long time ago; most likely this is now well known ...
6
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0answers
138 views

K theory as the fundamental group

There are several ways in which one can define $K$-theory for $C^*$-algebras: for $K_0(A)$ group two aproaches: algebraic (using idempotents) and topological (using projections, i.e. self-adjoint ...
3
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1answer
193 views

(Geometric) Proof for the projective bundle formula in K-theory

I'm trying to piece together a proof of the projective bundle formula from several incomplete sources. Here's the statement I'd like to prove: Projective bundle formula: Let $\pi: E \to X$ be a ...
12
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2answers
367 views

H-space structures on non-sphere suspensions?

It is well known that $S^n$ admits an H-space structure if and only if $n=0,1,3,7$. I'm interested in whether there are other suspensions $\Sigma X$ that admit H-space structures: Question 1 For ...
17
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3answers
1k views

How do you relate the number of independent vector fields on spheres and Bott Periodicity for real K-Theory?

The theory of Clifford algebras gives us an explicit lower bound for the number of linearly independent vector fields on the $n$-sphere, and Adams proved that this is actually always the best ...
6
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1answer
231 views

Closed formulas for topological K-theory?

Let $X$ be a compact manifold. I'm interested in whether any of the following cases admits a general closed formula for (complex)-$K$-theory. Let $E$ be a complex vector bundle with a given line ...
4
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37 views

Waldhausen's regular coherent groups: torsionfree non-examples and behaviour under taking products?

Waldhausen defined a group $G$ to be regular coherent, if for all regular noetherian rings $R$ the group algebra $RG$ is regular coherent. (see Waldhausen - Algebraic $K$-Theory of generalized free ...
5
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126 views

What kind of K-theory is this?

Suppose I have a triangulated category $T$, say the category of modules over a dg or $A_\infty$-algebra. Let me write $GL(T)$ for the groupoid whose objects are all finite collections of generators ...
12
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3answers
453 views

K-theory of non-compact spaces

This is a question on nomenclature of $K$-theory in the topological category. The $K$-theory of a compact space $X$ is defined as the Grothendieck group of the vectorbundles on $X$. The Atiyah-Jänich ...
11
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1answer
148 views

What does it tell us, if we know a unital C*-algebra has approximately inner (half-)flip?

This is a somewhat vague question, but I think it is not too open-ended and should admit well-circumscribed answers by specialists in operator algebras.$\newcommand{\Cst}{{\rm C}^*}$ It arises from ...
5
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1answer
421 views

A generalized K- theory via generalized idempotents

Edit After the answer by Neil Strickland, I add the word "a ring" in this new version. In the literature, there is a concept of generalized idempotent: an n- idempotent is an element $a$ of a Banach ...
4
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1answer
260 views

A Question About the Elliott-Natsume-Nest Proof of Bott Periodicity

In Wegge-Olsen’s book K-Theory and C$ ^{*} $-Algebras, there is an outline of a proof of Bott Periodicity (the proof is due to George Elliott, Toshikazu Natsume and Ryszard Nest). The first step of ...
3
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0answers
127 views

isomorphism of Chern character in kk-theory

Suppose we work with Fréchet algebras. Cuntz defined kk-theory for those algebras and hence we have the notions of K-theory and K-homology for those algebras. Now suppose Chern character is ...
5
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1answer
95 views

Isomorphism classes of differential rank $k$ vectors bundles over $S^q$ [closed]

Could anybody provide a motivated sketch of why the isomorphism classes of the differentiable rank $k$ real vector bundles over the sphere $S^q$ are given by$$\text{Vect}_k(S^q) \simeq \pi_{q - ...
8
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0answers
123 views

Characteristic classes for odd $K$-theory

There are different models of odd $K$-theory. In one case, one takes the group $U=\lim\limits_{\longrightarrow}U(n)$ as classifying space. Similarly, if $\mathcal U$ denotes the unitary group of a ...
11
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1answer
166 views

Atiyah-Singer index theorem, pairing between K-homology and K-theory and Chern character

There is a general (abstract) index theorem in noncommutative geometry: you take a K-theory class and K-homology class (which is represented by a triple $(A,H,F)$) and you pair them together. This ...
28
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0answers
478 views

Atiyah-Singer theorem-a big picture

So far I made several attempts to really learn Atiyah-Singer theorem. In order to really understand this result rather broad background is required: you need to know analysis (pseudodifferential ...
5
votes
1answer
196 views

a question about Bockstein spectral sequence

I find the following theorem for Bockstein spectral sequence at http://pages.vassar.edu/mccleary/files/2011/04/MC10.fin_.pdf, page 459: Question. for a fixed $k$, if $\beta$ does not hit ...
1
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0answers
162 views

A functor on the category of rings, algebras or compact Hausdorff topological space

Assume that $R$ is a unital ring or a complex or real (Banach or $C^{*}$) algebra. We define a relation $M$ on $R$ as follows: $$a\;M b \;\;\; \text{iff}\;\; a=xy,\;b=yx \;\; \text{for ...
6
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2answers
239 views

Generators of the $ K_{0} $-group of the non-commutative torus $ A_{\theta} $ with $ \theta \in \mathbb{Q} $ (i.e. rational rotation algebra)

I am studying the non-commutative torus $ A_{\theta} $. When $ \theta $ is irrational, $ {K_{0}}(A_{\theta}) $ is generated by $ [1] $ and $ [p_{\theta}] $. (Note: $ p_{\theta} $ is a projection in ...
3
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1answer
199 views

Generators K-theory of Cuntz algebras

The Cuntz algebra $O_n$ is the C*-algebra generated by n isometries $S_1$, ..., $S_n$ such that $S_i^* S_j=\delta_{i,j}$ and $\sum_{i=1}^nS_iS_i^*=1$. Cuntz proved that this algebra has the following ...
1
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1answer
356 views

Murray–von Neumann equivalence on C$^*$-algebra and von Neumann algebra

Let $H$ be a separable infinite dimensional Hilbert space, $M \subset B(H)$ a von Neumann algebra and $A \subset M$ a separable $C^*$-algebra such that $A''=M$. Let $p,q \in M_{\infty}(A)$ be ...
3
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1answer
219 views

$K$-Theory of finite dimensional Banach algebras

Is there a finite dimensional Banach algebra $A$ for which $K_{0}(A)$ is a finite group? I asked this question in MSE but I received no answer ...
8
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1answer
244 views

$K$ theory and singular cohomology

For cell complexes${}^1$ $X$ we have an isomorphism $$ K^*(X)\otimes \mathbb{Q}\cong H^{*}(X;\mathbb{Q}), $$ which is induced by the Chern character. What is the analogous statement for $KO(X)$? ...
11
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2answers
531 views

Intuition/idea behind a proof of the splitting principle?

The splitting principle is as follows. Given a vector bundle $E \to X$ with $X$ compact Hausdorff, there is a compact Hausdorff space $F(E)$ and a map $p: F(E) \to X$ such that the induced map ...
15
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5answers
4k views

Textbook or lecture notes in topological K-Theory

I am looking for a good introductory level textbook (or set of lecture notes) on classical topological K-Theory that would be suitable for a one-semester graduate course. Ideally, it would require ...
4
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0answers
182 views

Milnor's model of $EG$ and Kac-Moody groups

I am working with non-compact Kac-Moody groups $\mathcal{K}$. We can use Milnor's join model for $E\mathcal{K}=\varinjlim \mathcal{K}^{*n}$, where $\mathcal{K}^{*n}$ is the iterated join (see page 20 ...
7
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2answers
223 views

What is the “quaternionic” super Brauer group?

In addition to the two reasonably well-known categories $\mathrm{SuperVect}_{\mathbb R}$ and $\mathrm{SuperVect}_{\mathbb C}$ of real and complex super vector spaces, each of which is monoidally ...
8
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3answers
199 views

Index of a family of operators

In the usual setting of the Atiyah-Singer index theorem the situation is as follows: we have a closed smooth manifold $M$ without boundary and $D$ is some elliptic differential operator acting on ...
21
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1answer
820 views

Topology of categories, very basic facts surrounding Quillen's Higher Algebraic K-Theory I

In his paper Higher Algebraic K-Theory I (see [here][1]), Quillen introduces a topological space $BC$, called the classifying space of $C$, and tries to relate its topology to the categorical ...
1
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1answer
298 views

A (possible) equivalent relation on the space of vector bundles

Edit: According to the essential comment of Alex Degtyarev, we revise the question as follows; Assume that $\alpha$ and $\beta$ are two oriention preserving automorphism of Lie groups $O(n)$ and ...
3
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1answer
89 views

self-Whitney sum of the canonical vector bundle on Grassmannians

Let $G_{k}(\mathbb{R}^N)$ be the Grassmannian manifold consisting of $k$-subspaces in $\mathbb{R}^N$. There is a canonical $k$-dimensional vector bundle $$ \gamma_{k,N}: \mathbb{R}^k\longrightarrow ...
5
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1answer
272 views

Good references for K-theory of modular curves?

The title says it. I am looking for a good exposition on the K-theory of the curves $X_{i}(N)$, $Y_{i}(N)$, where $i\in\{0,1\}$. I have some background in $K$-theory and also some background in ...
5
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2answers
241 views

Relative Characteristic classes

A pair of vector bundles over a base space $X$ is a pair $(E,F)$ where $E$ is a vector bundle over $X$ and $F$ is a sub-bundle of $E$. Two pairs $(E_{1},F_{1})$ and $(E_{2}, F_{2})$ are ...
21
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5answers
3k views

What is the equivariant cohomology of a group acting on itself by conjugation?

This question makes sense for any topological group $G$, but I'd particularly like to know the answer for $G$ a compact, connected Lie group. $G$ acts on itself by conjugation. One has the ...
5
votes
1answer
295 views

reference request for mod p and p-adic K-theory

Is there a good reference that explains mod p K-theory and p-adic or p-complete K- theory? All I know about K-theory is the topological K-theory of "vector bundles and k-theory" in Switzer's book ...
7
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0answers
220 views

Do there exist “non-algebraic tensor products” for “algebraic” triangulated categories?

Let us call a triangulated category algebraic if it admits a differential graded enhancement (i.e., an enrichment in complexes of abelian groups). Certainly, there is a notion of a tensor product on ...
6
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2answers
442 views

K theory long exact sequence

(1) Suppose that $Z\subset X$ is a closed embedding, $U = X\setminus Z$ is the complement. If relevant, suppose that both $X, Z$ are smooth and even (if relevant) that the normal bundle of $Z\subset ...
9
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1answer
124 views

Compute the index of the Dirac operator on $C_0(R^2)$ to obtain Bott element in $K_0$

I am studying the paper of Baum-Connes-Higson to understand the Connes-Kasparov conjecture. In example 4.23, they discuss the case $G=\mathbb{R}^2$. I have constructed the Dirac operator, but I’m ...
13
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3answers
2k views

State of the art for Gersten's conjecture for K-theory?

Does anyone know (of a reference to) under what restrictions on the regular scheme $X$ it is known that we have an exact sequence $$0 \to \mathcal{K}_n(X) \to \bigoplus_{x \in X^{(0)}} K_n(k(x)) \to ...
6
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0answers
103 views

$K_0$ an $KH_0$ of a normal crossing variety

Let $k$ be a field (say, algebraically closed to fix the ideas) and let $X$ be a strict (aka simple) normal crossing variety over $k$, so that $X$ is union of regular varieties with intersection that ...
4
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1answer
138 views

covering map from spheres to projective spaces and the associated vector bundle

Let $S^n$ be the $n$-sphere and consider a $2$-sheeted covering $$ S^n\longrightarrow\mathbb{R}P^n. $$ We have an associated vector bundle $$ \xi: \mathbb{R}^2\longrightarrow ...
2
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0answers
95 views

Chern character of finite $CW$-complexes and rational Pontrjagin class of vector bundles

Let $K$ be a finite $CW$-complex. Could you give any references or explanations for the following two items? I do not understand. Thanks! (1). The Chern character from $\tilde{KO}^0(K)$ to the ...
7
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1answer
159 views

classifying maps of Whitney sums of vector bundles

For an $n$-dimensional vector bundle $\xi$ with structure group $G\leq O(n)$ over a $CW$-complex $B$, we have a classifying map up to homotopy $$ f(\xi): B\longrightarrow BG, $$ $f(\xi)\in [B;BG]$, ...
2
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0answers
86 views

order of elements in a mapping space

Let $B$ be a finite CW-complex and $\xi$ be a vector bundle over $B$ with structure group $\Sigma_n$, the $n$-th symmetric group. Then corresponding to $\xi$, we have a classifying map $$ g\in \tilde ...
4
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0answers
62 views

Is $U(C(\beta \mathbb{D}))/U_0(C(\beta \mathbb{D}))\simeq K_1(C(\beta \mathbb{D}))$?

Let $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$, $\beta\mathbb{D}$ be the Stone-Cech compactification of $\mathbb{D}$, $C(\beta\mathbb{D})$ be the ring of complex-valued continuous functions on ...
0
votes
1answer
116 views

Nonstable $K$-theory question

Let $Y$ be a compact, Hausdorff topological space, and $X$ be a locally compact, contractible, Hausdorff space which is homeomorphic to a dense subset of $Y$. Question A: Is ...
4
votes
0answers
142 views

Connectivity of the group of invertible elements of $C(S^{2})\otimes A$

For what type of $C^{*}$ algebras $A$, the group of invertible elements of $C(S^{2}) \otimes A$ is a connected group? All finite dimensional $A$ satisfy this property. Is it true to say ...