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

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63 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
39 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 ...
6
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139 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 ...
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256 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 ...
3
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1answer
194 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
368 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 ...
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 ...
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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
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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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 ...
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 ...
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 ...
5
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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 ...
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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 ...
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 ...
3
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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 ...
8
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1answer
245 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 ...
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 ...
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 ...
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
votes
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
votes
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 ...
5
votes
1answer
296 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 ...
21
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1answer
821 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 ...
7
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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 ...
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 ...
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 ...
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 ...
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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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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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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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
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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 ...
6
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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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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 ...
8
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2answers
826 views

Is every vector bundle over a noncompact finite-dimensional manifold a summand of a trivial bundle?

In the notes of Vector Bundles and K-theory by Prof Allen Hatcher, on page 12 he proved a Proposition that for each vector bundle $E\to B$ with $B$ compact hausdorff there exists a vector bundle ...
4
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1answer
226 views

A question on complex line bundle over $S^{2}$

Consider the trivial bundle $\epsilon_{2}=S^{2}\times \mathbb{C}^{2}$ with the standard Hermitian inner product $<(a,b), (c,d)>=a\bar{c}+b\bar{d}$. Assume that $\ell$ is a sub line bundle of ...
3
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0answers
122 views

Seeking an unpublished manuscript by Tetsuro Okuyama

Several papers in representation theory attribute the notion of relatively projective modules to Tetsuro Okuyama's manuscript "A generalization of projective covers of modules over finite group ...
5
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131 views

What is the etale sheafification of the (unramified) Milnor-Witt $K$-theory

I would like a reference/argument for the truth/falsity of the following statement: The etale sheafification of the unramified Milnor-Witt K-theory (Nisnevich) sheaves are the (etale sheafification ...
12
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1answer
203 views

Equivariant Fredholm operators classify equivariant K-theory

Let $\mathcal{F}$ be the space of Fredholm operators on a separable Hilbert space $H$ with the topology induced by the operator norm. If $X$ is compact, Atiyah-Jänich proved that ...
10
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259 views

Morava $K$-theory of $K( \mathbb{Z}/p^2)$

The $p$-adic completion of $K( \mathbb{F}_p)$ is known (by Quillen's calculation) to be $H \mathbb{Z}_p$; in particular, $K(\mathbb{F}_p)$ is acyclic with respect to all Morava $K$-theories $K(n), 0 ...
16
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527 views

How boundedly generated is $SL_3(\mathbb{Z})$?

The group $G = \mathrm{SL}_3(\mathbb{Z})$ is known to be boundedly generated, that is, there exists some $m \in \mathbb{N}$, and $g_1, \dots, g_m \in G$ such that we have the following equality of ...
5
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136 views

Non trivial rank 2 holomorphic vector bundles in complex dimensions greater than or equal 2

Does every compact complex manifold of complex dimension greater than or equal two possess a nontrivial rank 2 holomorphic vector bundle?
3
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140 views

Can one complete a morphism of commutative triangles to a “commutative cube” in a triangulated category?

This question is a continuation of Can one extend a morphism of commutative triangles to a morphism of octahedral diagrams?. I am deeply grateful for the contributions there; they roughly say that ...