**1**

vote

**0**answers

51 views

### The kernel of the residue map before passing to Milnor's K-theory

Let $F$ be a field of zero characteristic. All groups are taken modulo torsion.
Consider a residue map from the exterior algebra of the multiplicative group of the function field of the projective ...

**1**

vote

**0**answers

43 views

### What is the quotient $\mathbb{T}^3/\mathbb{Z}_2$? [migrated]

What is the quotient space $\mathbb{T}^3/\mathbb{Z}_2$, When $\mathbb{Z}_2$ acting on 3-dimensional torus $\mathbb{T}^3 := \mathbb{R}^3/\mathbb{Z}^3$ by sending $x$ to $-x$? Does anyone know how to ...

**9**

votes

**2**answers

628 views

### Symplectic K-theory

For a ring $R$ consider symplectic K-theory defined as follows: let $\operatorname{Sp}(R) = \lim_n \operatorname{Sp}_{2n}(R)$, let $\operatorname{ESp}(R)$ be the subgroup generated by elementary ...

**8**

votes

**0**answers

211 views

### Geometric vs combinatorial motives over Spec Z

Consider the category of reduced schemes of finite type over $\mathbb{Z}$. Take the Grothendieck group of this category, i.e. the free abelian group on isomorphism classes, modulo the usual "syzygy" ...

**6**

votes

**1**answer

197 views

### Linearized Waldhausen $K$-Theory

In Waldhausen's foundational $A(X)$ paper (in Springer LNM 1126) there are some brief remarks on p. 400 about how to define the "linearized" $K$-theory of a space $X$ using abelian group objects in ...

**3**

votes

**1**answer

145 views

### K-homology of Cantor set and abelian AF-algebras

This may be a standard question answered in a book, or article. I don't know. I know that there exist related results with $\lim^1$-sequences (Rosenberg and Schochet).
What is
...

**8**

votes

**1**answer

206 views

### When does the continuous Galois(=etale) cohomology of fields coincide with the naive one? Often true by the Bloch-Kato conjecture?

For a field $F$ I am interested in its $l$-adic (Galois=\'etale) cohomology; here $l$ is a prime distinct from the characteristic of $F$ (for simplicity one may assume that the latter is $0$).
For ...

**1**

vote

**0**answers

184 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 ...

**2**

votes

**1**answer

169 views

### Complexification of real k-theory gives index $2$ subgroup of complex k-theory

We have $\widetilde{KO}(S^4) \cong \mathbb{Z}$ and $\widetilde{K}(S^4) \cong \mathbb{Z}$. There is a map $i:\widetilde{KO}(S^4) \rightarrow \widetilde{K}(S^4)$ that takes a stable vector bundle to ...

**1**

vote

**0**answers

127 views

### The definition of $SK_1$ for an arbitrary ring

Let $R$ be a unitary associative ring. If $R$ is commutative, then one defines $SK_1(R)$ as the quotient $$SK_1(R)=SL(R)/E(R)$$ (Definition 2.8 of ...

**0**

votes

**1**answer

150 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 ...

**13**

votes

**0**answers

522 views

### What is operator tmf?

One of the many wonderful things about K-theory, relative to other generalized cohomology theories, is that it can be defined for not-necessarily-commutative C*-algebras. The resulting construction, ...

**3**

votes

**1**answer

78 views

### Local index formula for >ungraded< elliptic operators

Let $P\colon E \to F$ be an elliptic pseudodifferential operator over $M$. Assuming that $P$ defines a finitely summable Fredholm module, we may apply the Chern-Connes character to it to get a cyclic ...

**0**

votes

**1**answer

170 views

### A question on K- theory of non commutative $C^\star$ algebra

Edit: According to the comment of Andre Henriques I revise the question:
What is an example of a noncommutative unital $C^\star$ algebra $A$, which is not Morita equivalent to a commutative ...

**5**

votes

**2**answers

496 views

### Mayer-Vietoris sequence for topological K-theory

I'm reading the paper Loop groups and twisted K-theory I by Freed, Hopkins, and Teleman. They give some examples of computing (twisted) K groups using the Mayer-Vietoris sequence.
I'm a bit ...

**10**

votes

**1**answer

445 views

### Is there a category whose isomorphisms are precisely the simple homotopy equivalences?

If we start with the category of finite complexes and continuous maps, and then identify two morphisms iff they are homotopic, we get the homotopy category of finite complexes, and it is trivial to ...

**1**

vote

**1**answer

123 views

### What is the relationship between $BU$ and $\textrm{Fred}_0(H)$?

Let $U=\cup_{n=1}^{\infty} U(n)$ be endowed with the weak topology, let $BU$ be its classifying space. Then for any compact CW complex $X$, $[X,BU]$ classifies all the vector bundles on $X$ up to ...

**10**

votes

**0**answers

211 views

### What is the symmetric monoidal functor from Clifford algebras to invertible K-module spectra?

There ought to be a symmetric monoidal functor from the symmetric monoidal $2$-groupoid whose objects are Morita-invertible real superalgebras (precisely the Clifford algebras), morphisms are ...

**23**

votes

**2**answers

994 views

### What are the “correct” conventions for defining Clifford algebras?

I have three related questions about conventions for defining Clifford algebras.
1) Let $(V, q)$ be a quadratic vector space. Should the Clifford algebra $\text{Cliff}(V, q)$ have defining ...

**3**

votes

**1**answer

98 views

### Is the square diagram of index and exponential maps in $K$-theory of $C^*$-algebras anti-commutative?

Assume we have a $3\times 3$ grid with rows and columns being short exact sequences of $C^*$-algebras.
This gives a grid of 6-term exact sequences: 3 "horizontal" sequences and 3 "vertical" ...

**3**

votes

**1**answer

316 views

### Doing some homological algebra in triangulated categories

It's well known that chain complexes are an abelian category, and in particular we can consider chain complexes of chain complexes, i.e. double complexes. Given a double complex $A^{\bullet\bullet} ...

**3**

votes

**0**answers

67 views

### The 4-th generator of $K_1$ group for 3-dimensional NC tori algebra

An $n$-dimensional NC torus algebra $A_\theta^{(n)}$ is defined for any antisymmetric $n\times n$ matrix $\theta$ of real numbers as the universal $C^*$-algebra, generated by unitaries ...

**3**

votes

**0**answers

70 views

### Integral group rings on which stably free modules are free

Let $G$ be a torsion-free group and $ZG$ the integral group rings. Recall that a projective module $P$ over $ZG$ is stably free if there is an isomorphism $P \oplus ZG^n \cong ZG^m$. Are there known ...

**4**

votes

**1**answer

287 views

### Fundamental class in K-theory and orientability

In ordinary homology, the classical results give the following situation:
for a compact, connected, topological manifold $M$ of dimension $n$ we have, for each ring $R$, that $H_n(M,M \setminus ...

**2**

votes

**0**answers

133 views

### The space of Fredholm operators as a classifying space

Is it true that the space of Fredholm operators on a separable Hilbert space is the classifying space for K-theory in the category of paracompact spaces?
Everyone quotes the theorem of Atiyah-Janich ...

**2**

votes

**0**answers

124 views

### Varieties with Chow groups supported in positive codimension: examples and properties?

In their 1983 paper "Remarks on correspondences and algebraic cycles" Bloch and Srinivas proved several interesting properties of smooth proper varieties (over universal domains) whose Chow groups of ...

**1**

vote

**1**answer

83 views

### Equivalence of definitions of the Milnor $K$-groups

In Kurihara's paper: "The exponential homomorphisms for the Milnor $K$-groups and an explicit reciprocity law" he difines, in the first page, the $q$-th Milnor K-group for the ring $R$ as
...

**1**

vote

**0**answers

91 views

### Morita Equivalence of Full Corners in $C^*$-algebras

Suppose $\mathcal{A}$ is a $C^*$-algebra with a unique normalized trace and $p \in \mathcal{A}$ is a projection so that $\mathcal{B} = p\mathcal{A}p$ is a full corner.
Does $\mathcal{B}$ have a ...

**3**

votes

**0**answers

153 views

### Cancellation and splitting theorems for vector bundles etc over schemes

It is not too hard, in the theory of vector bundles over manifolds (or nice topological spaces, say locally contractible with finite covering dimension), to arrive at a splitting theorem. This ...

**3**

votes

**1**answer

116 views

### Why is SL(n,Z)[p] modulo the group normally generated by elementary matrices abelian?

I'm trying to understand part of Bass-Milnor-Serre's paper on the congruence subgroup problem. I'm pretty sure that the following statement is proved in there, but I'm having trouble find (and/or ...

**1**

vote

**0**answers

51 views

### Which groups may be obtained as $K$-homology groups?

Recently I asked the following question, about the separability of the underlying $C^*$-algebra in the definition of $K$-homology:
mathoverflow.net/questions/181361
As far as I understood, ...

**1**

vote

**1**answer

70 views

### Definition of homotopy between Kasparov modules

I'm trying to understand the definition of homotopy between Kasparov modules as presented in Blackadar's book on K-theory for operator algebras. $A,B$ will be C*-algebras, while $E$ will denote a ...

**1**

vote

**0**answers

211 views

### Two questions on canonical line bundle over $\mathbb{C}P^{n}$

The canonical line bundle over $\mathbb{C}P^{n}$ is denoted by $\ell_{n}$. It is well known that: $$(\ell_{1}\otimes \ell_{1})\oplus1 \simeq\ell_{1}\oplus\ell_{1}$$ Both sides are isomorphic ...

**5**

votes

**1**answer

104 views

### Separability of the C*-algebra in the definition of K-homology

There are (at least) two approaches to K-homoology: one is via the so called dual algebra which is due to Paschke. The second is via the Fredholm modules and is due to Kasparov. In Nigel Higson's book ...

**4**

votes

**0**answers

107 views

### The 'most general' papers on rational Borel-Moore motivic homology and K'-theory?

There are two ways to define Borel-Moore motivic homology (of schemes) with rational coefficients: one should either consider certain complexes of algebraic cycles, or the $\gamma$-filtrations of ...

**5**

votes

**1**answer

132 views

### “The” kronecker foliation or “a” kronecker foliation?

Consider the following two foliations of torus:
1)The Kronecker foliation with slope $\sqrt{2}$
2)The Kronecker foliation with slope $\pi$
As I learn from the literature, these two foliations are ...

**0**

votes

**0**answers

178 views

### K-theory of $\mathbb{RP}^\infty$

can anyone give some reference of K-theory and K-homology of $\mathbb{RP}^\infty$, both $K_0$ and $K_1$.
PS: also posted in stackexchange

**1**

vote

**3**answers

187 views

### K-theory for the $C^*-$algebra of the continuous functions on the $2-$torus and the Bott projection

I am trying to understand the K-theory for the $C^*-$algebra of the continuous functions on the $2-$dimensional torus $T^2$. In particular I am interested on the $K_0-$group. I have read that the ...

**5**

votes

**1**answer

143 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 ...

**5**

votes

**0**answers

127 views

### Topology for bounded operators quotiented by Schatten ideal

I saw this particular question on stackexchange. Since there has been zero answers and since I've been interested in this question myself I want to ask it here.
Given the $C^{\ast}$-algebra of bounded ...

**9**

votes

**1**answer

198 views

### Algebraic $K$-theory of algebras in symmetric spectra: reference

I want to use the technicalities of structured ring spectra for the first time in my life, and I am not really familiar with the relevant literature. I am looking for a reference that defines ...

**7**

votes

**0**answers

135 views

### Reference for analyticity of $K$-theory

This is a follow-up to my last question, Homotopy excision for structured ring spectra -- reference?. The immediate reason why I care about Blakers-Massey theorems for ring spectra is to prove that ...

**1**

vote

**1**answer

83 views

### On two notions of 'generators' for a 'large' triangulated category

Let $C$ be a triangulated category that is closed with respect to arbitrary small coproducts; let $D$ be some class of objects of $C$. Then it would be natural to say that $D$ generates $C$ either if
...

**4**

votes

**2**answers

277 views

### $K$-homology of $BG$

Let $G$ be a finite group. Atiyah proved that the $K$-cohomology of $BG$ vanishes in odd degrees and in even degrees is the completion of the representation ring of $G$ at the augmentation ideal.
...

**11**

votes

**0**answers

188 views

### Do quotients of amenable groups C*-algebras satisfy the UCT?

Let G be a discrete amenable group.
General Question: Let $J$ be an ideal of $C^*(G)$, the group C*-algebra of $G.$ Does $C^*(G)/J$ satisfy the universal
coefficient theorem (UCT)?
I am mainly ...

**7**

votes

**2**answers

213 views

### Covering Spaces and Vector Bundles

Suppose $f: Y \rightarrow X$ is a covering map between compact Hausdorff spaces $X$ and $Y$. Then $f$ induces a algebra homomorphism $f^*:C(X) \rightarrow C(Y)$ and gives $C(Y)$ the structure of a ...

**1**

vote

**0**answers

37 views

### How to compute $KR^{-q}(S^{p,0})$

I am reading Atiyah's paper on $KR$-theory, and wondering how to compute $KR^{-q}(S^{p,0})$. The cases $p\geqslant 3$ may be simply deduced from the following isomorphism in the paper,
$$
...

**5**

votes

**1**answer

363 views

### Dennis trace map K----> THH

I have some questions about Dennis trace map in algebraic K-Theory. I was wondering if there is some conceptual way to look at this map $K(-)\rightarrow THH(-)$ (natural transformation from K-Theory ...

**11**

votes

**1**answer

406 views

### Homotopy spheres with vanishing and non-vanishing $\alpha$-invariant

I'm unsure whether this question is appropriate for mathoverflow, so feel free to criticize.
All manifolds are closed, smooth and have dimensions $n\ge 5$.
The Atiyah-Shapiro-Bott-Orientation gives ...

**1**

vote

**0**answers

82 views

### Complex non stably trivial complex vector bundle with vanishing Chern classes

There have been some discussions of a similar question, but I'd like an example of a non-trivial class in complex K-theory of an orientable manifold, for which all Chern classes vanish. This does not ...