Questions tagged [kt.k-theory-and-homology]
Algebraic and topological K-theory, relations with topology, commutative algebra, and operator algebras
343
questions with no upvoted or accepted answers
50
votes
0
answers
12k
views
Atiyah's paper on complex structures on $S^6$
M. Atiyah has posted a preprint on arXiv on the non-existence of complex structure on the sphere $S^6$.
https://arxiv.org/abs/1610.09366
It relies on the topological $K$-theory $KR$ and in ...
48
votes
0
answers
17k
views
What is the current understanding regarding complex structures on the 6-sphere?
In October 2016, Atiyah famously posted a preprint to the arXiv, "The Non-Existent Complex 6-Sphere" containing a very brief proof $S^6$ admits no complex structure, which I immediately read and ...
37
votes
0
answers
5k
views
Homology of $\mathrm{PGL}_2(F)$
Update: As mentioned below, the answer to the original question is a strong No. However, the case of $\pi_4$ remains, and actually I think that this one would follow from Suslin's conjecture on ...
36
votes
0
answers
865
views
Chern character of a Representation
Let $G$ be a finite group. Under the identification of the representation ring $R_{\mathbb{C}}(G)$ with the equivariant K-theory $KU^0_G(\ast)$ of the point, followed by Atiyah-Segal completion-...
36
votes
0
answers
1k
views
Functor that maps to both $KO^n$ and $KO^{-n}$
(my question is also meaningful for complex K-theory, but since Kn(X) is always isomorphic to K-n(X), it's less interesting)
I start by recalling the analytic definition of KO-theory:
The following ...
27
votes
0
answers
1k
views
Computational complexity of topological K-theory
I am a novice with K-theory trying to understand what is and what is not possible.
Given a finite simplicial complex $X$, there of course elementary ways to quickly compute the cohomology of $X$ with ...
26
votes
0
answers
2k
views
derived category of equivariant coherent sheaves and fixed points
The K-group $K^T(X)$ of $T$(torus)-equivariant coherent sheaves on a variety $X$ is isomorphic to $K^T(X^T)$, that of the fixed point locus via the inclusion homomorphism, when we tensor the quotient ...
23
votes
0
answers
615
views
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 ...
23
votes
0
answers
569
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 ...
18
votes
0
answers
537
views
A curious switch in infinite dimensions
Let $V$ be a finite dimensional real vector space. Let $GL(V)$ be the set of invertible linear transformations, and $\Phi(V)$ be the set of all linear transformations. We can also characterize $\Phi(V)...
18
votes
0
answers
702
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 ...
18
votes
0
answers
866
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, ...
18
votes
0
answers
544
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 ...
18
votes
0
answers
812
views
Can we define spectral triples using the language of rigged Hilbert spaces?
The traditional mathematical approach to quantum mechanics,
as developed by von Neumann, is based on Hilbert spaces and unbounded self-adjoint operators.
Another approach, which more closely resembles ...
16
votes
0
answers
582
views
K-theory and homology of groups
It is known that for any ring $R$,
$$K_{1}(R)=H_{1}(GL_{\infty}(R), \mathbb{Z})$$
$$ K_{2}(R)= H_{2}(E_{\infty}(R),\mathbb{Z})$$
$$ K_{3}(R)= H_{3}(St_{\infty}(R),\mathbb{Z})$$
where $GL_{\infty}= ...
15
votes
0
answers
363
views
Dennis trace map for stable $\infty$-category, naively
I'm trying to get more intiution about higher K-theory, Hochschild homology and the trace map between by thinking about these objects from an informal $\infty$-categorical perspective, instead of ...
14
votes
0
answers
216
views
Hauptvermutung for non-manifolds
The Hauptvermutung proposes the following: if two finite simplicial complexes are homeomorphic then they are PL-homeomorphic, meaning that they have a common refinement.
People are mostly interested ...
14
votes
0
answers
516
views
Reference for equivariant Atiyah-Jänich theorem
The equivariant Atiyah-Jänich theorem is an isomorphism
$$
[X,F]_G \cong K_G^0(X),
$$
where $G$ is a compact Lie group, $X$ is a compact $G$-manifold, $F$ is the space of Fredholm operators on a ...
14
votes
1
answer
2k
views
Finite dimensional real division algebras
A celebrated theorem of Milnor and Kervaire asserts that any finite dimensional (not necessarily associative, unital) division algebra over the real numbers has dimension 1,2,4 or 8. This result is ...
12
votes
0
answers
364
views
Can Quillen-Lichtenbaum recover Borel's computation?
Borel famously used analysis on symmetric spaces to compute the rationalised algebraic $K$-theory groups of rings of integers $\mathcal{O}_F$ in number fields, e.g. $K_i(\mathbb{Z}) \otimes \mathbb{Q}...
12
votes
0
answers
380
views
Looking for an invariant similar to algebraic K-theory
I'm wondering if there is an invariant, similar to algebraic K-theory, topological hochshild homologic, topological cyclic homology etc... that has the following properties:
a) It attach to each small ...
12
votes
0
answers
492
views
The homotopy theory presented by a Waldhausen category
Waldhausen introduced his categories for the purposes of defining algebraic $K$-theory of suitable categories. From a modern perspective, it looks like he was really doing two things at once:
...
12
votes
0
answers
326
views
Homology of Gersten complex for singular schemes
It is one of the important facts in K-theory/motivic cohomology that the Gersten-type complexes (for Quillen K-theory, Milnor K-theory or more generally Rost's cycle modules) are exact for smooth ...
11
votes
0
answers
259
views
Criteria for a map of rings to induce an equivalence on K-theory?
Algebraic $K$-theory is Morita invariant, but surely it does not detect Morita equivalence. What are some examples of rings (or ring spectra) $R$ and $S$ that are not Morita equivalent, but ...
11
votes
0
answers
489
views
Chromatic Homotopy Theory and Physics
Chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work relating ...
11
votes
0
answers
878
views
Higher traces in Hochschild cohomology
Let $A$ be an associative algebra over a field $k$. Let $\rho:A \rightarrow \mathrm{End}(M)$ a left module, finite dimensional over $k$. Then the map $a \mapsto \mathrm{tr}_M \rho(a)$ is a well ...
11
votes
0
answers
264
views
Direct proof of the equivalence of symmetric monoidal $K$-theory and exact sequence $K$-theory?
When all exact sequences split in $C$, we have $\Omega B C \simeq K(C):=\Omega Q(C)$. Heuristically, this is because the space of upper-triangular matrices is contractible. Can this be made precise? I ...
11
votes
0
answers
747
views
Rosenberg's proof of Bass-Heller-Swan
I'm reading the proof the Bass-Heller-Swan Theorem in Rosenberg's book Algebraic K-Theory and Applications (Theorem 3.2.22), which asserts
$$K_1(R[t,t^{-1}]) \cong K_0(R) \oplus K_1(R) \oplus NK^+_1(R)...
10
votes
0
answers
309
views
Adams blue book lemma 17.14: computing a $\mathbb{F}_2$ basis for a filtration of $H\mathbb{Z}_*(bu \wedge bu)$
First off let me apologize for not being able to give all the context for this question. I'm learning how to do computations in stable homotopy theory and have been particularly spending a lot of time ...
10
votes
0
answers
203
views
Find a 4-manifold bounding a 3-torus with any abelian representation in SL_2
Fix $x,y,z\in \mathbb{C}^*$ and let $M=S^1\times S^1\times S^1$ with $\rho:\pi_1(M)\to \operatorname{SL}_2(\mathbb{C})$ mapping the three generators to diagonal matrices with entries $(x,x^{-1})$, $(y,...
10
votes
0
answers
208
views
Recover the field from its Milnor K-groups
For every field $F$, consider $K_n^M(F)$ the $n$-th Milnor K-group of $F$ for each $n \in \Bbb N$, and form the Milnor K-ring $K^M(F)=\oplus_{n \geq 0}K^M_n(F)$. For instance, $K_1(F)=F^{\times}$.
...
10
votes
0
answers
381
views
The term "absolute geometry"
My question concerns the so-called absolute geometry over the "field with one element" F_1 or over the spectrum $\mathrm{Spec}(F_1)$, cf. https://ncatlab.org/nlab/show/Borger%27s+absolute+geometry. I ...
10
votes
0
answers
313
views
Is there a spectral sequence of Atiyah's topological KR-theory that can be used to compute basic examples?
For Segal's complex $G$-equivariant $K$-theory, it is well-known that there is an Atiyah-Hirzebruch spectral sequence. If say $G$ a finite group and $X$ a finite CW-complex, the second page of this ...
10
votes
0
answers
6k
views
Atiyah's paper "Non-existent complex 6-sphere"
I'm trying to understand the main idea of Atiyah's proof (https://arxiv.org/abs/1610.09366). Although there were discussions on MO year ago I couldn't find answers to my questions.
Consider the ...
10
votes
0
answers
172
views
Baum Connes conjecture and abstract isomorphism
Baum-Connes conjecture states that for a locally compact group $G$ the so called assebly map $\mu$ between $G$-equivariant K-homology of the universal example for proper actions of $G$ and K-theory of ...
10
votes
0
answers
406
views
Segal-Freed-Hopkins-Teleman = Atiyah-Hirzebruch/Leray-Serre?
Freed-Hopkins-Teleman (section 3.7) generalise Segal's (Proposition 5.3) spectral sequence for equivariant K-theory to more general local quotient groupoids (that is, topological groupoids locally ...
10
votes
0
answers
461
views
Complex $K$-theory of extended powers of a Moore spectrum
Consider a Moore spectrum $S^n/p$. Has the $K$-theory of the extended powers of $S^n/p$ been computed?
For some context: equivalently, I'd like to have the free $E_\infty$-algebra over $KU$ on the $...
10
votes
0
answers
339
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" ...
10
votes
0
answers
259
views
Is the generation of rings by their units a question in K-theory?
Susan's question When can number rings be spanned (as $\mathbb{Z}$-modules) by units? smells like an algebraic K-theory question in disguise. I'll reformulate the question first:
Given an integral ...
10
votes
0
answers
319
views
H-space structure on the Calkin algebra
By the Atiyah-Jänich theorem the K-group $K^0(X)$ for a compact space $X$ may be represented as $[X, U(Q)]$, where $Q = B(H)/K(H)$ is the Calkin algebra and $H$ is a separable infinite dimensional ...
10
votes
0
answers
1k
views
Regarding the Gerstenhaber bracket on Hochschild cohomology and Morita equivalence
Associated to any $A_\infty$ $k$-algebra $A$ the Hochschild cochain complex $CH^*(A)$ has the structure of a dg-Lie algebra and a dg-algebra which are compatible enough that the cohomology is a ...
9
votes
0
answers
1k
views
Some questions about Clausen's third IHES lecture on Efimov K-theory
I have some questions about the last theorem stated by Clausen at https://youtu.be/2xNG4rHUC6U?si=yw9eYiygLegH0nQK&t=4319. I'm not very familar with the definitions, so please correct me about any ...
9
votes
0
answers
288
views
Why are projectionless $C^*$-algebras important (Kadison's conjecture)
It was considered an important result for a long time to show that the reduced $C^*$-algebra of the free group $C^*_r(F_2)$ has no nontrivial projections. I believe this is also known as Kadison's ...
9
votes
0
answers
337
views
Geometric motivation behind the Fredholm module definition
If $A$ is an involutive algebra over the complex numbers $\mathbb{C}$, then a Fredholm module over $A$ consists of an involutive representation of $A$ on a Hilbert space $H$, together with a self-...
9
votes
0
answers
218
views
Torsion in Atiyah Singer index formula
In the papers of Atiyah and Singer, they first show their index theorem and then derive some index formulas.
For the Fredholm index living in the integers, they use the fact that on spheres the Chern ...
9
votes
0
answers
317
views
Is there a citeable source for generators and relations of simplicial sets?
Simplicial sets can be specified using generators and relations,
in complete analogy with groups, rings, etc.
More precisely, a system of generators of relations for a simplicial set
consists of a ...
9
votes
0
answers
456
views
Beilinson regulators and Bloch's mythological algebraic intermediate Jacobians
In the paper introducing his motivic cycle complexes, Bloch outlines a project he says he was going to return to in the future:
Towards the end of page 270, he says, given a smooth projective variety ...
9
votes
0
answers
236
views
Chern Classes or Chern character classes in the Lichtenbaum-Quillen conjecture?
Let $F$ be a number field, $\mathcal{O}$ its ring of integers, $r>1$ an integer and $\ell$ a prime number different from $2$.
The Lichtenbaum-Quillen conjecture, now a theorem by Voevodsky, Rost, ...
9
votes
0
answers
366
views
Which of the physics dualities are closest in essence to the Spanier-Whitehead duality (with a subquestion)?
First of all, what I want to ask is slightly more elaborate than what stands in the title (hence the subquestion).
I am telling this since as it is, the title contains a meaningful question, but it ...
9
votes
0
answers
349
views
Derived functor of Malcev completion ( = "rational K-theory")
By a theorem of Keune (which is more of definition rather then theorem per se) K-groups of ring $R$ are Puppe's derived functors of pronilpotent completion of $GL(R)$. They are derived in the most ...