All Questions
Tagged with noncommutative-geometry at.algebraic-topology
22 questions
9
votes
1
answer
443
views
Hochschild cohomology of a group algebra
Let $K$ be a field and $G=\pi_1(\Sigma_g)$ the surface group of genus $\geq 2$. I want to know the Hochschild cohomology of the group algebra $A=K[G]$ with coefficients in $A$ and $A\otimes A$, namely,...
- 985
9
votes
0
answers
361
views
Bernoulli-like polynomials
Let $\psi_0 (x,t)=\frac{te^{xt}}{1-e^{-t}}$. Then
$$\psi_0(0,t)=\frac{t}{1-e^{-t}};$$
$$\psi_0(x,t)=1+\sum_{n=1}^\infty \frac{t^n}{n!} B_n(x)$$
where $B_n$ is a monic polynomial of degree $n.$
Now ...
- 401
11
votes
2
answers
1k
views
Good reference for topological Hochschild homology
I want to start reading topological Hochschild homology(THH) as well as topological cyclic homology (TC).
I have read the Hochschild homology and cyclic homology from the book Cyclic homology by J. ...
- 629
8
votes
1
answer
539
views
Vanishing of Hochschild homology of a category
Let $A$ be a dg- or $A_{\infty}$-category (with $\mathbb{Z}$-graded Hom sets, over a field of characteristic $0$). Let $HH_*(A)$ be the Hochschild homology of $A$.
Suppose that $HH_n(A)=0$ for all $n ...
- 937
2
votes
1
answer
386
views
Homotopy groups of noncommutative spaces
In the approach to noncommutative geometry of Alain Connes any Hausdorff compact space $X$ is replaced by its algebra of complex valued continuous functions $C^0(X)$, and one regard general (that is, ...
user95283
1
vote
0
answers
117
views
The holonomy groupoid of certain one dimensional foliations of 2 dimensional Euclidean regions
What Is the first fundamental group of each of the following $3$ dimensional Hausdorff manifolds? What about homology groups of these 3-manifolds? Is the first one a contractible manifold?
The ...
- 356
10
votes
1
answer
827
views
Baum Connes conjecture and Atiyah-Singer index theorem
Baum Connes conjecture is considered as a far generalisation of the Atiyah Singer index theorem (in K-theoretical formulation). I would like to understand how the latter follows from this conjecture. ...
- 9,330
2
votes
0
answers
254
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 ...
- 493
6
votes
2
answers
1k
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 $Z\subset X$...
- 7,952
23
votes
0
answers
463
views
Topological loops vs. algebro-geometric suspension in Hochschild homology
Let $k$ be a base commutative ring, and let $A$ be a (unital but not necessarily commutative) $k$-algebra. The cone on $A$ is the ring $CA$ of infinite matrices $(a_{ij})_{i,j \geq 1}$ that are ...
- 6,069
10
votes
2
answers
834
views
Analytical formula for topological degree
At the first page of the following article http://arxiv.org/pdf/1004.1018v1.pdf [edit: the formula on the arXiv differs from the formula in the published paper, and the formula displayed below is the ...
- 101
11
votes
1
answer
2k
views
A survey for various $K$-homology theories and their relationship
The ordinary Topological $K$ theory defined by Atiyah and Hirzebruch is a generalized cohomology theory (see wikipedia).There is the Bott spectrum associated to this generalized cohomology theory....
- 593
2
votes
0
answers
167
views
A noncommutative vector bundle associated with a codimension one foliation
Assume that we have a codimension one foliation of a manifold $M$ which is generated by a one form $\alpha$. So the following $\phi$ satisfies $\phi \circ \phi =0$:$$\phi:\Omega^{i}(M)\to \...
- 356
4
votes
0
answers
356
views
Flat Connections on Ring Spectra
So first I'll try to give a really quick reminder of the classical description of these things when one is doing non-commutative descent theory. In the setting of discrete algebra, if we have a ...
- 10.4k
6
votes
1
answer
392
views
The function algebra $C^{\infty}(M\#N)$ of the connected sum of two spaces
Operations such as taking union or Cartesian products of spaces have direct analogues in term of algebra of functions on them (direct sum and tensor product, respectively),
my question is:
Is there ...
- 309
3
votes
1
answer
182
views
Local index formula for >ungraded< elliptic operators
Let $P\colon E \to F$ be an elliptic pseudodifferential operator over $M$. Assuming that $P$ defines a finitely summable Fredholm module, we may apply the Chern-Connes character to it to get a cyclic ...
- 2,998
8
votes
1
answer
374
views
Do we have a "topological assembly map" in the Baum-Connes conjecture?
In the equivariant Atiyah-Singer index theorem, when $G$ is a compact group acting on a manifold $M$ and $R(G)$ is the representation ring of $G$. We have the analytic index
$$
\text{a-ind}: K^*_G(TM)\...
- 9,019
10
votes
3
answers
1k
views
The "right" $C^*$ algebraic proof of Bott Periodicity
In learning about the K-theory of $C^*$-algebras, I have encountered the following 3 proofs of Bott periodicity:
$\bullet$ An argument based on Moyal quantization found in "Elements of Noncommutative ...
- 1,010
3
votes
1
answer
817
views
dg-lie structure on $HH^*$ and Koszul duality
This is shamelessly close to my other question: A Question on Koszul duality and $B(\infty)$ structures on $HH^*$. Maybe this one will get a better response. Rather than rewrite that one, I am going ...
- 3,758
1
vote
0
answers
193
views
How to get countably many generators for $K_{j}^{G}(\beta G)$ ??
Hey
I am trying to find out how the Baum-Connes conjecture works over $GL(1)$ over local fiels.
I am just wondering if anybody knows how to get a countable many generators for in the L.H.S of the ...
- 85
7
votes
0
answers
297
views
Inner product on Hochschild homology in 2d TCFTs
This should be an easy question for some people. Take a compact $A(\infty)$ algebra with a cyclically symmetric non-degenerate inner product. In Kontsevich and Soibelman's article "Notes on $A(\infty)$...
- 3,758
5
votes
0
answers
517
views
A smooth twisted tensor product of dg algebras?
I want to consider a Z/2Z dg algebra. As an algebra, it is generated over $\mathbb{Q}$ by two elements where x is even and e is odd with the relations $xe=ex$ and $e^2=1$(this makes it in particular ...
- 3,758