All Questions
21 questions
2
votes
0
answers
101
views
A roof genus of high dimensional lens space
Let $p$ be a natural number, and for $i\in \{0,
..., p-1\}$,
denote the irreducible rank one complex representation of $\mathbb{Z}/p$. by $\rho_{i}$.
Let $a=(a_{1},\ldots a_{d}) $ ...
- 2,630
9
votes
1
answer
756
views
Does there exist a GRR-like generalization of the AS Index Theorem?
The Hirzebruch Riemann-Roch Theorem (HRR) expresses an analytic/algebraic invariant, namely the Euler-Poincaré characteristic of a vector bundle $V$ over a compact complex/algebraic manifold $X$, as ...
- 1,020
5
votes
1
answer
308
views
Compactly supported chern character
It is a standard result that for a CW complex $X$, the chern character
$$\text{ch}: K^*(X)\otimes_{\mathbb{Z}} \mathbb{Q}\to H^*(X,\mathbb{Q})$$
induces an isomorphism. Suppose now that $X$ is an open ...
- 988
5
votes
0
answers
297
views
Chern-Weil theory in the cohomological Atiyah-Singer theorem
I am interested in the following piece of data appearing in the cohomological Atiyah-Singer theorem. My reference is "The index of elliptic operators. III" by Atiyah and Singer.
Let $D:\...
- 2,247
6
votes
0
answers
230
views
Equivariant Venice Lemma
In the paper J. Simons and D. Sullivan. Structured vector bundles define differential K-theory, one of the key ideas is the so called Venice Lemma, which essentially can be stated as
Theorem: For ...
- 386
5
votes
1
answer
366
views
K-theory for a (geometric) stack
There is a notion of $K$-theory for a manifold $M$.
Is there a notion of $K$-theory for a stack $\mathcal{D}\rightarrow \text{Man}$ that is representable by a Lie groupoid $\mathcal{G}$; that is $...
- 6,023
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 ...
- 171
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 ...
- 2,995
5
votes
0
answers
238
views
Tensor product of "difference bundles" ( Atiyah construction)
There is a well-known in index theory "difference bundle" construction of Atiyah( see for example the original paper "Clifford modules"). And also there is a corresponding formula for the tensor ...
- 51
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 ...
- 9,870
4
votes
1
answer
384
views
Torsion In $K$ theory on simply connected manifolds
The usual construction for finding torsion elements on complex $K$ theory is using flat vector bundles. So is it still possible to find a simply connected compact space with a nonzero torsion in its $...
- 435
2
votes
0
answers
105
views
Multiplicativity of the analytic index (or of kernel bundle)
What I want to ask is the multiplicativity of the analytic index of a family of Dirac operators.
In the single operator case the analytic index of elliptic operator is multiplicative. This is proved ...
- 1,173
7
votes
0
answers
359
views
Aityah-Patodi-Singer theorem in odd dimensions and Maslov triple indices
Let $W$ be a compact manifold with boundary and $D^W$ a graded Dirac type operator on $W$, of product type near the boundary acting on a vector bundle $E\to W$. One obtains a graded Fredholm operator $...
- 186
62
votes
3
answers
6k
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 a rather broad background is required: you need
to know analysis (pseudodifferential ...
- 9,330
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
33
votes
2
answers
2k
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 ...
- 118k
18
votes
3
answers
2k
views
Can eta invariant be written in terms of topological data?
The eta invariant was introduced by Atiyah, Patodi, and Singer. It roughly measures the asymmetry of the spectrum of a self-adjoint elliptic operator with respect to the origin. In the paper "Exotic ...
- 875
4
votes
2
answers
633
views
filtration in K-theory and ordinary cohomology
I am going to ask a question, which could be a stupid one.
I am reading a paper "an index theorem in differential K-theory". The first paragraph of section 8.28 recalls a filtration of K-theory $K_{(...
- 1,173
18
votes
3
answers
3k
views
Atiyah-Patodi-Singer Eta invariant and Chern-Simons form
I am trying to understand the Atiyah-Patodi-Singer index theorem in the case of Dirac operators in four dimensions. I have three questions about the eta invariant:
1) Is eta a topological invariant (...
- 405
0
votes
0
answers
307
views
A modified version of K-theory for manifolds ?
If $X$ is a compact smooth manifold, $K^{0}(X)$ can be defined as the algebraic $K_{0}$-group of $C^{\infty}(X)$. In order to do that we use the following equivalence relation: we say that two ...
- 505
9
votes
3
answers
1k
views
Integration in equivariant K-theory
Let F be a smooth classifying space for K-theory (ordinary or equivariant). If X is a smooth compact manifold and W is a real vector space of dimension n, there is an integration map from the ...
- 226