All Questions
Tagged with dg.differential-geometry kt.k-theory-and-homology
13 questions
50
votes
0
answers
12k
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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
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
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
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
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
1
vote
0
answers
240
views
Smooth version of the splitting principle
Inspried by this MO question A manifold whose tangent space is a sum of line bundles and higher rank vector bundles we pose the following question as a possible smooth version of the splitting ...
- 356
18
votes
3
answers
3k
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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
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
5
votes
1
answer
772
views
Does bundle with torsion Chern classes admit flat connection?
I want to know something about torsion in topological k-theory. So, consider complex bundle with chern classes lying in torsion part of integer homologies and my question is : does it admit a flat ...
- 789
4
votes
1
answer
236
views
Hodge theory, conformal manifolds and Fredholm modules-understanding the proof of one Lemma
I would like to understand the proof of Lemma 1, page 339 in this book. Very briefly, the context is as follows: we have even dimensional oriented conformal manifold with the Hodge star operator ...
- 9,330
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
2
votes
2
answers
124
views
Invertible (isometric) sections of certain hom bundles over sphere
Assume that we have a vector bundle $E$ over $S^n$.
Is there a continuous family of invertible linear maps $T_x:E_x \to E_{-x}$?
Here continuity has the obvious meaning as soon as ...
- 356
1
vote
0
answers
172
views
Calculation about Chern character in a special setting
I'm confused with working out the Chern character in the following special setting.
Let $E$ be a spinor bundle
$$S=P_{Spin(2n)}(S^{2n})\times_\rho \mathbb{C}^{2n}$$
over sphere $S^{2n}$, where $\rho$ ...
- 563