0
votes
0answers
67 views

Ring structure for $K^{-1}$?

My questions are whether there exists a product structure for $K^{-1}(X)$? Here $K^{-1}$ is the odd topological $K$-group, and $X$ is a compact space (or a manifold), say. If such a ring structure ...
6
votes
2answers
382 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 ...
19
votes
2answers
581 views

Applications of Atiyah-Singer using pseudodifferential operators

Though the Atiyah-Singer index theorem holds for pseudodifferential operators, all the applications of the index theorem I know of only need it for Dirac-type operators. I know that pseudodifferential ...
2
votes
2answers
304 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 ...
1
vote
0answers
148 views

Equivariant $K$-theory, singular vectors, and flag manifolds

For a homogeneous space $M = G/B$, with $G$ a (complex) semi-simple Lie group, it is very well-known that equivariant vector bundles $E$ over $M$ correspond to representations $(V_{\lambda},\lambda)$ ...
4
votes
1answer
406 views

second fundamental form

Hi all, Currently I'm reading a paper about the geometry of Grassmannians: www.omup.jp/modules/papers/riemann/04Nagatomo.pdf In there, the author regards the second fundamental form of the ...
3
votes
1answer
294 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 ...
10
votes
2answers
1k 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 ...
0
votes
0answers
259 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 ...
5
votes
1answer
313 views

Splitting principle in equivariant cohomology

The following is a weaker version of what is called splitting principle in Appendix C, page 12, see also for a lighter version Brions Eq cohom and eq intersection theory, page 6: Let $G$ be a compact ...
8
votes
3answers
759 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 ...
22
votes
9answers
4k views

Looking for an introduction to orbifolds

Is there any source where the basic facts about orbifolds are written and proved in full detail? I found the article by Satake "The Gauss-Bonnet Theorem for V-manifolds", but I'd like to have a more ...