Timeline for What is the role of equivariance in the Atiyah-Singer index theorem?
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| Dec 18, 2011 at 14:49 | comment | added | Paul Siegel | The K-homology proof in Higson-Roe follows the overall scheme of the IEO proof, only it is organized around the ubiquitous "Kasparov product" which clarifies a lot of the subtle analysis. They prove the theorem for hypersurfaces in $\mathbb{R}^n$ first (as Ebert mentioned this requires no equivariant theory) and then explain the modifications needed to prove the theorem in the general case. | |
| Dec 18, 2011 at 14:44 | comment | added | Paul Siegel | Here's a few more references. The first place where it was explicitly realized that K-homology is generated by elliptic operators is in Atiyah's "Global theory of elliptic operators", and its worth looking at because it anticipates a lot of the more sophisticated developments in the theory that came later. There is also a textbook treatment of K-homology and the index theorem in the form of the book "Analytic K-homology" by Higson and Roe - if you're willing to take a bit on faith you can skip straight to chapter 11 (the proof of the index theorem). | |
| Dec 17, 2011 at 21:29 | comment | added | Akhil Mathew | @Johannes: Thanks for the additional references. | |
| Dec 17, 2011 at 18:26 | comment | added | Johannes Ebert | @Akhil: yes, K-homology is the homology theory associated to the K-theory spectrum. This itself is not very useful. It was Atiyah's idea that an elliptic operator on a closed manifold should represent an element in K-homology. A good starting point is the paper "Index theory, bordism and K-homology" by Baum and Douglas. This idea has been developped by G. Kasparov into the monumental ''KK-theory''. | |
| Dec 17, 2011 at 14:42 | comment | added | Akhil Mathew | (Is it supposed to be the homology theory associated to the K-theory spectrum? I remember reading that the space of Fredholm operators on an infinite-dimensional Hilbert space had the homotopy type of $BU \times \mathbb{Z}$.) | |
| Dec 17, 2011 at 14:40 | comment | added | Akhil Mathew | Thanks! I don't know anything about K-homology, but I suppose I'll look more at Atiyah's papers. | |
| Dec 16, 2011 at 20:55 | history | answered | Paul Siegel | CC BY-SA 3.0 |