Timeline for Chern character of the index bundle for a family of Dirac operators
Current License: CC BY-SA 4.0
7 events
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| Jun 30, 2018 at 19:05 | history | edited | Michael Albanese | CC BY-SA 4.0 |
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| Apr 2, 2011 at 9:19 | comment | added | Johannes Ebert | Yes, you are missing something. The Atiyah-Singer family index theorem holds for any compact Hausdorff base $X$. The definition of a smooth bundle and a fibrewise elliptic operator needs to be adjusted a bit. Only the homotopy type of $X$ and isomorphism class of the smooth fibre bundle matters. Of course, if $X$ is a closed manifold, then so is $Z$. An elliptic operator $D$ on $Z$ restricts to an elliptic operator $D_{fib}$ on the fibres. Of course $ind(D)$ and the index bundle of $D_{fib}$ are related. | |
| Apr 2, 2011 at 1:15 | comment | added | charris | Pontryagin classes only depend on the topological structure of the manifold. Is it ever possible that there could be relations between the Chern character of the index bundle and the Pontryagin classes on $X$, or am I still missing something? | |
| Apr 1, 2011 at 17:09 | answer | added | Johannes Ebert | timeline score: 4 | |
| Apr 1, 2011 at 14:33 | comment | added | charris | That's good to know. Thank you for your response! I will read up on it some more. | |
| Apr 1, 2011 at 9:04 | comment | added | Johannes Ebert | The answer to your second question is no. The smooth structure on the base space $X$ is completely irrelevant for index theory on fibre bundles on $X$. See Atiyah-Singer "Index of elliptic operators IV". | |
| Mar 31, 2011 at 20:08 | history | asked | charris | CC BY-SA 2.5 |