Timeline for Chern classes of generators of $K(S^{2n})$

Current License: CC BY-SA 4.0

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Jun 7, 2018 at 18:11	comment	added	Michael Albanese		There is an error in the proof of Theorem IV 1.4, it should be $\operatorname{ch}_2E$ not $\operatorname{ch}E$. Making this adjustment, the conclusion is $\operatorname{ind}(D^+) = \frac{2^n}{(n-1)!}c_n(E)$ which is not strong enough to prove the theorem. However, using the Dirac operator coupled to $E$ gives the stated index and the correct conclusion.
Jun 6, 2018 at 14:09	comment	added	Oliver Nash		Many apologies if, as it seems, I am spreading misinformation. Glancing quickly at L&M, I can't quite line up what you're saying though. Furthermore, is not Theorem IV 1.4 support for my remark? [I'll check carefully later --- for now I have to get back to the day job!]
Jun 6, 2018 at 13:59	comment	added	Michael Albanese		@OliverNash: I believe that index of the signature operator coupled to $E$ is $2^n$ times the integer you mention, see Theorem III.13.9 of Lawson and Michelsohn's Spin Geometry. On the other hand, the index of the Dirac operator coupled to $E$ has this integer as its index, see Theorem III.13.10.
Jun 6, 2018 at 13:14	comment	added	Oliver Nash		So, because the image of the Chern character is the integral cohomology, $c_n(E)/(n-1)!$ is an integer for any $E$. This raises the question: what is this integer that is intrinsically associated to the vector bundle E? In fact there is an answer: it is the index of the signature operator coupled to E.
Jun 6, 2018 at 13:13	history	edited	Michael Albanese	CC BY-SA 4.0	added 13 characters in body
Jun 6, 2018 at 12:36	history	edited	Michael Albanese	CC BY-SA 4.0	added 2 characters in body
Jun 6, 2018 at 2:42	history	edited	Michael Albanese	CC BY-SA 4.0	added 296 characters in body
Jun 6, 2018 at 2:35	history	edited	Michael Albanese	CC BY-SA 4.0	added 296 characters in body
Jun 6, 2018 at 2:27	history	answered	Michael Albanese	CC BY-SA 4.0