Timeline for K-theory as a generalized cohomology theory
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 17, 2010 at 22:08 | comment | added | Tilman | But the statement about cohomology theories being determined by their coefficients is not totally wrong -- if you have a natural transformation $K \to L$ of homology theories which induce an isomorphism on the point (or $\mathbf{S}^0$ in the case of reduced theories), then it's an isomorphism. But you do need to have a natural transformation in the first place. | |
| Mar 17, 2010 at 19:51 | comment | added | Charles Rezk | We're thinking of generalized cohomology theories as taking values in graded abelian groups. Thus $\tilde{K}^q(S^0)\not\approx \tilde{KO}^*(S^0)$, for instance if $q\equiv -1,-2,-6\mod 8$. | |
| Mar 17, 2010 at 19:36 | comment | added | roger123 | Reid, a last thing: Why doesn't the corrected version of 3 hold? Isn't it true that $\tilde K(S^0)=\widetilde{KO}(S^0)=\mathbb{Z}$? | |
| Mar 17, 2010 at 19:27 | comment | added | Andy Putman | It could easily be fixed by putting a link to the entry for the Atiyah-Hirzebruch spectral sequence as a way of making this precise. | |
| Mar 17, 2010 at 19:24 | comment | added | Reid Barton | Hmm, I see. Well, Wikipedia does use quotes around "determined by its values on a point" :) but I agree that this statement is misleading at best. | |
| Mar 17, 2010 at 19:14 | vote | accept | roger123 | ||
| Mar 17, 2010 at 19:14 | comment | added | roger123 | Thanks, Reid. Someone should change the "Generalized cohomology theories" section in Wikipedia's article on "Cohomology". | |
| Mar 17, 2010 at 18:59 | history | answered | Reid Barton | CC BY-SA 2.5 |