Timeline for Are generalized cohomology theories a set of complete homotopy invariants for spaces ?
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
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| Mar 3, 2011 at 2:06 | comment | added | Tyler Lawson | @Eric: Yes, you're correct - that's a ridiculous mistake on my part. I won't edit it since I don't want to bump it, but I suppose the second example will have to suffice - or any 2-cell complex formed by attaching a sphere where the attaching map becomes stably trivial. | |
| Mar 2, 2011 at 22:50 | comment | added | Eric Wofsey | $S^3\vee S^5$ and $\Sigma\mathbb{CP}^2$ do not agree on all cohomology theories--they are not stably equivalent. Specifically, any spectrum on which the Hopf map acts nontrivially (eg, the sphere spectrum) will give a cohomology theory that distinguishes them. | |
| Mar 2, 2011 at 21:30 | comment | added | David Roberts♦ | +1 and welcome to the 10k+ club! More seriously, if you also consider nonabelian cohomology $H^1(-,G)$ for systems of coefficients $G$, as well as all (abelian) cohomology theories, then there is a result of Artin-Mazur (I think!) on homotopy types of topoi that says that if all of these send a map to an isomorphism, then is is a (weak) homotopy equivalence. Grothendieck used this characterisation in his work on derivators, but I may be misremembering. | |
| Mar 2, 2011 at 20:59 | comment | added | Jeff Strom | Let $X$ be a noncontractible space such that $\Sigma X \simeq *$. Then every cohomology theory will see $X$, incorrectly, as contractible. There are such spaces. | |
| Mar 2, 2011 at 19:22 | comment | added | John Klein | I'd like to point out a special case of your plus construction example: how about $M^3 =$ Poincare's homology sphere? This has a degree one map $M^3 \to S^3$ which is a multiplicative cohomology isomorphism in any theory. | |
| Mar 2, 2011 at 18:00 | history | edited | Tyler Lawson | CC BY-SA 2.5 |
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| Mar 2, 2011 at 17:34 | history | edited | Qfwfq | CC BY-SA 2.5 |
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| Mar 2, 2011 at 17:28 | comment | added | Qfwfq | Thank you. Interesting! (I accept this answer, among the others, just because this is more expanded but the idea was present also in the others) | |
| Mar 2, 2011 at 17:25 | vote | accept | Qfwfq | ||
| Mar 2, 2011 at 17:11 | history | answered | Tyler Lawson | CC BY-SA 2.5 |