Timeline for "non natural" iso between homotopy and homology
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 8, 2010 at 21:50 | answer | added | Tilman | timeline score: 4 | |
| Oct 8, 2010 at 17:11 | comment | added | Manuel Rivera | @ Somnath - Yes, this is what "motivated" the question. I am one of the new guys at Sullivan's class at CUNY and after class last wednesday I was wondering if we can get any interesting information about the geometry of the spaces - not the geometric meaning of the isomorphisms- for which some of these isomorphisms are not the given by the natural Hurewicz map... | |
| Oct 8, 2010 at 16:21 | comment | added | Somnath Basu | @ Manuel - Just to give slightly different example : A space $X$ and the space $\prod K(\pi_i(X),i)$ have isomorphic homotopy groups. Unless you have a map from one to the other realizing these isomorphisms, there is no geometric meaning to these isomorphisms. The same goes for homology too - given a suitable $X$ you can cook up a product of Moore spaces which have homology isomorphic to $X$. But until you have a map one way or another that realizes these isomorphisms, it is not veru helpful. | |
| Oct 8, 2010 at 15:29 | comment | added | Manuel Rivera | Yes, certainly the question sounds like it lacks of motivation, since crazy maps between homotopy and cohomology and not particularly useful. I was just trying to think what is happening geometrically when you have these "non natural" isomorphisms. | |
| Oct 8, 2010 at 15:25 | history | edited | Manuel Rivera | CC BY-SA 2.5 |
edited title
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| Oct 8, 2010 at 15:19 | comment | added | Kevin H. Lin | The "right" viewpoint on invariants like $H_i$ and $\pi_i$ is that they are functorial invariants. It is somehow "wrong" to not think of them functorially. The Whitehead theorem gives some justification for this philosophy. | |
| Oct 8, 2010 at 15:11 | comment | added | Josh | Would we have to know, a priori, all of the homotopy groups of a space to answer this question? | |
| Oct 8, 2010 at 15:07 | comment | added | Ryan Budney | @Manuel, do you have a particular motivation for this problem? The answer is going to be some small class of CW-complexes but I'm not seeing a motivation for the question. In that regard the title is a little misleading because homotopy groups form a type of graded lie algebra and homology groups don't. Moreover, you're asking for a dimension-wise isomorphism of groups, not an equality of groups. | |
| Oct 8, 2010 at 14:39 | history | asked | Manuel Rivera | CC BY-SA 2.5 |