Timeline for Why is the definition of the higher homotopy groups the "right one"?
Current License: CC BY-SA 3.0
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| Oct 27, 2017 at 6:03 | comment | added | Tim Porter | Dan: your first sentence is highly contentious. Many 'spaces' arising in geometry (in algebraic geometry at least) are not of that form. I feel that the 'combinatorial' nature of many spaces, then, almost suggests that the spatial structure is not really what is being studied by the homotopy type invariants. I used to say that when a topologist say 'given a space, $X$' one should ask 'how is it given?'. If an analyst asks for invariants of a space occurring in their work, it may not be a CW-complex, but using Cech-type methods one can use CW-complex methods in its study. | |
| Oct 23, 2017 at 21:56 | comment | added | Dan Ramras | I think that since most of the "geometric" and "combinatorial" spaces that occur in mathematics - that is, manifolds, real and complex algebraic varieties, spaces built from combinatorial data like categories and posets - have the homotopy type of CW complexes, one can break the circularity so to speak by beginning with such objects as the motivating examples. Then they happen to fit into a nice class of objects - spaces of the homotopy type of a CW complex - for which higher homotopy groups are a very useful invariant. | |
| Oct 23, 2017 at 13:25 | comment | added | Steve Costenoble | Absolutely, regarding the complication. I didn't mention that, but stated the result carefully with it in mind. | |
| Oct 23, 2017 at 6:06 | comment | added | Tim Porter | There is one added complication, Steve, and that given two spaces that you know the homotopy groups of, and yeah they are isomorphic, you still have to construct a suitable map realising the isomorphisms. This realisation problem was there from the start in J. H. C. Whitehead's fundamental Combinatorial Homotopy papers (late 1949s, early 1950s) and have been explored a lot by Baues in his books. The circularity that you point out is one reason why CW-complexes are where the rapid expansion of algebraic topology concentrated at least initially. | |
| Oct 22, 2017 at 15:54 | comment | added | მამუკა ჯიბლაძე | @TylerLawson if I am not mistaken, homotopy groups of finite topological spaces are isomorphic to the homotopy groups of the geometric realizations of the corresponding preorders. | |
| Oct 21, 2017 at 23:45 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Oct 21, 2017 at 20:00 | comment | added | Tyler Lawson | I agree, and this is probably going to be a problem for somebody who (as the questioner said) has a background in arithmetic geometry and often encounters topological spaces of a very different type. | |
| Oct 21, 2017 at 19:25 | history | answered | Steve Costenoble | CC BY-SA 3.0 |