Timeline for How to get product on cohomology using the K(G, n)?
Current License: CC BY-SA 2.5
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| Mar 30, 2010 at 17:23 | history | edited | Dev Sinha | CC BY-SA 2.5 |
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| Mar 30, 2010 at 17:16 | comment | added | Tyler Lawson | I should point out Graeme Segal gave a very nice construction of a homology theory associated to (connective) K-theory where, instead of taking points of X parametrized by elements of an abelian group, you parametrize by linear subspaces of an infinite inner product space, and require that the subspaces for distinct points are orthogonal. When points collide, you take the sum of the vector spaces. This is one starting point for constructing (co)homology theories out of "Г-spaces". | |
| Mar 30, 2010 at 17:12 | history | edited | Dev Sinha | CC BY-SA 2.5 |
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| Mar 30, 2010 at 16:06 | comment | added | Dev Sinha | Yes - it is exactly (not just homotopy equivalent to) the free abelian group on the n-sphere. Thanks for pointing that out. | |
| Mar 30, 2010 at 12:16 | comment | added | Allen Hatcher | In case the abelian group A is infinite cyclic this labeled configuration space is the free abelian group generated by the points of the n-sphere (other than the basepoint, which gives the zero element of the group, corresponding to the boundary of the cube in Baez's description). This space was studied by Dold and Thom in their 1959 Annals paper on quasifibrations, where they showed it is a K(A,n). They also treated the case that A is finite cyclic, and I would guess that general abelian groups A could be handled similarly. | |
| Mar 30, 2010 at 6:01 | comment | added | Dev Sinha | These look like configuration spaces, but points can "collide and multiply." There is a filtration where the subquotients are one-point compactifications of configuration spaces, which must be known back-and-forth by Fred Cohen. One can use this to relate Dyer-Lashof and Steenrod algebras (also classical). The iterated bar construction, on which this is merely a geometric gloss, is much older. But this geometric gloss is not the standard vocabulary even for topologists who use these things every day. I first heard it in a lecture by Baez (and I also think it appeared in "This week's finds..") | |
| Mar 30, 2010 at 4:56 | comment | added | Ryan Budney | Is this labelled configuration space construction originally Baez's? I would have guessed it's much older. | |
| Mar 30, 2010 at 3:19 | history | edited | Dev Sinha | CC BY-SA 2.5 |
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| Mar 29, 2010 at 20:42 | history | edited | Dev Sinha | CC BY-SA 2.5 |
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| Mar 29, 2010 at 7:26 | history | answered | Dev Sinha | CC BY-SA 2.5 |