Talk:Künneth theorem

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Mathematics rating: Start Class Mid Priority  Field: Topology

Derived categories provide a very attractive way to hugely generalise the classical Künneth formula (applies to sheaf cohomology and in a relative setting) while at the same time even simplifying the formulation. The classical result is the special case where the sheaves sit on one-point spaces. I shall develop this generalization into the article at a later stage. Stca74 21:07, 16 May 2007 (UTC)

[edit] Rewrite

This article is just written extremely poorly and needs an overhaul from someone who knows the material far better than I do. There is inconsistent variation between singular and CW homology. There are no references cited; the general formula at the end appears to have been copied verbatim from Allen Hatcher's text. —Preceding unsigned comment added by 71.206.187.25 (talk) 03:16, 13 October 2007 (UTC)

What is "CW homology"? Charles Matthews 18:27, 14 October 2007 (UTC)
For a CW-complex one can build a chain complex using the combinatorics of the attaching maps. Then the homology of the this complex can be called the "CW cohomology". However, it is not essentially a new cohomology theory rather than one more method of computing the "ordinary" homology of a space that happens to be (homotopic to) a CW-complex. The CW-chain complex tends to be a simpler combinatorial object than a simplicial complex from a triangulation (need fewer cells than triangles), and has the benefit of being finitely generated (for finite CW complexes) compered to the singular chain complex. I don't have a reference at hand to verify if there were conditions (beyond homotopical equivalence to a CW complex) for a space to have isomorphic singular and CW homology. But this is a standard topic, see e.g., Bredon's Topology and Geometry for details.
As for the need for an overhaul, completely agree. No time to invest to this myself right now, I a'm afraid. Stca74 11:37, 15 October 2007 (UTC)