5

In Rudyak's book it is proved (3.22 page 160) that every additive cohomology theory (in the sense of Eilenberg-Steenrod) defined on the category of finite dimensional pointed CW complexes can be represented by a spectrum, and this representing spectrum is unique up to equivalence.

This implies the existence and uniqueness of extension of additive cohomology theories from this category to the category of pointed CW complexes.

I am interested in a slightly different setting, and wonder whether the existence and uniqueness of extension still hold:

Question: Let F be a cohomology theory defined on the category of finite dimensional, countable CW complexes, where a cohomology theory for me means an absolute theory satisfying a Mayer-Vietoris property. Is there a unique (up to iso) extension to an additive cohomology theory for CW complexes, and more generally, if the cohomology theory is an equivariant cohomology theory (Z graded, G compact Lie group) defined on the category of finite dimensional countable G-CW complexes, is there a unique (up to iso) extension to an additive cohomology theory for G-CW complexes.

Liviu Nicolaescu
  • 34,003
Haggai Tene
  • 216
  • 2
  • 2
  • 3
    An old theorem due to Adams, extending another one of Brown, says that any cohomology theory defined on finite expectra is represented by an essentially unique spectrum. In particular you can extend it in an essentially unique way. However, I'm afraid this doesn't even answer your first question. – Fernando Muro Nov 27 '13 at 15:39
  • How do you get uniqueness? – Qiaochu Yuan Nov 27 '13 at 19:43
  • Qiaochu, the proof is highly non-trivial, and there's not much homotopy theory in it, but a lot of disguised set theory. This is why there are few generalizations of this result, contrary to what happens with other results such as Brown's representability (for cohomology theories defined in all spectra). – Fernando Muro Nov 27 '13 at 20:22
  • @Fernando Thanks. Adams, in "A variant of EH Brown’s representability theorem", proves an existence theorem 1.6, but as far as I see he doesn't prove uniqueness. Therefore, I am not sure I understand your comment to Qiaochu. – Haggai Tene Nov 27 '13 at 21:15
  • 1
    For uniqueness, see Addendum 1.5 in that paper, and recall Whitehead's theorem. i understood Qiaochu's comment as a deeper interest in the proof. – Fernando Muro Nov 27 '13 at 21:22
  • You're welcome! But your question is still different. I'm really curious about the answer. – Fernando Muro Nov 27 '13 at 21:59
  • @Fernando: sorry, I didn't realize the question was ambiguous. It was addressed to the OP regarding the uniqueness claim in the second paragraph. This is probably standard but I don't know how the argument goes. I am worried about the failure of Brown representability (http://mathoverflow.net/questions/104866/brown-representability-for-non-connected-spaces). – Qiaochu Yuan Nov 28 '13 at 09:51
  • Qiaochu, no problem, man ;-) – Fernando Muro Nov 28 '13 at 10:12

0 Answers0