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In Rudyak's On Thom Spectra, Orientability, and Cobordism, two variants of Brown's representability theorem are presented: given a natural transformation $f^*: E^* \to F^*$ of cohomology theories, Brown's representability theorem asserts that we can lift $f^*$ uniquely to a map $f: E \to F$ of spectra, and we can also lift $f^*$ to map $f^i: \Omega^\infty \Sigma^i E \to \Omega^\infty \Sigma^i F$ of representing spaces -- but he does not assert that this second kind of lift is unique.

My feeling is that this second map becomes unique after stabilization. Is there an accessible example of such a natural transformation with two nonhomotopic representing maps on classifying spaces, illustrating nonuniqueness?

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The actual work being done here is by the Yoneda lemma. Brown's representability theorem tells you that these spaces represent the cohomology theory, turning natural transformations into morphisms is done by Yoneda.

That said, uniqueness holds.

Of course, one has to be a little bit more careful about what one means by "natural transformation" in this case. Natural transformation of what? If your natural transformation is of the whole cohomology theory in one go (aka a stable operation) then, indeed, you get a morphism of spectra. If your natural transformation acts on one particular level (aka an unstable operation) then you get a morphism of spaces.

Where you do not get uniqueness is if you have a family of unstable operations (aka a family of morphisms of spaces) which look as if they fit together to give a stable operation. You can get "phantom" morphisms, and there's a "$\lim^1$" term that controls this. A nice place to read about all of this is the papers by Boardman and Boardman, Johnson, and Wilson on stable and unstable cohomology operations (Handbook of algebraic topology, also available via Steve Wilson's homepage).


Update: To expand on that last point (I was deliberately vague because I didn't have Boardman's paper in front of me and couldn't remember which way the maps went): a stable operation (morphism of spectra) defines a compatible (under suspension) family of unstable operations (morphisms of spaces, indeed, morphisms of infinite loop spaces) but this assignment may not be injective (it is always surjective). There is a short exact sequence (attributed to Milnor, and (9.7) in Boardman's paper):

$$ 0 \to \lim_n{}^1 E^{k-1}(\underline{E}_n,o) \to E^k(E,o) \to \lim_n E^k(\underline{E}_n,o) \to 0 $$

The "o" means "pointed" and $\underline{E}_n$ is the $n$th component in the spectrum $E$. So if the $\lim^1$ term vanishes, you get an isomorphism (whence injectivity) but if not, then there may be "phantom" stable operations that are non-trivial but can't be detected in the unstable realm.


Update: And now I've been shown a copy of the book references in the question and have read what may well be the statements themselves. The second one is subtly different to the summary above. In the book, the statement is that a natural transformation of cohomology theories of spaces can be lifted to a morphism of the representing spectra. This is not necessarily unique. The basic reason being that the Yoneda lemma does not apply to this case because the representing object is in a different category. So cohomology of spaces represented by a spectrum is the composition of $\Sigma^\infty$ and the cohomology theory and thus not an honest representable functor. In general, there's no way (that I can think of) to pull-back or push-forward natural transformations along a functor.

Nonetheless, because spectra and spaces are closely related, it is possible in this case to construct a natural-transformation-of-cohomology-of-spaces from one of spectra. This works because the pieces of the cohomology theory are individually representable as spaces, and so again the Yoneda lemma comes into play. Then one can ask about this map, and the $\lim{}^1$ sequence says that it is always surjective but not necessary injective.

So, in summary:

  • $E^\ast(R)$: both spectram: Yoneda => $Nat(E^\ast,F^\ast) \cong \{E,F\}$
  • $E^k(X)$: both spaces: Yoneda => $Nat(E^k,F^k) \cong [\underline{E}_k, \underline{F}_k]$
  • $E^\ast(X)$: spectrum and space: Milnor => $\{E,F\} \to Nat(E^\ast,F^\ast)$ surjective
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Thanks a lot! That's just what I was looking for! –  Adam Hughes May 7 '10 at 19:03

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