Multiplicative cohomology theories and smash products

Question

In his student guide on page 154, Adams gives a construction of products for cohomology using "pairings" of spectra (now known as maps from $E\wedge E\to E$). But then he says

However, G. W. Whitehead did not say (because it is not true) that every product in generalised cohomology theory arises from a pairing in this sense.

On the other hand, if $E$ is a CW-spectrum (as in Adams' Chicago lecture notes), one can turn it into an $\omega$-spectrum in the category of CW-spectra using Milnor's theorem. Then each product in the corresponding generalized cohomology theory tautologically defines maps $E_k\wedge E_\ell\to E_{k+\ell}$.

Thus my questions: What examples have G. W. Whitehead and J. F. Adams been thinking of? In which situations would the naïve Argument above fail?

Answer 1

I do not claim that I know what Adams meant; for all I know his definition of "product" may include products which are not bilinear. Take the following with a grain of salt.

What I suspect is the following. Adams says, just before your quote:

"In order to construct such products, G.W. Whitehead introduced the notion of a 'pairing of spectra'. A pairing of spectra from $E$ and $F$ to $G$, in the sense of G.W. Whitehead, consists of maps $$ \mu_{n,m}: E_n \wedge F_m \to G_{n+m} $$ which satisfy suitable axioms..." (Said suitable axioms appearing in detail in Whitehead's paper "Generalized homology theories".)

Adams then describes the pairing on complex cobordism as arising from maps $MU_n \wedge MU_m \to MU_{n+m}$.

I suspect that what Adams meant is that, given spectra $E$, $F$, and $G$, there may be pairings from $E$ and $F$ to $G$ which do not arise from maps of the above specific type: for example, if you skipped the step of replacing $G$ with an $\Omega$-spectrum, or don't adopt the "cells now, maps later" methodology, some needed targets for maps from $E_n \wedge F_m$ may be missing. For example, the stable Hopf map $\eta \in \pi_2^{stable}(S^1)$ cannot be represented by a pairing from $\Sigma^\infty S^1$ and $\Sigma^\infty S^1$ to $\Sigma^\infty S^1$.