In his [student guide](http://www.cambridge.org/de/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-students-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](http://press.uchicago.edu/ucp/books/book/chicago/S/bo21302708.html)), 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?



Multiplicative cohomology theories and smash products
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](http://www.maths.ed.ac.uk/~aar/papers/gww9.pdf)".)


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 $\Sigma^\infty S^1$.


