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Suppose you are given an infinite loop space $T_0$. The $i$th deloopings, $T_i$ then form a prespectrum with evaluation maps maps $\Sigma T_i \to T_{i+1}$. In the two examples that I know of, the eilenberg mclane spectrum and the thom spectrum, there are maps from $K(G,n) \wedge K(G,m) \to K(G,m+n)$ and the same for the thom spectrum. The picture for the thom spectrum is especially illuminating - $T(\xi(O(n))) \wedge T(\xi(O(m)))$ is homeomorphic to $T(\xi(O(n))\times \xi(O(m)))$. And there is a map $T(\xi(O(n))\times \xi(O(m))) \to T(\xi(O(n+m)))$ which can be viewed as coming from the gauss map taking a point in an n-dimensional vector space and a point in the m-dimensional vector space to the point in the direct sum of these vector spaces taken in $\mathbb{R}^\infty \oplus \mathbb{R}^\infty=\mathbb{R}^\infty$. The picture for eilenberg mclane spaces is quite different,(and more general): If we let $K_m=K(G,m)$, then the map from $K_m \wedge K_n \to K_{m+n}$ is given by the the fundamental class $\iota={h^{-1}}{\in Hom(H_{m+n}(K_{m}\wedge K_n,\mathbb{Z}),\pi_{m+n}K_{m} \wedge K_n)=H^{m+n}(K_{m}\wedge K_n ,G)}$.

The only way I would know how to unify the two perspectives is by using a connectivity assumption on $T_0$.

Question: Is the question in the question box true and if not can you give a counterexample? (i.e. can my approach for my two examples above be generalized to any infinite loop space)

Edit: --- This approach works for the prespectrum $colim_n \Omega^n S^n$ (where the colimit is taken w.r.t. the maps $\Omega^n S^n \to \Omega^n (\Omega \Sigma) S^n=\Omega^{n+1} S^{n+1}$

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  • $\begingroup$ Take a discrete abelian group that doesn't have a ring structure. $\endgroup$ – Qiaochu Yuan Apr 1 '16 at 2:51

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