In essence my question can be stated as follows: fill in the analogy

$$ \text{cup product} \qquad\qquad \leftrightarrow \qquad \text{Samelson product} $$

$$ \updownarrow \qquad\qquad \qquad\qquad \qquad\qquad\updownarrow $$

$$ \quad \quad \text{cup}_i \text{-product} \quad \qquad \leftrightarrow \qquad \qquad \quad\qquad\text{?}\qquad \qquad\quad\quad $$ It is known that Samelson products (for a loop space) are Hilton-Eckmann dual to cup products (see e.g., Arkowitz, Martin: Commutators and cup products. Illinois J. Math. 8 1964 571–581. )

Taking this a bit further, the construction of the Steenrod algebra uses the fact that the reduced diagonal $X \to X\wedge X$ is $\Bbb Z/2$-equivariant.
It is not hard to show that the Samelson product also has an equivariance. This suggests to me that the graded Lie algebra structure on the homotopy groups of a space can be refined to take this into account.

Any ideas?

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    $\begingroup$ Your upper $\leftrightarrow$ is Koszul-Quillen duality between commutative and Lie algebras, therefore, since the $\smile_i$-products form an $E_\infty$-algebra, the lower one should be the Koszul-Quillen duality between $E_\infty$-algebras and $L_\infty$-algebras. Have a look at any reference book on rational homotopy theory, where this theory works simplest. $\endgroup$ May 15, 2012 at 15:38
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    $\begingroup$ @Fernando: In my understanding in the rational case, we can take a simplicial group model for the loop space, say $G$, and form its group ring over $\Bbb Q$ to get a simlpicial Hopf algebra. The degree-wise primitives then give a differential graded Lie algebra, which is a strict version of an $L_\infty$-algebra. However, if we work integrally (or maybe mod 2), are you saying that we get a non-strict $L_\infty$-algebra? Is this written down anywhere? $\endgroup$
    – John Klein
    May 15, 2012 at 17:18
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    $\begingroup$ Not really. In fact, the characteristic $>0$ case is not yet well understood or developed, but since you just asked about analogies I offered you the characteristic $0$ analogue ;-) $\endgroup$ May 16, 2012 at 7:35
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    $\begingroup$ @Fernando, can we take your suggestion a little further and ask whether there are interesting analogues of Steenrod $p$'th powers coming from the $\mathbb{Z}/p$-equivariant homology of the p-th term $L_{\infty}(p)$ of the $L_{\infty}$-operad? $\endgroup$ May 16, 2012 at 12:48
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    $\begingroup$ Looking for something else I've come up with this paper: Smirnov, V. A. E∞-structures on homotopy groups. (Russian) Mat. Zametki 61 (1997), no. 1, 152--156; translation in Math. Notes 61 (1997), no. 1-2, 127–130. It seems to contain an appropriate notion of $L_\infty$-algebras over $\mathbb{F}_p$. Unforturately the paper is very short and gives no proofs, and I haven't found references where the claims in this paper are proven. $\endgroup$ May 25, 2012 at 23:20


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