algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
symmetric monoidal (∞,1)-category of spectra
Milnor’s theorem describes the dual mod-$p$ Steenrod algebra in terms of generators and relations.
In the following, we use for $p = 2$ the notation
This serves to unify the expressions for $p = 2$ and for $p \gt 2$ in the following. Notice that for all $p$
$P^n$ has even degree $deg(P^n) = 2n(p-1)$;
$\beta$ has odd degree $deg(\beta) = 1$.
(Milnor’s theorem)
The dual mod $p$-Steenrod algebra $\mathcal{A}^\ast_{\mathbb{F}_p}$ (def. ) is, as an associative algebra, the free graded commutative algebra
on generators:
$\xi_n$ being the linear dual to $P^{p^{n-1}} P^{p^{n-2}} \cdots P^p P^1$;
$\tau_n$ being linear dual to $P^{p^{n-1}} P^{p^{n-2}} \cdots P^p P^1\beta$.
Moreover, the coproduct on $\mathcal{A}^\ast_{\mathbb{F}_p}$ is given by
and
where we set $\xi_0 \coloneqq 1$.
This is due to (Milnor 58). See for instance (Kochman 96, theorem 2.5.1)
John Milnor, The Steenrod algebra and its dual, Ann. of Math. 67 (1958), 150–171.
Stanley Kochman, section 2.5 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Last revised on January 25, 2021 at 15:31:10. See the history of this page for a list of all contributions to it.