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Milnor's theorem

Contents

Context

Algebraic topology

Algebra

Contents

Idea

Milnor’s theorem describes the dual mod-pp Steenrod algebra in terms of generators and relations.

Statement

In the following, we use for p=2p = 2 the notation

P nSq 2n P^n \coloneqq Sq^{2n}
βSq 1. \beta \coloneqq Sq^1 \,.

This serves to unify the expressions for p=2p = 2 and for p>2p \gt 2 in the following. Notice that for all pp

  • P nP^n has even degree deg(P n)=2n(p1)deg(P^n) = 2n(p-1);

  • β\beta has odd degree deg(β)=1deg(\beta) = 1.

Theorem

(Milnor’s theorem)

The dual mod pp-Steenrod algebra 𝒜 𝔽 p *\mathcal{A}^\ast_{\mathbb{F}_p} (def. ) is, as an associative algebra, the free graded commutative algebra

𝒜 𝔽 p *Sym 𝔽 p(ξ 1,ξ 2,,τ 0,τ 1,) \mathcal{A}^\ast_{\mathbb{F}_p} \simeq Sym_{\mathbb{F}_p}(\xi_1, \xi_2, \cdots, \;\tau_0, \tau_1, \cdots)

on generators:

  • ξ n\xi_n being the linear dual to P p n1P p n2P pP 1P^{p^{n-1}} P^{p^{n-2}} \cdots P^p P^1;

  • τ n\tau_n being linear dual to P p n1P p n2P pP 1βP^{p^{n-1}} P^{p^{n-2}} \cdots P^p P^1\beta.

Moreover, the coproduct on 𝒜 𝔽 p *\mathcal{A}^\ast_{\mathbb{F}_p} is given by

Ψ(ξ n)=k=0nξ nk p kξ k \Psi(\xi_n) = \underoverset{k = 0}{n}{\sum} \xi_{n-k}^{p^k} \otimes \xi_k

and

Ψ(τ n)=τ n1+k=0nξ nk p kξ nk p kτ k, \Psi(\tau_n) = \tau_n \otimes 1 + \underoverset{k=0}{n}{\sum} \xi_{n-k}^{p^k} \xi_{n-k}^{p^k}\otimes \tau_k \,,

where we set ξ 01\xi_0 \coloneqq 1.

This is due to (Milnor 58). See for instance (Kochman 96, theorem 2.5.1)

References

Last revised on January 25, 2021 at 10:31:10. See the history of this page for a list of all contributions to it.