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In algebraic topology, power operations are cohomology operations in multiplicative cohomology theory which are higher-degree analogs of cup product-squares symmetrized in the appropriate homotopy-theoretic sense.


At least to some extent, power operations may be understood as the higher algebra-generalization of the ordinary pp-power map () p(-)^p on a commutative ring, the one that appears in the definition of Fermat quotients, p-derivations and Frobenius morphisms.

See for instance Lurie, from remark 2.2.7 on for relation to the Frobenius homomorphism and see the example below. See Guillot 06, Morava-Santhanam 12 for further discussion and speculation in this direction.

For EE an E-∞ ring and XX a topological space (∞-groupoid, homotopy type), a map a:XEa\;\colon\;X \to E is a cocycle in the Whitehead-generalized cohomology of XX with coefficients in EE.

The nn-th cup product power of this aa is the composite

a n:X ×n(a,,a)E ×nμE, a^n \;\colon\; X^{\times n} \overset{\;\;\; (a,\cdots,a) \;\;\;}{\longrightarrow} E^{\times n} \overset{\;\; \mu \;\;}{\longrightarrow} E \,,

where the second map is given the multiplication operation in the ring spectrum EE. Since this is, by assumption, commutative up to coherent higher homotopy, this map factors through the homotopy quotient by the ∞-action of the symmetric group Σ n\Sigma_n:

a n:X×*Σ nX nΣ nE. a^n \;\colon\; X \times \ast \sslash \Sigma_n \longrightarrow X^n \sslash \Sigma_n \longrightarrow E \,.

The cohomology class of this EE-cocycle on X×BΣ nX \times B \Sigma_n is the nn-th (symmetric) power of aa.


Steenrod squares and Steenrod power operations

On ordinary cohomology over a topological space, the power operations are the Steenrod operations;

Specifically for n=2n = 2 and E=H 2E = H \mathbb{Z}_2, the second (symmetric) power of aH(X, 2)a \in H(X,\mathbb{Z}_2) is an element in H (P ×X, 2)H (X, 2)[x]H^\bullet(\mathbb{R}P^\infty \times X, \mathbb{Z}_2) \simeq H^\bullet(X,\mathbb{Z}_2)[x] and the coefficients of this polynomial in xx are the Steenrod operations on aa.

For p>2p \gt 2 there are the Steenrod power operations (e.g. Rognes 12, around theorem 3.3, quick exposition here).

Kudo-Araki-Dyer-Lashof operations

On an infinite loop space the power operations are the Kudo-Araki-Dyer-Lashof operations?

Adams operations

In the context of complex K-theory power operations are the Adams operations.

On K(1)K(1)-local KUKU-algebras

From this MO comment by Akhil Mathew:

Let RR be a K(1)-local E-∞ ring under (p-adic) complex K-theory KU. Then there exists a basic power operation θ:π 0Rπ 0R\theta \colon \pi_0 R \to \pi_0 R (see Hopkins) such that :

  • ψ(x)=defx p+pθ(x)\psi(x) \stackrel{\mathrm{def}}{=} x^p + p \theta(x) defines a ring homomorphism from π 0Rπ 0R\pi_0 R \to \pi_0 R.

  • θ\theta satisfies all the identities needed to make ψ\psi a ring-homomorphism after “division by pp.” For instance ψ(x+y)=ψ(x)+ψ(y)\psi(x+y) = \psi(x) + \psi(y) implies that

    θ(x+y)=θ(x)+θ(y)+x py p(x+y) pp, \theta(x+y) = \theta(x) + \theta(y) + \frac{x^p - y^p - (x+y)^p}{p} \,,

    where the last term is an integral polynomial in x,yx,y and is interpreted as such.

(see also Rezk 09, example 1.3)

This is a “θ\theta-algebra.”/p-derivation as in remark above.

Notice that ψ\psi is, in particular, a lift of the Frobenius homomorphism. There are generalizations of ψ,θ\psi, \theta at higher chromatic levels, too, and there is a modular interpretation of the resulting algebraic structure in (Rezk 09).

By (Strickland 98) we have that if GG is the formal group associated to a Morava E-theory, then Frobenius lifts (twhich corresponds to degree p kp^k subgroups of GG) are classified by maps into E 0(BΣ p r)/I trE^0(B \Sigma_{p^r})/I_{t r} where I trI_{t r} is the transfer ideal. So, for example, the map ψ\psi above corresponds to a universal map KU 0KU 0(BΣ p)/I trKU 0KU^0 \to KU^0(B \Sigma_p)/I_{t r} \simeq KU^0.


The basic idea is nicely described in

(from which some of the above text is adapted).

More technical surveys include

Lecture notes on the Steenrod squares and power operations include

The original articles are

More discussion in the generality of E-infinity arithmetic geometry is in

Discussion for K(1)K(1)-local E E_\infty-rings is in

and discussion of power operations in Morava E-theory is in

Comments on the analogy between power operations in homotopy theory and Lambda ring structure in Borger's absolute geometry are in

Last revised on December 1, 2022 at 08:27:19. See the history of this page for a list of all contributions to it.