nLab power operation

under construction

Idea

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.

Remark

At least to some extent, power operations may be understood as the higher algebra-generalization of the ordinary $p$-power map $(-)^p$ 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 $E$ an E-∞ ring and $X$ a topological space (∞-groupoid, homotopy type), a map $a\;\colon\;X \to E$ is a cocycle in the Whitehead-generalized cohomology of $X$ with coefficients in $E$.

The $n$-th cup product power of this $a$ is the composite

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

where the second map is given the multiplication operation in the ring spectrum $E$. 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 $\Sigma_n$:

$$a^n \;\colon\; X \times \ast \sslash \Sigma_n \longrightarrow X^n \sslash \Sigma_n \longrightarrow E \,.$$

The cohomology class of this $E$-cocycle on $X \times B \Sigma_n$ is the $n$-th (symmetric) power of $a$.

Examples

Steenrod squares and Steenrod power operations

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

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

For $p \gt 2$ 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 $K(1)$-local $KU$-algebras

From this MO comment by Akhil Mathew:

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

• $\psi(x) \stackrel{\mathrm{def}}{=} x^p + p \theta(x)$ defines a ring homomorphism from $\pi_0 R \to \pi_0 R$.

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

  $$\theta(x+y) = \theta(x) + \theta(y) + \frac{x^p - y^p - (x+y)^p}{p} \,,$$

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

(see also Rezk 09, example 1.3)

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

Notice that $\psi$ is, in particular, a lift of the Frobenius homomorphism. There are generalizations of $\psi, \theta$ 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 $G$ is the formal group associated to a Morava E-theory, then Frobenius lifts (twhich corresponds to degree $p^k$ subgroups of $G$) are classified by maps into $E^0(B \Sigma_{p^r})/I_{t r}$ where $I_{t r}$ is the transfer ideal. So, for example, the map $\psi$ above corresponds to a universal map $KU^0 \to KU^0(B \Sigma_p)/I_{t r} \simeq KU^0$.

Related concepts

• Adams operation

• logarithmic cohomology operation

• spectral symmetric algebra

References

The basic idea is nicely described in

• Charles Rezk, MO comment Nov 09

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

More technical surveys include

• Charles Rezk, Lectures on power operations (dvi) (2006)

• Charles Rezk, Power operations in Morava E-theory – a survey (2009) (pdf)

• Charles Rezk, Isogenies, power operations, and homotopy theory, article (pdf) and talk at ICM 2014 (pdf)

Lecture notes on the Steenrod squares and power operations include

• John Rognes, section 3 of The Adams spectral sequence, 2012 (pdf)

The original articles are

• Charles Rezk, The units of a ring spectrum and a logarithmic cohomology operation, J. Amer. Math. Soc. 19 (2006), 969-1014 (arXiv:math/0407022)

• Charles Rezk, Power operations for Morava E-theory of height 2 at the prime 2 (arXiv:0812.1320)

• Charles Rezk, The congruence criterion for power operations in Morava E-theory, Homology, Homotopy and Applications, 11(2), 327-379 (arXiv:0902.2499)

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

• Jacob Lurie, Rational and p-adic Homotopy Theory

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

• Michael Hopkins, $K(1)$-local $E_\infty$-Ring spectra (pdf)

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

• Matthew Ando, Isogenies of formal group laws and power operations in the cohomology theories $E_n$, Duke Math. J. Volume 79, Number 2 (1995), 423-485 (Euclid)

• Neil Strickland, Morava E-theory of symmetric groups (arXiv:math/9801125)

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

• Pierre Guillot, Adams operations in cohomotopy (arXiv:0612327)

• Jack Morava, Rakha Santhanam, Power operations and Absolute geometry, 2012 (pdf)