Steenrod Cohomology Operations

• Steenrod Cohomology Operations are universal, natural, and stable cohomology operations acting on mod p cohomology, defined via axioms like identity, instability, Cartan formula, and Adem relations.
• They extend to algebraic geometry through Wu formulas and cycle maps, linking étale and Chow cohomology to provide obstructions to algebraicity.
• These operations underpin vital results in topology and arithmetic geometry, offering refined obstructions to classical conjectures such as the Hodge and Tate conjectures.

Steenrod cohomology operations are a collection of universal, natural, and stable cohomology operations acting on the mod $p$ cohomology of topological spaces, algebraic varieties, and various generalized cohomology theories. These operations, and the algebraic structures they generate, play a foundational role in algebraic topology, algebraic geometry, equivariant homotopy theory, and arithmetic geometry. Their defining axioms and secondary relations encode deep structural properties of cohomological invariants and furnish powerful obstructions in both topology and arithmetic.

1. Classical Steenrod Algebra and Fundamental Axioms

The Steenrod algebra $\mathcal{A}_\ell$ is the graded algebra of stable cohomology operations acting on mod $\ell$ cohomology, where $\ell$ is a prime. For $\ell=2$, its generators are the Steenrod squares $\Sq^i:H^q(X;\mathbb{F}_2)\rightarrow H^{q+i}(X;\mathbb{F}_2)$; for odd $\ell$, the primary generators are the reduced power operations $$P^i:H^q(X;\mathbb{F}_\ell)\rightarrow H^{q+2i(\ell-1)}(X;\mathbb{F}_\ell)$$, together with the Bockstein $\beta$.

The classical axioms governing $\mathcal{A}_\ell$ (Benoist, 2022) include:

• Identity: $\Sq^0 = P^0 = \mathrm{id}$ (for $\ell=2$, $\beta = \Sq^1$).
• Instability: $\Sq^i(x)=0$ for $i>\deg x$ and $\Sq^{\deg x}(x)=x^2$; $P^i(x)=0$ for $2i>\deg x$ and $P^{\deg x/(2\ell-2)}(x)=x^\ell$.
• Cartan formula: For $x,y$ homogeneous,

$\Sq(x\cup y) = \sum_{i+j=*}\Sq^i(x)\cup\Sq^j(y); \qquad P(x\cup y) = P(x)\cup P(y); \qquad \beta(x\cup y) = \beta(x)\cup y + (-1)^{\deg x} x\cup\beta(y).$

• Adem relations: For $a<2b$ (resp. $a < b(\ell-1)$),

$\Sq^a \Sq^b = \sum_{i=0}^{\lfloor a/2\rfloor} \binom{b-i-1}{a-2i}\Sq^{a+b-i}\Sq^i;$

(analogous binomial formulas for $P^aP^b$ at odd $\ell$).

These relations ensure naturality, functoriality, and compatibility with spectral sequences and the algebraic "excess" filtration on cohomology.

2. Wu Formulae and Functoriality in Étale and Chow Cohomology

In algebraic geometry, a relative Wu theorem for étale cohomology asserts compatibility of Steenrod operations with pushforward (Gysin) maps involving the étale Stiefel–Whitney classes of virtual normal bundles. Given a proper morphism $f: Y \to X$ of regular schemes over $S_\ell$, the Wu formula expresses

$\Sq(f_*x) = f_*(\Sq(x)\cup w(N_f)), \quad (\ell=2); \qquad P(f_*x) = f_*(P(x)\cup w(N_f)), \quad (\ell>2),$

where $w(N_f)$ is the total étale Stiefel–Whitney class of the virtual normal bundle $N_f\in K_0(Y)$ [(Benoist, 2022), Thm.~2.3].

On the level of Chow groups, Brosnan showed that $\mathcal{A}_\ell$ (with all odd-degree elements acting trivially) acts on $\mathrm{CH}^*(X)/\ell$ for smooth $X$ over $k$. The corresponding Chow-Wu formula is

$\Sq(f_*x) = f_*(\Sq(x)\cdot w(N_f)),\ \quad P(f_*x) = f_*(P(x)\cdot w(N_f)).$

The cycle class map $$cl:\mathrm{CH}^c(X)/\ell \to H^{2c}(X,\mu_\ell^{\otimes c})$$ relates the two: for $\ell=2$,

$\Sq(cl(x)) = \sum_{i\geq0}(1+\omega)^{c-i}\cdot cl(\Sq^{2i}(x)),$

where $\omega$ is the class of $-1$ in $H^1(\operatorname{Spec}k, \mathbb{F}_\ell)$ [(Benoist, 2022), Thm.~3.4].

3. Cohomological Obstructions to Algebraicity

The compatibility theorem implies that the image of the cycle class map is stable under the mod $\ell$ Steenrod algebra if $k$ contains a primitive $\ell^2$-th root of unity. All odd-degree operations annihilate algebraic classes, reproducing Atiyah–Hirzebruch's obstruction to the integral Hodge and Tate conjectures (Benoist, 2022). Even without such roots, certain specific operators (e.g., $\Sq^2+(c(c-1)/2)\omega^2$ or $\Sq^3+(c+1)\omega\Sq^2+(c(c-1)/2)\omega^3$) send algebraic classes to algebraic classes or annihilate them.

These obstructions have been used to construct, for instance:

• A 5-dimensional variety over $\overline{\mathbb{Q}}_p$ violating the 2-adic integral Tate conjecture in codimension 2.
• For any $\ell\in\{2,3,5\}$, a $(2\ell+2)$-fold over $\overline{\mathbb{Q}}_p$ with a codimension-2 Tate class $x$ with $P^1(x)\neq 0\pmod{\ell}$ on reduction.
• For each $\ell\neq p$, a smooth $X$ over $\mathbb{F}_q$ of dimension $2\ell+3$ and an $x$ in the kernel of the Galois descent map such that $x$ is non-algebraic [(Benoist, 2022), Thm.~4.10].
• Over $\mathbb{R}$, explicit 4-folds with classes constructed from equivariant topology for which the composite operator $\Sq^3(x)+\omega\Sq^2(x)+\omega^3x$ is nonzero, giving counterexamples to real algebraicity not explained by classical Hodge-theoretic failures.

4. Algebraizability, Manifolds, and Extended Applications

The Borel–Haefliger cycle map relates the algebraic cycles to the cohomology of compact smooth manifolds $M$. By the theorem of Akbulut–King, the image $H^*_{alg}(M;\mathbb{F}_2)$ is stable under Steenrod operations. It follows that:

• For every $c\geq 2$ there exists $M$ and $x\in H^c(M;\mathbb{F}_2)$ which are not algebraizable, detected by non-integrality of their square or higher Steenrod images.
• There exist $M$ and $x\in H^5(M;\mathbb{F}_2)$ algebraizable but not in the subalgebra generated by Stiefel–Whitney classes and Poincaré duals, answering questions of Benedetti–Dedò and Kucharz [(Benoist, 2022), Thm.~5.6].

These arguments generalize early obstructions to algebraicity by combining Steenrod operations, cycle class maps, and topological fixed-point theorems.

5. Broader Impact: Cartan, Adem, Wu, and Future Directions

The deep formal structure of Steenrod operations—including the Cartan formula, Adem relations, and Wu pushforward formulas—extends to the context of algebraic geometry with precise control via étale and Chow-theoretic invariants. The connection between Chow-theoretic and étale Steenrod operations, via the cycle class map and the comprehensive compatibility theorems, offers refined obstructions not only to classical and integral Hodge and Tate conjectures, but also to finer real and arithmetic forms of algebraicity.

Avenues for further research highlighted include:

• Cohomology theories: Extension of these techniques to higher $K$-theory or cobordism operations (e.g., Quick’s $\ell$-adic cobordism).
• Motivic refinements: Investigation of integrality and torsion phenomena via motivic Steenrod operations (e.g., Voevodsky–Riou, Primozic (Primozic, 2019, Riou, 2012)).
• Interplay with unramified cohomology and analytic methods: Utilizing these operations to define and detect higher unramified cohomology obstructions and analytic cycles.

These directions underline the centrality of Steenrod operations for constructing and explaining non-algebraic cohomology phenomena across topology, algebraic geometry, and arithmetic geometry (Benoist, 2022).