Steenrod Square

The Steenrod square is a central concept in algebraic topology, describing how one can systematically “square” cohomology classes in a way compatible with the cup product and natural with respect to continuous maps. It forms part of the broader Steenrod algebra.

Definition

For any topological space $X$, the Steenrod squares are a family of natural transformations
$S{q}^i:H^n(X;{\mathbb{Z}}_2)\to H^{n+i}(X;{\mathbb{Z}}_2)$
called stable cohomology operations, defined for $i\ge 0$. Each $S{q}^i$ raises degree by $i$, with all computations made using mod 2 coefficients to guarantee additivity (since 2 = 0 in ${\mathbb{Z}}_2$).¹²³

Axioms

The Steenrod squares are uniquely characterized by five axioms:⁴¹

1. Naturality: $f^{\ast }(S{q}^i(x))=S{q}^i(f^{\ast }(x))$ for any continuous map $f:X\to Y$.
2. Identity: $S{q}^0=\text{id}$.
3. Normalization: If $x\in H^n(X;{\mathbb{Z}}_2)$, then $S{q}^n(x)=x⌣x$.
4. Dimension Bound: $S{q}^i(x)=0$ if $i>n$.
5. Cartan Formula: $S{q}^n(x⌣y)={\sum }_{i+j=n}S{q}^i(x)⌣S{q}^j(y)$.

Additionally, the Adem relations are algebraic identities among compositions of squares:
$S{q}^{a}S{q}^b={\sum }_c\left(\genfrac{}{}{0ex}{}{b-c-1}{a-2c}\right)S{q}^{a+b-c}S{q}^c$
valid when $a<2b$ (computed modulo 2).⁴

The Steenrod Algebra

The Steenrod algebra, denoted ${\mathcal{A}}_2$, is the graded algebra generated freely over ${\mathbb{Z}}_2$ by the symbols $S{q}^i$ subject to the Adem relations. For any space $X$, the mod 2 cohomology $H^{\ast }(X;{\mathbb{Z}}_2)$ is naturally a module over ${\mathcal{A}}_2$.¹⁴

Geometric Interpretation

If a cohomology class $\alpha \in H^{\ast }(X)$ is represented by a submanifold $f:Y\hookrightarrow X$, then
$S{q}^i(\alpha )=f_{\ast }(w_i({\nu }_{Y/X}))$
where $w_i({\nu }_{Y/X})$ is the $i$-th Stiefel–Whitney class of the normal bundle. This provides an intuitive geometric reason the higher Steenrod squares vanish after a certain degree.¹

Example

For the infinite real projective space ${\mathbb{RP}}^{\infty }$, whose cohomology ring is $H^{\ast }({\mathbb{RP}}^{\infty };{\mathbb{Z}}_2)={\mathbb{Z}}_2[\alpha ]$ with $|\alpha |=1$:
$S{q}^0(\alpha )=\alpha ,S{q}^1(\alpha )={\alpha }^2,S{q}^{i>1}(\alpha )=0$
.⁴

Extensions and Variants

• Steenrod reduced powers generalize these operations for primes $p>2$.
• Quantum Steenrod squares deform the classical construction using quantum cohomology on symplectic manifolds, preserving analogous Cartan-like relations.⁵⁶

Overall, Steenrod squares enrich cohomology theory by introducing a structured way to “multiply” classes beyond the cup product, revealing deeper algebraic and geometric invariants of spaces.

References

⁷⁸⁹

Footnotes

1. https://en.wikipedia.org/wiki/Steenrod_algebra ↩ ↩² ↩³ ↩⁴

2. https://encyclopediaofmath.org/wiki/Steenrod_square ↩

3. https://ncatlab.org/nlab/show/Steenrod+square ↩

4. https://people.math.binghamton.edu/malkiewich/steenrod.pdf ↩ ↩² ↩³ ↩⁴

5. https://research-information.bris.ac.uk/files/214985408/A_construction_of_the_quantum_Steenrod_squares_and_their_algebraic_relations.pdf ↩

6. https://www.maths.ox.ac.uk/node/26423 ↩

7. https://webhomes.maths.ed.ac.uk/~v1ranick/papers/singersq2.pdf

8. https://academic.oup.com/jtopol/article-abstract/7/3/817/924619

9. https://web.math.ku.dk/~jg/students/guldberg.bs.2009.pdf