group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
What are called the Steenrod squares is the system of cohomology operations on cohomology with coefficients in $\mathbb{Z}_2$ (the cyclic group of order 2) which is compatible with suspension (the “stable cohomology operations”). They are special examples of power operations.
The Steenrod squares together form the Steenrod algebra, see there for more.
We discuss the explicit construction of the Steenrod-operations in terms of chain maps of chain complexes of $\mathbb{F}_2$-vector spaces equipped with a suitable product. We follow (Lurie 07, lecture 2).
Write $\mathbb{F}_2 \coloneqq \mathbb{Z}/2\mathbb{Z}$ for the field with two elements.
For $V$ an $\mathbb{F}_2$-module, hence an $\mathbb{F}_2$-vector space, and for $n \in \mathbb{N}$, write
for the homotopy quotient of the $n$-fold tensor product of $V$ with itself by the action of the symmetric group. Explicitly this is presented, up to quasi-isomorphism by the ordinary coinvariants $D_n(V)$ of the tensor product of $V^{\otimes n}$ with a free resolution $E \Sigma_n^\bullet$ of $\mathbb{F}_2$:
This is called the $n$th extended power of $V$.
For instance
where on the right we have the, say, singular cohomology cochain complex of the homotopy quotient $\ast //\Sigma_2 \simeq B \Sigma_2 \simeq \mathbb{R}P^\infty$, which is the homotopy type of the classifying space for $\Sigma_2$.
is called a symmetric multiplication on $V$ (a shadow of an E-infinity algebra structure). The archetypical class of examples of these are given by the singular cohomology $V = C^\bullet(X, \mathbb{F}_2)$ of any topological space $X$, for instance of $B \Sigma_2$.
Therefore there is a canonical isomorphism
of the cochain cohomology of the extended square of the chain compplex concentrated on $\mathbb{F}_2$ in degree $n$ with the singular homology of this classifying space shifted by $2 n$.
Using this one gets for general $V$ and for each $i \leq n$ a map that sends an element in the $n$th cochain cohomology
represented by a morphism of chain complexes
to the element
represented by the chain map
If moreover $V$ is equipped with a symmetric product $D_2(V) \longrightarrow V$ as above, then one can further compose and form the element
represented by the chain map
This linear map
is called the $i$th Steenrod operation or the $i$th Steenrod square on $V$. By default this is understood for $V = C^\bullet(X,\mathbb{F}_2)$ the $\mathbb{F}_2$-singular cochain complex of some topological space $X$, as in the above examples, in which case it has the form
For $n \in \mathbb{N}$ write $B^n \mathbb{Z}_2$ for the classifying space of ordinary cohomology in degree $n$ with coefficients in the group of order 2 $\mathbb{Z}_2$ (the Eilenberg-MacLane space $K(\mathbb{Z}_2,n)$), regarded as an object in the homotopy category $H$ of topological spaces).
Notice that for $X$ any topological space (CW-complex),
is the ordinary cohomology of $X$ in degree $n$ with coefficients in $\mathbb{Z}_2$. Therefore, by the Yoneda lemma, natural transformations
correspond bijectively to morphisms $B^k \mathbb{Z}_2 \to B^l \mathbb{Z}_2$.
The following characterization is due to (SteenrodEpstein).
The Steenrod squares are a collection of cohomology operations
hence of morphisms in the homotopy category
for all $n,k \in \mathbb{N}$ satisfying the following conditions:
for $n = 0$ it is the identity;
if $n \gt deg(x)$ then $Sq^n(X) = 0$;
for $k = n$ the morphism $Sq^n : B^n \mathbb{Z}_2 \to B^{2n} \mathbb{Z}_2$ is the cup product $x \mapsto x \cup x$;
$Sq^n(x \cup y) = \sum_{i + j = n} (Sq^i x) \cup (Sq^j y)$;
An analogous definition works for coefficients in $\mathbb{Z}_p$ for any prime number $p \gt 2$. The corresponding operations are then usually denoted
Under composition, the Steenrod squares form an associative algebra over $\mathbb{F}_2$, called the Steenrod algebra. See there for more.
$Sq^1$ is the Bockstein homomorphism of the short exact sequence $\mathbb{Z}_2 \to \mathbb{Z}_4 \to \mathbb{Z}_2$.
The Steenrod squares are compatible with the suspension isomorphism.
Therefore the Steenrod squares are often also referred to as the stable cohomology operations
The composition of Steenrod square operations satisfies the following relations
for all $0 \lt i \lt 2 j$.
Here $\left( a \atop b \right) \coloneqq 0$ if $a \lt b$.
(Adem relation for postcomposition with the Bockstein homomorphism $Sq^1 = \beta$)
For $j \geq 2$ and $i =1$, the Adem relations (prop. 1) say that:
This gives rise to:
For odd $2n + 1 \in \mathbb{N}$ defines the integral Steenrod squares to be
By example 1 and by this example these indeed are lifts of the odd Steenrod squares:
in that we have
For $\phi \colon S^{k+n-1} \to S^k$, a map of spheres, the Steenrod square
(on the homotopy cofiber $cofib(\phi)\simeq S^k \underset{S^{k+n-1}}{\cup} D^{k+n}$)
is non-vanishing exactly for $n \in \{1,2,4,8\}$.
See at Hopf invariant one theorem.
The operations were first defined in
The axiomatic definition appears in
Lecture notes on Steenrod squares and the Steenrod algebra include
Jacob Lurie, 18.917 Topics in Algebraic Topology: The Sullivan Conjecture, Fall 2007. (MIT OpenCourseWare: Massachusetts Institute of Technology), Lecture notes
Lecture 2 Steenrod operations (pdf)
Lecture 3 Basic properties of Steenrod operations (pdf)
Lecture 4 The Adem relations (pdf)
Lecture 5 The Adem relations (cont.) (pdf)
See also