Special and general types
What are called the Steenrod squares is the system of cohomology operations on cohomology with coefficients in (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.
Construction in terms of extended squares
We discuss the explicit construction of the Steenrod-operations in terms of chain maps of chain complexes of -vector spaces equipped with a suitable product. We follow (Lurie 07, lecture 2).
Write for the field with two elements.
For an -module, hence an -vector space, and for , write
for the homotopy quotient of the -fold tensor product of with itself by the action of the symmetric group. Explicitly this is presented, up to quasi-isomorphism by the ordinary coinvariants of the tensor product of with a free resolution of :
This is called the th extended power of .
where on the right we have the, say, singular cohomology cochain complex of the homotopy quotient , which is the homotopy type of the classifying space for .
A chain map
is called a symmetric multiplication on (a shadow of an E-infinity algebra structure). The archetypical class of examples of these are given by the singular cohomology of any topological space , for instance of .
Therefore there is a canonical isomorphism
of the cochain cohomology of the extended square of the chain compplex concentrated on in degree with the singular homology of this classifying space shifted by .
Using this one gets for general and for each a map that sends an element in the th cochain cohomology
represented by a morphism of chain complexes
to the element
represented by the chain map
If moreover is equipped with a symmetric product as above, then one can further compose and form the element
represented by the chain map
This linear map
is called the th Steenrod operation or the th Steenrod square on . By default this is understood for the -singular cochain complex of some topological space , as in the above examples, in which case it has the form
For write for the classifying space of ordinary cohomology in degree with coefficients in the group of order 2 (the Eilenberg-MacLane space ), regarded as an object in the homotopy category of topological spaces).
Notice that for any topological space (CW-complex),
is the ordinary cohomology of in degree with coefficients in . Therefore, by the Yoneda lemma, natural transformations
correspond bijectively to morphisms .
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 satisfying the following conditions:
for it is the identity;
if is a manifold of dimension then ;
for the morphism is the cup product ;
An analogous definition works for coefficients in for any prime number . The corresponding operations are then usually denoted
Under composition, the Steenrod squares form an associative algebra over , called the Steenrod algebra. See there for more.
Relation to Bockstein homomorphism
is the Bockstein homomorphism of the short exact sequence .
Compatibility with suspension
The Steenrod squares are compatible with the suspension isomorphism.
Therefore the Steenrod squares are often also referred to as the stable cohomology operations
for all .
For , a map of spheres, the Steenrod square
(on the homotopy cofiber )
is non-vanishing exactly for .
(Adams 60, theorem 1.1.1).
See at Hopf invariant one theorem.
The operations were first defined in
- Norman Steenrod, Products of cocycles and extensions of mappings, Annals of mathematics (1947)
The axiomatic definition appears in
Lecture notes on Steenrod squares and the Steenrod algebra include
- Wen-Tsun Wu, Sur les puissances de Steenrod, Colloque de Topologie de Strasbourg, IX, La Bibliothèque Nationale et Universitaire de Strasbourg, (1952)