In algebraic topology, a Steenrod algebra was defined by Henri Cartan (1955) to be the algebra of stable cohomology operations for mod p {\displaystyle p} cohomology. For a given prime number p {\displaystyle p} , the Steenrod algebra A p {\displaystyle A_{p}} is the graded Hopf algebra over the field F p {\displaystyle \mathbb {F} _{p}} of order p {\displaystyle p} , consisting of all stable cohomology operations for mod p {\displaystyle p} cohomology. It is generated by the Steenrod squares introduced by Norman Steenrod (1947) for p = 2 {\displaystyle p=2} , and by the Steenrod reduced p {\displaystyle p} th powers introduced in Steenrod (1953a, 1953b) and the Bockstein homomorphism for p > 2 {\displaystyle p>2} . The term "Steenrod algebra" is also sometimes used for the algebra of cohomology operations of a generalized cohomology theory.
Cohomology operations
A cohomology operation is a natural transformation between cohomology functors. For example, if we take cohomology with coefficients in a ring R {\displaystyle R} , the cup product squaring operation yields a family of cohomology operations:
H n ( X ; R ) → H 2 n ( X ; R ) {\displaystyle H^{n}(X;R)\to H^{2n}(X;R)}
x ↦ x ⌣ x . {\displaystyle x\mapsto x\smile x.}
Cohomology operations need not be homomorphisms of graded rings; see the Cartan formula below. These operations do not commute with suspension—that is, they are unstable. (This is because if Y {\displaystyle Y} is a suspension of a space X {\displaystyle X} , the cup product on the cohomology of Y {\displaystyle Y} is trivial.) Steenrod constructed stable operations
S q i : H n ( X ; Z / 2 ) → H n + i ( X ; Z / 2 ) {\displaystyle Sq^{i}\colon H^{n}(X;\mathbb {Z} /2)\to H^{n+i}(X;\mathbb {Z} /2)}
for all i {\displaystyle i} greater than zero. The notation S q {\displaystyle Sq} and their name, the Steenrod squares, comes from the fact that S q n {\displaystyle Sq^{n}} restricted to classes of degree n {\displaystyle n} is the cup square. There are analogous operations for odd primary coefficients, usually denoted P i {\displaystyle P^{i}} and called the reduced p {\displaystyle p} -th power operations:
P i : H n ( X ; Z / p ) → H n + 2 i ( p − 1 ) ( X ; Z / p ) {\displaystyle P^{i}\colon H^{n}(X;\mathbb {Z} /p)\to H^{n+2i(p-1)}(X;\mathbb {Z} /p)}
The S q i {\displaystyle Sq^{i}} generate a connected graded algebra over Z / 2 {\displaystyle \mathbb {Z} /2} , where the multiplication is given by composition of operations. This is the mod 2 Steenrod algebra. In the case p > 2 {\displaystyle p>2} , the mod p {\displaystyle p} Steenrod algebra is generated by the P i {\displaystyle P^{i}} and the Bockstein operation β {\displaystyle \beta } associated to the short exact sequence
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