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Steenrod Algebra


The Steenrod algebra has to do with the cohomology operations in singular cohomology with integer mod 2 coefficients. For every n in Z and i in {0,1,2,3,...} there are natural transformations of functors

 Sq^i:H^n(-;Z_2)->H^(n+i)(-;Z_2)
(1)

satisfying:

1. Sq^i=0 for i>n.

2. Sq^n(x)=x cup x for all x in H^n(X,A;Z_2) and all pairs (X,A).

3. Sq^0=id_(H^n(-;Z_2)).

4. The Sq^i maps commute with the coboundary maps in the long exact sequence of a pair. In other words,

 Sq^i:H^*(-;Z_2)->H^(*+i)(-;Z_2)
(2)

is a degree i transformation of cohomology theories.

5. (Cartan relation)

 Sq^i(x cup y)=sum_(j+k=i)Sq^j(x) cup Sq^k(y).
(3)

6. (Adem relations) For i<2j,

 Sq^i degreesSq^j(x)=sum_(k=0)^(|_i/2_|)(j-k-1; i-2k)Sq^(i+j-k) degreesSq^k(x).
(4)

7. Sq^i degreesSigma=Sigma degreesSq^i where Sigma is the cohomology suspension isomorphism.

The existence of these cohomology operations endows the cohomology ring with the structure of a module over the Steenrod algebra A, defined to be T(F_(Z_2){Sq^i:i in {0,1,2,3,...}})/R, where F_(Z_2)(-) is the free module functor that takes any set and sends it to the free Z_2 module over that set. We think of F_(Z_2){Sq^i:i in {0,1,2,...}} as being a graded Z_2 module, where the ith gradation is given by Z_2·Sq^i. This makes the tensor algebra T(F_(Z_2){Sq^i:i in {0,1,2,3,...}}) into a graded algebra over Z_2. R is the ideal generated by the elements Sq^iSq^j+sum_(k=0)^(|_i/2_|)(j-k-1; i-2k)Sq^(i+j-k)Sq^k and 1+Sq^0 for 0<i<2j. This makes A into a graded Z_2 algebra.

By the definition of the Steenrod algebra, for any space (X,A), H^*(X,A;Z_2) is a module over the Steenrod algebra A, with multiplication induced by Sq^i·x=Sq^i(x). With the above definitions, cohomology with coefficients in the ring Z_2, H^*(-;Z_2) is a functor from the category of pairs of topological spaces to graded modules over A.


See also

Adem Relations, Cartan Relation, Cohomology, Graded Algebra, Ideal, Module, Topological Space

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Cite this as:

Weisstein, Eric W. "Steenrod Algebra." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SteenrodAlgebra.html

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