Steenrod Algebra
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30-5-2021
1702
Steenrod Algebra
The Steenrod algebra has to do with the cohomology operations in singular cohomology with integer mod 2 coefficients. For every 
and ![i in <span style=]()
{0,1,2,3,...}" src="https://mathworld.wolfram.com/images/equations/SteenrodAlgebra/Inline2.gif" style="height:15px; width:103px" /> there are natural transformations of functors
 |
(1)
|
satisfying:
1. 
for 
.
2. 
for all 
and all pairs 
.
3. 
.
4. The 
maps commute with the coboundary maps in the long exact sequence of a pair. In other words,
 |
(2)
|
is a degree 
transformation of cohomology theories.
5. (Cartan relation)
 |
(3)
|
6. (Adem relations) For 
,
 |
(4)
|
7. 
where 
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 
, defined to be ![T(F_(Z_2)<span style=]()
{Sq^i:i in {0,1,2,3,...}})/R" src="https://mathworld.wolfram.com/images/equations/SteenrodAlgebra/Inline15.gif" style="height:21px; width:210px" />, where 
is the free module functor that takes any set and sends it to the free 
module over that set. We think of ![F_(Z_2)<span style=]()
{Sq^i:i in {0,1,2,...}}" src="https://mathworld.wolfram.com/images/equations/SteenrodAlgebra/Inline18.gif" style="height:21px; width:156px" /> as being a graded 
module, where the 
th gradation is given by 
. This makes the tensor algebra ![T(F_(Z_2)<span style=]()
{Sq^i:i in {0,1,2,3,...}})" src="https://mathworld.wolfram.com/images/equations/SteenrodAlgebra/Inline22.gif" style="height:21px; width:191px" /> into a graded algebra over 
. 
is the ideal generated by the elements 
and 
for 
. This makes 
into a graded 
algebra.
By the definition of the Steenrod algebra, for any space 
, 
is a module over the Steenrod algebra 
, with multiplication induced by 
. With the above definitions, cohomology with coefficients in the ring 
, 
is a functor from the category of pairs of topological spaces to graded modules over 
.
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