Steenrod algebra

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In algebraic topology, a Steenrod algebra was defined by Henri Cartan (1955) to be the algebra of stable cohomology operations for mod cohomology.

For a given prime number , the Steenrod algebra is the graded Hopf algebra over the field of order , consisting of all stable cohomology operations for mod cohomology. It is generated by the Steenrod squares introduced by Norman Steenrod (1947) for , and by the Steenrod reduced th powers introduced in Steenrod (1953a, 1953b) and the Bockstein homomorphism for .

The term "Steenrod algebra" is also sometimes used for the algebra of cohomology operations of a generalized cohomology theory.

Cohomology operations

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A cohomology operation is a natural transformation between cohomology functors. For example, if we take cohomology with coefficients in a ring  , the cup product squaring operation yields a family of cohomology operations:

 
 

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   is a suspension of a space  , the cup product on the cohomology of   is trivial.) Steenrod constructed stable operations

 

for all   greater than zero. The notation   and their name, the Steenrod squares, comes from the fact that   restricted to classes of degree   is the cup square. There are analogous operations for odd primary coefficients, usually denoted   and called the reduced  -th power operations:

 

The   generate a connected graded algebra over  , where the multiplication is given by composition of operations. This is the mod 2 Steenrod algebra. In the case  , the mod   Steenrod algebra is generated by the   and the Bockstein operation   associated to the short exact sequence

 .

In the case  , the Bockstein element is   and the reduced  -th power   is  .

As a cohomology ring

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We can summarize the properties of the Steenrod operations as generators in the cohomology ring of Eilenberg–Maclane spectra

 ,

since there is an isomorphism

 

giving a direct sum decomposition of all possible cohomology operations with coefficients in  . Note the inverse limit of cohomology groups appears because it is a computation in the stable range of cohomology groups of Eilenberg–Maclane spaces. This result[1] was originally computed[2] by Cartan (1954–1955, p. 7) and Serre (1953).

Note there is a dual characterization[3] using homology for the dual Steenrod algebra.

Remark about generalizing to generalized cohomology theories

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It should be observed if the Eilenberg–Maclane spectrum   is replaced by an arbitrary spectrum  , then there are many challenges for studying the cohomology ring  . In this case, the generalized dual Steenrod algebra   should be considered instead because it has much better properties and can be tractably studied in many cases (such as  ).[4] In fact, these ring spectra are commutative and the   bimodules   are flat. In this case, these is a canonical coaction of   on   for any space  , such that this action behaves well with respect to the stable homotopy category, i.e., there is an isomorphism   hence we can use the unit the ring spectrum     to get a coaction of   on  .

Axiomatic characterization

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Norman Steenrod and David B. A. Epstein (1962) showed that the Steenrod squares   are characterized by the following 5 axioms:

  1. Naturality:   is an additive homomorphism and is natural with respect to any  , so  .
  2.   is the identity homomorphism.
  3.   for  .
  4. If   then  
  5. Cartan Formula:  

In addition the Steenrod squares have the following properties:

  •   is the Bockstein homomorphism   of the exact sequence  
  •   commutes with the connecting morphism of the long exact sequence in cohomology. In particular, it commutes with respect to suspension  
  • They satisfy the Adem relations, described below

Similarly the following axioms characterize the reduced  -th powers for  .

  1. Naturality:   is an additive homomorphism and natural.
  2.   is the identity homomorphism.
  3.   is the cup  -th power on classes of degree  .
  4. If   then  
  5. Cartan Formula:  

As before, the reduced p-th powers also satisfy the Adem relations and commute with the suspension and boundary operators.

Adem relations

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The Adem relations for   were conjectured by Wen-tsün Wu (1952) and established by José Adem (1952). They are given by

 

for all   such that  . (The binomial coefficients are to be interpreted mod 2.) The Adem relations allow one to write an arbitrary composition of Steenrod squares as a sum of Serre–Cartan basis elements.

For odd   the Adem relations are

 

for a<pb and

 

for  .

Bullett–Macdonald identities

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Shaun R. Bullett and Ian G. Macdonald (1982) reformulated the Adem relations as the following identities.

For   put

 

then the Adem relations are equivalent to

 

For   put

 

then the Adem relations are equivalent to the statement that

 

is symmetric in   and  . Here   is the Bockstein operation and  .

Geometric interpretation

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There is a nice straightforward geometric interpretation of the Steenrod squares using manifolds representing cohomology classes. Suppose   is a smooth manifold and consider a cohomology class   represented geometrically as a smooth submanifold  . Cohomologically, if we let   represent the fundamental class of   then the pushforward map

 

gives a representation of  . In addition, associated to this immersion is a real vector bundle call the normal bundle  . The Steenrod squares of   can now be understood — they are the pushforward of the Stiefel–Whitney class of the normal bundle

 

which gives a geometric reason for why the Steenrod products eventually vanish. Note that because the Steenrod maps are group homomorphisms, if we have a class   which can be represented as a sum

 

where the   are represented as manifolds, we can interpret the squares of the classes as sums of the pushforwards of the normal bundles of their underlying smooth manifolds, i.e.,

 

Also, this equivalence is strongly related to the Wu formula.

Computations

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Complex projective spaces

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On the complex projective plane  , there are only the following non-trivial cohomology groups,

 ,

as can be computed using a cellular decomposition. This implies that the only possible non-trivial Steenrod product is   on   since it gives the cup product on cohomology. As the cup product structure on   is nontrivial, this square is nontrivial. There is a similar computation on the complex projective space  , where the only non-trivial squares are   and the squaring operations   on the cohomology groups   representing the cup product. In   the square

 

can be computed using the geometric techniques outlined above and the relation between Chern classes and Stiefel–Whitney classes; note that   represents the non-zero class in  . It can also be computed directly using the Cartan formula since   and

 

Infinite Real Projective Space

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The Steenrod operations for real projective spaces can be readily computed using the formal properties of the Steenrod squares. Recall that

 

where   For the operations on   we know that

 

The Cartan relation implies that the total square

 

is a ring homomorphism

 

Hence

 

Since there is only one degree   component of the previous sum, we have that

 

Construction

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Suppose that   is any degree   subgroup of the symmetric group on   points,   a cohomology class in  ,   an abelian group acted on by  , and   a cohomology class in  . Steenrod (1953a, 1953b) showed how to construct a reduced power   in  , as follows.

  1. Taking the external product of   with itself   times gives an equivariant cocycle on   with coefficients in  .
  2. Choose   to be a contractible space on which   acts freely and an equivariant map from   to   Pulling back   by this map gives an equivariant cocycle on   and therefore a cocycle of   with coefficients in  .
  3. Taking the slant product with   in   gives a cocycle of   with coefficients in  .

The Steenrod squares and reduced powers are special cases of this construction where   is a cyclic group of prime order   acting as a cyclic permutation of   elements, and the groups   and   are cyclic of order  , so that   is also cyclic of order  .

Properties of the Steenrod algebra

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In addition to the axiomatic structure the Steenrod algebra satisfies, it has a number of additional useful properties.

Basis for the Steenrod algebra

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Jean-Pierre Serre (1953) (for  ) and Henri Cartan (1954, 1955) (for  ) described the structure of the Steenrod algebra of stable mod   cohomology operations, showing that it is generated by the Bockstein homomorphism together with the Steenrod reduced powers, and the Adem relations generate the ideal of relations between these generators. In particular they found an explicit basis for the Steenrod algebra. This basis relies on a certain notion of admissibility for integer sequences. We say a sequence

 

is admissible if for each  , we have that  . Then the elements

 

where   is an admissible sequence, form a basis (the Serre–Cartan basis) for the mod 2 Steenrod algebra, called the admissible basis. There is a similar basis for the case   consisting of the elements

 ,

such that

 
 
 
 

Hopf algebra structure and the Milnor basis

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The Steenrod algebra has more structure than a graded  -algebra. It is also a Hopf algebra, so that in particular there is a diagonal or comultiplication map

 

induced by the Cartan formula for the action of the Steenrod algebra on the cup product. This map is easier to describe than the product map, and is given by

 
 
 .

These formulas imply that the Steenrod algebra is co-commutative.

The linear dual of   makes the (graded) linear dual   of A into an algebra. John Milnor (1958) proved, for  , that   is a polynomial algebra, with one generator   of degree  , for every k, and for   the dual Steenrod algebra   is the tensor product of the polynomial algebra in generators   of degree     and the exterior algebra in generators τk of degree    . The monomial basis for   then gives another choice of basis for A, called the Milnor basis. The dual to the Steenrod algebra is often more convenient to work with, because the multiplication is (super) commutative. The comultiplication for   is the dual of the product on A; it is given by

  where  , and
  if  .

The only primitive elements of   for   are the elements of the form  , and these are dual to the   (the only indecomposables of A).

Relation to formal groups

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The dual Steenrod algebras are supercommutative Hopf algebras, so their spectra are algebra supergroup schemes. These group schemes are closely related to the automorphisms of 1-dimensional additive formal groups. For example, if   then the dual Steenrod algebra is the group scheme of automorphisms of the 1-dimensional additive formal group scheme   that are the identity to first order. These automorphisms are of the form

 

Finite sub-Hopf algebras

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The   Steenrod algebra admits a filtration by finite sub-Hopf algebras. As   is generated by the elements [5]

 ,

we can form subalgebras   generated by the Steenrod squares

 ,

giving the filtration

 

These algebras are significant because they can be used to simplify many Adams spectral sequence computations, such as for  , and  .[6]

Algebraic construction

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Larry Smith (2007) gave the following algebraic construction of the Steenrod algebra over a finite field   of order q. If V is a vector space over   then write SV for the symmetric algebra of V. There is an algebra homomorphism

 

where F is the Frobenius endomorphism of SV. If we put

 

or

 

for   then if V is infinite dimensional the elements   generate an algebra isomorphism to the subalgebra of the Steenrod algebra generated by the reduced p′th powers for p odd, or the even Steenrod squares   for  .

Applications

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Early applications of the Steenrod algebra were calculations by Jean-Pierre Serre of some homotopy groups of spheres, using the compatibility of transgressive differentials in the Serre spectral sequence with the Steenrod operations, and the classification by René Thom of smooth manifolds up to cobordism, through the identification of the graded ring of bordism classes with the homotopy groups of Thom complexes, in a stable range. The latter was refined to the case of oriented manifolds by C. T. C. Wall. A famous application of the Steenrod operations, involving factorizations through secondary cohomology operations associated to appropriate Adem relations, was the solution by J. Frank Adams of the Hopf invariant one problem. One application of the mod 2 Steenrod algebra that is fairly elementary is the following theorem.

Theorem. If there is a map   of Hopf invariant one, then n is a power of 2.

The proof uses the fact that each   is decomposable for k which is not a power of 2; that is, such an element is a product of squares of strictly smaller degree.

Michael A. Mandell gave a proof of the following theorem by studying the Steenrod algebra (with coefficients in the algebraic closure of  ):

Theorem. The singular cochain functor with coefficients in the algebraic closure of   induces a contravariant equivalence from the homotopy category of connected  -complete nilpotent spaces of finite  -type to a full subcategory of the homotopy category of [[ -algebras]] with coefficients in the algebraic closure of  .

Connection to the Adams spectral sequence and the homotopy groups of spheres

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The cohomology of the Steenrod algebra is the   term for the (p-local) Adams spectral sequence, whose abutment is the p-component of the stable homotopy groups of spheres. More specifically, the   term of this spectral sequence may be identified as

 

This is what is meant by the aphorism "the cohomology of the Steenrod algebra is an approximation to the stable homotopy groups of spheres."

See also

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References

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  1. ^ "at.algebraic topology – (Co)homology of the Eilenberg–MacLane spaces K(G,n)". MathOverflow. Retrieved 2021-01-15.
  2. ^ Adams (1974), p. 277.
  3. ^ Adams (1974), p. 279.
  4. ^ Adams (1974), p. 280.
  5. ^ Mosher & Tangora (2008), p. 47.
  6. ^ Ravenel (1986), pp. 63–67.

Pedagogical

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Motivic setting

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References

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