词条 | Steenrod algebra |
释义 |
In algebraic topology, a Steenrod algebra was defined by {{harvs|txt|last=Cartan|first=Henri|authorlink=Henri Cartan|year=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 {{harvs|txt|last=Steenrod|first=Norman|authorlink=Norman Steenrod|year=1947}} for , and by the Steenrod reduced th powers introduced in {{harvtxt|Steenrod|1953}} 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 operationsA 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 . Axiomatic characterization{{harvs|txt|last1=Steenrod|first1=Norman|authorlink1=Norman Steenrod|last2=Epstein|first2=David B. A. |authorlink2=David B. A. Epstein|year=1962}} showed that the Steenrod squares are characterized by the following 5 axioms:
In addition the Steenrod squares have the following properties:
Similarly the following axioms characterize the reduced -th powers for .
As before, the reduced p-th powers also satisfy Ádem relations and commute with the suspension and boundary operators. Ádem relationsThe Ádem relations for were conjectured by {{harvs|txt|last=Wu|first=Wen-tsün|authorlink=Wu Wenjun|year=1952}} and established by {{harvs|txt|authorlink=José Ádem|first=José|last=Ádem|year=1952}}. They are given by for all such that . (The binomial coefficients are to be interpreted mod 2.) The Ádem relations allow one to write an arbitrary composition of Steenrod squares as a sum of Serre–Cartan basis elements. For odd the Ádem relations are for a<pb and for a≤pb Bullett–Macdonald identities{{harvs|txt|last1=Bullett|first1=Shaun R.|last2=Macdonald|first2=Ian G.|authorlink2=Ian G. Macdonald|year=1982}} reformulated the Ádem relations as the following identities.For put then the Ádem relations are equivalent to For put then the Ádem relations are equivalent to the statement that is symmetric in and . Here is the Bockstein operation and . ComputationsInfinite Real Projective SpaceThe Steenrod operations for real projective space can be readily computed using the formal properties of the Steenrod squares. Recall that where For the operations on we know that Using the operation we note that the Cartan relation implies that is a ring morphism. Hence Since there is only one degree component of the previous sum, we have that ConstructionSuppose 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 . {{harvtxt|Steenrod|1953}} showed how to construct a reduced power in , as follows.
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 . The structure of the Steenrod algebra{{harvs|txt|last=Serre|first=Jean-Pierre|authorlink=Jean-Pierre Serre|year=1953}} (for ) and {{harvs|txt|authorlink=Henri Cartan|last=Cartan|first=Henri|year=1954|year2=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 Ádem 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 sequenceis 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. There is a similar basis for the case consisting of the elements such that Hopf algebra structure and the Milnor basisThe Steenrod algebra has more structure than a graded Fp-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. It 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 ψ makes the (graded) linear dual A* of A into an algebra. {{harvtxt|Milnor|1958}} proved, for p = 2, that A* is a polynomial algebra, with one generator ξk of degree 2k − 1, for every k, and for p > 2 the dual Steenrod algebra A* is the tensor product of the polynomial algebra in generators ξk of degree 2pk - 2 (k ≥ 1) and the exterior algebra in generators τk of degree 2pk - 1 (k ≥ 0). The monomial basis for A* 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 A* is the dual of the product on A; it is given by where ξ0=1, and if p>2 The only primitive elements of A* for p=2 are the , and these are dual to the (the only indecomposables of A). Relation to formal groupsThe 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 p=2 then the dual Steenrod algebra is the group scheme of automorphisms of the 1-dimensional additive formal group scheme x+y that are the identity to first order. These automorphisms are of the form Algebraic construction{{harvtxt|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 homomorphismwhere F is the Frobenius endomorphism of SV. If we put or for f∈SV then if V is infinite dimensional the elements Pi 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 Sq2i for p = 2. ApplicationsThe most famous early applications of the Steenrod algebra to outstanding topological problems were the solutions by J. Frank Adams of the Hopf invariant one problem and the vector fields on spheres problem. Independently Milnor and Bott, as well as Kervaire, gave a second solution of the Hopf invariant one problem, using operations in K-theory; these are the Adams operations. One application of the mod 2 Steenrod algebra that is fairly elementary is the following theorem. Theorem. If there is a map S2n - 1 → Sn of Hopf invariant one, then n is a power of 2. The proof uses the fact that each Sqk 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. Connection to the Adams spectral sequence and the homotopy groups of spheresThe cohomology of the Steenrod algebra is the E2 term for the (p-local) Adams spectral sequence, whose abutment is the p-component of the stable homotopy groups of spheres. More specifically, the E2 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
ReferencesPedagogical
References
2 : Algebraic topology|Hopf algebras |
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