Steenrod algebra

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

For a given prime number $p$ , the Steenrod algebra $A_p$ is the graded Hopf algebra over the field $\mathbb {F} _{p}$ of order $p$ , consisting of all stable cohomology operations for mod $p$ cohomology. It is generated by the Steenrod squares introduced by Norman Steenrod (1947) for $p=2$ , and by the Steenrod reduced $p$ th powers introduced in Steenrod (1953) and the Bockstein homomorphism for $p>2$ .

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, the cup product squaring operation yields a family of cohomology operations:

H^{n}(X;R)\to H^{{2n}}(X;R)

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

Sq^{i}:H^{n}(X;\mathbb {Z} /2)\to H^{n+i}(X;\mathbb {Z} /2)

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

P^{i}\colon H^{n}(X;\mathbb {Z} /p)\to H^{n+2i(p-1)}(X;\mathbb {Z} /p)

The $Sq^{i}$ generate a connected graded algebra over $\mathbb {Z} /2$ , where the multiplication is given by composition of operations. This is the mod 2 Steenrod algebra. In the case $p>2$ , the mod $p$ Steenrod algebra is generated by the $P^{i}$ and the Bockstein operation $\beta$ associated to the short exact sequence

0\to \mathbb {Z} /p\to \mathbb {Z} /p^{2}\to \mathbb {Z} /p\to 0.

In the case $p=2$ , the Bockstein element is $Sq^{1}$ and the reduced $p$ -th power $P^{i}$ is $Sq^{2i}$ .

Axiomatic characterization

Norman Steenrod and David B. A. Epstein (1962) showed that the Steenrod squares $Sq^{n}\colon H^{m}\to H^{m+n}$ are characterized by the following 5 axioms:

Naturality: $Sq^{n}:H^{m}(X;\mathbb {Z} /2)\to H^{m+n}(X;\mathbb {Z} /2)$ is an additive homomorphism and is functorial with respect to any $f:X\to Y.$ so $f^{*}(Sq^{n}(x))=Sq^{n}(f^{*}(x))$ .
$Sq^{0}$ is the identity homomorphism.
$Sq^{n}(x)=x\smile x$ for $x\in H^{n}(X;\mathbb {Z} /2)$ .
If $n>\deg(x)$ then $Sq^{n}(x)=0$
Cartan Formula: $Sq^{n}(x\smile y)=\sum _{{i+j=n}}(Sq^{i}x)\smile (Sq^{j}y)$

In addition the Steenrod squares have the following properties:

$Sq^{1}$ is the Bockstein homomorphism $\beta$ of the exact sequence $0\to \mathbb {Z} /2\to \mathbb {Z} /4\to \mathbb {Z} /2\to 0.$
$Sq^{i}$ commutes with the connecting morphism of the long exact sequence in cohomology. In particular, it commutes with respect to suspension $H^{k}(X;\mathbb {Z} /2)\cong H^{k+1}(\Sigma X;\mathbb {Z} /2)$
They satisfy the Ádem relations, described below

Similarly the following axioms characterize the reduced $p$ -th powers for $p>2$ .

Naturality: $P^{n}:H^{m}(X,\mathbb {Z} /p\mathbb {Z} )\to H^{m+2n(p-1)}(X,\mathbb {Z} /p\mathbb {Z} )$ is an additive homomorphism and natural.
$P^0$ is the identity homomorphism.
$P^{n}$ is the cup $p$ -th power on classes of degree $2n$ .
If $2n>\dim(X)$ then $P^{n}(x)=0$
Cartan Formula: $P^{n}(x\smile y)=\sum _{{i+j=n}}(P^{i}x)\smile (P^{j}y)$

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

Ádem relations

The Ádem relations for p=2 were conjectured by Wen-tsün Wu (1952) and proved by José Ádem (1952) and are given by

Sq^{i}Sq^{j}=\sum _{k=0}^{\lfloor i/2\rfloor }{j-k-1 \choose i-2k}Sq^{i+j-k}Sq^{k}

for all i, j > 0 such that i < 2j. (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 p the Ádem relations are

P^{{a}}P^{{b}}=\sum _{i}(-1)^{{a+i}}{(p-1)(b-i)-1 \choose a-pi}P^{{a+b-i}}P^{i}

for a<pb and

P^{{a}}\beta P^{{b}}=\sum _{i}(-1)^{{a+i}}{(p-1)(b-i) \choose a-pi}\beta P^{{a+b-i}}P^{i}+\sum _{i}(-1)^{{a+i+1}}{(p-1)(b-i)-1 \choose a-pi-1}P^{{a+b-i}}\beta P^{i}

for a≤pb

Bullett–Macdonald identities

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

For $p=2$ put

P(t)=\sum _{{i\geq 0}}t^{i}{\text{Sq}}^{i}

then the Ádem relations are equivalent to

P(s^{2}+st)\cdot P(t^{2})=P(t^{2}+st)\cdot P(s^{2})

For $p>2$ put

P(t)=\sum _{{i\geq 0}}t^{i}{\text{P}}^{i}

then the Ádem relations are equivalent to the statement that

(1+s{\text{Ad}}\beta )P(t^{p}+t^{{p-1}}s+\cdots +ts^{{p-1}})P(s^{p})

is symmetric in $s$ and $t$ . Here $\beta$ is the Bockstein operation and $(\operatorname {Ad} \beta )P=\beta P-P\beta$ .

Computations

Infinite Real Projective Space

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

H^{*}(\mathbb {RP} ^{\infty };\mathbb {Z} /2)\cong \mathbb {Z} /2[x],

where $\deg(x)=1.$ For the operations on $H^{1}$ we know that

{\begin{aligned}Sq^{0}(x)&=x\\Sq^{1}(x)&=x^{2}\\Sq^{k}(x)&=0&&{\text{ for any }}k>1\end{aligned}}

Using the operation

Sq:=Sq^{0}+Sq^{1}+Sq^{2}+\cdots

we note that the Cartan relation implies that

Sq:H^{*}(X)\to H^{*}(X)

is a ring morphism. Hence

Sq(x^{n})=(Sq(x))^{n}=(x+x^{2})^{n}=\sum _{i=0}^{n}{n \choose i}x^{n+i}

Since there is only one degree $n+i$ component of the previous sum, we have that

Sq^{i}(x^{n})={n \choose i}x^{n+i}

Construction

Suppose that π is any degree n subgroup of the symmetric group on n points, u a cohomology class in H^q(X, B), A an abelian group acted on by π, and c a cohomology class in H_i(π, A). Steenrod (1953) showed how to construct a reduced power uⁿ/c in H^kq−i(X, (A ⊗ B ⊗ ... ⊗ B) / π) as follows.

Taking the external product of u with itself n times gives an equivariant cocycle on Xⁿ with coefficients in B ⊗ ... ⊗ B.
Choose E to be a contractible space on which π acts freely and an equivariant map from E × X to Xⁿ. Pulling back uⁿ by this map gives an equivariant cocyle on E × X and therefore a cocycle of E / π × X with coefficients in B ⊗ ... ⊗ B.
Taking a slant product with c in H_i(E / π, A) gives a cocycle of X with coefficients in H₀(π, A ⊗ B ⊗ ... ⊗ B)

The Steenrod squares and reduced powers are special cases of this construction where π is a cyclic group of prime order p=n acting as a cyclic permutation of n elements, and the groups A and B are cyclic of order p, so that H₀(π, A ⊗ B ⊗ ... ⊗ B) is also cyclic of order p.

The structure of the Steenrod algebra

Jean-Pierre Serre (1953) (for $p=2$ ) and Henri Cartan (1954, 1955) (for $p>2$ ) described the structure of the Steenrod algebra of stable mod $p$ 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 sequence

i_{1},i_{2},\ldots ,i_{n}

is admissible if for each $j$ , we have that $i_{j}>2i_{j+1}$ . Then the elements

Sq^{I}=Sq^{i_{1}}\cdots Sq^{i_{n}},

where $I$ 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 $p>2$ consisting of the elements

Sq_{p}^{I}=Sq_{p}^{i_{1}}\cdots Sq_{p}^{i_{n}},

such that

i_{j}\geq pi_{{j+1}}

i_{j}\equiv 0,1{\bmod 2}(p-1)

Sq_{p}^{{2k(p-1)}}=P^{k}

Sq_{p}^{{2k(p-1)+1}}=\beta P^{k}

Hopf algebra structure and the Milnor basis

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

\psi :A\to A\otimes A.

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

\psi (Sq^{k})=\sum _{{i+j=k}}Sq^{i}\otimes Sq^{j}

\psi (P^{k})=\sum _{{i+j=k}}P^{i}\otimes P^{j}

\psi (\beta )=\beta \otimes 1+1\otimes \beta .

The linear dual of ψ makes the (graded) linear dual A_* of A into an algebra. Milnor (1958) proved, for p = 2, that A_* is a polynomial algebra, with one generator ξ_k of degree 2^k − 1, for every k, and for p > 2 the dual Steenrod algebra A_* is the tensor product of the polynomial algebra in generators ξ_k of degree 2p^k - 2 (k ≥ 1) and the exterior algebra in generators τ_k of degree 2p^k - 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

\psi (\xi _{n})=\sum _{{i=0}}^{n}\xi _{{n-i}}^{{p^{i}}}\otimes \xi _{i}.

where ξ₀=1, and

\psi (\tau _{n})=\tau _{n}\otimes 1+\sum _{{i=0}}^{n}\xi _{{n-i}}^{{p^{i}}}\otimes \tau _{i}

if p>2

The only primitive elements of A_* for p=2 are the $\xi _{1}^{{2^{i}}}$ , and these are dual to the $Sq^{{2^{i}}}$ (the only indecomposables of A).

Relation to formal groups

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 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

x\rightarrow x+\xi _{1}x^{2}+\xi _{2}x^{4}+\xi _{3}x^{8}+\cdots

Algebraic construction

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

{\begin{cases}P(x):SV[[x]]\to SV[[x]]\\P(x)(v)=v+F(v)x=v+v^{q}x&v\in V\end{cases}}

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

P(x)(f)=\sum P^{i}(f)x^{i}\qquad p>2

or

P(x)(f)=\sum Sq^{2i}(f)x^{i}\qquad p=2

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.

Applications

The 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 S^{2n - 1} → Sⁿ of Hopf invariant one, then n is a power of 2.

The proof uses the fact that each Sq^k 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 spheres

The 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

\mathrm {Ext} _{A}^{s,t}(\mathbb {F} _{p},\mathbb {F} _{p}).

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

References

Pedagogical

Malkiewich, Cary, The Steenrod Algebra (PDF), Archived from the original on 2017-08-15

References

Ádem, José (1952), "The iteration of the Steenrod squares in algebraic topology", Proceedings of the National Academy of Sciences of the United States of America, 38: 720–726, Bibcode:1952PNAS...38..720A, doi:10.1073/pnas.38.8.720, ISSN 0027-8424, JSTOR 88494, MR 0050278, PMC 1063640
Bullett, S. R.; Macdonald, I. G. (1982), "On the Adem relations", Topology. An International Journal of Mathematics, 21 (3): 329–332, doi:10.1016/0040-9383(82)90015-5, ISSN 0040-9383, MR 0649764
Cartan, Henri (1954), "Sur les groupes d'Eilenberg-Mac Lane. II", Proceedings of the National Academy of Sciences of the United States of America, 40: 704–707, Bibcode:1954PNAS...40..704C, doi:10.1073/pnas.40.8.704, ISSN 0027-8424, JSTOR 88981, MR 0065161, PMC 534145
Cartan, Henri (1955), "Sur l'itération des opérations de Steenrod", Commentarii Mathematici Helvetici, 29 (1): 40–58, doi:10.1007/BF02564270, ISSN 0010-2571, MR 0068219
Allen Hatcher, Algebraic Topology. Cambridge University Press, 2002. Available free online from the author's home page.
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Malygin, S.N.; Postnikov, M.M. (2001) [1994], "Steenrod square", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
May, J. Peter (1970), "A general algebraic approach to Steenrod operations", The Steenrod Algebra and its Applications (Proc. Conf. to Celebrate N. E. Steenrod's Sixtieth Birthday, Battelle Memorial Inst., Columbus, Ohio, 1970) (PDF), Lecture Notes in Mathematics, 168, Berlin, New York: Springer-Verlag, pp. 153–231, doi:10.1007/BFb0058524, MR 0281196
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Steenrod algebra

Cohomology operations

Axiomatic characterization

Ádem relations

Bullett–Macdonald identities

Computations

Infinite Real Projective Space

Construction

The structure of the Steenrod algebra

Hopf algebra structure and the Milnor basis

Relation to formal groups

Algebraic construction

Applications

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

See also

References

Pedagogical

References