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Steenrod square in nLab

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In algebraic topology, what are called the Steenrod squares is the system of cohomology operations on ordinary cohomology with coefficients in ℤ 2\mathbb{Z}_2 (the cyclic group of order 2) which is compatible with suspension (the “stable cohomology operations”). They are special examples of power operations.

The collection of Steenrod squares for all degrees forms the Steenrod algebra, see there for more.

Definition

Construction in terms of extended squares

We discuss the explicit construction of the Steenrod-operations in terms of chain maps of chain complexes of 𝔽 2\mathbb{F}_2-vector spaces equipped with a suitable product. We follow (Lurie 07, lecture 2).

Write 𝔽 2≔ℤ/2ℤ\mathbb{F}_2 \coloneqq \mathbb{Z}/2\mathbb{Z} for the field with two elements.

For VV an 𝔽 2\mathbb{F}_2-module, hence an 𝔽 2\mathbb{F}_2-vector space, and for n∈ℕn \in \mathbb{N}, write

V hΣ n ⊗n∈𝔽 2Mod V^{\otimes n}_{h \Sigma_n} \in \mathbb{F}_2 Mod

for the homotopy quotient of the nn-fold tensor product of VV with itself by the action of the symmetric group. Explicitly this is presented, up to quasi-isomorphism by the ordinary coinvariants D n(V)D_n(V) of the tensor product of V ⊗nV^{\otimes n} with a free resolution EΣ n •E \Sigma_n^\bullet of 𝔽 2\mathbb{F}_2:

V hΣ n ⊗n≃D n(V)≔(V ⊗n⊗EΣ n) Σ n. V^{\otimes n}_{h \Sigma_n} \simeq D_n(V) \coloneqq (V^{\otimes n} \otimes E\Sigma_n)_{\Sigma_n} \,.

This is called the nnth extended power of VV.

For instance

D 2(𝔽 2[−n])≃𝔽 2[−2n]⊗C •(BΣ 2), D_2( \mathbb{F}_2[-n]) \simeq \mathbb{F}_2[-2n] \otimes C^\bullet(B \Sigma_2) \,,

where on the right we have the, say, singular cohomology cochain complex of the homotopy quotient *//Σ 2≃BΣ 2≃ℝP ∞\ast //\Sigma_2 \simeq B \Sigma_2 \simeq \mathbb{R}P^\infty, which is the homotopy type of the classifying space for Σ 2\Sigma_2.

A chain map

D 2(V)⟶V D_2(V) \longrightarrow V

is called a symmetric multiplication on VV (a shadow of an E-infinity algebra structure). The archetypical class of examples of these are given by the singular cohomology V=C •(X,𝔽 2)V = C^\bullet(X, \mathbb{F}_2) of any topological space XX, for instance of BΣ 2B \Sigma_2.

Therefore there is a canonical isomorphism

H k(D 2(𝔽 2[−n]))≃H 2n−k(BΣ 2,𝔽 2)e 2n H^k(D_2(\mathbb{F}_2[-n])) \simeq H_{2n - k}(B \Sigma_2, \mathbb{F}_2) e_{2n}

of the cochain cohomology of the extended square of the chain compplex concentrated on 𝔽 2\mathbb{F}_2 in degree nn with the singular homology of this classifying space shifted by 2n2 n.

Using this one gets for general VV and for each i≤ni \leq n a map that sends an element in the nnth cochain cohomology

[v]∈H n(V) [v] \in H^n(V)

represented by a morphism of chain complexes

v:𝔽 2[−n]⟶V v \;\colon\; \mathbb{F}_2[-n] \longrightarrow V

to the element

Sq¯ i(v)∈H n+1(D 2(V)) \overline{Sq}^i(v) \in H^{n+1}(D_2(V))

represented by the chain map

𝔽 2[−n−i]⟶1C •(BΣ 2,𝔽 2)⟶≃D 2(𝔽 2[−n])⟶D 2(v)D 2(V). \mathbb{F}_2[-n-i] \stackrel{1}{\longrightarrow} C^\bullet(B \Sigma_2, \mathbb{F}_2) \stackrel{\simeq}{\longrightarrow} D_2(\mathbb{F}_2[-n]) \stackrel{D_2(v)}{\longrightarrow} D_2(V) \,.

If moreover VV is equipped with a symmetric product D 2(V)⟶VD_2(V) \longrightarrow V as above, then one can further compose and form the element

Sq i(v)∈H n+1(V) {Sq}^i(v) \in H^{n+1}(V)

represented by the chain map

𝔽 2[−n−i]⟶1C •(BΣ 2,𝔽 2)⟶≃D 2(𝔽 2[−n])⟶D 2(v)D 2(V)⟶V. \mathbb{F}_2[-n-i] \stackrel{1}{\longrightarrow} C^\bullet(B \Sigma_2, \mathbb{F}_2) \stackrel{\simeq}{\longrightarrow} D_2(\mathbb{F}_2[-n]) \stackrel{D_2(v)}{\longrightarrow} D_2(V) \longrightarrow V \,.

This linear map

Sq i:H •(V)⟶H •+i(V) Sq^i \;\colon\; H^\bullet(V) \longrightarrow H^{\bullet + i}(V)

is called the iith Steenrod operation or the iith Steenrod square on VV. By default this is understood for V=C •(X,𝔽 2)V = C^\bullet(X,\mathbb{F}_2) the 𝔽 2\mathbb{F}_2-singular cochain complex of some topological space XX, as in the above examples, in which case it has the form

Sq i:H •(X,𝔽 2)⟶H •+i(X,𝔽 2). Sq^i \;\colon\; H^\bullet(X, \mathbb{F}_2) \longrightarrow H^{\bullet+i}(X,\mathbb{F}_2) \,.

Axiomatic characterization

For n∈ℕn \in \mathbb{N} write B nℤ 2B^n \mathbb{Z}_2 for the classifying space of ordinary cohomology in degree nn with coefficients in the group of order 2 ℤ 2\mathbb{Z}_2 (the Eilenberg-MacLane space K(ℤ 2,n)K(\mathbb{Z}_2,n)), regarded as an object in the homotopy category HH of topological spaces).

Notice that for XX any topological space (CW-complex),

H n(X,ℤ 2)≔H(X,B nℤ 2) H^n(X, \mathbb{Z}_2) \coloneqq H(X, B^n \mathbb{Z}_2)

is the ordinary cohomology of XX in degree nn with coefficients in ℤ 2\mathbb{Z}_2. Therefore, by the Yoneda lemma, natural transformations

H k(−,ℤ 2)→H l(−,ℤ 2) H^{k}(-, \mathbb{Z}_2) \to H^l(-, \mathbb{Z}_2)

correspond bijectively to morphisms B kℤ 2→B lℤ 2B^k \mathbb{Z}_2 \to B^l \mathbb{Z}_2.

The following characterization is due to (SteenrodEpstein).

Definition

The Steenrod squares are a collection of cohomology operations

Sq n:H k(−,ℤ 2)⟶H k+n(−,ℤ 2), Sq^n \;\colon\; H^k(-, \mathbb{Z}_2) \longrightarrow H^{k+n}(-, \mathbb{Z}_2) \,,

hence of morphisms in the homotopy category

Sq n:B kℤ 2⟶B k+nℤ 2 Sq^n \;\colon\; B^k \mathbb{Z}_2 \longrightarrow B^{k + n} \mathbb{Z}_2

for all n,k∈ℕn,k \in \mathbb{N} satisfying the following conditions:

  1. for n=0n = 0 it is the identity;

  2. if n>deg(x)n \gt deg(x) then Sq n(x)=0Sq^n(x) = 0;

  3. for k=nk = n the morphism Sq n:B nℤ 2→B 2nℤ 2Sq^n : B^n \mathbb{Z}_2 \to B^{2n} \mathbb{Z}_2 is the cup product x↦x∪xx \mapsto x \cup x;

  4. Sq n(x∪y)=∑ i+j=n(Sq ix)∪(Sq jy)Sq^n(x \cup y) = \sum_{i + j = n} (Sq^i x) \cup (Sq^j y);

An analogous definition works for coefficients in ℤ p\mathbb{Z}_p for any prime number p>2p \gt 2. The corresponding operations are then usually denoted

P n:B kℤ p⟶B k+nℤ p. P^n \;\colon\; B^k \mathbb{Z}_p \longrightarrow B^{k+n} \mathbb{Z}_{p} \,.

Under composition, the Steenrod squares form an associative algebra over 𝔽 2\mathbb{F}_2, called the Steenrod algebra. See there for more.

Properties

Relation to Bockstein homomorphism

Sq 1Sq^1 is the Bockstein homomorphism of the short exact sequence ℤ 2→ℤ 4→ℤ 2\mathbb{Z}_2 \to \mathbb{Z}_4 \to \mathbb{Z}_2.

Compatibility with suspension

The Steenrod squares are compatible with the suspension isomorphism.

Therefore the Steenrod squares are often also referred to as the stable cohomology operations

Relation to Massey products

See at Massey product, Relation to Steenrod squares

Adem relations

Proposition

(Adem relations)

The composition of Steenrod square operations satisfies the following relations

Sq i∘Sq j=∑ 0≤k≤i/2(j−k−1i−2k) mod2Sq i+j−k∘Sq k Sq^i \circ Sq^j = \sum_{0 \leq k \leq i/2} \left( { { j - k - 1 } \atop { i - 2k } } \right)_{mod 2} Sq^{i + j -k} \circ Sq^k

for all 0<i<2j0 \lt i \lt 2 j.

Here (ab)≔0\left( a \atop b \right) \coloneqq 0 if a<ba \lt b.

Example

(Adem relation for postcomposition with the Bockstein homomorphism Sq 1=βSq^1 = \beta)

For j≥2j \geq 2 and i=1i =1, the Adem relations (prop. ) say that:

Sq 1∘Sq j =(j−11) mod2⏟(j−1) mod2Sq j+1 ={Sq j+1 | jeven 0 | jodd \begin{aligned} Sq^1 \circ Sq^j & = \underset{ (j-1)_{mod 2} }{ \underbrace{ \left( { {j - 1 } \atop 1 } \right)_{mod 2} }} Sq^{j + 1} \\ & = \left\{ \array{ Sq^{j+1} &\vert& j \, \text{even} \\ 0 &\vert& j \, \text{odd} } \right. \end{aligned}

This gives rise to:

Example

(integral Steenrod squares)

For odd 2n+1∈ℕ2n + 1 \in \mathbb{N} defines the integral Steenrod squares to be

Sq ℤ 2n+1≔β∘Sq 2n. Sq^{2n + 1}_{\mathbb{Z}} \;\coloneqq\; \beta \circ Sq^{2n} \,.

By example and by this example these indeed are lifts of the odd Steenrod squares:

(mod2)∘Sq ℤ 2n+1=Sq 2n+1, (mod\, 2) \circ Sq^{2n + 1}_{\mathbb{Z}} \;=\; Sq^{2n+1} \,,

in that we have

Sq ℤ 2n+1 : B •(ℤ/2ℤ) ⟶Sq 2n B •+2n(ℤ/2ℤ) ⟶β B •+2n+1ℤ ↓ id ↓ id ↓ B k+2n+1(mod2) Sq 2n+1 : B •(ℤ/2ℤ) ⟶Sq 2n B •+2n(ℤ/2ℤ) ⟶Sq 1 B •+2n+1(ℤ/2ℤ) \array{ Sq^{2n+1}_{\mathbb{Z}} &\colon& B^{\bullet} (\mathbb{Z}/2\mathbb{Z}) &\overset{Sq^{2n}}{\longrightarrow}& B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z}) &\overset{ \beta }{\longrightarrow}& B^{\bullet + 2n + 1} \mathbb{Z} \\ && \downarrow^{ id } && \downarrow^{ id } && \downarrow^{\mathrlap{B^{k + 2 n + 1}(mod\, 2)}} \\ Sq^{2n+1} &\colon& B^{\bullet} (\mathbb{Z}/2\mathbb{Z}) &\underset{Sq^{2n}}{\longrightarrow}& B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z}) &\underset{ Sq^1 }{\longrightarrow}& B^{\bullet + 2n + 1} (\mathbb{Z}/2\mathbb{Z}) }

Examples

Hopf invariant

Proposition

For ϕ:S k+n−1→S k\phi \colon S^{k+n-1} \to S^k, a map of spheres, the Steenrod square

Sq n:H k(cofib(ϕ),𝔽 2)⟶H k+n(cofib(ϕ),𝔽 2) Sq^n \colon H^k(cofib(\phi), \mathbb{F}_2) \longrightarrow H^{k+n}(cofib(\phi),\mathbb{F}_2)

(on the homotopy cofiber cofib(ϕ)≃S k∪S k+n−1D k+ncofib(\phi)\simeq S^k \underset{S^{k+n-1}}{\cup} D^{k+n})

is non-vanishing exactly for n∈{1,2,4,8}n \in \{1,2,4,8\}.

(Adams 60, theorem 1.1.1).

See at Hopf invariant one theorem.

References

The operations were first defined in

  • Norman Steenrod, Products of cocycles and extensions of mappings, Annals of Mathematics Second Series, Vol. 48, No. 2 (Apr., 1947), pp. 290-320 (jstor:1969172)

The axiomatic definition appears in

Lecture notes on Steenrod squares and the Steenrod algebra include

See also

  • Wen-Tsun Wu, Sur les puissances de Steenrod, Colloque de Topologie de Strasbourg, IX, La Bibliothèque Nationale et Universitaire de Strasbourg, (1952)

  • Rocio Gonzalez-Diaz, Pedro Real, A Combinatorial Method for Computing Steenrod Squares, Journal of Pure and Applied Algebra 139 (1999) 89-108 (arXiv:math/0110308)

Discussion in homotopy type theory:

Last revised on January 17, 2023 at 05:54:31. See the history of this page for a list of all contributions to it.