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integral Steenrod square

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Definition

Definition

(integral Steenrod squares)

For odd 2n+12n + 1 \in \mathbb{N} the integral Steenrod square Sq 2n+1Sq^{2n + 1}_{\mathbb{Z}} is the composition of the mod-2 Steenrod square Sq 2nSq^{2n} with the Bockstein homomorphism β\beta associated with the sequence 2mod2/2\mathbb{Z} \overset{\cdot 2}{\to} \mathbb{Z} \overset{mod\, 2}{\longrightarrow} \mathbb{Z}/2\mathbb{Z}:

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

Properties

Proposition

The odd-degree integral Steenrod squares from def. are indeed integral lifts of the mod-2 Steenrod squares in that

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

This follows from the relation of the Bockstein homomorphism to the first Steenrod square

(mod,2)β=Sq 1 (mod, 2) \circ \beta = Sq^1

(this example) together with the first Adem relation

Sq 1Sq 2n=Sq 2n+1 Sq^1 \circ Sq^{2n} = Sq^{2n+1}

(this example):

Sq 2n+1 : B +2n(/2) Sk 2n B +2n(/2) β B +2n+1 id id B k+2n+1(mod2) Sq 2n+1 : B +2n(/2) Sk 2n B +2n(/2) Sq 1 B +2n+1(/2) \array{ Sq^{2n+1}_{\mathbb{Z}} &\colon& B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z}) &\overset{Sk^{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 + 2n} (\mathbb{Z}/2\mathbb{Z}) &\underset{Sk^{2n}}{\longrightarrow}& B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z}) &\underset{ Sq^1 }{\longrightarrow}& B^{\bullet + 2n + 1} (\mathbb{Z}/2\mathbb{Z}) }
Proposition

(integral Steenrod square in terms of Bockstein homomorphism for exponential sequence)

The integral Steenrod squares (def. ) may equivalently be written in terms of the Bockstein homomorphism δ\delta of the exponential sequence 2πmod2πU(1)\mathbb{Z} \overset{\cdot 2\pi}{\longrightarrow} \mathbb{R} \overset{mod\, 2 \pi}{\longrightarrow} U(1) as

(1)Sq 2n+1:B (/2)Sq 2nB +2n(/2)ιB +2nU(1)δB +2n+1. Sq^{2n+1}_{\mathbb{Z}} \;\colon\; B^{\bullet} (\mathbb{Z}/2\mathbb{Z}) \overset{ Sq^{2 n} }{\longrightarrow} B^{\bullet + 2n } (\mathbb{Z}/2\mathbb{Z}) \overset{ \iota }{\longrightarrow} B^{\bullet + 2n} U(1) \overset{\delta}{\longrightarrow} B^{\bullet + 2n + 1} \mathbb{Z} \,.
Proof

Since β=δι\beta = \delta \circ \iota, by this example.

Examples

Example

(integral Steenrod square refined to ordinary differential cohomology)

Let G^ 2n+2:XB 2n+1U(1) conn\hat G_{2n+2} \colon X \to \mathbf{B}^{2n+1} U(1)_{conn} be a cocycle in ordinary differential cohomology of degree 2n+22n + 2.

By inserting the bottom triangle of the ordinary differential cohomology hexagon (this diagram) into the factorization in (1) we obtain a canonical refinement of the integral Steenrod square Sq 2n+1[G 2n+2]Sq^{2n+1}_{\mathbb{Z}} [G_{2n + 2}] to a cocycle Sq^ 2n+1G^ 2n+2\widehat{Sq}^{2n+1}_{\mathbb{Z}} \hat G_{2n+2} in ordinary differential cohomology, which happens to be flat

B 4n+2U(1) B 4n+2U(1) conn ιSq 2n δ χ X G 2n+2 B 2n+2 Sq 2n+1 B 4n+3. \array{ && && \mathbf{B}^{4n+2} \flat U(1) &\longrightarrow& \mathbf{B}^{4n+2}U(1)_{conn} \\ && & {}^{\mathllap{ \iota \circ Sq^{2n} }}\nearrow & & {}_{\mathllap{\delta}}\searrow & \downarrow^{\chi} \\ X &\underset{G_{2n + 2}}{\longrightarrow}& B^{2n+2} \mathbb{Z} && \underset{ Sq^{2n+1}_{\mathbb{Z}} }{\longrightarrow} && B^{4n+3} \mathbb{Z} } \,.

If one moreover asks that the integral Steenrod square vanishes

[Sq 2n+1G 2n+2]=0H 4n+3(X,) [ Sq^{2n+1}_{\mathbb{Z}} G_{2n+2}] \;=\; 0 \;\in\; H^{4n+3}(X,\mathbb{Z})

(as in Diaconescu-Moore-Witten 00, around (6.9) for n=1n = 1) then the curvature exact sequence and characteristic class exact sequence in ordinary differential cohomology (this prop.) imply that the class of Sq^ 2n+1G^ 2n+2\widehat{Sq}^{2n+1}_{\mathbb{Z}} \hat G_{2n + 2} is identified with a class in de Rham cohomology in degree 4n+34n+3:

H diff 2n+2(X)| Sq 2n+1=0Sq^ 2n+1H dR 4n+3(X). H^{2n+2}_{diff}(X)|_{Sq^{2n+1}_{\mathbb{Z}} = 0} \overset{\widehat{Sq}_{\mathbb{Z}}^{2n+1}}{\longrightarrow} H^{4n+3}_{dR}(X) \,.

References

The third integral Steenrod square Sq 3Sq^3_{\mathbb{Z}} plays a central role in the discussion of the supergravity C-field in

Last revised on March 20, 2018 at 08:20:34. See the history of this page for a list of all contributions to it.