nLab integral Steenrod square

Definition

Definition

(integral Steenrod squares)

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

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

Properties

Proposition

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

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

Proof

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

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

(this example) together with the first Adem relation

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

(this example):

$$\array{ Sq^{2n+1}_{\mathbb{Z}} &\colon& B^{\bullet} (\mathbb{Z}/2\mathbb{Z}) &\overset{Sk^{2n}}{\longrightarrow}& B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z}) &\overset{ \beta }{\longrightarrow}& B^{\bullet + 2n + 1} \mathbb{Z} \\ && \downarrow^{ id } && \downarrow^{ id } && \downarrow^{\mathrlap{B^{k + 2 n + 1}(mod\, 2)}} \\ Sq^{2n+1} &\colon& B^{\bullet} (\mathbb{Z}/2\mathbb{Z}) &\underset{Sk^{2n}}{\longrightarrow}& B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z}) &\underset{ Sq^1 }{\longrightarrow}& B^{\bullet + 2n + 1} (\mathbb{Z}/2\mathbb{Z}) }$$

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 $\delta$ of the exponential sequence $\mathbb{Z} \overset{\cdot 2\pi}{\longrightarrow} \mathbb{R} \overset{mod\, 2 \pi}{\longrightarrow} U(1)$ as

(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 $\beta = \delta \circ \iota$, by this example.

Examples

Example

(integral Steenrod square refined to ordinary differential cohomology)

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

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 $Sq^{2n+1}_{\mathbb{Z}} [G_{2n + 2}]$ to a cocycle $\widehat{Sq}^{2n+1}_{\mathbb{Z}} \hat G_{2n+2}$ in ordinary differential cohomology, which happens to be flat

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

If one moreover asks that the integral Steenrod square vanishes

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

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

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

Related concepts

• integral Stiefel-Whitney classes

• integral Wu structure

References

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

• Gregory Moore, Edward Witten, $E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory, Adv. Theor. Math. Phys. 6:1031-1134, 2003 (arXiv:hep-th/0005090)