nLab co-binary Sullivan differential is Whitehead product

Idea

We discuss (Prop. below) how the rationalization of the Whitehead product is the co-binary part of the Sullivan differential in rational homotopy theory.

Notation and conventions

We make explicit some notation and normalization conventions that enter the statement.

In the following, for $W$ a $\mathbb{Z}$-graded module, we write

$$W \wedge W \;\coloneqq\; Sym^2(W) \;\coloneqq\; \big( W \otimes W \big) / \big( \alpha \otimes \beta \sim (-1)^{ n_\alpha n_\beta } \beta \otimes \alpha \big) \,,$$

where on the right $\alpha, \beta \in W$ are elements of homogeneous degree $n_\alpha, n_\beta \in \mathbb{Z}$, respectively. The point is just to highlight that “$(-)\wedge(-)$” is not to imply here a degree shift of the generators (as it typically does in the usual notation for Grassmann algebras).

Let $X$ be a simply connected topological space with Sullivan model

(1)$$CE( \mathfrak{l} X ) \;=\; \big( Sym^\bullet\big(V^\ast\big), d_X \big)$$

for $V^\ast$ the graded vector space of generators, which is the $\mathbb{Q}$-linear dual graded vector space of the graded $\mathbb{Z}$-module (=graded abelian group) of homotopy groups of $X$:

$$V^\ast \;\coloneqq\; Hom_{Ab}\big( \pi_\bullet(X), \mathbb{Q} \big) \,.$$

Declare the wedge product pairing to be given by

(2)$$\array{ V^\ast \wedge V^\ast &\overset{\Phi}{\longrightarrow}& Hom_{Ab} \big( \pi_\bullet(X) \wedge \pi_\bullet(X) , \mathbb{Q} \big) \\ (\alpha, \beta) &\mapsto& \Big( v \wedge w \;\mapsto\; (-1)^{ n_\alpha \cdot n_\beta } \alpha(v)\cdot \beta(w) + \beta(v)\cdot \alpha(w) \Big) }$$

where $\alpha$, $\beta$ are assumed to be of homogeneous degree $n_\alpha, n_\beta \in \mathbb{N}$, respectively.

(Notice that the usual normalization factor of $1/2$ is not included on the right. This normalization follows Andrews-Arkowitz 78, above Thm. 6.1.)

Finally, write

(3)$$[-]_2 \;\colon\; Sym^\bullet\big(V^\ast\big) \longrightarrow V^\ast \wedge V^\ast$$

for the linear projection on quadratic polynomials in the graded symmetric algebra.

Statement

Proposition

(co-binary Sullivan differential is Whitehead product)

Let $X$ be a simply connected topological space of rational finite type, so that it has a Sullivan model with Sullivan differential $d_X$ (1).

Then the co-binary component (3) of the Sullivan differential equals the $\mathbb{Q}$-linear dual map of the Whitehead product $[-,-]_X$ on the homotopy groups of $X$:

$$[d_X \alpha]_2 \;=\; [-,-]_X^\ast \,.$$

More explicitly, the following diagram commutes:

$$\array{ V^\ast &\overset{ [-]_2\circ d_X }{\longrightarrow}& V^\ast \wedge V^\ast \\ \big\downarrow^{ \mathrlap{=} } && \big\downarrow^{ \mathrlap{\Phi} } \\ Hom_{Ab} \big( \pi_\bullet(X), \mathbb{Q} \big) & \underset{ Hom_{Ab}\big( [-,-]_X , \; \mathbb{Q} \big) }{ \longrightarrow } & Hom_{Ab} \big( \pi_\bullet(X) \wedge \pi_\bullet(X), \; \mathbb{Q} \big) } \,,$$

where the wedge product on the right is normalized as in (2).

(Andrews-Arkowitz 78, Thm. 6.1, see also Félix-Halperin-Thomas 00, Prop. 13.16)

Examples

Hopf fibrations

For $X = S^2$ the 2-sphere, consider the following two elements of its homotopy groups (of spheres, as it were):

1. $id_{S^2} \in \pi_2\big( S^2 \big)$ (represented by the identity function $S^2 \to S^2$)

2. $h_{\mathbb{C}} \in \pi_3\big( S^3 \big)$ (represented by the complex Hopf fibration)

Then the Whitehead product satisfies

(4)$$\big[ id_{S^2}, \; id_{S^2} \big] \;=\; 2 \cdot h_{\mathbb{C}} \,.$$

(by this Example).

Now let

$$vol_{S^2},\; vol_{S^3} \;\in\; CE\big( \mathfrak{l}S^2 \big)$$

be the two generators of the Sullivan model of the 2-sphere, normalized such that they correspond to the volume forms of the 2-sphere and (after pullback along the complex Hopf fibration $h_{\mathbb{C}}$) of the 3-sphere, respectively.

This means that the Sullivan differential is

(5)$$d_{S^2} vol_{S^3} \;=\; c \cdot vol_{S^2} \wedge vol_{S^2}$$

for some rational number $c \in \mathbb{Q}$.

Notice that with the normalization in (2) we have

$$\begin{aligned} \Phi(vol_{S^2}, vol_{S^2})\big( id_{S^2} \wedge id_{S^2} \big) & = (-1)^{2 \cdot 2} vol_{S^2}\big( id_{S^2}\big) \cdot vol_{S^2}\big( id_{S^2}\big) + vol_{S^2}\big( id_{S^2}\big) \cdot vol_{S^2}\big( id_{S^2}\big) \\ & = 2 \end{aligned}$$

Therefore Prop. gives

$$\array{ \big\{ vol_{S^3} \big\} &\overset{[-]_2\circ d_{S^2}}{\longrightarrow}& c' \cdot ( vol_{S^2} \otimes vol_{S^2} ) \\ \big\downarrow && \big\downarrow^{\mathrlap{\Phi}} \\ \big\{ h_{\mathbb{H}} \big\} &\overset{ [-,-]_X^\ast }{\longrightarrow}& \big\{ id_{S^2} \wedge id_{S^2} \mapsto 2 \big\} }$$

where in the bottom row we used the Whitehead product (4).

Hence $c = 1$:

$$d_{S^2} vol_3 \;=\; - vol_{S^2} \wedge vol_{S^2} \,.$$

See also Félix-Halperin-Thomas 00, Example 1 on p. 178.

References

Under the general relation between the Sullivan model and the original Quillen model of rational homotopy theory, the statement comes from

• Daniel Quillen, section I.5 of Rational Homotopy Theory, Annals of Mathematics Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 (jstor:1970725)

It is made fully explicit in

• Peter Andrews, Martin Arkowitz, Theorem 6.1 Sullivan’s Minimal Models and Higher Order Whitehead Products, Canadian Journal of Mathematics, 30(5), 961-982, 1978 (doi:10.4153/CJM-1978-083-6)

where the result is attributed to

• Pierre Deligne, Phillip Griffiths, John Morgan, Dennis Sullivan, Real homotopy theory of Kähler manifolds, Invent Math (1975) 29: 245 (doi:10.1007/BF01389853)

(which however just touches on it in passing)

and in

• Francisco Belchí, Urtzi Buijs, José M. Moreno-Fernández, Aniceto Murillo, Prop. 3.1 of: Higher order Whitehead products and $L_\infty$ structures on the homology of a DGL, Linear Algebra and its Applications, Volume 520 (2017), pages 16-31 (arXiv:1604.01478, doi:10.1016/j.laa.2017.01.008)

Textbook accounts:

• Yves Félix, Steve Halperin, J.C. Thomas, Prop. 13.16 in Rational Homotopy Theory, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000.