co-binary Sullivan differential is Whitehead product



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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 WW a \mathbb{Z}-graded module, we write

WWSym 2(W)(WW)/(αβ(1) n αn ββα), 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 α,βW\alpha, \beta \in W are elements of homogeneous degree n α,n βn_\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 XX be a simply connected topological space with Sullivan model

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

for V *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 XX:

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

Declare the wedge product pairing to be given by

(2)V *V * Φ Hom Ab(π (X)π (X),) (α,β) (vw(1) n αn βα(v)β(w)+β(v)α(w)) \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 α\alpha, β\beta are assumed to be of homogeneous degree n α,n βn_\alpha, n_\beta \in \mathbb{N}, respectively.

(Notice that the usual normalization factor of 1/21/2 is not included on the right. This normalization follows Andrews-Arkowitz 78, above Thm. 6.1.)

Finally, write

(3)[] 2:Sym (V *)V *V * [-]_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.



(co-binary Sullivan differential is Whitehead product)

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

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

[d Xα] 2=[,] X *. [d_X \alpha]_2 \;=\; [-,-]_X^\ast \,.

More explicitly, the following diagram commutes:

V * [] 2d X V *V * = Φ Hom Ab(π (X),) Hom Ab([,] X,) Hom Ab(π (X)π (X),), \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)


Hopf fibrations

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

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

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

Then the Whitehead product satisfies

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

(by this Example).

Now let

vol S 2,vol S 3CE(𝔩S 2) 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 h_{\mathbb{C}}) of the 3-sphere, respectively.

This means that the Sullivan differential is

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

for some rational number cc \in \mathbb{Q}.

Notice that with the normalization in (2) we have

Φ(vol S 2,vol S 2)(id S 2id S 2) =(1) 22vol S 2(id S 2)vol S 2(id S 2)+vol S 2(id S 2)vol S 2(id S 2) =2 \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

{vol S 3} [] 2d S 2 c(vol S 2vol S 2) Φ {h } [,] X * {id S 2id S 22} \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=1c = 1:

d S 2vol 3=vol S 2vol S 2. d_{S^2} vol_3 \;=\; - vol_{S^2} \wedge vol_{S^2} \,.

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


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

where the result is attributed to

(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 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.

Last revised on August 26, 2020 at 07:17:48. See the history of this page for a list of all contributions to it.