Contents
Context
Homotopy theory
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
Lie theory
∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
Contents
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 a -graded module, we write
where on the right are elements of homogeneous degree , respectively. The point is just to highlight that “” is not to imply here a degree shift of the generators (as it typically does in the usual notation for Grassmann algebras).
Let be a simply connected topological space with Sullivan model
(1)
for the graded vector space of generators, which is the -linear dual graded vector space of the graded -module (=graded abelian group) of homotopy groups of :
Declare the wedge product pairing to be given by
(2)
where , are assumed to be of homogeneous degree , respectively.
(Notice that the usual normalization factor of is not included on the right. This normalization follows Andrews-Arkowitz 78, above Thm. 6.1.)
Finally, write
(3)
for the linear projection on quadratic polynomials in the graded symmetric algebra.
Statement
Proposition
(co-binary Sullivan differential is Whitehead product)
Let be a simply connected topological space of rational finite type, so that it has a Sullivan model with Sullivan differential (1).
Then the co-binary component (3) of the Sullivan differential equals the -linear dual map of the Whitehead product on the homotopy groups of :
More explicitly, the following diagram commutes:
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 the 2-sphere, consider the following two elements of its homotopy groups (of spheres, as it were):
-
(represented by the identity function )
-
(represented by the complex Hopf fibration)
Then the Whitehead product satisfies
(4)
(by this Example).
Now let
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 ) of the 3-sphere, respectively.
This means that the Sullivan differential is
(5)
for some rational number .
Notice that with the normalization in (2) we have
Therefore Prop. gives
where in the bottom row we used the Whitehead product (4).
Hence :
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
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 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.