What are some good examples of simply connected manifolds with interesting Whitehead Lie algebras over R? Most of the manifolds that one thinks about if one is pretty naive are not so interesting--- Lie groups have abelian Whitehead algebras and homogeneous spaces have no higher product structure.
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2$\begingroup$ Connect sums of $\mathbb CP^2$'s are a decent example. The complement of a Cantor set in $\mathbb R^3$ would be a similar one. $\endgroup$– Ryan BudneyCommented Feb 16, 2010 at 2:27
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1$\begingroup$ The simplest interesting example is the Whitehead product for $S^2$, i.e., $\pi_2(S^2)\times\pi_2(S^2)\to\pi_3(S^2)$ is precisely given by sending $(1,1)$ to $2$. This is equivalent to the Hopf invariant of the Hopf map being 1, which is further equivalent to the linking number of fibres in the Hopf map being $2$. $\endgroup$– Somnath BasuCommented Feb 21, 2010 at 1:14
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I guess that by the Whitehead Lie algebra, you mean the homotopy group Lie algebra $\pi_*(\Omega X)\simeq \pi_{*-1}(X)$ maybe tensored by the reals $R$. In that case there is a theorem of Felix-Halperin-THomas, called the dichotomy theorem which tells you that either this Lie algebra is finite-dimensional (and the space is said to be "elliptic"), or it is very big in the sense that the ranks of $\pi_k(X)$ grows exponentially with k (and the space is then called "hyperbolic"). If the Euler characteristic of the manifold is negative then the space is always hyperbolic/ Moreover when the space is hyperbolic the Whitehead Lie algebra is very far from being abelian: actually its radical is finite dimensional. Therefore any manifold with negative euler characteristic has an non abelian infinite dimensional homotopy Lie algebra.
To generalize what Ryan says, actually any connected sum of two simply connected manifolds $M$ and $N$ is hyperbolic unless the cohomology of both $M$ and $N$ are truncatated polynomial algebras on a single genrator (like the sphere or $CP(n)$). In particular the connected sum of 3 or more closed manifolds not having the rational homotopy type of a sphere is hyperbolic.
Another example of a non abelian Whitehead Lie algebra but finite dimensional, is the one associated to a manifold $M$ obtained as an $S^5$-bundle with base $S^3\times S^3$ and where the euler class of the bundle is the fundamental class of the base (or any non zero multiple of it). In that case the Whitehead rational Lie algebra $\pi_*(M)\otimes Q$ is of dimension $3$ with basis $x,y,[x,y]$ where $x$ and $y$ are in degree $3$ and $[x,y]$ is in degree $5$. Thus this manifold M is elliptic. Interestingly enough, the cohomology algebra of M is isomorphic to that of the connected sum $W$ of two copies of $S^3\times S^8$, but $W$ is hyperbolic.
