When are the homology and cohomology Hopf algebras of topological groups equal?

Question

Suppose we have a topological group $G$, then the multiplication map $\mu$ and the diagonal map $\Delta$ provide the cohomology $H^\ast(G;R)$ (with Pontryagin coproduct and cup coproduct) and homology $H_\ast(G;R)$ (with Pontryagin product and diagonal coproduct) of $G$ with the structure of a Hopf algebra (assuming vanishing of the Tor-terms in the Kunneth theorem). If both satisfy some mild conditions, these are in fact dual. A nice historical explanation of this can be found in Cartier's "Primer on Hopf Algebras" (http://inc.web.ihes.fr/prepub/PREPRINTS/2006/M/M-06-40.pdf). My general question is the following: when we know they are in fact isomorphic as Hopf algebras?

The Samelson theorem says that if $G$ is a Lie group and $R$ is a field of characteristic zero, then $H^\ast(G;R)$ is commutative, associative and coassociative and an exterior algebra on primitive generators of odd degree. I think this implies that $H^\ast(G;R)$ is isomorphic to $H_\ast(G;R)$ as a Hopf algebra. Is this correct? To what extent does Samelson's result extend to non-zero characteristic?

My goal is to limit the amount of calculation when I do calculations of the homology and cohomology of Lie groups.

Answer 1

The mod $2$ cohomology of $\mathrm{SO}_n$ is not an exterior algebra as soon as $n\geq3$ (there is a unique non-zero element in degree $1$ whose square is non-zero, think of the case $\mathrm{SO}_{3}$ which is $3$-dimensional real projective space) while the homology ring is an exterior algebra.

Another example is $K(\mathbb Z,2)$ (aka $\mathbb C\mathbb P^\infty$), its integral cohomology ring is a polynomial ring on a degree $2$-generator while its homology ring is the free divided power algebra on a degree $2$-generator.

Answer 2

The example of $\mathrm{SO}_n$ that Torsten Ekedahl gives is quite instructive. Taking the union over all $n$, the infinite-dimensional group $\mathrm{SO}$ has mod $2$ cohomology a polynomial ring with one generator in each odd degree and mod $2$ homology an exterior algebra with one generator in each positive degree. Restricting to a finite dimensional $\mathrm{SO}_n$ has the effect of restricting the homology and cohomology algebras to a finite number of generators and truncating the polynomial algebra by relations that say a power of a generator is zero, where the exponent is a certain power of $2$. (Details can be found in Section 3.D of my algebraic topology book.)

The guiding principle here is that the dual of a polynomial algebra is a divided polynomial algebra. Over $\mathbb Q$ a divided polynomial algebra is just a polynomial algebra, but this is not true over $\mathbb Z$ or $\mathbb{Z}/p$. Over $\mathbb{Z}/2$ a divided polynomial algebra is in fact an exterior algebra, as in the case of $\mathrm{SO}$.

Answer 3

Perhaps the following example can also be of some use. Consider the looped suspension of infinite-dimensional complex projective space, $\Omega\Sigma \,\mathbb{CP}^\infty$. By Bott-Samelson its homology is the associative ring over the integers in countably many indeterminates, also known as the algebra of non-symmetric functions, NSymm. It is a Hopf algebra (but not primitively generated). It's dual is the cohomology of this space and is the so-called algebra of quasi-symmetric functions, QSymm; also a Hopf algebra of course. As an algebra over the integers QSymm is commutative free polynomial. See arXiv:math/0410366 for a proof and an explicit set of free polynomial generators.

The corresponding commutative situation is provided by of the homology and cohomology Hopf algebras of the classifying space BU. These are dual and isomorphic, and both equal to the free commutative algebra in countably many indeterminates, one in each double dimension, also known as Symm, the algebra of symmetric functions over the integers.