Talk:Cohomology ring

Bialgebra? Superalgebra?
Another patent-crazy question: does the cohomology ring, taken together with e.g. Poincare duality, satisfy the axioms for being a bialgebra or a Hopf algebra or a Lie superalgebra?

Here's what I mean: The cup product of two cohomology classes can be defined as a pullback of comultiplication on the tensor products of chain complexes (i.e. the tensor product on chain complexes goes up to a tensor product on the cohomology groups, and the comultiplication puts it all back together again.) Poincare duality then seems to tell one how to go the other way. Is the result a bialgebra? Because these things are skew-commutative and graded, etc. is the result ever a super Lie algebra? I'm not sure I understand what I'm asking; I'm just trying to figure out how a cup product actually works. linas 16:14, 21 December 2006 (UTC)