Dreams on the (Co)Homology of Manifolds
The Context. Let be an -dimensional manifold (a
topological space that locally looks like
) and let be a ring.
The following should be a list of spaces and their duals;
there ought to be no (co)homology beyond that list:
There should be an ``intersection pairing''
induced from the intersection pairing of submanifolds which ought to
In particular, there should be a pairing
so with some further optimism,
ought to be the same as
. (And why not call that ``Poincaré Duality''?)
There should be a ``cap product''
There should be a ``cup product''
ought to be a ring!
Jules Henri Poincaré
Warning. Dreams are based on reality. Often,
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