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Dreams on the (Co)Homology of Manifolds

This document in PDF: ManifoldDreams.pdf

The Context. Let $ M$ be an $ n$-dimensional manifold (a topological space that locally looks like $ {\mathbb{R}}^n$) and let $ R$ be a ring.

Dream 1   The following should be a list of spaces and their duals; there ought to be no (co)homology beyond that list:

$\displaystyle \xymatrix{
H_0(M) & H_1(M) & H_2(M) & \cdots & H_{n-1}(M) & H_n(M) \\
H^0(M) & H^1(M) & H^2(M) & \cdots & H^{n-1}(M) & H^n(M)

Dream 2   There should be an ``intersection pairing''

$\displaystyle H_k\times H_l \to H_{k+l-n}, $

induced from the intersection pairing of submanifolds which ought to satisfy $ \partial(\sigma\cap\lambda) =
(\partial\sigma)\cap\lambda + \sigma\cap(\partial\lambda)$.

Dream 3   In particular, there should be a pairing

$\displaystyle H_k\times H_{n-k} \to H_0=R, $

so with some further optimism, $ H_k$ ought to be the same as $ (H_{n-k})^\star=H^{n-k}$. (And why not call that ``Poincaré Duality''?)

Dream 4   $ H_n$ should be $ R$ (and hence $ H^n$ should be $ R$ as well).

Dream 5   There should be a ``cap product''

$\displaystyle \cap:H_k\times H^l \to H_{k-l}. $

Dream 6   There should be a ``cup product''

$\displaystyle \cup:H^k\times H^l \to H^{k+l}, $

and so $ H^\star:=\oplus_k H^k$ ought to be a ring!

Jules Henri Poincare

Jules Henri Poincaré

(from http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Poincare.html)

Warning. Dreams are based on reality. Often, distorted reality.

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Dror Bar-Natan 2005-03-31