So, I studied differential topology from Kosinski's book and cellular homology from Geoghegan's book and, for a long time, I thought handlebody homology was the be-all and end-all for homology theories for smooth manifolds. Then that changed, basically with the Asian professor from video series embedded in my Week 01 posts in the 'blog, and I began to think that DeRham cohomology and "DeRham homology" were the be-all and the end-all homology and cohomology theories, at least for orientable, closed, connected, (smooth) Riemannian manifolds.
So, I've been toying around with a construction for a while, based on an idea from a paper by my dissertation advisor, Craig Guilbault. In a paper he co-authors with Fred Tinsley, they discuss homology (and cohomology, implicitly) with "twisted integer" coefficients.
This gave me the idea to try DeRham cohomology (and homology, in a meaningful sense of the phrase, using Poincare duals of differential forms) with "twisted real" coefficients, that is H*DR(M; ℝ [ℤ2]) with "group field" coefficients; one essentially works in the double-cover of the manifold -- which is 2 disjoint copies of the manifold, if the initial manifold is orientable, or the orientation double-cover, if the initial manifold is non-orientable -- and does all of one's calculations there, then one "divides by 2".
I talked with Craig, and in Hatcher's book, there is a notion of homology with "twisted integer" coefficients that is different from both the ordinary, untwisted homology and from the twisted "group field" coefficients homology of the double cover; evidently, all 3 homologies fit into a long exact sequence. Davis and Kirk's book gives a nice treatment of the topic of homology with twisted coefficients in general. This coincides with the notion of De Rham cohomology with coefficients in the canonical line bundle over a manifold (which gives the cohomology with twisted coefficients from Davis and Kirk in the case M is non-orientable and the usual cohomology of M in the case M is orientable - just what the doctor ordered!) from Bott and Tu's book.
So, I discovered a way to rescue an orientation form in the case that M is non-orientable with De Rham cohomology. Now, I just need a way to recover torsion from De Rham cohomology, and I'll be in business with an all-purpose version of De Rham cohomology.
