Changes between Version 1 and Version 2 of Ticket #29581, comment 93


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Timestamp:
09/21/21 20:07:00 (3 months ago)
Author:
gh-mjungmath
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  • Ticket #29581, comment 93

    v1 v2  
    66> Perhaps that is a bit special. The torus is a little more interesting example. Mainly what I am thinking is if can we compute the Betti numbers without necessarily knowing what the manifold is in advance.
    77
    8 I don't see how that should be possible with this implementation. Characteristic cohomology classes usually don't tell you anything about the cohomology of your underlying base space. For once, the characteristic cohomology classes usually don't generate the cohomology groups. Moreover, if a characteristic cohomology class has torsion in the ZZ-cohomology, its corresponding class in the de Rham cohomology vanishes. So you lose a lot of information on the way.
     8I don't see how that should be possible with this implementation. Characteristic cohomology classes usually don't tell you anything about the cohomology of your underlying base space. For once, the characteristic cohomology classes in general don't generate the cohomology groups. Moreover, if a characteristic cohomology class has torsion in the ZZ-cohomology, its corresponding class in the de Rham cohomology vanishes. So you lose a lot of information on the way.
    99
    1010> > > Perhaps as a followup ticket, a thematic tutorial on how to compute with the cohomology classes of the manifold might be good (using the module-level doc as a starting point).