Timeline for Geometric sets determined by chains (for integration and Stokes' theorem)
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Dec 28, 2021 at 21:30 | comment | added | Bence Racskó | @JHM The emphasis is on subsets . If a given subset is represented by two different smooth chains, all forms should have the same integral on them. | |
| Dec 28, 2021 at 21:29 | comment | added | Bence Racskó | @JHM I have just skimmed through VBC, and no, I don't see anything in that paper that would help me. Or maybe there is, but I don't undertand it, as I have said, I don't know much alg top. What I am looking for is extremely basic. If anything like "fundamental classes" are involved, it's overcomplication. I am not interested in doing any topology or cohomology theory or anything of the sort here. Literally all I want is to know what conditions to put on smooth $k$-chains such that they fit together nicely into subsets on which I can integrate differential forms. | |
| Dec 28, 2021 at 20:40 | comment | added | JHM | @Bence Racsko: Doesn't the topological singular chain theory, like defined by Eilenberg-Greenberg, and reviewed in Gromov's "VBC", have everything you need? | |
| Dec 28, 2021 at 20:19 | comment | added | Bence Racskó | @JHM I'll check out the references, but I specifically want to avoid geometric measure theory. What I want is actually very simple. I want to use "semi-regular" chain elements (i.e. those which are injective immersions on the interior) as basic building blocks to construct sets on one can integrate. To do that I need some conditions on chains, since a general chain has multiplicities, can be too degenerate, can intersect itself, can be stacked in a way that integrals on interior boundaries amplify rather than cancel etc. I am looking for the most optimal conditions to avoid these issues. | |
| Dec 28, 2021 at 19:46 | comment | added | JHM | Sounds like you're talking about Federer's geometric integration, and the problem of representing currents via subsets. I remember finding this paper pretty interesting in this direction : Sullivan, Dennis. "Cycles for the dynamical study of foliated manifolds and complex manifolds." Inventiones mathematicae 36.1 (1976): 225-255. You might also consider Gromov's "Volume and Bounded Cohomology" and integrating cochains over "averaged chains". | |
| Dec 28, 2021 at 16:25 | history | edited | Bence Racskó | CC BY-SA 4.0 |
edited title
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| Dec 28, 2021 at 16:07 | comment | added | Bence Racskó | @BenMcKay I know that, but this is not what the question is about. My question is about which chains can be thought of as "geometric"; corresponding to subsets of manifolds in a faithful manner. | |
| Dec 28, 2021 at 16:06 | comment | added | Ben McKay | The unit cube is an oriented manifold with corners. Define integration on oriented manifolds with corners, by partition of unity. Then pull back forms to your unit cube and integrate. I think the rest is an exercise. | |
| Dec 28, 2021 at 15:50 | history | edited | Bence Racskó | CC BY-SA 4.0 |
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| Dec 28, 2021 at 15:41 | history | asked | Bence Racskó | CC BY-SA 4.0 |