Author: geodude Format: TextI have no idea whether any of the things below could be done. But they would definitely settle the question about the legitimacy and importance of the nPOV.
1) Define a sort of complex for higher homotopy groupoids. Define the same thing for "forms" (for 1-forms: connections/holonomy functors are already there).
2) Prove that the two complexes above, defined in a certain way, are dual (for holonomy functors and 1-homotopy it's already true). "Cohomotopy".
3) Prove that the reduction to a point (monoid) of the complexes above is abelian and isomorphic to the homotopy groups, and that the central extension of whatever you find is homology.
4) "Fundamental theorem of topology": Let M and N be locally diffeomorphic. Then they are globally diffeomorphic IF AND ONLY IF their higher homotopy complexes (see above) are isomorphic.
The 4) is probably the dream of many. I think that a sufficient and necessary condition for diffeomorphisms, homology and homotopy is not there simply because the classic higher homotopy groups are "not very nice".
What are your "dream-results"?
I have no idea whether any of the things below could be done. But they would definitely settle the question about the legitimacy and importance of the nPOV.
1) Define a sort of complex for higher homotopy groupoids. Define the same thing for "forms" (for 1-forms: connections/holonomy functors are already there). 2) Prove that the two complexes above, defined in a certain way, are dual (for holonomy functors and 1-homotopy it's already true). "Cohomotopy". 3) Prove that the reduction to a point (monoid) of the complexes above is abelian and isomorphic to the homotopy groups, and that the central extension of whatever you find is homology. 4) "Fundamental theorem of topology": Let M and N be locally diffeomorphic. Then they are globally diffeomorphic IF AND ONLY IF their higher homotopy complexes (see above) are isomorphic.
The 4) is probably the dream of many. I think that a sufficient and necessary condition for diffeomorphisms, homology and homotopy is not there simply because the classic higher homotopy groups are "not very nice".