this is a rather stupid question regarding preliminaries for the definition of boundaries(adsbygoogle = window.adsbygoogle || []).push({});

the question is whether every closed n-1 dim. closed submanifold [itex]C[/itex] of an arbitrary n-dim. manifold defines a volume [itex]V[/itex]; i.e. whether [itex]\partial V = C[/itex] can be turned around such that V is defined as the "interior" of [itex]C[/itex] (instead of defining the boundary [itex]\partial V[/itex] in terms of V)

my observation was that this seems to fail for the torus [itex]T^2[/itex] and a closed loop [itex]\gamma[/itex] on [itex]T^2[/itex] if [itex]\gamma[/itex] has a non-trivial winding number, i.e. if the winding number [itex]w(\gamma) \neq (0,0)[/itex]; a loop with [itex](1,0)[/itex] does not define an "interior" on the torus (the interior is the whole torus) and therefore seems not to be the boundary of a volume (the torus itself has no boundary);

so my question is which conditions for a closed manifold must hold such that it can be used as a boundary of a volume in the sense of integration of differential firms and stokes theorem

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# Definition of boundary, Stokes' theorem

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