Diffeological vector pseudo-bundles

Abstract

We consider a diffeological counterpart of the notion of a vector bundle (we call this counterpart a pseudo-bundle, although in the other works it is called differently; among the existing terms there are a "regular vector bundle" of Vincent and "diffeological vector space over X" of Christensen-Wu). The main difference of the diffeological version is that (for reasons stemming from the independent appearance of this concept elsewhere), diffeological vector pseudo-bundles may easily not be locally trivial (and we provide various examples of such, including those where the underlying topological bundle is even trivial). Since this precludes using local trivializations to carry out many typical constructions done with vector bundles (but not the existence of constructions themselves), we consider the notion of diffeological gluing of pseudo-bundles, which, albeit with various limitations that we indicate, provides when applicable a substitute for said local trivializations. We quickly discuss the interactions between the operation of gluing and typical operations on vector bundles (direct sum, tensor product, taking duals) and then consider the notion of a pseudo-metric on a diffeological vector pseudo-bundle. MSC (2010): 53C15 (primary), 57R35 (secondary).

Pseudo-bundles of Exterior Algebras as Diffeological Clifford Modules

“…On the level of the underlying topological 8 spaces, diffeological gluing (introduced in [10]) is just the usual topological gluing. The result is endowed with a canonical diffeology, called the gluing diffeology.…”

“…The gluing of these pseudo-bundles consists in the already-defined operations of gluing V 1 to V 2 alongf , gluing X 1 to X 2 along f , and π 1 to π 2 along (f , f ). It is easy to check (see [10]) that the resulting map π 1 ∪ (f ,f ) π 2 : V 1 ∪f V 2 → X 1 ∪ f X 2 is a diffeological vector pseudo-bundle for the gluing diffeologies on V 1 ∪f V 2 and X 1 ∪ f X 2 ; in particular, the vector space structure on its fibres is inherited from either V 1 or V 2 (more precisely, it is inherited from V 1 on fibres over the points in i 1 (X 1 \ Y ), and from V 2 on fibres over the points in i 2 (X 2 )).…”

“…Diffeological gluing of pseudo-bundles is relatively well-behaved with respect to the usual operations on vector bundles. More precisely, it commutes with the direct sum and the tensor product, while he situation is somewhat more complicated for the dual pseudo-bundles, see [10] (the facts needed are recalled below).…”

“…The basic object for us, though, is not just a diffeological space, but a diffeological vector pseudobundle ( [4], [17], [5], [10]). The difference with respect to the standard notion is not only in the fact that the smooth structure is replaced by a diffeological one (under some respects this would be a minor difference), but also in that it does not have to be locally trivial, although in many contexts we do add this assumption, see the discussion of pseudo-metrics, which replace the usual Riemannian metricsdiffeological pseudo-bundles frequently do not carry the latter.…”

“…For the basic operations on diffeological vector pseudo-bundles, the behavior under diffeological gluing was considered in [10] (see also [11] for some details); for tensor algebras and pairs of given Clifford modules, in [8]. What is lacking is a study of gluing of pseudo-bundles of the exterior algebras, in particular, the covariant version.…”