General Geometry/Manifolds

Definition (manifold):

Let be a site, and let be a subcategory of . A manifold of type consists of the aforementioned site together with a class of isomorphisms of , where the 's are open in and the are objects of , such that

  1. for all there is a covering of such that for each , there exists an with and
  2. whenever st. and consitute a covering and and belong to the class , the maps and guaranteed by the universal property of the pullback are isomorphisms onto their respective images and is a morphism of .


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Proposition (fundamental induction lemma):

Let   be a logical statement, whose arguments are a topological manifold   and a closed subset  . Suppose that the following are true:

  1. Whenever   is compact and convex, where   is a coördinate chart on  , then   is true
  2. Whenever   and   are true, then   is true
  3. Whenever   is a descending chain of compact subsets of  ,   is true for all  , then   is true
  4. Whenever   is true for all relatively compact, open  , then   is true

Then for all closed  ,   is true.

Proof: First we prove by induction on   that for all sets of the type   for certain compact convex   ( ), the statement   is true. We proceed by induction on  . For  , the statement is implied by the first assumption. Suppose now that   is true. Note that also   is true by the first assumption. Also

 ,

and  , where   is compact and convex as the intersection of two compact and convex sets. Thus, since the sets   are only   many, by induction on  , we may also conclude that   holds. By 2., we conclude  .

Now, we prove that   is true whenever   is a compact subset of a set of the type  , where   is compact and convex. Indeed, for each  , cover   by all the cubes of sidelength   centered at the points of   that intersect it. Then set

 , so that  .

By the second assumption,   holds for each  , and hence by the third assumption   holds.

Now we claim by induction on   that whenever   are compact subsets of sets of the type   (  compact and convex), then   holds. For  , this follows from what we just proved. For the induction step, suppose that   holds. Note that also   holds by what we just proved. Then we have

 ,

and since   is a compact subset of  ,   holds by what we just proved, and therefore, by induction,   holds. Hence, by 2., we get that   is true.

Now we are ready to prove that   is true whenever   is compact. Indeed, let   be compact. Then cover   by sets  , where   are some charts and   is compact and convex. By compactness of  , we may extract a finite subcover  . By intersecting with   and retaining compactness (as the intersection of two compact sets is compact), we may assume that   are contained within   and in particular

 .

Thus, by the previous step,   holds.

Finally, let   be an arbitrary closed subset of  , and let   be a relatively compact open subset. Then   is true since   is compact as a closed subset of a compact set. Hence, by the fourth assumption,   is true.  


Definition (vector bundle):

Let   be a topological ring, and let   be a manifold. An  -dimensional  -vector bundle over   is a manifold   together with a morphism of manifolds   such that:

  1. for each  , the set   (which is called the fibre of  ), is a finite-dimensional topological vector space over  
  2. for each  , there exists a neighbourhood   of  , an   and a map   which is a fibre-wise TVS isomorphism such that the diagram
     
    commutes.