Fiber Bundles and Vector Bundles I
6.4 Sheaves,Germs and Jets
In this section we introduce some formalism that will not only provide some convenient language but also provides conceptual tools. The objects we intro-duce here form an whole area of mathematics and may be put to uses far more sophisticated than we do here. Some professional mathematicians may find it vaguely offensive that we introduce these concepts without an absolute neces-sity for some specific and central problems. However, the conceptual usefulness of the ideas go a long way toward helping us organize our thoughts on several issues. So we persist.
There is a interplay in geometry and topology between local and global data.
To see what the various meanings of the word local might be let consider a map z:C∞(M)→C∞(M) which is not necessarily linear. For anyf ∈C∞(M) we have a functionz(f) and its value (z(f))(p)∈Rat some pointp∈M. Let us consider in turn the following situations:
1. It just might be the case that wheneverf andg agree on some neighbor-hood ofpthen (z(f))(p) = (z(g))(p). So all that matters for determining (z(f))(p) is the behavior off in any arbitrarily small open set containing p. To describe this we say that (z(f))(p) only depends on the “germ” of f at p.
2. Certainly iff andgagree on some neighborhood ofpthen they have the same Taylor expansion at p. The reverse is not true however. Suppose that whenever two functionsf andghave Taylor series which agree up to and including terms of order |x|k then (z(f))(p) = (z(g))(p). Then we say that (z(f))(p) depends only on thek−jet off at p.
3. If (z(f))(p) = (z(g))(p) whenever df|p = dg|p we have a special case of the previous situation since df|p=dg|pexactly whenf andghave Taylor series which agree up to and including terms of order 1. So we are talking about the “1−jet”.
4. Finally, it might be the case that (z(f))(p) = (z(g))(p) exactly whenp.
Of course it is also possible that none of the above hold at any point. Notice as we go down the list we are saying that the information needed to deter-mine (z(f))(p) is becoming more and more local; even to the point of being infinitesimal.
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This gives us a family of sections; on for each open set U. To reverse the situation, suppose that we have a family of sectionsσU :U →EwhereU varies over the open sets (or just a cover ofM). When is it the case that such a family is just the family of restrictions of some (global) section σ: M →E? This is one of the basic questions of sheaf theory. .
Definition 6.14 A presheaf of abelian groups (resp. rings etc.) on a manifold (or more generally a topological space M is an assignment M(U) to each open set U ⊂M together with a family of abelian group (resp. ring etc.) homomorphismsrVU :M(U)→ M(V)for each nested pair V ⊂U of open sets and such that
Presheaf 1 rWV ◦rUV =rUW wheneverW ⊂V ⊂U.
Presheaf 2 rVV = idV for all openV ⊂M.
Definition 6.15 Let M be a presheaf and Ra presheaf of rings. If for each open U ⊂ M we have that M(U) is a module over the ring R(U) and if the multiplication mapR(U)×M(U)→ M(U)commutes with the restriction maps rUW then we say thatM is a presheaf of modules overR.
The best and most important example of a presheaf is the assignmentU 7→
Γ(E, U) for some vector bundle E →M and where by definition rVU(s) = s|V
for s∈Γ(E, U). In other words rVU is just the restriction map. Let us denote this presheaf bySE so thatSE(U) = Γ(E, U).
Definition 6.16 A presheaf homomorphism h : M1 → M2 is an is an as-signment to each open set U ⊂M an abelian group (resp. ring, module, etc.) morphismhU :M1(U)→ M2(U)such that wheneverV ⊂U then the following
Note we have used the same notation for the restriction maps of both presheaves.
Definition 6.17 We will call a presheaf a sheaf if the following properties hold wheneverU =S
Uα∈UUα for some collection of open setsU. Sheaf 1 Ifs1, s2∈ M(U)andrUUαs1=rUUαs2 for allUα∈ U thens1=s2. Sheaf 2 Ifsα∈ M(Uα)and whenever Uα∩Uβ6=∅ we have
rUUα∩Uβsα=rUUα∩Uβsβ
then there exists a s∈ M(U)such thatrUUαs=sα.
If we need to indicate the space M involved we will write MM instead of M.
It is easy to see that all of the following examples are sheaves. In each case the mapsrUUα are just the restriction maps.
Example 6.4 (Sheaf of smooth functions) CM∞(U) = C∞(U). Notice that CM∞ is a sheaf of modules over itself. Also CM∞(U) is sometimes denoted by FM(U).
Example 6.5 (Sheaf of holomorphic functions ) Sheaf theory really shows its strength in complex analysis. This example is one of the most studied. How-ever, we have not yet defined the notion of a complex manifold and so this example is for those readers with some exposure to complex manifolds. Let M be a complex manifold and let OM(U)be the algebra of holomorphic functions defined onU.Here too,OM is a sheaf of modules over itself. Where as the sheaf CM∞always has global sections, the same is not true forOM . The sheaf theoretic approach to the study of obstructions to the existence of global sections has been very successful.
Example 6.6 (Sheaf of continuous functions) C(U) =C(U)
Example 6.7 (Sheaf of smooth sections) SME(U) = Γ(U, E) for a vector bundle E→M. HereSME is a sheaf of modules overCM∞.
Remark 6.1 ( Notational convention !) Even though sheaf theory is a deep subject, one of our main purposes for introducing it is for notational (and con-ceptual) convenience. Each of the above presheaves is a presheaf of sections.
As we proceed into the theory of differentiable manifolds we will meet many in-stances where theorems stated for sections are for both globally defined sections and for sections over opens sets and many times the results will be natural in one way or another with respect to restrictions. For this reason we might simply writeSE∞,C∞ etc. whenever we are proving something that would work globally or locally in a natural way. In other words, when the reader seesSME orCM∞ he or she should think about both SE(U) andSE(M)or both C∞(U)andC∞(M) and so on.
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