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Chapter 9 Partitions of Unity, Covering Maps ~ Chapter 9 Partitions of Unity, Covering Maps ~ 9.1 Partitions of Unity To study manifolds, it is often necessary to construct var- ious objects such as functions, vector fields, Riemannian metrics, volume forms, etc., by gluing together items con- structed on the domains of charts. Partitions of unity are a crucial technical tool in this glu- ing process. 505 506 CHAPTER 9. PARTITIONS OF UNITY, COVERING MAPS ~ The first step is to define “bump functions”(alsocalled plateau functions). For any r>0, we denote by B(r) the open ball n 2 2 B(r)= (x1,...,xn) R x + + x <r , { 2 | 1 ··· n } n 2 2 and by B(r)= (x1,...,xn) R x1 + + xn r , its closure. { 2 | ··· } Given a topological space, X,foranyfunction, f : X R,thesupport of f,denotedsuppf,isthe closed set! supp f = x X f(x) =0 . { 2 | 6 } 9.1. PARTITIONS OF UNITY 507 Proposition 9.1. There is a smooth function, b: Rn R, so that ! 1 if x B(1) b(x)= 2 0 if x Rn B(2). ⇢ 2 − See Figures 9.1 and 9.2. 1 0.8 0.6 0.4 0.2 K3 K2 K1 0 1 2 3 Figure 9.1: The graph of b: R R used in Proposition 9.1. ! 508 CHAPTER 9. PARTITIONS OF UNITY, COVERING MAPS ~ > Figure 9.2: The graph of b: R2 R used in Proposition 9.1. ! Proposition 9.1 yields the following useful technical result: 9.1. PARTITIONS OF UNITY 509 Proposition 9.2. Let M be a smooth manifold. For any open subset, U M, any p U and any smooth ✓ 2 function, f : U R, there exist an open subset, V , ! with p V and a smooth function, f : M R, de- fined on2 the whole of M, so that V is compact,! e V U, supp f U ✓ ✓ and e f(q)=f(q), for all q V. 2 e 510 CHAPTER 9. PARTITIONS OF UNITY, COVERING MAPS ~ If X is a (Hausdor↵) topological space, a family, U↵ ↵ I, { } 2 of subsets U↵ of X is a cover (or covering)ofX i↵ X = ↵ I U↵. 2 S Acover, U↵ ↵ I,suchthateachU↵ is open is an open cover. { } 2 If U↵ ↵ I is a cover of X,foranysubset,J I,the { } 2 ✓ subfamily U↵ ↵ J is a subcover of U↵ ↵ I if X = { } 2 { } 2 ↵ J U↵,i.e., U↵ ↵ J is still a cover of X. 2 { } 2 S Given a cover Vβ β J,wesaythatafamily U↵ ↵ I is { } 2 { } 2 a refinement of Vβ β J if it is a cover and if there is a function, h: I {J,sothat} 2 U V ,forall↵ I. ! ↵ ✓ h(↵) 2 Afamily U↵ ↵ I of subsets of X is locally finite i↵for every point,{ p} 2 X,thereissomeopensubset,U,with p U,sothat2U U = for only finitely many ↵ I. 2 \ ↵ 6 ; 2 9.1. PARTITIONS OF UNITY 511 Aspace,X,isparacompact i↵every open cover has an open locally finite refinement. Remark: Recall that a space, X,iscompact i↵it is Hausdor↵and if every open cover has a finite subcover. Thus, the notion of paracompactness (due to Jean Dieudonn´e)is a generalization of the notion of compact- ness. Recall that a topological space, X,issecond-countable if it has a countable basis, i.e., if there is a countable family of open subsets, Ui i 1,sothateveryopensubsetofX { } ≥ is the union of some of the Ui’s. Atopologicalspace,X,iflocally compact i↵it is Haus- dor↵and for every a X,thereissomecompactsubset, K,andsomeopensubset,2 U,witha U and U K. 2 ✓ As we will see shortly, every locally compact and second- countable topological space is paracompact. 512 CHAPTER 9. PARTITIONS OF UNITY, COVERING MAPS ~ It is important to observe that every manifold (even not second-countable) is locally compact. Definition 9.1. Let M be a (smooth) manifold. A par- tition of unity on M is a family, fi i I,ofsmooth functions on M (the index set I may{ be} 2 uncountable) such that (a) The family of supports, supp fi i I,islocallyfinite. { } 2 (b) For all i I and all p M,wehave0 fi(p) 1, and 2 2 f (p)=1, for every p M. i 2 i I X2 Note that condition (b) implies that supp fi i I is a { } 2 cover of M.If U↵ ↵ J is a cover of M,wesaythat { } 2 the partition of unity fi i I is subordinate to the cover { } 2 U↵ ↵ J if supp fi i I is a refinement of U↵ ↵ J. { } 2 { } 2 { } 2 When I = J and supp fi Ui,wesaythat fi i I is ✓ { } 2 subordinate to U↵ ↵ I with the same index set as the partition of unity{ . } 2 9.1. PARTITIONS OF UNITY 513 In Definition 9.1, by (a), for every p M,thereissome open set, U,withp U and U meets2 only finitely many 2 of the supports, supp fi. So, f (p) =0foronlyfinitelymanyi I and the infinite i 6 2 sum i I fi(p)iswelldefined. 2 P Proposition 9.3. Let X be a topological space which is second-countable and locally compact (thus, also Hausdor↵). Then, X is paracompact. Moreover, ev- ery open cover has a countable, locally finite refine- ment consisting of open sets with compact closures. 514 CHAPTER 9. PARTITIONS OF UNITY, COVERING MAPS ~ Remarks: 1. Proposition 9.3 implies that a second-countable, lo- cally compact (Hausdor↵) topological space is the union of countably many compact subsets. Thus, X is count- able at infinity,anotionthatwealreadyencountered in Proposition 5.11 and Theorem 5.14. 2. Amanifoldthatiscountableatinfinityhasacount- able open cover by domains of charts. It follows that M is second-countable. Thus, for manifolds, second- countable is equivalent to countable at infinity. Recall that we are assuming that our manifolds are Haus- dor↵and second-countable. 9.1. PARTITIONS OF UNITY 515 Theorem 9.4. Let M be a smooth manifold and let U↵ ↵ I be an open cover for M. Then, there is a { } 2 countable partition of unity, fi i 1, subordinate to { } ≥ the cover U↵ ↵ I and the support, supp fi, of each fi is compact.{ } 2 If one does not require compact supports, then there is a partition of unity, f↵ ↵ I, subordinate to the cover { } 2 U↵ ↵ I with at most countably many of the f↵ not identically{ } 2 zero. (In the second case, supp f U .) ↵ ✓ ↵ We close this section by stating a famous theorem of Whitney whose proof uses partitions of unity. Theorem 9.5. (Whitney, 1935) Any smooth mani- fold (Hausdor↵and second-countable), M, of dimen- sion n is di↵eomorphic to a closed submanifold of R2n+1. For a proof, see Hirsch [26], Chapter 2, Section 2, Theo- rem 2.14. 516 CHAPTER 9. PARTITIONS OF UNITY, COVERING MAPS ~ 9.2 Covering Maps and Universal Covering Manifolds Covering maps are an important technical tool in alge- braic topology and more generally in geometry. We begin with covering maps. Definition 9.2. Amap,⇡ : M N,betweentwo smooth manifolds is a covering map!(or cover)i↵ (1) The map ⇡ is smooth and surjective. (2) For any q N,thereissomeopensubset,V N, so that q 2V and ✓ 2 1 ⇡− (V )= Ui, i I [2 where the U are pairwise disjoint open subsets, U i i ✓ M,and⇡ : Ui V is a di↵eomorphism for every i I.Wesaythat! V is evenly covered. 2 The manifold, M,iscalledacovering manifold of N. See Figure 9.3. 9.2. COVERING MAPS AND UNIVERSAL COVERING MANIFOLDS 517 U1 U π -1 (q) i 3 Ui 2 π -1(q) U2 U i π π q V q V Figure 9.3: Two examples of a covering map. The left illustration is ⇡ : R S1 with ⇡(t)=(cos(2⇡t), sin(2⇡t)), while the right illustration is the 2-fold antipodal! covering of 2 RP by S2. It is useful to note that a covering map ⇡ : M N is ! alocaldi↵eomorphism(whichmeansthatd⇡p : TpM T N is a bijective linear map for every p M) ! ⇡(p) 2 518 CHAPTER 9. PARTITIONS OF UNITY, COVERING MAPS ~ A homomorphism of coverings, ⇡1 : M1 N and ⇡ : M N,isasmoothmap,φ: M !M ,sothat 2 2 ! 1 ! 2 ⇡ = ⇡ φ, 1 2 ◦ that is, the following diagram commutes: φ M1 / M2 . ⇡1 ⇡2 ! N } We say that the coverings ⇡1 : M1 N and ⇡ : M N are equivalent i↵there! is a homomorphism, 2 2 ! φ: M1 M2,betweenthetwocoveringsandφ is a di↵eomorphism.! 1 As usual, the inverse image, ⇡− (q), of any element q N is called the fibre over q,thespaceN is called the 2base and M is called the covering space. 9.2. COVERING MAPS AND UNIVERSAL COVERING MANIFOLDS 519 As ⇡ is a covering map, each fibre is a discrete space. 1 Note that a homomorphism maps each fibre ⇡1− (q)inM1 1 to the fibre ⇡− (φ(q)) in M ,foreveryq M . 2 2 2 1 Proposition 9.6. Let ⇡ : M N be a covering map. ! 1 If N is connected, then all fibres, ⇡− (q), have the 1 same cardinality for all q N. Furthermore, if ⇡− (q) is not finite then it is countably2 infinite. When the common cardinality of fibres is finite it is called the multiplicity of the covering (or the number of sheets). For any integer, n>0, the map, z zn,fromtheunit circle S1 = U(1) to itself is a covering7! with n sheets. The map, t: (cos(2⇡t), sin(2⇡t)), 7! is a covering, R S1,withinfinitelymanysheets.
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