Chapter 9 Partitions of Unity, Covering Maps ~

Home , Covering space, Local diffeomorphism, Partition of unity

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 gluing 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

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.

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 V J,wesaythatafamily U↵ ↵ I is { } 2 { } 2 a reﬁnement of V 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 Dieudonné)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 partition 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, every open cover has a countable, locally finite refinement 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, locally compact (Hausdor↵) topological space is the union of countably many compact subsets. Thus, X is countable 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 manifold (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,: M !M ,sothat 2 2 ! 1 ! 2 ⇡ = ⇡ , 1 2 that is, the following diagram commutes:

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 ! : M1 M2,betweenthetwocoveringsand 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 ⇡ ((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. ! 520 CHAPTER 9. PARTITIONS OF UNITY, COVERING MAPS ~ It is also useful to note that a covering map, ⇡ : M N, is a local di↵eomorphism (which means that ! d⇡p : TpM T⇡(p)N is a bijective linear map for every p M). ! 2 The crucial property of covering manifolds is that curves in N can be lifted to M,inauniqueway.

Deﬁnition 9.3. Let ⇡ : M N be a covering map, and let P be a Hausdor↵topological! space. For any map : P N,alift of through ⇡ is a map : P M so that! ! e = ⇡ , as in the following commutativee diagram.

M = ⇡ ✏ / P e N 9.2. COVERING MAPS AND UNIVERSAL COVERING MANIFOLDS 521 We would like to state three propositions regarding covering spaces.

However, two of these propositions use the notion of a simply connected manifold.

Intuitively, a manifold is simply connected if it has no “holes.”

More precisely, a manifold is simply connected if it has a trivial fundamental group.

Afundamentalgroupisahomotopicloopgroup.

Therefore, given topological spaces X and Y ,weneed to deﬁne a homotopy between two continuous functions f : X Y and g : X Y . ! ! 522 CHAPTER 9. PARTITIONS OF UNITY, COVERING MAPS ~ Deﬁnition 9.4. Let X and Y be topological spaces, f : X Y and g : X Y be two continuous functions, and let!I =[0, 1]. We! say that f is homotopic to g if there exists a continuous function F : X I Y (where X I is given the product topology)⇥ such! that F (x, 0) = ⇥f(x)andF (x, 1) = g(x)forallx X.The map F is a homotopy from f to g,andthisisdenoted2 f F g.Iff and g agree on A X,i.e. f(a)=g(a) whenever⇠ a A,wesayf is homotopic✓ to g relative A,andthisisdenoted2 f g rel A. ⇠F

Ahomotopyprovidesameansofcontinuouslydeforming f into g through a family f of continuous functions { t} ft : X Y where t [0, 1] and f0(x)=f(x)and f (x)=!g(x)forallx 2X. 1 2 9.2. COVERING MAPS AND UNIVERSAL COVERING MANIFOLDS 523

For example, let D be the unit disk in R2 and consider two continuous functions f : I D and g : I D. ! !

Then f F g via the straight line homotopy F : I I D,where⇠F (x, t)=(1 t)f(x)+tg(x). ⇥ !

Proposition 9.7. Let X and Y be topological spaces and let A X. Homotopy (or homotopy rel A) is an equivalence✓ relation on the set of all continuous functions from X to Y .

The next two propositions show that homotopy behaves well with respect to composition.

Proposition 9.8. Let X, Y , and Z be topological spaces and let A X. For any continuous functions f : X Y , g : X✓ Y , and h: Y Z, if f g rel ! ! ! ⇠F A, then h f h F h g rel A as maps from X to Z. ⇠ f / h X Y / Z. g / 524 CHAPTER 9. PARTITIONS OF UNITY, COVERING MAPS ~ Proposition 9.9. Let X, Y , and Z be topological spaces and let B Y . For any continuous functions f : X Y ,✓g : Y Z, and h: Y Z, if ! ! 1 ! g G h rel B, then g f F h f rel f B, where F (⇠x, t)=G(f(x),t). ⇠

g f / / X Y / Z. h

In order to deﬁne the fundamental group of a topological space X,werecallthedeﬁnitionofaloop.

Deﬁnition 9.5. Let X be a topological space, p be a point in X,andletI =[0, 1]. We say ↵ is a loop based at p = ↵(0) if ↵ is a continuous map ↵: I X with ↵(0) = ↵(1). ! 9.2. COVERING MAPS AND UNIVERSAL COVERING MANIFOLDS 525 Given a topological space X,chooseapointp X and form S,thesetofallloopsinX based at p. 2

By applying Proposition 9.7, we know that the relation of homotopy relative to 0, 1 is an equivalence relation on S.Thisleadstothefollowingdeﬁnition.{ }

Deﬁnition 9.6. Let X be a topological space, p be a point in X,andlet↵ be a loop in X based at p.The set of all loops homotopic to ↵ relative to 0, 1 is the homotopy class of ↵ and is denoted ↵ . { } h i

Deﬁnition 9.7. Given two loops ↵ and in a topological space X based at p,theproduct ↵ is a loop in X based at p deﬁned by · ↵(2t)0t 1 ↵ (t)=   2 · (2t 1) 1

We leave it the reader to check that the multiplication of homotopy classes is well deﬁned and associative, namely ↵ = ↵ whenever ↵, ,and are loops inh X· ibased·h i athp.i·h · i

Let e be the homotopy class of the constant loop in h i 1 X based at p,anddeﬁnetheinverseof ↵ as ↵ = 1 1 h i h i ↵ ,where↵ (t)=↵(1 t). h i With these conventions, the product operation between homotopy classes gives rise to a group. In particular, 9.2. COVERING MAPS AND UNIVERSAL COVERING MANIFOLDS 527 Proposition 9.10. Let X be a topological space and let p be a point in X. The set of homotopy classes of loops in X based at p is a group with multiplication given by ↵ = ↵ h i·h i h · i

Deﬁnition 9.8. Let X be a topological space and p a point in X.ThegroupofhomotopyclassesofloopsinX based at p is the fundamental group of X based at p, and is denoted by ⇡1(X, p).

If we assume X is path connected, we can show that ⇡1(X, p) ⇠= ⇡1(X, q)foranypointsp and q in X.There- fore, when X is path connected, we simply write ⇡1(X).

Deﬁnition 9.9. If X is path connected topological space and ⇡1(X)= e ,(whichisalsodenotedas⇡1(X)=(0)), we say X is simplyh i connected.

In other words, every loop in X can be shrunk in a continuous manner within X to its basepoint. 528 CHAPTER 9. PARTITIONS OF UNITY, COVERING MAPS ~ Examples of simply connected spaces include Rn and Sn whenever n 2. On the other hand, the torus and the circle are not simply connected. See Figures 9.4 and 9.5.

p q

Figure 9.4: The torus is not simply connected. The loop at p is homotopic to a point, but the loop at q is not. 9.2. COVERING MAPS AND UNIVERSAL COVERING MANIFOLDS 529

p p F

α p α

Figure 9.5: The unit sphere S2 is simply connected since every loop can be continuously deformed to a point. This deformation is represented by the map F : I I S2 where F (x, 0) = ↵ and F (x, 1) = p. ⇥ !

We now state without proof the following results: 530 CHAPTER 9. PARTITIONS OF UNITY, COVERING MAPS ~ Proposition 9.11. If ⇡ : M N is a covering map, then for every smooth curve,!↵: I N, in N (with 0 I) and for any point, q M,! such that ⇡(q)= ↵(0)2 , there is a unique smooth2 curve, ↵: I M, lifting ↵ through ⇡ such that ↵(0) = q. See! Figure 9.6. e e M

~ q = α (0)

~ α

0 π (q) = α(0) I α

Figure 9.6: The lift of a curve ↵ when ⇡ : R S1 is ⇡(t)=(cos(2⇡t), sin(2⇡t)). ! 9.2. COVERING MAPS AND UNIVERSAL COVERING MANIFOLDS 531 Proposition 9.12. Let ⇡ : M N be a covering map and let : P N be a smooth! map. For any p P , ! 0 2 any q0 M and any r0 N with ⇡(q0)=(p0)=r0, the following2 properties hold:2 (1) If P is connected then there is at most one lift, : P M, of through ⇡ such that (p )=q . ! 0 0 (2) If P is simply connected, then such a lift exists. e e

M q0 7 3⇡ ✏ p P e / N r 0 2 3 0 532 CHAPTER 9. PARTITIONS OF UNITY, COVERING MAPS ~ Theorem 9.13. Every connected manifold, M, pos- sesses a simply connected covering map, ⇡ : M M, that is, with M simply connected. Any two simply! connected coverings of N are equivalent. f f In view of Theorem 9.13, it is legitimate to speak of the simply connected cover, M,ofM,alsocalleduniversal covering (or cover)ofM. f Given any point, p M,let⇡1(M,p)denotethefunda- mental group of M 2with basepoint p.

If : M N is a smooth map, for any p M,ifwe write q =!(p), then we have an induced group2 homomorphism

: ⇡1(M,p) ⇡1(N,q). ⇤ ! 9.2. COVERING MAPS AND UNIVERSAL COVERING MANIFOLDS 533 Proposition 9.14. If ⇡ : M N is a covering map, for every p M, if q = ⇡(p),! then the induced homo- 2 morphism, ⇡ : ⇡1(M,p) ⇡1(N,q), is injective. ⇤ !

Proposition 9.15. Let ⇡ : M N be a covering map ! and let : P N be a smooth map. For any p0 P , any q M and! any r N with ⇡(q )=(p )=2 r , 0 2 0 2 0 0 0 if P is connected, then a lift, : P M, of such ! that (p0)=q0 exists i↵ e e (⇡1(P, p0)) ⇡ (⇡1(M,q0)), ⇤ ✓ ⇤ as illustrated in the diagram below

=M ⇡1(M,q0) 6 ⇡ ⇡ ✏ ✏ ⇤ / i↵ P e N / ⇡1(P, p0) ⇡1(N,r0) ⇤

Basic Assumption:Foranycovering,⇡ : M N,if N is connected then we also assume that M is connected.! 534 CHAPTER 9. PARTITIONS OF UNITY, COVERING MAPS ~ Using Proposition 9.14, we get

Proposition 9.16. If ⇡ : M N is a covering map and N is simply connected, then! ⇡ is a di↵eomorphism (recall that M is connected); thus, M is di↵eomorphic to the universal cover, N, of N.

The following propositione shows that the universal covering of a space covers every other covering of that space. This justiﬁes the terminology “universal covering.”

Proposition 9.17. Say ⇡1 : M1 N and ⇡ : M N are two coverings! of N, with N con- 2 2 ! nected. Every homomorphism, : M1 M2, between these two coverings is a covering map.! As a conse- quence, if ⇡ : N N is a universal covering of N, ! then for every covering, ⇡0 : M N, of N, there is a ! covering, : Ne M, of M. ! e 9.2. COVERING MAPS AND UNIVERSAL COVERING MANIFOLDS 535 The notion of deck-transformation group of a covering is also useful because it yields a way to compute the fundamental group of the base space.

Deﬁnition 9.10. If ⇡ : M N is a covering map, a deck-transformation is any di!↵eomorphism, : M M,suchthat⇡ = ⇡ ,thatis,thefollowing diagram! commutes:

M / M . ⇡ ⇡ ! N }

Note that deck-transformations are just automorphisms of the covering map.

The commutative diagram of Definition 9.10 means that adecktransformationpermuteseveryfibre.Itisimme- diately verified that the set of deck transformations of a covering map is a group denoted ⇡ (or simply, ), called the deck-transformation group of the covering. 536 CHAPTER 9. PARTITIONS OF UNITY, COVERING MAPS ~ Observe that any deck transformation, ,isaliftof⇡ through ⇡.Consequently,ifM is connected, by Proposi- tion 9.12 (1), every deck-transformation is determined by its value at a single point.

So, the deck-transformations are determined by their ac- 1 tion on each point of any fixed fibre, ⇡ (q), with q N. 2 1 Since the fibre ⇡ (q)iscountable,isalsocountable, that is, a discrete Lie group.

1 Moreover, if M is compact, as each fibre, ⇡ (q), is compact and discrete, it must be finite and so, the deck- transformation group is also finite.

The following proposition gives a useful method for de- termining the fundamental group of a manifold.

Proposition 9.18. If ⇡ : M M is the universal covering of a connected manifold,! M, then the deck- transformation group, , isf isomorphic to the fundamental group, ⇡1(M), of M. e 9.2. COVERING MAPS AND UNIVERSAL COVERING MANIFOLDS 537

Remark: When ⇡ : M M is the universal covering of M,itcanbeshownthatthegroup! actssimplyand 1 transitively on every fibre,f ⇡ (q). e 1 This means that for any two elements, x, y ⇡ (q), 2 there is a unique deck-transformation, suchthat (x)=y. 2 e So, there is a bijection between ⇡1(M) ⇠= andthefibre 1 ⇡ (q). e Proposition 9.13 together with previous observations implies that if the universal cover of a connected (compact) manifold is compact, then M has a finite fundamental group.

We will use this fact later, in particular, in the proof of Myers’ Theorem. 538 CHAPTER 9. PARTITIONS OF UNITY, COVERING MAPS ~