Manifolds, sheaves, and cohomology

Manifolds, sheaves, and cohomology Torsten Wedhorn These are the lecture notes of my 3rd year Bachelor lecture in the winter semester 2013/14 in Paderborn. This manuscript differs from the lecture: It does not contain any pictures, and the lecture is in German. Contents 0 Categories 2 (A) The language of categories.......................... 2 (B) Functors.................................... 4 (C) Limits and Colimits............................. 6 1 Sheaves 7 (A) Presheaves and sheaves........................... 8 (B) Stalks of sheaves............................... 10 (C) Sheaves attached to presheaves....................... 13 (D) Direct and inverse images of sheaves.................... 14 (E) Ringed spaces................................. 17 2 Manifolds 18 (A) Premanifolds................................. 18 (B) Topological properties............................ 20 (C) Topological groups.............................. 23 (D) Manifolds................................... 24 3 Local theory of manifolds: tangent spaces and submanifolds 28 (A) Tangent spaces................................ 28 (B) Morphisms of constant rank......................... 31 (C) Regular submanifolds............................ 33 4 Local-Global: Torsors and non abelian Čech cohomology 36 (A) Group actions................................. 36 (B) G-manifolds.................................. 38 (C) Torsors.................................... 40 (D) Non-abelian Čech cohomology........................ 42 (E) First term sequence of cohomology..................... 46 5 Vector bundles 49 (A) O X -modules.................................. 49 (B) Locally free modules and vector bundles.................. 52 (C) Tangent bundle................................ 56 (D) Digression: Exterior power.......................... 58 (E) Differential forms and de Rham complex.................. 60 1

6 Soft sheaves 63 (A) Definition and examples of soft sheaves................... 63 (B) Soft sheaves and partitions of sections................... 66 (C) Acyclicity of soft sheaves........................... 67 7 Cohomology of Sheaves 68 (A) Homotopy category of modules....................... 68 (B) Diagram chases................................ 70 (C) Injective and K-injective modules...................... 71 (D) Cohomology of complexes of sheaves.................... 73 (E) Examples of acyclic sheaves......................... 81 (F) Applications.................................. 83 8 Proper base change theorem 84 (A) Inverse image and cohomology....................... 84 (B) Direct image and cohomology........................ 85 (C) Proper base change.............................. 86 9 Cohomology of sheaves of locally constant functions 87 (A) Contractible spaces.............................. 87 (B) Singular cohomology............................. 89 (C) Cohomology and Singular Cohomology................... 92 (D) Homotopy invariance of cohomology.................... 93 (E) Applications.................................. 94 0 Categories (A) The language of categories Definition 0.1. A category (German: Kategorie) C consists of (a) a class 1 Ob(C) of objects, (b) for any two objects X and Y a set Hom C (X, Y ) = Hom(X, Y ) of morphisms from X to Y, (c) for every object X an element id X Hom(X, X), the identity of X, (d) for any three objects X, Y, Z a composition map Hom(X, Y ) Hom(Y, Z) Hom(X, Z), (f, g) g f such that (1) the composition of morphisms is associative (i.e., for all objects X, Y, Z, W and for all f Hom(X, Y ), g Hom(Y, Z), h Hom(Z, W ) one has h (g f) = (h g) f, (2) the elements id X are neutral elements with respect to composition (i.e., for all objects X and Y and morphisms f Hom(X, Y ) one has f id X = f = id Y f. A category is called small if Ob(C) is a set. 1 We refer to [Sch] 3.3 for the usage of a class. Here we ignore all set-theoretic problems. 2

Definition 0.2. A morphism f : X Y in a category is called an isomorphism if there exists a morphism g : Y X such that f g = id Y and g f = id X. We often write f : X Y to indicate that f is an isomorphism. We also write X = Y and say that X and Y are isomorphic if there exists an isomorphism X Y. Definition 0.3. A subcategory of a category C is a category C such that every object of C is an object of C and such that Hom C (X, Y ) Hom C (X, Y ) for any pair (X, Y ) of objects of C, compatibly with composition of morphisms and identity elements. The subcategory C is called full if Hom C (X, Y ) = Hom C (X, Y ) for all objects X and Y of C. Example 0.4. (1) Category of sets (Sets): Objects are sets, for two sets X and Y a morphism is a map X Y, composition in the category is the usual composition of maps, the identity of a set X is the usual identity map id X. An isomorphism in (Sets) is simply a bijective map. (2) (Grp) the catagory of groups: Objects are groups, morphisms are group homomorphisms, composition is the usual composition of group homomorphisms, the identity is the usual identity. An isomorphism in (Grp) is a group isomorphism. (3) (Ab) is the full subcategory of (Grp) of abelian groups: Objects are abelian groups, morphisms are group homomorphisms, composition is the usual composition of group homomorphisms, the identity is the usual identity. (4) (Ring) the category of rings: Objects are rings, morphisms are ring homomorphisms,... (5) (Top) the category of topological spaces: Objects are topological spaces, morphisms are continuous maps,... An isomorphism is a homeomorphism. (6) (Toppt) the category of pointed topological spaces: Objects are pairs (X, x), where X is a topological spaces and x X, morphisms f : (X, x) (Y, y) are continuous maps f : X Y such that f(x) = y,... (7) Let k be a field. (k-vec) the category of k-vector spaces: Objects are k-vector spaces, morphisms are k-linear maps,... An isomorphism is an isomorphism of k-vector spaces (tautology!). (8) Let A be a ring. (A-Mod) the category of left A-modules for a ring A: Objects are... (9) Let A be a commutative ring. (A-Alg) the category of commutative A-algebras: Objects are... Definition 0.5. For every category C the opposite category, denoted by C opp, is the category with the same objects as C and where for two objects X and Y of C opp we set Hom C opp(x, Y ) := Hom C (Y, X) with the obvious composition law. Definition 0.6. Let I be a set. (1) A relation on I is called partial preorder or simply preorder, if i i for all i I and i j, j k imply i k for all i, j, k I. (2) A preorder is called partial order or simply orderif i j and j i imply i = j for all i, j I. 3

(3) A preorder is called filtered if I and if for all i, j I there exists a k I with i k and j k. (4) A partial order is called total order if for all i, j I one has i j or j i. Example 0.7. (1) The real numbers together with the usual ordering form a totally ordered set. (2) For n, m N we write n m if there exists k N such that m = kn. Then is a filtered partial order on N. (3) Let X be a a topological space. Let Open(X) be the set of open subsets of X (i.e., the topology of X). Then is a filtered partial order on Open(X). Remark 0.8. Every preordered set I can be made into a category, again denoted by I, whose objects are the elements of I and for two elements i, j I the set of morphisms Hom I (i, j) consists of one element if i j and is empty otherwise. There is a unique way to define a composition law in this category. (B) Functors Definition 0.9. Given categories C and D, a (covariant) functor (German: (covarianter) Funktor) (resp. a contravariant functor (German: kontravarianter Funktor) F : C D is given by attaching to each object C of C an object F (C) of D, and to each morphism f : C C in C a morphism F (f): F (C) F (C ) (resp. a morphism F (f): F (C ) F (C)), compatible with composition of morphisms, i.e., F (g f) = F (g) F (f) (resp. F (g f) = F (f) F (g)) and identity elements, i.e., F (id C ) = id F (C). A contravariant functor C D is the same as a covariant functor C opp D. If F is a functor from C to D and G a functor from D E, we write G F : C E for the composition. Remark 0.10. Let F : C D be a covariant or a contravariant functor. Let C and C be objects in C and let f : C C be an isomorphism in C. Then F (f) is an isomorphism. Indeed, let us assume that F is covariant... Example 0.11. A simple example is the functor which forgets some structure. An example is the functor F : (Grp) (Sets) that attaches to every group the underlying set and that sends every group homomorphism to itself (but now considered as a map of sets), Example 0.12. Let X be a topological space, x X. Recall that π 1 (X, x) denotes the fundamental group of (X, x): Elements in π 1 (X, x) are homotopy classes of closed paths γ : [0, 1] X with γ(0) = γ(1) = x. Multiplication of paths is by concatenation (i.e. by first running through the first path and then through the second path, each time with double velocity). Whenever f : (X, x) (Y, y) is a morphism in (Toppt), then γ f γ defines a group homomorphism π 1 (f): π 1 (X, x) π 1 (Y, y). It is easy to check that we obtain a functor π 1 : (Toppt) (Grp). 4

In particular, if f : (X, x) (Y, y) is a homeomorphism with f(x) = y (i.e., an isomorphism in (Toppt), then π 1 (f) is an isomorphism of groups. Definition 0.13. For two functors F, G: C D we call a family of morphisms α(s): F (S) G(S) for every object S of C functorial in S if for every morphism f : T S in C the diagram F (T ) α(t ) G(T ) commutes. F (f) G(f) F (S) α(s) G(S) Example 0.14. Let A be a commutative ring. (1) For every A-module M let M := Hom A (M, A) be its dual module and for every A-linear map u: M N let u : N M, λ λ u be its dual homomorphism. We obtain a contravariant functor ( ) : (A-Mod) (A-Mod). (2) For every A-module M one has the biduality homomorphism of A-modules ι M : M (M ), m (λ λ(m)), m M, λ M. Moreover, if u: M N is a homomorphism of A-modules, then the diagram M u N ι M (M ) (u ) ι N (N ) is commutative. In other words, M ι M is a morphism of functors id (A-Mod) ( ). Example 0.15. Consider the functors π 1 : (Toppt) (Grp) and H 1 (, Z): (Toppt) (Grp), (X, x) H 1 (X, Z). Then there is a morphism of functors h: π 1 H 1 (, Z) such that for every object (X, x) in (Toppt) the group homomorphism h X,x : π 1 (X, x) H 1 (X, Z) is surjective. Definition 0.16. Let F : C D be a functor. (1) F is called faithful (resp. fully faithful) if for all objects X and Y of C the map Hom C (X, Y ) Hom D (F (X), F (Y )), f F (f) is injective (resp. bijective). (2) F is called essentially surjective if for every object Y of D there exists an object X of C and an isomorphism F (X) = Y. (3) F is called an equivalence of categories if it is fully faithful and essentially surjective. There are analogous notions for contravariant functors. A contravariant functor which is an equivalence of categories is sometimes also called an anti-equivalence of categories. Theorem and Definition 0.17. A functor F : C D is an equivalence of categories if and only if there exists a quasi-inverse functor G, i.e., a functor G: D C such that G F = id C and F G = id D. 5

Proof. [McL] Chap. IV, 4, Theorem 1. Definition 0.18. Let C and D be categories and let F : C D and G: D C be functors. Then G is said to be right adjoint to F and F is said to be left adjoint to G if for all objects X in C and Y in D there exists a bijection which is functorial in X and in Y. Hom C (X, G(Y )) = Hom D (F (X), Y ) Example 0.19. If F : C D is an equivalence of categories, then a quasi-inverse functor G: D C is right adjoint and left adjoint to F. Indeed for all objects X in C and Y in D we have bijections, functorial in X and in Y Hom C (X, G(Y )) Hom C (G(F (X)), G(Y )) Hom D (F (X), Y ). This shows that F is left adjoint to G. A similar argument shows that F is also right adjoint to G. Example 0.20. Let k be a field and let F : (Vec(k)) (Sets) be the forgetful functor. Then F is right adjoint to the functor G which sends a set I to the k-vector space k (I) = { (x i ) i I ; x i k, x i = 0 for all but finitely many i} and which sends a map a: I J to the unique k-linear map G(a): k (I) k (J) such that G(a)(e i ) = e a(i) where (e i ) i I (resp. (e j ) j J is the standard basis of k (I) (resp. k (J) ). Indeed for every set I and every k-vectorspace V we have a bijection (C) Hom (Vec(k)) (k (I), V ) Hom (Sets) (I, V ), f (i f(e i )). Limits and Colimits Definition 0.21. Let C be a category. (1) An inductive system in C (resp. a projective system in C) is a functor X : I C (resp. a functor X : I opp (Sets)), where I is a category 2. Then I is also called the index category. We often write X i instead of X(i) for an object in I. Hence for a morphism ϕ: i j in I we have a morphism X(ϕ): X i X j (resp. X(ϕ): X j X i ). (2) A limit of a projective system X : I opp C is an object lim X = I opp lim X i I opp i = lim in C together with morphisms p i : lim I opp X X i in C for all objects i I such that (a) for every morphism ϕ: i j in I one has p i = X(ϕ) p j, 2 To avoid set-theoretical difficulties one should in addition assume that I is equivalent to a small category. i I X i 6

(b) for every object Z in C and for all morphism q i : Z X i such that for all morphism ϕ: i j in I one has q i = X(ϕ) q j there exists a unique morphism q : Z lim I opp X such that q i = p i q for all i Ob(I). (3) A colimit of an inductive system X : I C is an object colim I X = colim i I X i = lim X i i I in C together with morphisms s i : X i lim I X in C for all objects i I such that (a) for every morphism ϕ: i j in I one has s i = s j X(ϕ), (b) for every object Z in C and for all morphism t i : X i Z such that for all morphism ϕ: i j in I one has t i = t j X(ϕ) there exists a unique morphism q : colim I X Z such that t i = t s i. Remark 0.22. Limits and colimits are (if they exist) unique up to unique isomorphism by the uniqueness requirement in the definition. Example 0.23. Let I be a filtered ordered set and let I be the attached category. Let X : I (Sets) be an inductive system. Then colim I X exists in (Sets) and we have colim X = ( X i )/, I i I where ( i I X i) is the disjoint union of the sets X i and where is the equivalence relation x i x j k I, ϕ: i k, ψ : j k : X(ϕ)(x i ) = X(ψ)(x j ) for x i X i and x j X j. Example 0.24. Let I be a set with the discrete order (i.e., i j i = j). Consider I as a category and let X : I C be a diagram. (1) Then i I X i := lim I X is called the product of X (if it exists). If C = (Sets) (resp. C = (Top)), then i I X i is the usual product of sets (resp. of topological spaces). (2) Then i I X i := colim I X is called the coproduct of X or the direct sum of X (if it exists). If C = (Sets) (resp. C = (Top)), then i I X i is the disjount union of the sets X i (resp. of the topological spaces X i ). 1 Sheaves Notation: Let X be a topological space. 7

(A) Presheaves and sheaves Definition 1.1. A presheaf F on X consists of the following data. (a) For every open set U of X a set F (U), (b) for each pair of open sets U V a map res V U : F (V ) F (U), called restriction map, such that the following conditions hold (1) res U U = id F (U) for every open set U X, (2) for U V W open sets of X, res W U = resv U resw V. Let F 1 and F 2 be presheaves on X. A morphism of presheaves ϕ: F 1 F 2 is a family of maps ϕ U : F 1 (U) F 2 (U) (for all U X open) such that for all pairs of open sets U V in X the following diagram commutes F 1 (V ) ϕ V F 2 (V ) res V U F 1 (U) ϕ U res V U F 2 (U). Composition of morphisms of presheaves is defined in the obvious way and we obtain the category (PSh(X)) of presheaves on X. If U V are open sets of X and s F (V ) we will often write s U instead of res V U (s). The elements of F (U) are called sections of F over U. Very often we will also write Γ(U, F ) instead of F (U). Remark 1.2. We can also describe presheaves as follows. Let (Ouv X ) be the category whose objects are the open sets of X and, for two open sets U, V X, Hom(U, V ) is empty if U V, and consists of the inclusion map U V if U V (composition of morphisms being the composition of the inclusion maps). Then a presheaf is the same as a contravariant functor F from the category (Ouv X ) to the category (Sets) of sets. Remark and Definition 1.3. By replacing (Sets) in this definition by some other category C (e.g. the category of abelian groups, the category of rings, or the category of R-algebras, R a fixed ring) we obtain the notion of a presheaf F with values in C (e.g. a presheaf of abelian groups, a presheaf of rings, a presheaf of R-algebras. This signifies that F (U) is an object in C for every open subset U of X and that the restriction maps are morphisms in C. A morphism F 1 F 2 of presheaves with values in C is then simply a morphism of functors. Example 1.4. Let X be a topological space. (1) Let Y be a topological space. For U X open set C X;Y (U) := { f : U Y ; f continuous}. For U V X let res V U : C X;Y (V ) C X;Y (U) be the usual restriction f f U. Then C X;Y is a presheaf of sets. If Y = K (K = R or K = C), then C X;K (U) is a commutative K-algebra und C X;K is a presheaf of K-algebras. 8

(2) Let V be a finite-dimensional R-vector space and let X be an open subspace of V. Let α N 0 { }. For U X open set C α X;R (U) := C α X(U) := { f : U R ; f is a C α -map}. Again let res V U : C X α(v ) C X α(u) be the usual restriction f f U for U V X. This makes CX α into a presheaf of R-algebras. Let α 1. For X = R and U R open let d U : CR α α 1 (U) CR (U) be the derivative f f. Then (d U ) U is a morphism of presheaves of R-vector spaces. (3) Let V be a finite-dimensional C-vector space and let X be an open subspace of V. For U X open set O X (U) := O hol X (U) := { f : U C ; f holomorphic}. Then O X (with the usual restriction maps) is a presheaf of C-algebras. (4) Let E be a set. For U X open set F (U) := {f : U E constant map} with the usual restriction. Then F is a presheaf of sets. If E is an abelian group (resp. a ring), then F is a presheaf of abelian groups (resp. of rings). (5) Let B X be the Borel-σ-algebra of X (i.e., the σ-algebra generated by the open subsets of X). For U X open set M X (U) := { f : U R ; f measurable}. Then M X (with the usual restriction maps) is a presheaf of sets. Definition 1.5. Let X be a topological space. The presheaf F on X is called a sheaf, if for all open sets U in X and every open covering U = i U i the following condition holds: (Sh) Given s i F (U i ) for all i such that s i Ui U j = s j Ui U j for all i, j. Then there exists a unique s F (U) such that s Ui = s i. A morphism of sheaves is a morphism of presheaves. We obtain the category of sheaves on the topological space X, which we denote by (Sh(X)). In the same way we can define the notion of a sheaf of abelian groups, a sheaf of rings, a sheaf of R-modules, or a sheaf of R-algebras. Remark 1.6. Let X be a topological space, F a sheaf on X, let U X be open and let U = i U i be an open covering. Let s, s F (U) such that s Ui = s U i for all i I. Then s = s. Remark 1.7. Note that if F is a sheaf on X, F ( ) is a set consisting of one element (use the covering of the empty set with empty index set). In particular, if X consists of one point, a sheaf F on X is already uniquely determined by F (X) and sometimes we identify F with F (X). Example 1.8. Notations of Example 1.4. The presheaves C X;Y, C α X, and O X are sheaves. The presheaf of measurable functions on X is in general not a sheaf. The presheaf of constant functions with values in some set is in general not a sheaf. 9

(B) Stalks of sheaves Let X be a topological space, F be a presheaf on X, considered as a functor F : Open(X) opp (Sets), and let x X be a point. Let U(x) := { U X open ; x U } be the set of open neighborhoods of x, ordered by inclusion. We consider U(x) as a full subcategory of Open(X). Be restricting F to U(x) we obtain a functor F : U(x) opp (Sets). Note that U(x) opp is filtered. Definition and Remark 1.9. (1) The colimit F x := colim U(x) F is called the stalk of F in x. More concretely (Example 0.23), one has F x = { (U, s) ; U open neighborhood of x, s F (U)}/, where two pairs (U 1, s 1 ) and (U 2, s 2 ) are equivalent, if there exists an open neighborhood V of x with V U 1 U 2 such that s 1 V = s 2 V. (2) For each open neighborhood U of x we have a canonical map (1.9.1) F (U) F x, s s x which sends s F (U) to the class of (U, s) in F x. We call s x the germ of s in x. (3) If ϕ: F G is a morphism of presheaves on X, we have an induced map ϕ x := colim ϕ U : F x G x U x of the stalks in x. We obtain a functor F F x from the category of presheaves on X to the category of sets. For every x X and every open neighborhood one has a commutative diagram F (U) s sx F x ϕ U ϕ x G (U) t tx G x. Remark 1.10. If F is a presheaf on X with values in C, where C is the category of abelian groups, of rings, or any category in which filtered inductive limits exist, then the stalk F x is an object in C and we obtain a functor F F x from the category of presheaves on X with values in C to the category C. Example 1.11. Let X = C and O C be the sheaf of holomorphic functions on X. Fix z 0 C. Then two holomorphic functions f 1 and f 2 defined in open neighborhoods U 1 10

and U 2, respectively, of z 0 agree on some open neighborhood V U 1 U 2 if and only if they have same Taylor expansion around z 0. Therefore O C,z0 = a n (z z 0 ) n power series with positive radius of convergence, n 0 and the identity theorem says precisely that for a connected open neighborhood U of z 0, the natural map O(U) O C,z0 is injective. Example 1.12. Let X be a topological space, let C X = C X;R be the sheaf of continuous R-valued functions on X, and let x X. Then C X,x = { (U, f) ; x U X open, f : U R continuous}/, where (U, f) (V, g) if there exists x W U V open such that f W = g W. In particular we see that for x V U open and f : U R continuous one has (U, f) (V, f V ). As C X is a sheaf of R-algebras, C X,x is an R-algebra. For instance, addition is defined as follows. Let s, t C X,x, represented by (U, f) and (V, g). Choose x W U V open and replace (U, f) by (W, f W ) and (V, g) by (W, g W ). Hence we may assume that U = V = W. Then define the sum s + t as the equivalence class of (W, f + g). Of course one can also define in the same way the multiplication of two elements of C X,x. If s C X,x is represented by (U, f), then s(x) := f(x) R is independent of the choice of the representative (U, f). We obtain a surjective R-algebra homomorphism ev x : C X,x R, s s(x). Let m x := ker(ev x ) = { s C X,x ; s(x) = 0 }, then m x is a maximal ideal (because C X,x /m x = R is a field). Let s CX,x \ m x be represented by (U, f). Then f(x) 0. By shrinking U we may assume that f(y) 0 for all y Y (because f is continuous). Hence 1/f exists and s is a unit in C X,x. This shows that C X,x is a local ring. A similar remark holds of course also for sheaves of differentiable or holomorphic functions. Proposition 1.13. Let X be a topological space, F and G presheaves on X, and let ϕ, ψ : F G be two morphisms of presheaves. (1) Assume that F is a sheaf. Then the induced maps on stalks ϕ x : F x G x are injective for all x X if and only of ϕ U : F (U) G (U) is injective for all open subsets U X. (2) If F and G are both sheaves, the maps ϕ x are bijective for all x X if and only if ϕ U is bijective for all open subsets U X. (3) If F and G are both sheaves, the morphisms ϕ and ψ are equal if and only if ϕ x = ψ x for all x X. 11

Proof. For U X open consider the map F (U) x U F x, s (s x ) x U. We claim that this map is injective if F is a sheaf. Indeed let s, t F (U) such that s x = t x for all x U. Then for all x U there exists an open neighborhood V x U of x such that s Vx = t Vx. Clearly, U = x U V x and therefore s = t by Remark 1.6. Using the commutative diagram F (U) ϕ U G (U) x U F x x ϕx x U G x, we see that (3) and the necessity of the condition in (1) are implied by the above claim. Moreover, a filtered colimit of injective maps is always injective again: Let s 0, t 0 F x such that ϕ x (s 0 ) = ϕ x (t 0 ). Let s 0 be represented by (s, U) and t 0 by (t, V ). By shrinking U and V we may assume U = V. As ϕ U (s) x = ϕ x (s 0 ) = ϕ x (t 0 ) = ϕ U (t) x, there exists an open neighborhood x W U such that ϕ W (s W ) = ϕ U (s) W = ϕ U (t) W = ϕ W (t W ). As ϕ W is injective, we find s W = t W and hence s 0 = s x = t x = t 0. Therefore the condition in (1) is also sufficient. Hence we are done if we show that the bijectivity of ϕ x for all x U implies the surjectivity of ϕ U. Let t G (U). For all x U we choose an open neighborhood U x of x in U and s x F (U x ) such that (ϕ U x(s x )) x = t x. Then there exists an open neighborhood V x U x of x with ϕ V x(s x V x) = t V x. Then (V x ) x U is an open covering of U and for x, y U ϕ V x V y(sx V x V y) = t V x V y = ϕ V x V y(sy V x V y). As we already know that ϕ V x V y is injective, this shows sx V x V y = sy V x V y and the sheaf condition (Sh) ensures that we find s F (U) such that s V x = s x for all x U. Clearly, we have ϕ U (s) x = t x for all x U and hence ϕ U (s) = t. Definition 1.14. We call a morphism ϕ: F G of sheaves injective (resp. surjective, resp. bijective) if ϕ x : F x G x is injective (resp. surjective, resp. bijective) for all x X. Remark 1.15. Let ϕ: F G be a morphism of sheaves. (1) Then ϕ is injective (resp. bijective) if and only if ϕ U : F (U) G (U) is injective for all open U X. 12

(2) ϕ is surjective if and only if for all open subsets U X and every t G (U) there exist an open covering U = i U i (depending on t) and sections s i F (U i ) such that ϕ Ui (s i ) = t Ui, i.e. locally we can find a preimage of t. But the surjectivity of ϕ does not imply that ϕ U : F (U) G (U) is surjective for all open sets U of X Example 1.16. Let O C be the sheaf of holomorphic functions on C. For every open subset U C and f O C (U) we let D U (f) := f be the derivative. We obtain a morphism D : O C O C of sheaves of C-vector spaces. Then D is surjective, because locally every holomorphic function has a primitive. But D U is only surjective if every connected component of U is simply connected. (C) Sheaves attached to presheaves Proposition 1.17. Let F be a presheaf on a topological space X. Then there exists a pair ( F, ι F ), where F is a sheaf on X and ι F : F F is a morphism of presheaves, such that the following holds: If G is a sheaf on X and ϕ: F G is a morphism of presheaves, then there exists a unique morphism of sheaves ϕ: F G with ϕ ιf = ϕ. The pair ( F, ι F ) is unique up to unique isomorphism. Moreover, the following properties hold: (1) For all x X the map on stalks ι F,x : F x F x is bijective. (2) For every presheaf G on X and every morphism of presheaves ϕ: F G there exists a unique morphism ϕ: F G making the diagram (1.17.1) F ι F F ϕ ϕ G ι G G commutative. In particular, F F is a functor from the category of presheaves on X to the category of sheaves on X. Definition 1.18. The sheaf F is called the sheaf associated to F or the sheafification of F. We can reformulate the first part of Proposition 1.17 by saying that sheafification is the left adjoint functor to the inclusion functor of the category of sheaves into the category of presheaves. Proof. For U X open, elements of F (U) are by definition families of elements in the stalks of F which locally give rise to sections of F. More precisely, we define F (U) := { (s x ) x U F x ; x U: an open neighborhood W U of x, and t F (W ): w W : s w = t w }. For U V the restriction map F (V ) F (U) is induced by the natural projection x V F x x U F x. Then it is easy to check that F is a sheaf. For U X open, 13

we define ι F,U : F (U) F (U) by s (s x ) x U. The definition of F shows that, for x X, Fx = F x and that ι F,x is the identity. Now let G be a presheaf on X and let ϕ: F G be a morphism. Sending (s x ) x F (U) to (ϕ x (s x )) x G (U) defines a morphism F G. By Proposition 1.13 (3) this is the unique morphism making the diagram (1.17.1) commutative. If we assume in addition that G is a sheaf, then the morphism of sheaves ι G : G G, which is bijective on stalks, is an isomorphism by Proposition 1.13 (2). Composing the morphism F G with ι 1 G, we obtain the morphism ϕ: F G. Finally, the uniqueness of ( F, ι F ) is a formal consequence. Remark 1.19. If F is a presheaf of (abelian) groups, of rings, of R-modules, or of R-algebras, its associated sheaf is a sheaf of (abelian) groups, of rings, of R-modules, or of R-algebras. Remark 1.20. From Proposition 1.13 (2), we get the following characterization of the sheafification: Let F be a presheaf and G be a sheaf. Then G is isomorphic to the sheafification of F if and only and if there exists a morphism ι: F G such that ι x is bijective for all x X. Example 1.21. Let E be a set and denote by E X the sheaf of locally constant functions, i.e. E X (U) = {f : U E locally constant map}. This is a sheaf. Let F be the presheaf of constant functions f : U E, U X open. As every constant function is also locally constant, one has a map ι: F E X of presheaves. For every f E X (U) and every x U there exists an open neighborhood V of x such that f V is constant. This shows that ι x is bijective for all x X. Hence E X is the sheafification of F. The sheaf E X is called the constant sheaf with value E (D) Direct and inverse images of sheaves Definition 1.22. Let f : X Y be a continuous map of topological spaces. Let F be a presheaf on X. We define a presheaf f F on Y by (for V Y open) (f F )(V ) = F (f 1 (V )) the restriction maps given by the restriction maps for F. Then f F is called the direct image of F under f. Whenever ϕ: F 1 F 2 is a morphism of presheaves, the family of mapsf (ϕ) V := ϕ f 1 (V ) for V Y open is a morphism f (ϕ): f F 1 f F 2. We obtain a functor f from the category of presheaves on X to the category of presheaves on Y. The following properties are immediate. Remark 1.23. (1) If F is a sheaf on X, f F is a sheaf on Y. Therefore f also defines a functor f : (Sh(X)) (Sh(Y )). (2) If g : Y Z is a second continuous map, there exists an identity g (f F ) = (g f) F which is functorial in F. 14

Definition 1.24. let f : X Y be a continuous map and let G be a presheaf on Y. Define a presheaf on X by (1.24.1) U lim G (V ), V f(u), V Y open the restriction maps being induced by the restriction maps of G. Momentarily we denote this presheaf by f + G. Let f 1 G be the sheafification of f + G. We call f 1 G the inverse image of G under f. If f is the inclusion of a subspace X of Y, we also write G X instead of f 1 G and we write G (X) := (f 1 (G ))(X). Note that even if G is a sheaf, f + G is not a sheaf in general. Remark 1.25. Let f : X Y be a continuous map of topological spaces. (1) Let G be a presheaf on Y. The construction of f + G and hence of f 1 G is functorial in G. Therefore we obtain a functor f 1 : (PSh(Y )) (Sh(X)). (2) Let g : Y Z be a second continuous map and let H be a presheaf on Z. Fix an open subset U in X. An open subset W Z contains g(f(u)) if and only if it contains a subset of the form g(v ), where V Y is an open set containing f(u). This implies that f + (g + H ) = (g f) + H and we can deduce an isomorphism (1.25.1) f 1 (g 1 H ) = (g f) 1 H, which is functorial in H. (3) If x is a point of X and i: {x} X is the inclusion, the definition (1.24.1) shows that i 1 F = F x for every presheaf F on X (more precisely: i 1 (F )({x}) = F x ). (4) In particular, (1.25.1) implies for each presheaf G on Y the identity (1.25.2) (f 1 G ) x = G f(x). Remark 1.26. Let X, Y be a topological spaces. (1) Let U Y be an open subspace. If V U is open, then V is open in Y and one has (F U )(V ) = G (V ). (2) Let G be a sheaf on Y, let f : X Y be a continuous map of topological spaces, and let U X be an open subset. Then the construction of the sheafification in the proof of Proposition 1.17 shows that (f 1 G )(U) is the set of s = (s x ) x U x U G f(x) such that for all x U the following condition holds: There exist x W U open, V Y open with f(w ) V, and t G (V ) such that t f(w) = s f(w) for all w W. 15

Direct image and inverse image are functors which are adjoint to each other. More precisely: Proposition 1.27. Let f : X Y be a continuous map of topological spaces, let F be a sheaf on X and let G be a presheaf on Y. Then there is a bijection which is functorial in F and G. Hom (Sh(X)) (f 1 G, F ) Hom (PreSh(Y )) (G, f F ), ϕ ϕ, ψ ψ Proof. Let ϕ: f 1 G F be a morphism of sheaves on X, and let t G (V ), V Y open. Since f(f 1 (V )) V, we have a map G (V ) f + G (f 1 (V )), and we define ϕ V (t) as the image of t under the map G (V ) f + G (f 1 (V )) f 1 G (f 1 (V )) ϕ f 1 (V ) F (f 1 (V )) = f F (V ). Conversely, let ψ : G f F be a morphism of sheaves on Y. To define the morphism ψ it suffices to define a morphism of presheaves f + G F, which we call again ψ. Let U be open in X, and s f + G (U). If V is some open neighborhood of f(u), U is contained in f 1 (V ). Let V be such a neighborhood such that there exists s V G (V ) representing s. Then ψ V (s V ) f F (V ) = F (f 1 (V )). Let ψ U (s) F (U) be the restriction of the section ψ V (s V ) to U. Clearly, these two maps are inverse to each other. Moreover, it is straightforward albeit quite cumbersome to check that the constructed maps are functorial in F and G. Remark 1.28. We will almost never use the concrete description of f 1 G in the sequel. Very often we are given f, F, and G as in the proposition, and a morphism of sheaves ψ : G f F. Then usually it is sufficient to understand for each x X the map ψ x : G f(x) (1.25.2) = (f 1 G ) x F x induced by ψ : f 1 G F on stalks. The proof of the proposition shows that we can describe this map in terms of ψ as follows: For every open neighborhood V Y of f(x), we have maps G (V ) ψ V F (f 1 (V )) F x, and taking the limit over all V we obtain the map ψ x : G f(x) F x. Remark 1.29. Note that if F is a sheaf of rings (or of R-modules, or of R-algebras) on X, f F is a sheaf on Y with values in the same category. A similar statements holds for the inverse image. Finally, Proposition 1.27 holds (with the same proof) if we consider morphisms of sheaves of rings (or of R-modules, etc.). 16

(E) Ringed spaces We fix a commutative ring R (e.g., R = R, R = C, or R = Z). Definition 1.30. (1) An R-ringed space is a pair (X, O X ), where X is a topological space and where O X is a sheaf of commutative R-algebras on X. If (X, O X ) and (Y, O Y ) are ringed spaces, we define a morphism of R-ringed spaces (X, O X ) (Y, O Y ) as a pair (f, f ), where f : X Y is a continuous map and where f : O Y f O X is a homomorphism of sheaves of R-algebras on Y. (2) A locally R-ringed space is an R-ringed space (X, O X ), such that O X,x is a local ring for all x X. We then denote by m x the maximal ideal and by κ(x) := O X,x /m x its residue field. A morphism of locally R-ringed spaces (X, O X ) (Y, O Y ) is a morphism (f, f ) of ringed spaces such that the homomorphism of local rings f x : O Y,f(x) O X,x is local (i.e., f x(m f(x) ) m x ). For R = Z we simply say ringed space instead of Z-ringed space. Remark 1.31. (1) The datum of f is equivalent to the datum of a homomorphism of sheaves of R-algebras f : f 1 O Y O X on X by Proposition 1.27. Often we simply write f instead of (f, f ) or (f, f ). (2) The composition of morphisms of (locally) R-ringed spaces is defined in the obvious way (using Remark 1.23 (2)), and we obtain the category of (locally) R-ringed spaces. We call O X the structure sheaf of (X, O X ). Often we simply write X instead of (X, O X ). In general, f is an additional datum for a morphism. We will usually consider the simpler case that the structure sheaf is a sheaf of certain functions on open subsetes of X and then f is given by composition with f as in the following example. Example 1.32. Let X and Y be open subsets of finite-dimensional R-vector spaces. Let α N 0 { }. Then (X, CX α ) is a locally R-ringed space (cf. Example 1.12). For x X we have m x = { f CX,x α ; f(x) = 0 } and κ(x) = R. (1) Every C α -map f : X Y defines a morphism (f, f ): (X, CX α ) (Y, Cα Y ) of locally R-ringed spaces by f V : C α Y (V ) f (C α X)(V ) = C α X(f 1 (V )), t t f for V Y open. (2) Conversely, let (f, f ): (X, CX α ) (Y, Cα Y ) be any morphism of R-ringed spaces. We claim: (a) (f, f ) is automatically a morphism of locally R-ringed spaces. (b) For all V Y open the R-algebra homomorphism fv : Cα Y (V ) Cα X (f 1 (V )) is automatically given by the map t t f. Note that then f is automatically C α (consider for t projections to some coordinates). 17

To show (a) let x X. Set ϕ := fx, B := CX,x α, and A := Cα Y,f(x). Then ϕ: A B is a homomorphism of local R-algebras such that A/m A = R and B/m B = R. We claim that ϕ is automatically local ( ϕ 1 (m B ) is a maximal ideal of A). Indeed, ϕ induces an injective homomorphism of R-algebras A/ϕ 1 (m B ) B/m B = R. As a homomorphim of R-algebras, it is automatically surjective, hence A/ϕ 1 (m B ) = R is a field. Let us show (b). Let V Y be open and x f 1 (V ). Consider the diagram of R-algebra homomorphisms CY α(v ) fv CX α (f 1 (V )) t t f(x) CY,f(x) α f x C α X,x s s x ev f(x) : t t(f(x)) R R. ev x : s s(x) The upper rectangle is commutative and the evaluation maps are surjective. Hence there exists a homomorphism of R-algebras ι: R R making the lower rectangle commutative if and only if fx(ker(ev f(x) )) ker(ev x ), i.e., if and only if fx is local. Moreover, as a homomorphism of R-algebras, one must have ι = id R. Therefore we find fv (t)(x) = t(f(x)) which shows (b). Remark and Definition 1.33. Let (X, O X ) be a (locally) R-ringed space and let U X be open. Then (U, O X U ) is a (locally) R-ringed space. Such a (locally) R-ringed space is called open subspace of (X, O X ). A morphism f : (Z, O Z ) (X, O X ) of (locally) R-ringed spaces is called open embedding if U := f(z) is open in X and f induces an isomorphism (Z, O Z ) (U, O X U ). 2 Manifolds (A) Premanifolds Definition 2.1. (1) Let α N 0 { }. A locally R-ringed space (X, O X ) is called (real) C α -premanifold if there exists an open covering X = i I U i such that for all i I there exist n N 0, an open subspace Y of R n, and an isomorphism of locally R-ringed spaces (U i, O X Ui ) (Y, CY α ) (called chart). Here Y and n depend on i. In this case the structure sheaf is denoted by CX α and is called the sheaf of C α -functions on X. A family of charts Φ i : (U i, O X Ui ) (Y i, CY α i ) is called atlas if i I U i = X. We call a real C -premanifold a smooth premanifold. 18

(2) A morphism of C α -premanifolds (X, O X ) (Y, O Y ) is defined as a morphism of locally R-ringed spaces. Such a morphism is also called a C α -map. C -maps are called smooth. We obtain the category of C α -premanifolds. An isomorphism in the category of C α -premanifolds is called a C α -diffeomorphism. Every such morphism is given by a continuous map f : X Y such that for all V Y open and for all t O Y (V ) one has t f O X (f 1 (V )) (by Example 1.32). Let U R n and V R m be open subspaces. Then a map f : U V is a morphism of C α -premanifolds (U, C α U ) (V, Cα V ) if and only if f is a Cα -Abbildung (Example 1.32). Remark and Definition 2.2. Let X = (X, C α X ) be a Cα -premanifold, let x X. An integer n 0 is called dimension of X in x (denoted dim x (X)) if there exists an open neighborhood U of x and a chart Φ: U Y, where Y R n is open. The dimension is uniquely determined: If Ψ: V Z R m is a second chart, then the change of charts Φ U V Ψ 1 Ψ(U V ) : Ψ(U V ) Φ(U V ) is a C α -diffeomorphism of an open subset of R m onto an open subset of R m. Hence m = n (for α 1 this is clear because the derivative of a C α -diffeomorpism in a point is an R-linear isomorphism R m R n ; for α = 0 this is more difficult but we might see it later). The function X N 0, x dim x (X) is locally constant. If it is constant = n, then we say that X has dimension n and write n = dim X. Remark 2.3. Let X be a topological space such that there exists an open covering X = i I U i and for all i I homeomorphisms Φ i : U i Yi with Y i R n i open satisfying the following condition. For all i, j I the homeomorphism (change of charts) Φ i (U i U j ) Φ 1 i Φ j U i U j Φ j (U i U j ) is a C α -diffeomorphism. For V X open define C α X(V ) := { f : V R ; i: f Ui V Φ 1 i : Φ i (U i V ) R is C α }. This defines a sheaf of R-algebras on X, and (X, CX α ) is a Cα -premanifold. (U i, Φ i ) i I is an atlas. Then Definition 2.4. A C-ringed space (X, O X ) is called complex manifold if there exists an open covering X = i I U i such that for all i I there exist n N 0, an open subspace Y of C n, and an isomorphism of locally C-ringed spaces (U i, O X Ui ) (Y, OY hol). A morphism of complex premanifolds (X, O X ) (Y, O Y ) is defined as a morphism of locally C-ringed spaces. Such a morphism is also called holomorphic. We obtain the category of complex premanifolds. An isomorphism in the category of complex premanifolds is called a biholomorphic map. 19

Every such morphism of complex premanifolds is given by a continuous map f : X Y such that for all V Y open and for all t O Y (V ) one has t f O X (f 1 (V )) (cf. Example 1.32). (B) Topological properties Definition 2.5. Let X be a topological space. (1) Let x X be a point. A set B x of subsets of X which all contain x is called a neighborhood basis of x if every neighborhood of x contains some element of B x. (2) X is called first countable if every point of x has a countable neighborhood basis. (3) X is called second countable if the topology has a countable basis. Example 2.6. Let X be a topological space (1) Every metrisable space X (i.e. X is a topological space whose topology is given by a metric d) is first countable: For x X the sets { B 1/n (x) := { y X ; d(x, y) < 1/n } ; n N } { B 1/n (x) := { y X ; d(x, y) 1/n } ; n N } are countable neighborhood bases of x. (2) Every premanifold is first countable (locally it is homeomorphic to a metrisable space). (3) Let X be a second countable space and let B be a countable basis. Then X is first countable (for every point { B B ; x B } is a countable neighborhood basis of x) and X contains a countable dense subset Q (for every nonempty B B choose x B B and set Q := { x B ; B B }). (4) R n is second countable. Choose a norm on R n. Then a countable basis is given by { B 1/n (x) ; x Q n, n N }. (5) Let X be second countable topological space, B X a countable basis of the topology. Then every subspace Z of X is second countable: { B Z ; B B X } is a countable basis. In particular, every subspace of R n is second countable. (6) Let X = n U n, where (U n ) n is a countable open covering of X such that the subspace U n is second countable for all n. Then X is second countable. In particular, every premanifold that has a countable atlas is second countable. (7) Let X be an uncountable set endowed with the discrete topology. Then X is a zero-dimensional premanifold which is not second countable. Definition 2.7. Let X be a topological space (1) An open covering of X is a family (U i ) i I of open subsets of X such that X = i I U i. (2) An open covering V is a subcover of an open covering (U i ) i I if there exists a subset J I such that V = (U i ) i J. (3) An open covering (V j ) j J is a refinement of an open covering (U i ) i I if there for all j J there exists i I with V j U i. 20

(4) A family (A i ) i I of subsets A i X is called locally finite if every x X has an open neighborhood V such that { i I ; A i V } is finite. Note: Any subcover is a refinement. Definition 2.8. A topological space is called a Lindelöf space if every open covering of X has a countable subcovering. Lemma 2.9. Every second countable topological space X is a Lindelöf space. The converse is true if X is metrisable. Proof. Reelle Analsis: Exercise 52. Converse: Let X be Lindelöf. Fix n N. Choose a countable subcover U n of { B 1/n (x) ; x X }. Then the union of all U n is a countable basis of X. Definition 2.10. Let X be a Hausdorff topological space. (1) X is called locally compact if every point has a compact neighborhood. (2) X is called paracompact if every open covering has a locally finite refinement. (3) X is called σ-compact if it is the union of countably many compact subspace. Remark 2.11. (1) An open nonempty subset of normed R-vector space V is locally compact if and only if dim R (V ) < (see Analysis 2). (2) A premanifold X is locally compact if and only if X is Hausdorff. (3) Every compact space is paracompact. (4) One can show that every metrisable space is paracompact ([BouTG] IX, 4, Théorème 4). Proposition 2.12. Let X be a locally compact and second countable space. Then X has the following properties. (1) X is σ-compact. (2) X is paracompact. (3) X is metrisable. We will only prove (1) and (2) and refer to [Bre1] Chap. 1, Theorem 12.12 for Assertion (3). Proof. Construction of a countable base B such that B compact for all B B. Let B be a countable base of X, C x compact neighborhood of x X (automatically closed in X because X is Hausdorff). Define B := { V W ; V B and W B with W C x for some x X}. Then B countable basis (because X = x C x), V W C x compact. Write B = {B 1, B 2,... }. Proof of (1). Construct K n compact and i(n) N 0 with i(n) > i(n 1) such that B 1 B i(n) K n Kn+1 for all n N 0 (then n K n i B i = X). Set C m := m j=1 B j (then C m compact). Define K n and i(n) inductively. K 0 := and i(0) := 0. If K n 1 = X, we are done. Otherwise choose l N with B l (X \ K n 1 ). 21

Let i(n) N with i(n) > i(n 1) and C l K n 1 1 j n(i) B j (exists because C l K n 1 is compact) and set K n := 1 j i(n) B j. Then K n compact and K n 1 C l K n 1 1 j i(n) B j K n. Proof of (2). Let U = (U i ) i be an open covering of X. Let W = (W j ) j J be the refinement of U consisting of those W B with W U i for some i. Fix n N and set W j,n := W j (Kn+2 \ K n 1). Then W j,n K n+1 \ Kn. }{{} j J compact Let J(n) J be finite such that j J(n) W j,n K n+1 \ Kn. Let V = (V l ) l be the open covering consisting of W n,j for n N and j J(n) (it is a covering because n K n = X). Moreover: V is countable and a refinement of U by construction. W n,j K n+2 and hence V l compact for all l. V is locally finite: For x X let n N with x K n. Then Kn+1 is an open neighborhood of x which intersects only the finitely many W j,m with j J(m) and m n + 1 (by the definition of W j,m ). Theorem 2.13. (Urysohn s theorem/tietze extension theorem) Let X be a topological space. Then the following assertions are equivalent. (i) For any two closed subsets A, B X with A B = there exists a continuous function f : X [0, 1] such that f(a) = 0 for all a A and f(b) = 1 for all b B. (ii) For all disjoint closed sets A and B of X there exist open disjoint sets U and V such that A U and B V. (iii) For every closed subspace A of X and every continuous map f : A R there exists a continuous function f : X R such that f A = f. Proof. [BouTG] Chap. IX, 4. Note: (i) (ii) is easy: Take U = {f < 1/2} and V = {f > 1/2}. Definition 2.14. A topological space X is called normal if it is Hausdorff and satisfies the equivalent conditions of Theorem 2.13. Proposition 2.15. Every metrisable topological space is normal. Proof. Reelle Analysis: A, B X closed, A B =. Define f : X [0, 1], f(x) = d(x,a) d(x,a)+d(x,b). Proposition 2.16. Every paracompact space is normal. 22

Proof. Let X be a paracompact space, A, B X be closed with A B =. Claim: Let Y, Z X be closed with Y Z =. If for all y Y there exists y V y X open and Z W y X open with V y W y =, then there exists open neighborhood T of Y and open neighborhood U of Z with U T =. Claim X normal. X Hausdorff may apply claim to Y = A and Z = {x} for some x B. Hence there exist open neighborhood T of A and U of x with T U =. Then we can apply claim again to Z = A and Y = B to see that there exist open neighborhoods Ũ of A and Ṽ of B with Ũ Ṽ =. Hence X is normal. Proof of Claim. Let (T i ) i a locally finite refinement of the open cover of X formed of X \ Y and of the V y for y Y. Then: Y T i y i Y : T i V yi. Let T be the union of the T i with Y T i. Then T X open and Y T. It remains to find U: For all z Z there exists z S z X open such that S z intersects only finitely many T i (because (T i ) i is locally finite). In particular J z := { i I ; T i S z, T i Y } is finite. Set U z := S z i J z W yi. Then z U z X open, and U z T = (because W yi T i = ). Set U := z Z U z. Lemma 2.17. (Shrinking lemma) Let X be a normal space. Then for every locally finite open covering (U i ) i I there exists an open covering (V i ) i I such that V i U i. Proof. [BouTG] Chap. IX, 4.3, Cor. 1 de Théorème 3. Definition and Remark 2.18. Let X be a topological space and let Z X be a subset. Then Z is called locally closed if the following equivalent conditions are satisfied. (i) There exists a family (U i ) i I of open subsets U i X such that Z i I U i and such that Z U i is closed in U i. (ii) Z is an open subset of its closure Z in X. (iii) Z is the intersection of an open and a closed subset of X. Proof. (ii) (iii) (i) is clear. (i) (ii) : Let (U i ) i as in (i). As Z U i is closed in U i, we have Z U i = Z U i, which is open in Z. Hence Z = i (Z U i) is open in Z. Example 2.19. Every submanifold of R n (in the sense of Analysis 2) is a locally closed subspace in R n. (C) Topological groups Definition 2.20. Let X be a topological space, let Y be a set, and let π : X Y be a surjective map. Equivalently, let be an equivalence relation on X and let π : X Y := X/ be the projection. Define a topology on Y : V Y is open if π 1 (V ) is open. This topology is called the quotient topology. 23

Note: If Z is a topological space and there is a commutatiove diagram of maps X f Z π Y, f then f is continuous if and only if f is continuous. Definition 2.21. A topological group is a set G endowed with the structure of a topological space and of a group such that the maps G G G, (g, g ) gg, G G, g g 1 are continuous (where we endow G G with the product topology). Remark 2.22. G topological group, a G. Then left translation g ag and right tranlation g ga are homeomorphisms G G. Proposition 2.23. G topological group, H G subgroup. We endow G/H with the quotient topology (i.e. if π : G G/H is the projection, U G/H is defined to be open if and only if π 1 (U) is open). (1) The closure H is a subgroup of G. (2) If H is locally closed in G, then H is closed in G. (3) H is open in G (resp. closed in G) if and only if G/H is discrete (resp. Hausdorff). (4) If H is a normal subgroup, G/H is a topological group. Proof. Let a: G G G be the continuous map (g, h) gh 1. Then a(h H) = a(h H) a(h H) = H. This shows (1). To prove (2) we may assume that H is open and dense in G (by replacing G by the subgroup H). Then for g G the two cosets gh and H have nonempty intersection hence they are equal, i.e. g H. (3): G/H discrete gh open in G for all g G H open in G. If G/H is Hausdorff, then eh G/H is a closed point and its inverse image H in G is closed. Conversely, if H is closed, then H = HeH is a closed point in the quotient space H\G/H. Hence its inverse image under the continuous map G/H G/H H\G/H, (g 1 H, g 2 H) Hg2 1 g 1H is closed. But this is the diagonal of G/H G/H. (4) is clear. (D) Manifolds Definition 2.24. A C α -manifold (resp. a complex manifold) is a C α -premanifold (resp. a complex premanifold) whose underlying topological space is Hausdorff and second countable. 24