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Introduction to Algrbraic Topology

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Introduction to Algrbraic Topology Introduction to Algrbraic Topology Ingo Waschkies September 27, 2010 Abstract Lecture notes of a course given at the Université de Nice - Sophia Antipolis in fall 2010 Contents 1 Category Theory 2 1.1 Categories and Functors . 2 1.2 Cartesian squares . 5 2 The fundamental groupoid 9 2.1 Connected spaces . 9 2.2 Construction of the fundamental groupoid . 11 2.3 Homotopy invariance . 14 3 Coverings 16 3.1 Fiber bundles . 16 3.2 Coverings . 19 3.3 Group actions and coverings . 22 3.4 Universal coverings and fundamental groupoid . 24 1 1 Category Theory 1.1 Categories and Functors Definition 1.1. A category C is given by the following data: (i) a class of objects Ob C, (ii) for any two objects X, Y ∈ Ob C a set HomC (X, Y) whose elements are called the morphisms (or arrows) from X to Y (iii) and for any three objects X, Y, Z ∈ Ob C a map ◦XYZ : HomC (X, Y) × HomC (Y, Z) − HomC (X, Z);(f, g) 7 g ◦XYZ f which is called composition map (the index XYZ→will be immediately omitted in the→ notations). such that the following two axioms are satisfied: (C1) For any object X ∈ Ob C there exists a morphism idX ∈ HomC (X, X) such that for any object Y and any morphisms f ∈ HomC (X, Y) we have f ◦ idX = f and idY ◦ f = f. (C2) The composition of morphisms is associative, i.e. for any objects X, Y, Z, W ∈ Ob C and any three morphisms f ∈ HomC (X, Y), g ∈ HomC (Y, Z) h ∈ HomC (Z, W) we have h ◦ (g ◦ f) = (h ◦ g) ◦ f. Let C be a category and X, Y ∈ Ob C two objects. Instead of f ∈ HomC (X, Y) we will usually write f : X Y, the object X is called the source of the morphism f and Y is called the target of f. An endomorphism is a morphism which has the same source and target. A morphism f : X Y is→ called an isomorphism if there exixts a morphism g : Y X such that g ◦ f = idX −1 and f◦g = idY . If such a morphism g exists it is unique and called the inverse morphism f of f. → → Let C be a category and X ∈ Ob C be an object. Then HomC (X, X) is a set endowed with a binary operation ◦ (the composition of morphisms) which is associative and has a unit element idX. Such a structure is called a monoid in basic algebra. It is not necessarily a group since mor- phisms need not be invertible. A category with only a single object ∗ is completely determined by the monoid HomC (∗, ∗), hence categories with one object are the same as monoids. In this sense a category is a generalization of the algebraic notion of monoid. We have already mentioned that in order to have a group structure on HomC (X, X) we need to impose that every endomorphism of X is invertible. So similarly to the monoid situation we can state that a category with a single object such that every morphism is an isomorphism is the same as a group. This leads to the following generalization of the algebraic notion of group: Definition 1.2. A groupoid is a category such that every morphism in G is an isomorphism. We will see soon that in some sense a groupoid is actually just a collection of groups, a collection that need not be a set. It will be convenient to introduce the opposite category C◦ of a category C. The category C◦ has the same objects as C but source and target of morphisms are formally exchanged, i.e. a morphism f : X Y in C◦ is a morphism f : Y X in C. More precisely: → →2 Definition 1.3. Let C be a category. Its opposite category C◦ is given by the following data: (i) Ob C◦ = Ob C, ◦ (ii) for any two objects X, Y ∈ Ob C we set HomC◦ (X, Y) = HomC (Y, X) (iii) and for any three objects X, Y, Z ∈ Ob C◦ the map op ◦ : HomC◦ (X, Y) × HomC◦ (Y, Z) − HomC◦ (X, Z) is given by g ◦op f = f ◦ g (note that f, g are morphism f : Y X, g : Z Y in C so the → composition is well defined). Note that if we have a monoid (i.e. a category with just a single object),→ then the→ opposite cate- gory is given by the monoid with the opposite structure. Definition 1.4. Let C, D be two categories. A functor F : C D is given by the following data (i) for each object X ∈ Ob C an object F(X) ∈ Ob D, → (ii) for any two objects X, Y ∈ Ob C a map (for which we will always omit the index XY ) FXY : HomC (X, Y) − HomD (F(X),F(Y)) such that the following axioms hold → (F1) for any X ∈ Ob C we have F(idX) = idF(X) (F2) for any X, Y, Z ∈ Ob C and any morphisms f : X Y, g : Y Z we have F(g) ◦ F(f) = F(g ◦ f) → → The axioms (F1) and (F2) are often called the functoriality axioms, and a functor is often just referred to as a correspondance F : Ob C Ob D ; X 7 F(X) which is functorial (or natural) in X. This always means that there are maps as in (ii) which satisfy (F1) and (F2). In practical situations, correspondances→F : Ob C Ob→D ; X 7 F(X) will sometimes be natural in X but will invert the direction of arrows (and therefore are not functors in the sense of the above definition). The perhaps easiest example→ is the duality→ for vector spaces: a linear ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ map f : V W defines a linear map f : W V with idV = idV and (g ◦ f) = f ◦ g . This is what is called a contravariant functor. → → Definition 1.5. A contravariant functor F : C D is a functor F : C◦ D. This means, that it is given by the following data (i) for each object X ∈ Ob C an object F(X) ∈ Ob→D, → (ii) for any two objects X, Y ∈ Ob C a map F : HomC (X, Y) − HomD (F(Y),F(X)) such that the following axioms hold → (F’1) for any X ∈ Ob C we have F(idX) = idF(X) (F’2) for any X, Y, Z ∈ Ob C and any morphisms f : X Y, g : Y Z we have F(f) ◦ F(g) = F(g ◦ f) → → 3 If we want to emphasize that a functor is not contravariant we sometimes say that it is covari- ant. The category Set whose objects are the sets and whose morphisms are maps between sets plays a special role in category theory. Every category C is automatically equipped with a collection of Hom-functors indexed by the objects of C HX = HomC (X, · ): C − Set ; Y 7 HomC (X, Y) 0 On morphisms HX is defined as follows: for any morphism f : Y Y we define HX(f) as the composition with f, more precisely → → 0 → HX(f): HomC (X, Y) − HomC (X, Y ); g 7 HX(f)(g) = f ◦ g It is easy to check the functor axioms. Obviously we can also put the X in the second variable and get the so-called Yoneda-functors→ → YX = HomC ( · ,X): C − Set ; Y 7 HomC (Y, X) On morphisms YX is again defined as the composition, but beware this time we get a con- travariant functor. → → Given two functors F : C D and G : D E we can define the composition G ◦ F : C E by (G ◦ F)(X) = G(F(X)) for any object X ∈ Ob C and (G ◦ F)(f) = G(F(f)) for any morphism f from C. It is easily checked→ that G ◦ F satisfies→ the functor axioms F1 and F2. We may also easily→ see that the identity maps on objects and morphisms are functorial and define the identity functor IdC : C C, and we can form the category Cat of all categories (which leads to some set-theoretical problems as the collection of functors between two categories is not a set). Therefore we get→ the notion of an isomorphism between categories as a functor F : C D such that there exists a functor G : D C with G ◦ F = IdC and F ◦ G = IdD. This means just that F is an isomorphism of categories if and only if it is bijective on objects and morphisms.→ It turns out that this notion is of little importance→ because isomorphisms between categories arise only very rarely. This is because many constructions that can be formalized in category theory are only unique up to isomorphism, hence we encounter more frequently the situation that we have functors F : C D. G : D C such that for any object X ∈ Ob C the object GF(X) is not equal but isomorphic to X. A good example is the dual functor defined on the category of finite-dimensional vector→ spaces: → f f ∗ ∗ : Vect (k) − Vect (k); V 7 V = Homk (V, k) It is well-known that V V∗∗ is an isomorphism, but it is not an equality, so ∗ is not an iso- morphism of categories, it is what→ is called an equivalence→ of categories. To make a precise definition we need the notion→ of morphisms between functors: Definition 1.6. Let F, F0 : C D be functors. A morphism of functors φ : F G is a collection of morphisms φX indexed by all objects X ∈ Ob C → 0 → φX : F(X) F (X) 0 such that for every morphism f : X Y in C we get F (f) ◦ φX = φX 0 ◦ F(f).
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