1. MA5209 Algebraic Topology Wayne Lawton Department of Mathematics National University of Singapore S14-04-04, matwml@nus.edu.sgmatwml@nus.edu.sg http://math.nus.edu.sg/~matwml Lecture 4. Category Theory and Homotopy Theory (1, 4 September 2009)

2. Categories A category C consists of the following three mathematical entities: A class ob(C), whose elements are called objects;class A class hom(C), whose elements are called morphisms or maps or arrows.morphismsmaps Each morphism f has a unique source object a and target object b. We write f: a → b, and we say "f is a morphism from a to b". We write Hom(a, b), or Mor(a, b), to denote the hom-class of all morphisms from a to b. A binary operation, called composition of morphisms, such that for any three objects a, b, and c,binary operation we have Hom(a, b) × Hom(b, c) → Hom(a, c). The composition of f: a → b and g: b → c is written as gf and is governed by two axioms: AssociativityAssociativity: If f : a → b, g : b → c and h : c → d then h(gf) = (hg)f, and IdentityIdentity: For every object x, there exists a morphism 1 x : x → x called the identity morphism for x, such that for every morphism f : a → b, we haveidentity morphism ob(C) = Groups, Mor(a,b) = { homomorphisms f : a  b} ob(C) = Sets, Mor(a,b) = { functions f : a  b} ob(C) = Top Spaces, Mor(a,b) = { maps f : a  b} ob(C) = Top Spaces, Mor(a,b) = [a,b] = homotopy ob(C) = Pointed Top Spaces, Mor((X,p),(Y,q)) ={ maps f : a  b such that f(p) = q} classes of maps f : a  b with [f] [g] = [f g]

3. Commutative Diagram for Composition of Morphisms

4. Properties of Morphisms A morphism f : a → b is: a monomorphism (or monic) if f g1 = f g2 implies g1 = g2monomorphism for all morphisms g1, g2 : x → a. an epimorphism (or epic) if g1 f = g2 f implies g1 = g2epimorphism for all morphisms g1, g2 : b → x. an isomorphism if there exists a morphism g : b → aisomorphism with f g = 1b and g f = 1a. an endomorphism if a = b. End(a) denotes the class of endomorphisms of a.endomorphism an automorphism if f is both an endomorphism and an isomorphism. Aut(a) denotes the class of automorphisms of a.automorphism

5. Homotopy Equivalence Definition Topological Spaces X and Y are homotopy equivalent if [X, Y] contains an isomorphism. This means that there exist maps f : X  Y and g : Y  X such that g f and f g Question 1. Explain what isomorphisms are in the following categories: sets, topological spaces with maps as morphisms, groups. Question 3. Use Q2 to prove that homotopy equivalence is an equivalence relation on the class of topological spaces. Question 2. Show that in any category the composition of isomorphisms is an isomorphism. Question 4. Prove thatandare homotopy equivalent.

6. Relative Homotopy Definition Let X, Y be topological spaces and A X. Maps f, g : X  Y with homotopic relative to A, written are if there exists a homotopy from f to g such that Definition A groupoid is a category in which every morphism is an isomorphism. Question 5 Show that a group is a groupoid with exactly one object.

7. Contractible Spaces with f(0) = g(0) and f(1) = g(1) then Lemma X is contractible iff there exists Definition A topological space X is contractible Theorem If X is contractible and f, g : [0,1]  X and a map if it is homotopy equivalent to a single point space. such that Proof Constructby

8. Geometry of Theorem

9. Connectedness Definition A topological space X is disconnected otherwise it is connected. A subset Y of a topological space X is d. / c.. if it is d. / c. when it is given the subspace topology. Result The connected subsets of R with the usual topology are exactly the intervals. The image of a connected set under a map is connected. The product of connected subsets is connected. if any of the following equivalent conditions hold:

10. Path Connectedness Definition In a topological space X a point p is path connected to a point q if there exists a map f : [0,1]  X such that f(0) = p and f(1) = q. Lemma Path conn. is an equivalence relation. Proof. p is connected to p by the constant map f : [0,1]  X defined by f(x) = p, x in X. If p is connected to q by a path f : [0,1]  X then q is connected to p by the path g : [0,1]  X defined by g(s) = f(1-s), s in [0,1]. If p is connected to q by a path a and q is connected to r by a path b then p is connected to r by the path ba defined by Definition X is path connected if X is an equiv. class

11. Local Path Connectedness Definition X is locally path connected if for every x in X and every open O containing x there exists an open U containing x and U subset O such that every pair of points in U is path connected. Theorem In a locally path connected space path connected equiv. classes are open and closed. Proof Let Corollary If X is locally path connected be an equiv. class and Thenandis open hence there exists an open setwithand each point in is path connected toand hence Ifthere exists a path connected open and connected then X is path connected. withhence

12. Fundamental Groupoid of a pathwise connected topological space X is the following category: the objects are the points of X; for p, q in X the morphisms Mor(p,q) are the equivalence classes of maps f : [0,1]  X with f(0) = p, f(1) = q with respect to homotopy relative to {0,1}, and for p, q, r in X the composition of [a] in Mor(p,q) with [b] in Mor(q,r) (a, b : [0,1]  X with a(0) = p, a(1) = b(0) = q and b(1) = r), is given by [b][a] = [ba] where [b] and [a] denote the homotopy (relative to {0,1}) equivalence classes containing a and b respectively. Question 6. Prove that Remark The objects are not sets! is a groupoid.

13. Fundamental Group Fix a topological space X. For every p in X, the set Mor(p,p) together with the composition defined in the groupoid on the previous page, is a group, called the fundamental group of X based at p, and denoted by Question 7. Show that if p,q in X, then every [a] in Mor(p,q) defines an isomorphism by the formula

14. Covariant Functors Let C and D be categories. A covariant functor F from C to D is a function that associates to each object X in C an object F(X) and to each morphism f : X  Y in C a morphism F(f) : F(X)  F(Y) such that ifcategories Example Fix a topological space X and let C be the associated groupoid category, let D be the category of groups, and for p, q in X and [a] in Mor(p,q) define Question 8. Show that this describes a covariant functor from the categoryto the category of groups. thenand

15. The Fundamental Group Functor Question 9. Show that Consider the rule that associates to each pointed space defined by Definition The category of pointed topological spaces has objects (X,p) and Mor((X,p),(Y,q)) = { maps f : X  Y with f(p) = q }. the groupand to each pointed map the function Question 10. Show that is well defined. is a group homomorphism. Question 11. Show thatis a functor. Question 12. Show that if is a homotopy equivalence then is an isomorphism for all

16. Category of Groupoids The objects of this category are groupoids and for every pair of groupoids a and b, Mor(a,b) is the set of (covariant) functors from a to b. Question 13. Explain how to define composition of morphisms in this category and show that it actually produces a category.

17. The Groupoid Functor Definition For each topological space X let letbe the functor from the be its fundamental groupoid and for each map categoryto the category defined by: (here) Question 14. Show this defines a functor from the CAT = topological spaces to CAT = groupoids Question 15. Show thatfunctors andare homotopic (naturally equiv.)

18. Simply Connected Spaces Theorem The trivial group is the one having a single element, namely, its identity element. Lemma If a space X is path connected, then Definition A space is simply connected if it is path connected and its fundamental group is trivial. Question 17. Prove this and then show that every convex subspace of an affine space and the Lemma Every contractible space is simply conn. subspacesare simp. conn. Question 16. Prove this.

19. Simply Connected Spaces Theorem If are open subsets that are simply is a topological space and then Corollary is simply connected. Proof connected (regarded as subspaces) and such thatis path connected and is clearly path connected.Let and then show that is path connected.

20. Simply Connected Spaces Proof (Theorem) Clearly based at there exists an integer such that It suffices to show that and loops and a loop namely based at is pathwise connected. To show that it is simply connected we choose Proof By Lebesgue’s lemma there exists with each each where and is contained inorJoin p to each Construct and inif by a path respectively.

21. Covering Spaces Definition Let is a covering map such that covering space of X) andbe topological spaces. admissible neighborhood is a disjoint union of open sets A map if every index set such that for everythe restriction is a homeomorphism. Theare called sheets over and the mapsare (local) sections of since (and E is a has an open

22. Covering Spaces Lemma If has the quotient topology. 1. Each fibreis a discrete subspace. is a local homeomorphism. is a covering map then Question 18 Prove these statements. 2. 3.is surjective and Lemma Letbe a topological group, be a discrete subgroup, maponto the set X of left cosets, and give X the quotient topology. Then p is a c.m. be the Question 19 Prove this lemma. Question 20 Prove that

23. Fibre Bundles are 4-tuples exists an open cover space consisting of a total a fiber (space) and a projection such that there of a homeomorphism Example (trivial) such that and for each a base space Example a covering, connected, for anyQuestion 21. Prove this.

24. Lifting Definition Ifis a map,is a space, then a map or, equivalently, the diagram is a lift of a map if commutes. Question 22. Show that if there exists a lift of then iff there exists a nowhere vanishing tangent vector field onThen what can you conclude? is the projection on to the first column of

25. Fibrations Definition A surjective map is a then there exists fibration if it satisfies the following If commute. that makes homotopy lifting property: Remark: is a homotopy fromto where andis a lift of

26. Fibrations Theorem If Algebraic Topology (requires much work). is a fiber bundle and Proof Corollary 14 on page 96 in Spanier’s is paracompact then Definition A paracompact space is one in which every open cover has a locally finite refinement. Hence every compact space is paracompact is a fibration. Question 23. Show that if F is not discrete then the liftmay not be uniquely determined by and

27. Unique Lifting Theorem Ifis a covering,is conn.&loc conn. and then lift a map such that and Proof LetSince andis connected, it suffices to show that is both open and closed. Since we assume that all spaces are Hausdorff, it is closed (why?). For choose admissble open is a disjoint union of open sheets and let sheet that contains be the loc. conn.  open&closed inHence injective andon

28. Coverings are Fibrations Theorem (Covering Homotopy Theorem 3 on page 66 in Singer and Thorps Lecture Notes …) If then commute. Proof. Let that makes the diagram is a covering andis compact and open cover of be an admissible Then cover and find such that each for someThen use glueing and induction.

29. Implications of Unique Homotopy Lifting Corollary 1.(p. 67 in S&T)If Corollary 2.(p. 67 in S&T) is injective. then thereand exists unique lift with there exists a ‘natural’ correspondence and the set of cosetsbetween the fiber is a covering Ifis a covering Corollary 3.(p. 68 in S&T)If is a covering Theorem 4.(p. 71 in S&T)path.&loc. path. conn. loc. simply conn.

30. Galois Theory of Covering Transformations Definition If is called a covering transformation if Question 24 Show that these form a group G, that this group acts freely on each fiber. is a normal subgroup of and a homeomorphism then Theorem 9.(p. 76 in S&T)* If is the quotient space ofunder the action of Remark* E is pathwise connected, p is surjective.

31. Assignments Read Handouts: [2] pages 131-140 in Armstrong’s Basic Topology [3] pages 62-77 in Singer and Thorpe’s Lecture Notes on Elementary Topology and Geometry Learn how to compute the fundamental group of a simplicial complex using the algorithm in [2] and describe the group by its generators and relations as described in [1]. Learn van Kampen’s theorem. Do Problems 20-24 on page 140 of [2]. The ‘dunce cap’ in problem 22 is the polyhedron of a 2-simplex. [1] Appendix: Generators and relations

32. Projects Write a 5-10 page report and give a 20 minute talk on either 5,9 October on 1 of following topics: 1. classification of 2-dimensional manifolds 2. Jordan curve theorem3. Riemann surfaces 5. Differential topology theory (Milnor’s book) 9. Gauss-Bonnet theorem 6. Higher homotopy groups7. Knot theory 8. Lie group topology 10. Fibrations and Fiber Bundles 12. Groupoids and van Kampen’s theorem or another topic related to algebraic topology 4. Covering spaces (and their Galois theory) 11. Graph theory

33. Homotopy Problem 1. Let X be a topological space, let be a path from by define to anddefine by Construct a homotopyRELATIVE TO fromto the constant path defined by

34. Homotopy Solution 1. We need to construct a map with the following 3 properties: Strategy: constructas a composition of simple maps and are continuous. Glueing lemma  where

35. Covering Maps and definehave the topology as subspaces of Problem 2. Let by Solution i. Draw a ‘picture’ of E and X and p.

36. Covering Maps and definehave the topology as subspaces of Problem 2. Let by Solution: p is a 3-1 C.M.ii. Is p a covering map?

37. Covering Maps iii. Compute Solution Free group generated by a and b.

38. Covering Maps iv. Compute Solution Free groups generated by and respectively.

39. Covering Maps v. Compute Solution Free subgroups of and respectively. generated by These 3 subgroups are conjugate to each other!

40. Cofibrations Definition A mapis a then there exists cofibration if it satisfies the following that makes the solid diagram below commute homotopy extension property: that makes the augmented if there exist diagram below commute. Compare with fibrations on slide 25.

41. Contravariant Functors Given categories C,D. A contracovariant functor F from C to D is a function that associates to each object X in C an object F(X) and to each morphism f : X  Y in C a morphism F(f) : F(Y)  F(X) such that if Example Let C be a category, A in Obj(C) and define Question 24. Show this describes a covariant functor from the category C to the category SETS thenand

42. Natural Transformations Given categories C, D and functors F,G from C to D. A natural transformation from F to G is a function Such that if F, G are covariant (contravariant) the left (right) diagram below commutes for every Question 25. Show why NT are also called homotopies. Suggestion. Let be the two object groupoid and derive a correspondence between NT and functors H to from such that