Notes for Advanced Algebraic Topology

Transcription

1 Harvard University Math 231br Notes for Advanced Algebraic Topology Reuben Stern Spring Semester 2017

2 Contents 0.1 Preliminaries Administrative Stuff January 23, Overview of Course January 25, Some facts about the Category Spaces Example: Locality and Sheafiness Example: Products of Spaces Example: Gluings (or Quotient Spaces) Example: Space of Functions Pairs of Spaces January 27, The Punchline from Last Class Perpsectives on the Fundamental Group January 30, Higher Homotopy Groups Exact Sequences of Spaces February 1, Finishing Things from Last Time Dual Results: Exact Sequences Relative Homotopy Groups February 3, The Action of the Fundamental Group February 6, Fibrations February 8, Fiber Bundles and Examples February 10, CW Complexes February 13, The Homotopy Theory of CW Complexes Homotopy Excision and Corollaries

3 Math 231br CONTENTS 10.3 Introduction to Stable Homotopy February 15, Why Model Categories? Definitions, Examples, and Basic Properties Homotopy Relations February 17, The Homotopy Category of a Model Category Brief Introduction to Derived Functors February 22, The Homotopy Theory of CW Complexes, II Eilenberg-MacLane Spaces February 24, Brown Representability February 27, Spectra The Category of Spectra March 1, Co/homology Theories from Spectra March 3, The Hurewicz Theorem Spectral Sequences March 6, Obstruction Theory March 8, A Complicated Example March 10, Smash Products March 20, The Serre Spectral Sequence March 22, More About the Serre Spectral Sequence The Transgressive Differential Lemma Multiplicative Structure Reuben Stern 2 Spring 2017

4 Math 231br CONTENTS 23 March 24, More Spectral Sequences Multiplicative Extensions Computing H (K(Z/2, 1); F 2 ) March 27, G-bundles and Fiber Bundles Properties of Chern Classes The Steenrod Algebra and the Bar Spectral Sequence April 10, The Steenrod Algebra, part II April 12, A Serre Spectral Sequence Trick Serre Classes April 14, Serre s Method April 17, The Adams Spectral Sequence A Fun Computation A Collected Homework Problems 71 A.1 Problem Set A.2 Problem Set A.3 Problem Set A.4 Problem Set A.5 Problem Set B Notes from Switzer 79 References 80 Reuben Stern 3 Spring 2017

5 Math 231br 0.1 Preliminaries 0.1 Preliminaries These notes were taken during the Spring semester of 2017, in Harvard s Math 231br, Advanced Algebraic Topology. The course was taught by Eric Peterson, and met Monday/Wednesday/Friday from 2 to 3 pm. Allow me to elucidate the process for taking these notes: I take notes by hand during lecture, which I transfer to L A TEX at night. It is an unfortunate consequence of this method that these notes do not capture the unique lecturing style of the professor. Indeed, I take full responsibility for any errors in exposition or mathematics, but all credit for genuinely clever remarks, proofs, or exposition will be due to the professor (and not to the scribe). In an appendix at the end of this document, you will find the collected homework problems (with solutions). I make no promises regarding the correctness of these solutions; consider yourself warned. Please send any and all corrections to reuben_stern@college.harvard.edu. They will be most appreciated. A rough syllabus for the course is, in Eric s own words, to cover some initial segment of the book. To explain, the course plans to discuss general homology and cohomology theories, bordism homology and cohomology, stable homotopy groups of spheres, and spectral sequences as computational tools for homotopy groups. 0.2 Administrative Stuff The grading of the course is as follows: weekly-ish problem sets constitute about one third, a midterm paper is another third, and a final paper will be the final third. Reuben Stern 4 Spring 2017

6 Math 231br January 23, January 23, 2017 Didn t attend lecture. Went over course logistics and outlined the focus of the class. 1.1 Overview of Course The goal of this class is to give an introduction to homotopy theory (of spaces). This is four-part: 1. Decomposition of spaces (co/fiber sequences). 2. Invariants constructed from decompositions (H, π ) and their properties (theorems of Whitehead and Hurewicz, etc.). 3. Representability theorems (Brown, Adams) and the stable category (HZ, KU and KO, S, HG, etc.) 4. Computation (characteristic classes, Bott periodicity, the Steenrod operations, the Adams and Serre spectral sequences, etc.) As an example of the kind of analysis we can perform with these tools at our disposal, start with your favorite simply connected space, like S n 2. Its homotopy groups are notoriously difficult and important to compute. Theorem 1.1. (Hurewicz) If X is (n 1)-connected (that is, π k X = 0 for all k < n), then H n (X; Z) = π n X. We also have a decomposition by a fiber sequence such that X[n + 1, ] X K(π n X, n) π n X k = n π k K(π n X, n) = 0 otherwise, π π k X n + 1 k kx[n + 1, ] = 0 otherwise. Then we can also plug the data of H X and H K(π n X, n) into a gadget known as the Serre spectral sequence to get H X[n + 1, ]. Reuben Stern 5 Spring 2017

7 Math 231br January 25, January 25, 2017 This is the first lecture I attended. I noticed that the class was 75% graduate students, and got a little frightened. Need not fear though, as Eric is a warm and inviting lecturer! 2.1 Some facts about the Category Spaces Eric: I can t make it through a lecture without saying category at least five times. The goal of this lecture is to give a couple of technical lemmas and some basic constructions, so that we don t have to mention them explicitly ever again. There will be four examples in this class: a sheaf condition on a decomposition of a topological space, product spaces, gluings (or quotient spaces), and the space Y X of functions X Y. 2.2 Example: Locality and Sheafiness Let X be a topological space and X = j A j a decomposition of X into a collection of locally finite closed subsets. Then a continuous function f : X T (for an arbitrary space T ) is the same data as continuous functions f j : A j T agreeing on the overlap. We picture this as follows: Equivalently, we demand f i Ai A j = f j Ai A j. This condition is called locality or a sheaf condition on the collection {f i }. For intution on where the term sheaf condition comes from, here is an aside on sheaves! Make picture of overlap Aside 2.1. (Sheaves.) A sheaf is an assignment from subsets (usually taken to be open) of a space X to arbitrary sets, together with restriction maps: if B A X, we get a map res A,B : F (A) F (B). (Example: take F (A) = {continuous functions A T }.) We ask that these data satisfy: The restriction maps commute nicely: if C B A X, then res B,C res A,B = res A,C. For a cover {A j } of X, we think about the following diagram: F (X) res X,Aj j F (A j ) res Aj,A k A j res Aj,A j A l k,l F (A k A l ) The sheaf condition is then equivalent to the leftmost map equalizing the right hand side. And before we move on, an aside on equalizers! As Eric keeps pointing out, this class will be very category-heavy. Reuben Stern 6 Spring 2017

8 Math 231br 2.3 Example: Products of Spaces Aside 2.2. (Equalizers.) An equalizer (of sets) E S of S on which f and g restrict to the same function. f g T is the maximal subset Lemma 2.3. Suppose that F h S is some function such that f h = g h. Then there is a unique factorization of h as F h S! E i To conclude our previous example, the assignment F (A) = {continuous functions f : A T } forms a sheaf. 2.3 Example: Products of Spaces For any two spaces X and Y, there is a space X Y with the following property: any map T X Y is the same data as pairs of maps f : T X and g; T Y. If we write Spaces for the category of spaces 1, then we write Spaces(T, X Y ) to mean the set of continuous functions T X Y. The main idea about the space X Y, called the product space, is that we have a natural 2 bijection = Spaces(T, X Y ) Spaces(T, X) Spaces(T, Y ). 2.4 Example: Gluings (or Quotient Spaces) Given an equivalence relation R on a topological space X, the set of equivalence classes X/R under the relation forms a topological space with the natural map X X/R continuous and open. Lemma 2.4. If f : X Y is continuous such that xrx implies f(x) = f(x ), then there is a factorization of continuous maps X f Y X/R! 1 This is Eric s notation; I much prefer writing Top for this category, but I will follow his anyway because I am the student and he is the master. 2 In the categorical sense; we will talk about this more later Reuben Stern 7 Spring 2017

9 Math 231br 2.5 Example: Space of Functions Special Case: let A X, and take the maximum equivalence relation on A. Extend this by the identity relation on X to get a relation R where xrx if and only if x, x A. The quotient with respect to this equivalence relation is then called X/A (edge case 3 : X/ = X { }). Suppose A i X f Y, and f i is constant. Then the diagram commutes: A i X f Y X/A f Lemma 2.5. Suppose R is a relation on X and S is a relation on Y ; form the quotient space X Y/R S. Then there s a map X Y R S X R Y S which is always a continuous bijection. Unfortunately, this is the best we can say in general. Lemma 2.6. If Y is locally compact and S is the identity relation, then this map is a homeomorphism. Example 2.7. Take Y = [0, 1], the unit interval, and S to be the identity relation. Then X α I = X I α id. This shows up a lot. 2.5 Example: Space of Functions Definition 2.8. For spaces X and Y, we let Y X denote the space 4 of continuous functions X Y. 3 This is mostly a convention 4 With the compact-open topology; the exchange went like this: me: What s the topology on Y X? Eric: The compact-open topology, but the short answer is it doesn t matter. Reuben Stern 8 Spring 2017

10 Math 231br 2.6 Pairs of Spaces Lemma 2.9. If X is locally compact, then Y X X and Z are additionally Hausdorff, then ev Y is continuous. If X This is perhaps familiar as currying. Y Z X = (Y Z ) X. Corollary There is a natural bijection Spaces(X Z, Y ) = Spaces(X, Y Z ). It is good to imagine Z as fixed (i.e., we re considering an adjoint pair of functors ( Z) ( ) Z ), and X, Y changing. For example, given g : Y Y, then the square Spaces(X Z, Y ) Spaces(X, Y Z ) = g g Spaces(X Z, Y ) Spaces(X, (Y ) Z ) = commutes. 2.6 Pairs of Spaces Thinking back to homotopy, the fundamental group, relative homology, and other constructions in introductory algebraic topology, we often care a lot about pairs of spaces (A X) or spaces with a choice of basepoint ({x 0 } X). Maps of such objects are continuous functions f : X Y with the added condition that f A : A B. That is to say, f(a) B. You can do basically all of the above with pairs; we will only touch on a few particularly interesting ones. Lemma The product of two pairs (X, A) (Y, B) in the category of pairs is (X, A) (Y, B) = (X Y, (X B) (A Y )). This satisfies the universal property of product in the category of pairs. Proof. Exercise. The interesting thing about this construction is the distinguished subset of X Y begin given by (X B) (A Y ); this allows us to consider an interesting and important example: Reuben Stern 9 Spring 2017

11 Math 231br 2.6 Pairs of Spaces Example Let (X, {x 0 }) and (Y, {y 0 }) be pointed spaces. Then the product is (X, {x 0 }) (Y, {y 0 }) = (X Y, (X {y 0 }) ({x 0 } Y )). But this distinguished subset of the product should be familiar: it is exactly the wedge product of X and Y at x 0 and y 0! At the end of class, we began going through a presentation of a natural bijection which would lead well into the definition of the smash product; we didn t have enough time to complete it: Eric: 3 pm. I ll let you go. Fuck. We will complete that definition next time. Reuben Stern 10 Spring 2017

12 Math 231br January 27, January 27, The Punchline from Last Class To assign some notation, let Spaces / denote the category of spaces with a chosen basepoint. We have an inclusion of categories i : Spaces / Pairs that sends (X, x 0 ) to the pair (X, {x 0 }). In the other direction, there is a functor quot : Pairs Spaces / sending (X, A) to (X/A, ), where is the preferred point to which A is collapsed. We want to use this pair of functors to study the induced adjunction. We have a square (X, {x 0 }), (Y, {y 0 }) (X Y, X Y ) i (X, x 0 ), (Y, y 0 ) quot ( X Y X Y, ) The (pointed) space (X Y/X Y, ) is interesting because it participates in an exponential adjunction: Spaces / (X, Y Z ) = Spaces / (X Z, Y ). The space X Z is called the smash product. As a remark, So there are two products in Spaces /. X Z (X, x 0 ) (Y, y 0 ). Aside 3.1. X Y is the categorical coproduct in Spaces /. 3.2 Perpsectives on the Fundamental Group Recall the path space of X is the function space X I = X [0,1]. The set π 0 (X) of path components of X is thus the set of connected components of X I. We will write [Y, X] to denote the set of homotopy classes of maps Y X. In this notation, π 0 (X) = [, X], as is clear by the homotopy relation (a homotopy between maps f : X and g : X is a path in X). If we are considering pointed spaces, then π 0 (X, x 0 ) = [S 0, X], where S 0 = {±1}. Note that we will more often than not suppress basepoints in pointed spaces 5. 5 Also, for the foreseeable future, all spaces will be pointed unless explicitly stated otherwise. Reuben Stern 11 Spring 2017

13 Math 231br 3.2 Perpsectives on the Fundamental Group Remark 3.2. [Y, X] can be expressed as π 0 (X Y ). Recall that π 1 (X, x 0 ) is the collection of based loops in X, taken up to homotopy. With our notation, this is [S 1, (X, x 0 )]. Note that S 1 = S 0 S 1 : S 0 S 1 is two copies of S 1, and then we collapse down S 0 S 1 = { } S 1 to the basepoint on the other copy of S 1, leaving us with just the other copy. Therefore [S 1, X] = [S 0 S 1, X] = [S 0, X S1 ], which is π 0 (X S1 ). The space X S1 is the function space of based loops in X. It is so frequently used that it gets its own name: it is the loop space of X, and we denote it by Ω(X). Recall as well that π 1 (X) is a group. Question 3.3. Expressing π 1 as above, what is special about the functors π 0 Ω( ) or [S 1, ] that makes these functors group-valued? Eric: Naturally a, we re going to address this problem with category theory. a No pun intended Definition 3.4. A group is a set G with maps µ : G G G, η : G, and χ : G G such that the following diagrams commute: 1 µ G G G G G G = G G G G = G 1 η η 1 µ 1 µ µ 1 G G G G µ 1 χ 1 G G G G G G G G η G µ 1 χ η Lemma 3.5. If G is a group, then Sets(X, G) forms a group, for any other set X. Proof. We first have to define the group law on Sets(X, G): to do this, first note that there is a canonical isomorphism (whatever that means) Sets(X, G) Sets(X, G) = Sets(X, G G) Reuben Stern 12 Spring 2017

14 Math 231br 3.2 Perpsectives on the Fundamental Group ( two maps from one space into another is the same thing as a map into the cartesian product of the codomains ). We thus consider the composition µ µ : Sets(X, G) Sets(X, G) Sets(X, G G) Sets(X, G), where µ is postcomposition by µ. We do a similar thing with η and χ to define η and χ. Now, we have to check that all the relevant diagrams commute. But this is actually automatic: we just apply the functor Sets(X, ) to the commutative diagrams 6 we already have! Back to Question 3.3, what does this definition have to do with our problem? Answer 1: It turns out that ΩX is a group object in the homotopy category of pointed spaces. That is, we have maps µ : ΩX ΩX ΩX, η : ΩX, and χ : ΩX ΩX making all the relevant diagrams commute (note that in this case, is the terminal/initial object in the category htop ). In order to show this, we have to show that all the diagrams commute, but we ve done this before in 131/231a! This implies that [S 0, Ω(X)] = Sets(S 0, Ω(X)) is automatically a group ( that s cute! ). In fact, [Y, ΩX] is automatically a group for any set Y. Answer 2: Adjoint-ly, it turns out that S 1 is a cogroup object in 7 htop. Eric: It is somewhat unlikely that you ve heard the word cogroup aloud, so we ll define it now. Definition 3.6. A cogroup is a group with all the arrows turned around and all the products converted to coproducts. That is (understanding that X Y is the coproduct of X and Y in Spaces), we have maps µ : C C C, η : C, and χ : C C, satisfying properties of coassociativity, counitality, and coinversion. For instance, coassociativity states that the diagram C C C µ 1 C C 1 µ µ C C C µ commutes. An equivalent definition is that a cogroup is a group object in the opposite category. 6 A functor applied to a commutative diagram yields a commutative diagram 7 This is the homotopy category of pointed spaces, where morphisms are homotopy classes of maps Reuben Stern 13 Spring 2017

15 Math 231br 3.2 Perpsectives on the Fundamental Group Example 3.7. What do the cogroup operations look like for S 1? We have a pinch map µ : S 1 S 1 S 1, a collapsing map η : S 1, and a flipping map χ : S 1 S 1. Eric: I wish that I had three hands. We then show that the diagrams commute (this is easy but laborious; left as an exercise). Remark 3.8. S 1 ( ) is also a cogroup in htop. We can see this more or less algebraically: (Z (X Y )) = ((Z X) (Z Y )), so we can work things out. The functor S 1 ( ) is called the (reduced) suspension, and is written Σ( ). It is adjoint to the loop space functor. Lemma 3.9. The natural map [ΣX, Y ] = [X, ΩY ] is an isomorphism of group objects. Proof. Let f, g : ΣX Y. Denote their images in [X, ΩY ] by f, g : X ΩY. Then the composition µ ΣX f g ΣX ΣX Y Y Y defines f g, where the map is called the folding map. This map is defined by sending Y Y (y, y 0 ) y and Y Y (y 0, y) y. We can also use the composition f X X X g µ ΩY ΩY ΩY to define f g. So we have f, g, fg [ΣX, Y ] and f, g f g [X, ΩY ], and we also have (fg) [X, ΩY ]. The question then becomes, is f g = (f g)? In some sense, you just have to work this out from the definitions. It is not hard, just laborious. Reuben Stern 14 Spring 2017

16 Math 231br January 30, January 30, 2017 In this lecture, we continue our investigations of last time by looking at higher homotopy groups of spaces, and proceed to discuss co/exact (i.e., co/fiber) sequences of spaces. 4.1 Higher Homotopy Groups Definition 4.1. We define the homotopy groups of a space X by π n (X, x 0 ) = [Σ n S 0, X]. By the suspension-loop space adjunction, we have the equality of spaces [Σ n S 0, X] = [Σ n 1 S 0, ΩX] = = [S 0, Ω n X]. In the case n = 0, 1, there are no intermediate groups. But when n 2, we have some legitimately interesting structure: namely, the intermediate groups have two group structures on them directly, rather than through isomorphism. The following lemma will tell us that those structures are the same. Lemma 4.2. (Eckmann-Hilton) Let S be a set with two products, and, that share a unit. Suppose further that (x x ) (y y ) = (x y) (x y ). Then and agree, and both are associative and commutative. Proof. By computation: x y = (x e) (e y) = (x e) (e y) = x y. x y = (e x) (e y) = (e y) (e x) = y x. Associativity is similar. Also, The plan is thus to check that the products on [Σ n 1 S 0, ΩX] induced by the cogroup and group structures, respectively, share this special assumption, so that we may apply the Eckmann-Hilton argument. Corollary 4.3. [Σ n 1 S 0, ΩX] has only one product, and it is commutative. Proof. This is a proof by diagram chase and verbal diarrhea. The diagram is the following: µ µ (f f ) (g g ) fold fold K K (K K) (K K) (L L) (L L) L L K K (K ) (K ) ( K) ( K) µ 1 T 1 (f f ) (g g ) (L ) (L ) ( L) ( L) (f g) 1 T 1 (f g ) µ µ K K (K K) (K K) (L L) (L L) L L fold L Remark 4.4. There is a homeomorphism S n+1 = ΣS n. Reuben Stern 15 Spring 2017

17 Math 231br 4.2 Exact Sequences of Spaces 4.2 Exact Sequences of Spaces Let us begin by recalling that an exact sequence of groups N f G g K is a pair of maps such that g 1 (e) = im f. A natural question to ask is the following: Question 4.5. When does a sequence of spaces A B C induce an exact sequence 8 of sets 1. [, A] [, B] [, C] or 2. [A, ] [B, ] [C, ]? What does it even mean for a sequence of sets to be exact? Definition 4.6. A sequence of pointed sets A f B g C is exact if g 1 ( ) = im f. If condition 1 from the question is satisfied, we call the sequence of pointed spaces an exact sequence or a fiber sequence. If condition 2 is satisfied, we say it is a coexact sequence or cofiber sequence. Lemma 4.7. Any map of spaces X f Y extends to a coexact sequence f X Y Z. Proof/construction. Fix some test space T. We ultimately want a sequence [X, T ] [Y, T ] [?, T ]. For this sequence to be exact means that a map g : Y T which induces a null-homotopic map f g : X T is equal to the composition Y Z T for some map Z T. Let us define Z = Y f CX = Y CX f(x) (x, 1), where the cone on a topological space CX is defined to be X I/(X {0} {x 0 } I). Technically, this is the definition of the reduced cone, but we drop words all the time anyways, so what s the problem? Note that a nullhomotopy of a map X T is the same data as a map CX T 9, so a nullhomotopy of f γ is a map CX T such that CX T X f γ 8 Eric: Do you still call them air quotes when you write them on the board? 9 Exercise: convince yourself of this. Reuben Stern 16 Spring 2017

18 Math 231br 4.2 Exact Sequences of Spaces commutes. The sheaf condition then implies the existence of a map Y f CX T, the image of whose pushforward contains (f ) 1 ( ). For the other inclusion, we see that pushforward of, say, γ by this map (let s call it α) gives a map with nullhomotopic pushforward f (g α) : X T. Aside 4.8. The space Y f CX is equivalently the pushout (fiber coproduct) of X CX along f : X Y : X CW f Y Y f CW. Remark 4.9. We can iterate this process to get a sequence f i j X Y Y f CX (Y f CX) i CY ((Y f CX) i CY ) j C(Y f CX). These look nasty, but they really aren t! We can see that by the two following lemmas, given without proof: Lemma The map is a homotopy equivalence. (Y f CX) i CY ((Y f CX) i CY )/CY Lemma For A X, there is a homeomorphism (X i CA)/CA = X/A. Corollary (Y f CX) i CY = ((Y f CX) i CY )/CY = (Y f CX)/Y = ΣX. In general, that horribly messy sequence is just an infinitely long coexact sequence f X Y Y f CX ΣX ΣY Σ(Y f CX) Σ 2 X This gives rise to the exact sequence [X, T ] [Y, T ] [Cf, T ] [ΣX, T ] [ΣY, T ] [Σ(Cf), T ] [Σ 2 X, T ]. We get some interesting fringe phenomena at the maps [ΣX, T ] [Cf, T ] and [Σ 2 X, T ] [Σ(Cf), T ], where the target set has less structure than the initial. Reuben Stern 17 Spring 2017

19 Math 231br February 1, February 1, Finishing Things from Last Time Recall from last time: for a map X get a sequence f Y, we can iteratively construct Y f CX =: Cf to f X Y Cf ΣX ΣY Σ(Cf). Applying the functor [, T ] for some test space T, we get something interesting happening at Hom htop / (ΣX, T ) Hom htop / (Cf, T ) : there is a natural action of [ΣX, T ] on [Cf, T ]. Lemma 5.1. There is a factorization Hom htop / (Cf, T ) Hom htop / (Y, T ) Hom htop / (Cf,T ) Hom htop / (ΣX,T ) Proof. Chase some diagram. 5.2 Dual Results: Exact Sequences Last time, we needed to know that a nullhomotopy of f : X Y is equivalent data to a map H : CX Y where the diagram CX Y X f commutes. Since CX = XI, we can think of this as a map X Y I. Definition 5.2. For a map X f Y, the path space P f of f is P f = {(x, γ) X Y I : γ(1) = f(x)}. The next two lemmas follow by duality from things proved last time: Reuben Stern 18 Spring 2017

20 Math 231br 5.3 Relative Homotopy Groups pr Lemma 5.3. The sequence P X f f X Y is exact in Spaces /. Lemma 5.4. Iterating this construction yields a long exact sequence in Spaces / : Ω 2 X Ω 2 f Y ΩP f ΩX ΩY P f X Y. Applying π 0, it follows that π 1 P f π 1 X π 1 Y π 0 P f π 0 X π 0 Y is an exact sequence of homotopy groups. The major question then becomes: Question 5.5. What is this P f object, and how can we calculate anything about its homotopy groups? Knowing the homotopy groups can give us a significant amount of information about those of X and Y. 5.3 Relative Homotopy Groups Let us restrict our attention to the inclusion i : A X. Then (X, A, {x 0 }) is an object of Pair /. The space P i is then the mapping space By the exponential adjunction, we find that P i = (X, A, {x 0 }) (I, I,0). π n 1 P i = [(S n 1, ), (P i, γ 0 )] = [(D n, S n 1 ), (X, A)], which we define to be the relative homotopy group π n (X, A). Corollary 5.6. There is a long exact sequence π 2 (X, A) π 1 A π 1 X π 1 (X, A) π 0 A π 0 X. Definition 5.7. A pair (X, A) is called n-connected if π n (X, A) = 0. A map i : A X is a weak equivalence if it is -connected. Reuben Stern 19 Spring 2017

21 Math 231br 5.3 Relative Homotopy Groups Remark 5.8. We can actually convert any map f : Y X into an inclusion as follows: there is a homotopy equivalence X M f of X into the (reduced) mapping cylinder of f, and an inclusion Y i M f by sending Y to Y {0}. The diagram i M f Y f X commutes up to homotopy, as is immediate. Thus we can define connectedness for an arbitrary map f. Consider a pair of inclusions B A X. This gives us three pairs: (X, B), (X, A), and (A, B). Each of these pairs has a long exact sequence in homotopy groups associated to it; we can stitch these together into the following diagram: π n+1 (X, A) π n (A, B) π n 1 B π n 1 X π n A π n (X, B) π n 1 A π n B π n X π n (X, A) π n 1 (A, B) Lemma 5.9. The highlighted sequence is exact. Reuben Stern 20 Spring 2017

22 Math 231br February 3, February 3, The Action of the Fundamental Group Maybe you recall that there is an action of π 1 X on π n X. If not, you may remember that a path γ : I X induces an isomorphism Γ : π 1 (X, γ(0)) π 1 (X, γ(1)) by γ 1 ω γ. Definition 6.1. A group G is said to act compatibly on another group A when (i) G acts on A: there is a map α : G A A. (ii) The multiplication map on A is G-equivariant with the A-action. Explicitly, this means g (a 1 a 2 ) = (g a 1 )(g a 2 ). Example 6.2. Any group G acts compatibly on itself by conjugation. If A is abelian, then a compatible G-action is identical information to a Z[G]-module structure on A. Theorem 6.3. S 1 coacts compatibly a on S n for all n 1. a Or perhaps we should say coacts mpatibly. From this theorem, we get the useful corollary: Corollary 6.4. π 1 X acts compatibly on π n X for all n 1. Implicitly, we are stating that if K coacts compatibly on L, then [K, T ] acts compatibly on [L, T ]. Proof of theorem. This involves drawing a lot of pictures. Reuben Stern 21 Spring 2017

23 Math 231br February 6, February 6, Fibrations Remember that we had this special object P f, fitting into an exact sequence P f j X f Y where P f was defined as P f := {(γ, x) Y I X : γ(1) = f(x)}., We have a homotopy equivalence ΩY P j := {(α, (γ, x)) X I P f : α(1) = x, γ(1) = f(x)}. The picture is enough to see this: Example 7.1. Given X and Y spaces, π n (X Y ) = π n X π n Y. Indeed, the sequence make picture π n Y π n (X Y ) π n X is exact. Definition 7.2. A map p : E B has Reuben Stern 22 Spring 2017

24 Math 231br February 8, February 8, Fiber Bundles and Examples Reuben Stern 23 Spring 2017

25 Math 231br February 10, February 10, CW Complexes Reuben Stern 24 Spring 2017

26 Math 231br February 13, February 13, The Homotopy Theory of CW Complexes Let X be a space. The map g : S n 1 X gives a way of attaching one n-cell to X to make the space X g CS n 1. It is natural to ask: if g 1, g 2 are both representatives for the same homotopy class [g] π n 1 (X), do they both give homotopy equivalent spaces? That is to say, given a homotopy H : g 1 g 2, do we get a homotopy equivalence X g1 CS n 1 H X g2 CS n 1? The answer, of course, is yes. A schematic of this is as follows (and in the diagram): slide g 1 over via H. Part of the cone is homeomorphic to I S n 1 ; this fills in the part of the homotopy given by sliding g 1 (of course, all of this is in very imprecise language). We can then pull the top of the cone down over the rest of it, to give us the attachment on g 2. Ultimately, our goal is to become familiar with the behavior of X g CS n 1 as a homotopy type. make diagram Lemma For (X, A) a relative CW complex, the pair (X, (X, A) n ), where (X, A) n is the n-skeleton of (X, A), is n-connected: π n (X, (X, A) n ) = 0. Corollary The inclusion X n X is n-connected (or, the pair (X, X n ) is). Proof. Use seriously the simplicial approximation theorem from last time. Corollary π <n S n = 0. Proof. We use the CW structure of S n as one n-cell glued to a point. Then (S n ) n 1 =, and S n = CS n 1. By the lemma, π <n (S n, ) = 0. Lemma If (X, A) is n-connected, then there exists a pair (X, A ), homotopy equivalent to (X, A), such that (X, A ) n = A. This means that X has no new cells in it until at least dimension n + 1. Corollary For X an n-connected space and Y an m-connected space, the space X Y is (n + m + 1)-connected. As Eric says, this is a little more connected than you might expect Homotopy Excision and Corollaries 10.3 Introduction to Stable Homotopy Reuben Stern 25 Spring 2017

27 Math 231br February 15, February 15, 2017 Lecture by Jun Hou. The focus of the next two lectures is to give an introduction to the theory of model categories, a sort of axiomatic homotopy theory first introduced by Quillen in the 1970s Why Model Categories? We begin with a lot of motivation (nearly 25 minutes worth!). Our motivation will be split into four parts. I: Localizing Categories Let C be a category, and W Mor(C ). We want to construct a category with the morphisms in W formally inverted (i.e., become isomorphisms). Example Take C = Spaces, W = {weak equivalences}. This is a slightly naïve approach to things, but because it seems reasonable, we run with it for now. We construct a category C [W 1 ] which has the same objects as C, but where morphisms between X and Y are zig-zag morphisms going back and forth between regular morphisms and formally inverted equivalences: C [W 1 ](X, Y ) = {X Note that the backwards arrows are inversions of arrows in W. Y } Problem Generally, C [W 1 ] is not a (locally small) category. We call C [W 1 ] the Gabriel-Zisman localization of C at W, and it has some set-theoretic problems. To solve these problems, we want to find an equivalent category with better properties. By equipping (C, W ) with a model structure, we ll define the homotopy category ho(c ), which is better behaved. Proposition There exists a simplicial category L H C called the hammock localization that is initial among all simplicial categories with W inverted, i.e., there is a dashed arrow making the diagram commute: C D L H C Reuben Stern 26 Spring 2017

28 Math 231br 11.1 Why Model Categories? Aside A simplicial category is a category enriched over the category of simplicial sets. There are a lot of words to define here, so I ll have to cite a few sources. We say that a category C is enriched over a category D if the hom-sets C (X, Y ) are naturally objects of D. Many categories are already enriched over themselves: Top is enriched over itself, if we endow the sets Top(X, Y ) with the compact-open topology; vec k is endowed over itself, where vec k (V, W ) has the obvious vector space structure. All small and locally small categories are (by definition) enriched over Set. Let us define the simplex category to be the category with one object [n] for each natural number n, where [n] = {0, 1,..., n} is a finite ordinal. Morphisms in this category are (weakly) order-preserving maps. For example, there is a morphism [n] [n + 1] sending 0 1, 1 2,..., n n + 1. This category will be denoted. A simplicial set is then a presheaf of sets on, i.e., a contravariant functor X : op Sets. In general, one may define a simplicial object in a category C as a contravariant functor X : op C. We think of the set X[n] as the set of n-simplices of the simplicial set X. Definition We now define the hammock localization L H C. The objects ob L H C are just ob C. Because L H C is enriched over sset, morphisms must form simplicial sets. We let 0-simplices be zig-zags as before, in the morphisms of C [W 1 ]. The 1-simplices will be hammocks : X Y In general, the n-simplices are wider hammocks. Fact π 0 Mor(L H C ) = Mor(C [W 1 ]). II: Derived Functors Suppose we are given a functor F : C D, and both categories have homotopy categories. When can we get a functor F : ho C ho D such that the diagram C F D ho C F ho D Reuben Stern 27 Spring 2017

29 Math 231br 11.2 Definitions, Examples, and Basic Properties commutes? The naïve thing to do is to use Kan extensions, but there are issues with that. Model categories solve the problems, by telling us what kinds of functors you can derive, and exactly how to do it. Aside Let s take the time to define Kan extensions. Definition Let F : C E and K : C D be functors. A left Kan extension of F along K is a functor Lan K F : D E, together with a natural transformation η : F Lan K F K satisfying the following universal property: given other pair (G : D E, γ : F G K), there is a unique natural transformation α : Lan K F G such that γ = α η. This can be summarized in the following diagrams: F C E C E K D = γ G η Lan K F Right Kan extensions are defined dually, with the natural transformation arrows reversed. K F D α K III: Abstract Homotopy Theory The theory of model categories is a theory of homotopy theories, in the sense that every model category carries with it a homotopy theory. IV: (, 1)-categories One may want to ask if there exists a homotopy theory of homotopy theories, i.e., a model category of model categories. One way of formalizing this is in the theory of -categories: there is an -category of -categories, and model categories present -categories Definitions, Examples, and Basic Properties Definition Suppose we are given a diagram as follows, where the outer square commutes: A B i p X We say that i has the left lifting property with respect to p or p has the right lifting property with respect to i if the dashed arrow exists, and the whole diagram commutes. We write this i llp(p) or p rlp(i). Reuben Stern 28 Spring 2017 Y

30 Math 231br 11.2 Definitions, Examples, and Basic Properties Definition A model category C is a category with 3 distinguished wide subcategories 10 W, called weak equivalences, denoted fib, called fibrations, denoted cof, called cofibrations, dennoted satisfying the following axioms: (MC1) C is bicomplete (has all small limits and colimits) (MC2) W has the 2-out-of-3 property: if f and g are composable arrows, and two of {f, g, g f} are weak equivalences, then so is the third. (MC3) W, fib, and cof are closed under retracts. (MC4) cof llp(w fib) and fib rlp(w fib). (MC5) There exist functorial factorizations f and f Remark Recall that a map f is a retract of g if there is a commuting diagram A B A f id A X Y X g id X Axioms (MC4) and (MC5) make a category into a weak factorization system. Qullen s original formulation of model categories required only that C be finitely bicomplete and that factorizations need not be functorial. What we give as the definition for a model category is what Quillen originally called a closed model category. Remark The model category axioms are dual: if C is a model category, so is C op, with weak equivalences the same, and fibrations and cofibrations flipped. The upshot of this is that we only need to prove half of the theorems! Remark We call the category W fib the class of acyclic fibrations, and W cof is the class of acyclic cofibrations. 10 A wide subcategory is a subcategory that includes all objects and all identity morphisms. f Reuben Stern 29 Spring 2017

31 Math 231br 11.2 Definitions, Examples, and Basic Properties Definition Because C is bicomplete, it has an intial object and a final object. We say that A is a cofibrant object if A is a cofibration, and a fibrant object if A is a fibration. Example (1a) Take C = Spaces. Then we let W = {weak htpy equivalences}, fib = {Serre fibrations}, cof = {retracts of cell complexes} (1b) Again take C = Spaces, with W = {weak homotopy equivalences}, but this time let fib = {Hurewicz fibrations} and cof = {closed Hurewicz cofibrations}. By a theorem we will prove later, it doesn t really matter if you know what these are or not. (2a) Take chain complexes Ch R of R-modules. Let W = {quasiisomorphisms} (that is, maps that induce isomorphisms on homology), fib = {degree-wise epis in positive degrees}, and cof = {degre-wise monos in all degrees with projective cokernels}. This is called the projective model structure. (2b) C = Ch R again, and W the same. This time, take fibrations to be degree-wise epis in all degrees with injective kernel, and cofibrations to be degree-wise monos in positive degrees. This is called the injective model structure. 3 The category ssets of simplicial sets has a model structure. In general, given any abelian category, one may form the simplicial category sa, and endow it with a model structure. Exercise: check the model category axioms against some of these examples. Proposition cof = llp(w fib) W cof = llp(fib) fib = rlp(w cof) W fib = rlp(cof) Proof. As an example to get a taste for how these proofs go, we show cof = llp(w fib). Note that we only need to prove the direction. Suppose f llp(w fib). Factor f as A i B f r p X Y Reuben Stern 30 Spring 2017

32 Math 231br 11.3 Homotopy Relations so we can get a lift r. Redraw this as A A A f i f r X B X This is a retract diagram, so (MC3) gives that f is a cofibration. p Proposition The subcategories W cof and cof are stable under cobase change (i.e., pushout) and fib and W fib are stable under base change (i.e., pullback). Proof. An exercise in working with diagrams, much in the same way as the above Homotopy Relations Fix a model category C. Definition A (good) cylinder object for A ob C is a factorization of the fold map : A A A as A A cyl(a) A. Two maps f, g : A X are left homotopic (written f l g) if there exists a cylinder object for A and a map cyl(a) H X such that the diagram A f i 0 i 1 A A cyl(a) X H A g commutes. Lemma With notation as above, if A is cofibrant, then i 0 and i 1 are cofibrations. Reuben Stern 31 Spring 2017

33 Math 231br 11.3 Homotopy Relations Proof. We have a factorization A i 0 cyl(a) η 0 A A Why is η 0 a cofibration? The square A A η 0 A A is a pushout square, so η 0 deserves a tail. Proposition If A is cofibrant, then l is an equivalence relation on maps C (A, X). Reuben Stern 32 Spring 2017

34 Math 231br February 17, February 17, The Homotopy Category of a Model Category Last time, we defined a cylinder object to be a factorization of the fold map, and defined two maps to be left homotopic (f l g) if there exists a map cyl(a) H X such that the diagram A f A A cyl(a) X H A g commutes. From this, it is clear that l is symmetric and reflexive; the tough part is to show transitivity. Proposition Left homotopy is an equivalence relation. Proof. Suppose f l g via an object cyl(a) and a map H, and g l h via an object cyl(a) and a map H. The relevant diagram to chase is A i 0 A cyl(a) i A 1 cyl(a) C A The object C comes from taking the pushout. All maps not involving C are given in the definitions of the cylinder objects. The universal property of pushout gives us the dotted map, which is a weak equivalence by the two-out-of-three axiom. We also have the following Reuben Stern 33 Spring 2017

35 Math 231br 12.1 The Homotopy Category of a Model Category diagram commuting: A cyl(a) f C X A cyl(a) g which gives us the left homotopy relation. We thus define π l (A, X) to be the set of equivalence classes C (A, X)/ l. Lemma Suppose X is fibrant, and f l g : A X. Then if h : A A is a map, we have fh l gh. Proof. Step 1: There exists a map H : cyl(a) X witnessing f l g. Factor H (by the axiom MC5) as A A cyl(a) A cyl(a ) Step 2: How do we know cyl(a ) interpolates f and h? Because X is fibrant, we have the diagram H cyl(a) X H cyl(a ) which gives us a lift H. We can thus produce a good cylinder object. Step 3: Because cyl(a ) is a cylinder object for A, we get the diagram h h A A A A cyl(a) k cyl(a ) A A h Reuben Stern 34 Spring 2017

36 Math 231br 12.1 The Homotopy Category of a Model Category which commutes, giving us a lift k. We thus have the diagram A h A f k cyl(a ) cyl(a) X g H giving H k as a left homotopy. A h A Definition A good path object is a factorization of the diagonal map as X cocyl(x) X X. Two maps f r g : A X are right homotopic if f X A cocyl(x) g X Lemma Suppose f, g : A X. (a) If A is cofibrant, then f l g f r g. (b) If X is fibrant, then f r g f l g. Proof. More clever lifting things. The two equivalence relations thus coincide if A is cofibrant and X is fibrant. In this case, we define π(a, X) := π l (X, A) = π r (X, A). With this, we can state the Whitehead theorem, one of the most important theorems in homotopy theory. One consequence of this theorem is that a map between CW complexes is a weak equivalence (i.e., induces isomorphisms on homotopy) if and only if it is a homotopy equivalence. Let us say an object is bifibrant if it is both cofibrant and fibrant. Reuben Stern 35 Spring 2017

37 Math 231br 12.1 The Homotopy Category of a Model Category Theorem Let f : A X be a map between bifibrant objects. Then f is a weak equivalence if and only if f is a homotopy equivalence. Proof. Suppose f : A X. Factor f as A q f C p X Note that C is bifibrant: the composition A C is a cofibration, and the composition C X is a fibration. Thus, we have the diagram C s p X X where s lifts p, i.e., ps = id X. Lemma If A is cofibrant, and p : C X is an acylic fibration, then p : π l (A, C) π l (A, X) is a bijection. Assuming the lemma, we see that sp l id C. Dually, we find a homotopy inverse r for q, and rs is a homotopy inverse for f = pq. In the other direction, factor f as A q f C p X We want to show that f is a weak equivalence. By assumption, f has a homotopy inverse g : X A, i.e., there exists a map H : cyl(x) X witnessing fg id X. We have the diagram g q X A C i 0 cyl(x) X. H H p Reuben Stern 36 Spring 2017

38 Math 231br 12.1 The Homotopy Category of a Model Category Set s = Hi 1. Then ps = id X. We also have a homotopy inverse for q, say r. As pq = f, pqr = fr p fr, so sp qgp qgfr qr id C. Aside If X is any object of a model category C, we can make X into a cofibrant object via the factorization X QX We call the object QX a cofibrant replacement for X, because it is cofibrant and weakly equivalent to X. Dually, we define a fibrant replacement RX for X via the factorization X RX. Definition Let C be a model category. The homotopy category of C, ho(c ), is the category with objects the same as C, and morphisms ho(c )(X, Y ) = π(rqx, RQY ). Remark Note in general that RQX is not the same thing as QRX. Thus one may ask: does using QRX instead of RQX in the definition of the homotopy category make a different category? The next theorem will answer that: both satisfy the same universal property, and thus are equivalent. We have a canonical functor γ : C = ho(c ) (this is why we require our factorization to be functorial). Theorem The functor γ is a localization of C at W, i.e., (i) γ takes W to isomorphisms (ii) It is the initial such functor Reuben Stern 37 Spring 2017

39 Math 231br 12.2 Brief Introduction to Derived Functors 12.2 Brief Introduction to Derived Functors Definition Let F : C D be a functor between model categories. The total left derived functor LF : ho(c ) ho(d) is the terminal functor with respect to C F D γ C γ D ho(c ) LF D We can (more or less) obviously dualize this definition, to arrive at that of a total right derived functor. A natural thing to ask, then, is: what functors can we derive? One class of such functors is Quillen functors: Definition F is a left Quillen functor if it is a left adjoint and preserves cofibrations and acyclic cofibrations. It is a right Quillen functor if it is a right adjoint and preserves fibrations and acyclic fibrations. Lemma (Ken Brown s Lemma). Let C be a model category, and D a category with a subcategory of weak equivalences, satisfying the two-out-of-three condition. If F : C D is a functor that takes acyclic cofibrations between cofibrant objects to weak equivalences, then F takes all weak equivalences between cofibrant objects to weak equivalences. Proof. (This proof is adapted from [Hov98].) Let f : A B be a weak equivalence between cofibrant objects. We can factor the map f id B : A B B as A B q f id B C p B Because A and B are cofibrant, the pushout square B A A B tells us that the inclusions i 0 : A A B and i 1 : B A B are cofibrations (cofibrations are closed under pushouts). By the two out of three axiom (MC2), we see that both q i 0 Reuben Stern 38 Spring 2017

40 Math 231br 12.2 Brief Introduction to Derived Functors and q i 1 are weak equivalences, and thus acyclic cofibrations of cofibrant objects. By assumption then, both F (q i 0 ) and F (q i 1 ) are weak equivalences. As F (p q i 1 ) = F (id B ) is a weak equivalence, the two out of three axiom again gives that F (p) is a weak equivalence, and thus that F (f) = F (p q i 0 ) is a qeak equivalence. The upshot of this is that a left Quillen functor preserves weak equivalences between cofibrant objects. Example Take the category of chain complexes, Ch R. Fix some M Ch R. We have an adjunction of functors M R Hom R (M, ). We get the derived functors M L R and RHom R (M, ). Taking homology gives Tor and Ext, respectively. 2. There is an adjunction between the topological realization functor : ssets Spaces and the singular simplex functor Sing : Spaces ssets. This is a Quillen equivalence, i.e., ho(ssets) = ho(spaces). 3. Homology is the left derived functor of abelianization, whatever the hell that means. In summary, the homotopy category, derived adjunctions, and derived natural transformations give the data of a pseudo-2-functor between ModelCat ho Cat ad. Reuben Stern 39 Spring 2017

41 Math 231br February 22, February 22, The Homotopy Theory of CW Complexes, II Remember from last time, we classified CW complexes (up to homotopy) with 0, 1, and 2 cells. Lemma For all n-equivalences B Y, and commutative diagrams B Y ω D n ω S n 1 D n 13.2 Eilenberg-MacLane Spaces Reuben Stern 40 Spring 2017

42 Math 231br February 24, February 24, 2017 Recall that yesterday we proved the following: Lemma There exist spaces K(A, n) with π K(A, n) = A if = n, and 0 otherwise. Lemma If π <n X and π >n Y = 0, then [X, Y ] [π n X, π n Y ]. The idea of the second lemma is that X has no new cells of dimension less than n. From this we conclude Corollary The K(A, n) are unique up to homotopy. A further corollary is: Corollary ΩK(A, n) K(A, n 1). Collections of spaces with this property are super interesting! We should read this as saying: the space K(A, n 1) has a space sitting above it of which it is the loop space. We call the space K(A, n) the delooping of K(A, n 1) Brown Representability Remark The functor Spaces / (, T ) describes a sheaf, i.e., the pasting lemma holds. Theorem Brown. Take a functor F : htop op / Sets / which satisfies 1. Wedge axiom: F ( α X α ) = α F (X α ) 2. Sheaf condition (sometimes called the Mayer-Vietoris axiom): if X = A 1 A 2, and f 1 F (A 1 ) and f 2 F (A 2 ) such that f 1 A1 A 2 = f 2 A1 A 2, then there exists some f F (X) such that f 1 = f A1 and f 2 = f A2. Then there is a complex Y and an element u F (Y ) (u for universal ) such that = [X, Y ] F (X) via f f u is a natural bijection. Also, for τ : F G between t two such functors, there is a unique map t : Y F Y G such that [X, Y F ] [X, YG ] is compatible. Reuben Stern 41 Spring 2017