Harvard University Math 231br

Notes for Advanced Algebraic Topology

Reuben Stern

Spring Semester 2017 Contents

0.1 Preliminaries ...... 4 0.2 Administrative Stuff ...... 4

1 January 23, 2017 5 1.1 Overview of Course...... 5

2 January 25, 2017 6 2.1 Some facts about the Category Spaces ...... 6 2.2 Example: Locality and Sheafiness...... 6 2.3 Example: Products of Spaces ...... 7 2.4 Example: Gluings (or Quotient Spaces)...... 7 2.5 Example: Space of Functions...... 8 2.6 Pairs of Spaces ...... 9

3 January 27, 2017 11 3.1 The Punchline from Last Class ...... 11 3.2 Perpsectives on the Fundamental Group ...... 11

4 January 30, 2017 15 4.1 Higher Homotopy Groups ...... 15 4.2 Exact Sequences of Spaces...... 16

5 February 1, 2017 18 5.1 Finishing Things from Last Time ...... 18 5.2 Dual Results: Exact Sequences ...... 18 5.3 Relative Homotopy Groups...... 19

6 February 3, 2017 21 6.1 The Action of the Fundamental Group...... 21

7 February 6, 2017 22 7.1 Fibrations...... 22

8 February 8, 2017 23 8.1 Fiber Bundles and Examples...... 23

9 February 10, 2017 24 9.1 CW Complexes...... 24

10 February 13, 2017 25 10.1 The Homotopy Theory of CW Complexes ...... 25 10.2 Homotopy Excision and Corollaries...... 25

1 Math 231br CONTENTS

10.3 Introduction to Stable Homotopy...... 25

11 February 15, 2017 26 11.1 Why Model Categories? ...... 26 11.2 Definitions, Examples, and Basic Properties ...... 28 11.3 Homotopy Relations ...... 31

12 February 17, 2017 33 12.1 The Homotopy Category of a Model Category...... 33 12.2 Brief Introduction to Derived Functors...... 38

13 February 22, 2017 40 13.1 The Homotopy Theory of CW Complexes, II...... 40 13.2 Eilenberg-MacLane Spaces...... 40

14 February 24, 2017 41 14.1 Brown Representability...... 41

15 February 27, 2017 44 15.1 Spectra ...... 44 15.2 The Category of Spectra...... 45

16 March 1, 2017 47 16.1 Co/homology Theories from Spectra ...... 47

17 March 3, 2017 51 17.1 The Hurewicz Theorem...... 51 17.2 Spectral Sequences ...... 51

18 March 6, 2017 54 18.1 Obstruction Theory...... 54

19 March 8, 2017 55 19.1 A Complicated Example...... 55

20 March 10, 2017 56 20.1 Smash Products...... 56

21 March 20, 2017 57 21.1 The Serre Spectral Sequence...... 57

22 March 22, 2017 58 22.1 More About the Serre Spectral Sequence...... 58 22.2 The Transgressive Differential Lemma...... 58 22.3 Multiplicative Structure ...... 58

Reuben Stern 2 Spring 2017 Math 231br CONTENTS

23 March 24, 2017 59 23.1 More Spectral Sequences...... 59 23.2 Multiplicative Extensions...... 59 ∗ 23.3 Computing H (K(Z/2, 1); F2) ...... 59

24 March 27, 2017 60 24.1 G-bundles and Fiber Bundles ...... 60

25 Properties of Chern Classes 64

26 The Steenrod Algebra and the Bar Spectral Sequence 65

27 April 10, 2017 66 27.1 The Steenrod Algebra, part II...... 66

28 April 12, 2017 67 28.1 A Serre Spectral Sequence Trick...... 67 28.2 Serre Classes ...... 67

29 April 14, 2017 68 29.1 Serre’s Method ...... 68

30 April 17, 2017 69 30.1 The Adams Spectral Sequence...... 69 30.2 A Fun Computation ...... 69

A Collected Homework Problems 71 A.1 Problem Set 1...... 71 A.2 Problem Set 2...... 72 A.3 Problem Set 3...... 73 A.4 Problem Set 4...... 73 A.5 Problem Set 5...... 77

B Notes from Switzer 79

References 80

Reuben Stern 3 Spring 2017 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 Mon- day/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 LATEX 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 [email protected]. 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 Math 231br January 23, 2017

1 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 Sn≥2. Its homotopy groups are notoriously difficult and important to compute.

Theorem 1.1. (Hurewicz) If X is (n − 1)-connected (that is, πkX = 0 for all k < n), ∼ then Hn(X; Z) = πnX. We also have a decomposition

X[n + 1, ∞] X K(πnX, n)

by a fiber sequence such that   πnX k = n πkX n + 1 ≤ k πkK(πnX, n) = , πkX[n + 1, ∞] = . 0 otherwise 0 otherwise

Then we can also plug the data of H∗X and H∗K(πnX, n) into a gadget known as the Serre spectral sequence to get H∗X[n + 1, ∞].

Reuben Stern 5 Spring 2017 Math 231br January 25, 2017

2 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 con- structions, 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 S Let X be a topological space and X = j Aj 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 fj : Aj → T agreeing on the overlap. We picture this as follows: Make pic- Equivalently, we demand fi|Ai∩Aj = fj|Ai∩Aj . This condition is called locality or a sheaf condition on the collection {fi}. For intution on where the term “sheaf condition” ture of comes from, here is an aside on sheaves! 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 resA,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 resB,C ◦ resA,B = resA,C .

• For a cover {Aj} of X, we think about the following diagram:

res res Aj ,Ak∩Aj X,Aj Q Q F (X) j F (Aj) k,` F (Ak ∩ A`) res Aj ,Aj ∩A`

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 Math 231br 2.3 Example: Products of Spaces

f Aside 2.2. (Equalizers.) An equalizer (of sets) E ⊆ S T is the maximal subset g of S on which f and g restrict to the same function.

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

i ∃! E

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 spaces1, 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 natural2 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 xRx0 implies f(x) = f(x0), then there is a factorization of continuous maps

f X Y

∃! X/R

1This 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. 2In the categorical sense; we will talk about this more later

Reuben Stern 7 Spring 2017 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 xRx0 if and only if x, x0 ∈ A. The quotient with respect to this equivalence relation is then called X/A (edge case3: i f X/∅ = X q {∗}). Suppose A X Y , and f ◦ i is constant. Then the diagram commutes: f A i X Y

fe X/A

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 X Y R×S R × 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 X × I × I ∼= . α 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 space4 of continuous functions X → Y .

3This is mostly a convention 4With 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 Math 231br 2.6 Pairs of Spaces

Lemma 2.9. If X is locally compact, then Y X × X −−−−−→ev Y is continuous. If X and Z are additionally Hausdorff, then

Y Z×X ∼= (Y Z )X .

This is perhaps familiar as currying.

∼ Corollary 2.10. 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) a (−)Z ), and X,Y changing. For example, given g : Y → Y 0, then the square

∼ Spaces(X × Z,Y ) = Spaces(X,Y Z )

g◦ g◦

∼ Spaces(X × Z,Y 0) = Spaces(X, (Y 0)Z ) commutes.

2.6 Pairs of Spaces Thinking back to homotopy, the fundamental group, relative homology, and other construc- tions in introductory algebraic topology, we often care a lot about pairs of spaces (A ⊆ X) or spaces with a choice of basepoint ({x0} ⊆ 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 2.11. 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 Math 231br 2.6 Pairs of Spaces

Example 2.12. Let (X, {x0}) and (Y, {y0}) be pointed spaces. Then the product is

(X, {x0}) × (Y, {y0}) = (X × Y, (X × {y0}) ∪ ({x0} × Y )).

But this distinguished subset of the product should be familiar: it is exactly the wedge

product of X and Y at x0 and y0!

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 Math 231br January 27, 2017

3 January 27, 2017

3.1 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, x0) to the pair (X, {x0}). 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, {x0}), (Y, {y0}) (X × Y,X ∨ Y )

i quot

 X×Y  (X, x0), (Y, y0) X∨Y , ∗ The (pointed) space (X × Y/X ∨ Y, ∗) is interesting because it participates in an exponential adjunction:

Z ∼ Spaces∗/(X,Y ) = Spaces∗/(X ∧ Z,Y ).

The space X ∧ Z is called the smash product. As a remark,

X ∧ Z 6= (X, x0) × (Y, y0).

So there are “two products” in Spaces∗/.

Aside 3.1. X ∨ Y is the categorical coproduct in Spaces∗/.

3.2 Perpsectives on the Fundamental Group

I [0,1] Recall the path space of X is the function space X = X . The set π0(X) of path components of X is thus the set of connected components of XI . We will write [Y,X] to denote the set of homotopy classes of maps Y → X. In this notation, π0(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 0 0 π0(X, x0) = [S ,X], where S = {±1}. Note that we will more often than not suppress basepoints in pointed spaces5.

5Also, for the foreseeable future, all spaces will be pointed unless explicitly stated otherwise.

Reuben Stern 11 Spring 2017 Math 231br 3.2 Perpsectives on the Fundamental Group

Y Remark 3.2. [Y,X] can be expressed as π0(X ).

Recall that π1(X, x0) is the collection of based loops in X, taken up to homotopy. With 1 1 0 1 0 1 1 our notation, this is [S , (X, x0)]. Note that S = S ∧ S : S × S is two copies of S , and then we collapse down S0 ∨ S1 = {∗} q S1 to the basepoint on the other copy of S1, leaving 1 0 1 0 S1 S1 us with just the other copy. Therefore [S ,X] = [S ∧ S ,X] = [S ,X ], which is π0(X ). The space XS1 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 π1(X) is a group.

Question 3.3. Expressing π1 as above, what is special about the functors π0Ω(−) or [S1, −] that makes these functors group-valued?

Eric: “Naturallya, we’re going to address this problem with category theory.” aNo 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 1×η η×1 G × G × G G × G G ∼= G × ∗ G × G G ∼= ∗ × G

µ×1 µ µ 1 1

G × G µ G G

χ×1 1×χ G ∆ G × G G × G G × G ∆ G

µ

η η ∗ G ∗

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 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 µ0 : Sets(X,G) × Sets(X,G) ' Sets(X,G × G) ∗ Sets(X,G),

0 where µ∗ is postcomposition by µ. We do a similar thing with η and χ to define η and χ0. 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 diagrams6 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 [S0, Ω(X)] = Sets(S0, Ω(X)) is automatically a group (“that’s cute!”). In fact, [Y, ΩX] is automatically a group for any set Y . 1 7 Answer 2: “Adjoint-ly”, it turns out that S is a cogroup object in 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

µ0 : C → C ∨ C, η0 : C → ∗, and χ0 : C → C, satisfying properties of “coassociativity”, “counitality”, and “coinversion.” For instance, coassociativity states that the diagram

1∨µ0 C ∨ C ∨ C C ∨ C

µ0∨1 µ0

C ∨ C C µ0 commutes. An equivalent definition is that a cogroup is a group object in the opposite category.

6A functor applied to a commutative diagram yields a commutative diagram 7This is the homotopy category of pointed spaces, where morphisms are homotopy classes of maps

Reuben Stern 13 Spring 2017 Math 231br 3.2 Perpsectives on the Fundamental Group

Example 3.7. What do the cogroup operations look like for S1? We have a “pinch” map µ0 : S1 → S1 ∨S1, a collapsing map η0 : S1 → ∗, and a “flipping” map χ0 : S1 → S1. Eric: “I wish that I had three hands.” We then show that the diagrams commute (this is easy but laborious; left as an exercise).

1 Remark 3.8. S ∧(−) 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 S1 ∧ (−) 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 0, g0 : X → ΩY . Then the composition µ0 f∨g 0 ΣX ΣX ∨ ΣX Y ∨ Y ∆ Y defines f ◦ g, where the map ∆0 is called the folding map. This map is defined by sending

Y ∨ Y 3 (y, y0) 7→ y and Y ∨ Y 3 (y0, y) 7→ y. We can also use the composition

f 0×g0 µ X ∆ X × X ΩY × ΩY ΩY to define f 0 ◦ g0. So we have f, g, fg ∈ [ΣX,Y ] and f 0, g0f 0g0 ∈ [X, ΩY ], and we also have (fg)0 ∈ [X, ΩY ]. The question then becomes, is f 0 ◦ g0 = (f ◦ g)0? In some sense, you just have to work this out from the definitions. It is not hard, just laborious.

Reuben Stern 14 Spring 2017 Math 231br January 30, 2017

4 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

n 0 Definition 4.1. We define the homotopy groups of a space X by πn(X, x0) = [Σ S ,X]. By the suspension-loop space adjunction, we have the equality of spaces

[ΣnS0,X] = [Σn−1S0, ΩX] = ··· = [S0, ΩnX].

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 ∗ x0) ◦ (y ∗ y0) = (x ◦ y) ∗ (x0 ◦ y0). Then ∗ and ◦ agree, and both are associative and commutative.

Proof. By computation: x ◦ y = (x ∗ e) ◦ (e ∗ y) = (x ◦ e) ∗ (e ◦ y) = x ∗ y. Also, x ◦ y = (e ∗ x) ◦ (e ∗ y) = (e ◦ y) ∗ (e ◦ x) = y ∗ x. Associativity is similar. The plan is thus to check that the products on [Σn−1S0, 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. [Σn−1S0, 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:

0 (f∨f ) fold × × µ0×µ0 (g∨g0) K × K (K ∨ K) × (K ∨ K) (L ∨ L) × (L ∨ L) fold L × L ∆

(K×∗)×(K×∗) (f×f 0)∨(g×g0) (L×∗)×(L×∗) K ∆K ∪ ∪ L (∗×K)×(∗×K) (∗×L)×(∗×L) fold (f×g) µ0 1×T ×1 ∨ 1×T ×1 (f 0×g0) µ∨µ K ∨ K ∆×∆ (K × K) ∨ (K × K) (L × L) ∨ (L × L) L ∨ L

Remark 4.4. There is a homeomorphism Sn+1 ∼= ΣSn.

Reuben Stern 15 Spring 2017 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

Y q CX Z = Y ∪ CX = , f f(x) ∼ (x, 1) where the cone on a topological space CX is defined to be X × I/(X × {0} ∪ {x0} × 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

f∗γ X

8Eric: “Do you still call them air quotes when you write them on the board?” 9Exercise: convince yourself of this.

Reuben Stern 16 Spring 2017 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 4.10. The map

(Y ∪f CX) ∪i CY −→ ((Y ∪f CX) ∪i CY )/CY

is a homotopy equivalence.

∼ Lemma 4.11. For A ⊆ X, there is a homeomorphism (X ∪i CA)/CA = X/A.

∼ ∼ ∼ Corollary 4.12. (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 2 X Y Y ∪f CX ΣX ΣY Σ(Y ∪f CX) Σ X ···

This gives rise to the exact sequence

[X,T ] [Y,T ] [Cf, T ] [ΣX,T ] [ΣY,T ] [Σ(Cf),T ] [Σ2X,T ] ··· .

We get some interesting fringe phenomena at the maps [ΣX,T ] → [Cf, T ] and [Σ2X,T ] → [Σ(Cf),T ], where the target set has less structure than the initial.

Reuben Stern 17 Spring 2017 Math 231br February 1, 2017

5 February 1, 2017

5.1 Finishing Things from Last Time

f Recall from last time: for a map X −→ Y , we can iteratively construct Y ∪f CX =: Cf to get a sequence

f X Y Cf ΣX ΣY Σ(Cf) ··· .

Applying the functor [−,T ] for some test space T , we get something interesting happening at

HomhTop∗/ (ΣX,T ) HomhTop∗/ (Cf, T ): there is a natural action of [ΣX,T ] on [Cf, T ].

Lemma 5.1. There is a factorization

HomhTop∗/ (Cf, T ) HomhTop∗/ (Y,T )

Hom (Cf,T ) hTop∗/ Hom (ΣX,T ) hTop∗/

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

f X commutes. Since CX = XI, we can think of this as a map X → Y I .

f Definition 5.2. For a map X −→ Y , the path space Pf of f is

I Pf = {(x, γ) ∈ X × Y : γ(1) = f(x)}.

The next two lemmas follow by duality from things proved last time:

Reuben Stern 18 Spring 2017 Math 231br 5.3 Relative Homotopy Groups

prX f Lemma 5.3. The sequence Pf X Y is exact in Spaces∗/.

Lemma 5.4. Iterating this construction yields a long exact sequence in Spaces∗/:

2 2 f ··· Ω X Ω Y ΩPf ΩX ΩY Pf X Y.

Applying π0, it follows that

π1Pf π1X π1Y π0Pf π0X π0Y is an exact sequence of homotopy groups. The major question then becomes:

Question 5.5. What is this Pf 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, {x0}) is an object of Pair∗/. The space Pi is then the mapping space

(I,∂I,0) Pi = (X, A, {x0}) .

By the exponential adjunction, we find that

n−1 n n−1 πn−1Pi = [(S , ∗), (Pi, γ0)] = [(D ,S ), (X,A)], which we define to be the relative homotopy group πn(X,A).

Corollary 5.6. There is a long exact sequence

··· π2(X,A) π1A π1X π1(X,A) π0A π0X.

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 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,−→ Mf of X into the (reduced) mapping cylinder of f, i and an inclusion Y ,−→ Mf by sending Y to Y × {0}. The diagram

Mf i

f Y 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:

··· πn+1(X,A) πn(A, B) πn−1B πn−1X ···

πnA πn(X,B) πn−1A

··· πnB πnX πn(X,A) πn−1(A, B) ···

Lemma 5.9. The highlighted sequence is exact.

Reuben Stern 20 Spring 2017 Math 231br February 3, 2017

6 February 3, 2017

6.1 The Action of the Fundamental Group

Maybe you recall that there is an action of π1X on πnX. If not, you may remember that a −1 path γ : I → X induces an isomorphism Γ: π1(X, γ(0)) → π1(X, γ(1)) by γ · 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 · (a1a2) = (g · a1)(g · a2).

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. S1 coacts compatiblya on Sn for all n ≥ 1. aOr perhaps we should say “coacts mpatibly”.

From this theorem, we get the useful corollary:

Corollary 6.4. π1X acts compatibly on πnX 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 Math 231br February 6, 2017

7 February 6, 2017

7.1 Fibrations j f Remember that we had this special object Pf , fitting into an exact sequence Pf X Y , where Pf was defined as

I Pf := {(γ, x) ∈ Y × X : γ(1) = f(x)}. I We have a homotopy equivalence ΩY ' Pj := {(α, (γ, x)) ∈ X × Pf : α(1) = x, γ(1) = f(x)}. The picture is enough to see this: make picture

Example 7.1. Given X and Y spaces, πn(X ×Y ) = πnX ×πnY . Indeed, the sequence

πnY πn(X × Y ) πnX

is exact.

Definition 7.2. A map p : E → B has

Reuben Stern 22 Spring 2017 Math 231br February 8, 2017

8 February 8, 2017

8.1 Fiber Bundles and Examples

Reuben Stern 23 Spring 2017 Math 231br February 10, 2017

9 February 10, 2017

9.1 CW Complexes

Reuben Stern 24 Spring 2017 Math 231br February 13, 2017

10 February 13, 2017

10.1 The Homotopy Theory of CW Complexes Let X be a space. The map g : Sn−1 → X gives a way of attaching one n-cell to X n−1 to make the space X ∪g CS . It is natural to ask: if g1, g2 are both representatives for the same homotopy class [g] ∈ πn−1(X), do they both give homotopy equivalent spaces? That is to say, given a homotopy H : g1 ' g2, do we get a homotopy equivalence n−1 He n−1 X ∪g1 CS −−→ X ∪g2 CS ? The answer, of course, is yes. A schematic of this is as follows (and in the diagram): slide g1 over via H. Part of the cone is homeomorphic to make n−1 I × S ; this “fills in” the part of the homotopy given by sliding g1 (of course, all of this is dia- in very imprecise language). We can then “pull the top of the cone” down over the rest of gram it, to give us the attachment on g2. n−1 Ultimately, our goal is to become familiar with the behavior of X ∪g CS as a homotopy type.

Lemma 10.1. For (X,A) a relative CW complex, the pair (X, (X,A)n), where (X,A)n n is the n-skeleton of (X,A), is n-connected: π≤n(X, (X,A) ) = 0.

Corollary 10.2. The inclusion Xn ,−→ X is n-connected (or, the pair (X,Xn) is).

Proof. Use seriously the simplicial approximation theorem from last time.

n Corollary 10.3. π

Proof. We use the CW structure of Sn as one n-cell glued to a point. Then (Sn)n−1 = ∗, n n−1 n and S = ∗ ∪∗ CS . By the lemma, π

Lemma 10.4. If (X,A) is n-connected, then there exists a pair (X0,A0), homotopy equivalent to (X,A), such that (X0,A0)n = A0. This means that X0 has no new cells in it until at least dimension n + 1.

Corollary 10.5. 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.”

10.2 Homotopy Excision and Corollaries 10.3 Introduction to Stable Homotopy

Reuben Stern 25 Spring 2017 Math 231br February 15, 2017

11 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.

11.1 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 11.1. 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 → · ←−· → · ←−· · · → · ←− Y }

Note that the backwards arrows are inversions of arrows in W .

Problem 11.1.1. 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 11.2. There exists a simplicial category LH 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

LH C

Reuben Stern 26 Spring 2017 Math 231br 11.1 Why Model Categories?

Aside 11.3. 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;

veck is endowed over itself, where veck(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 7→ 1, 1 7→ 2, . . . , n 7→ 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 11.4. We now define the hammock localization LH C . The objects ob LH C are just ob C . Because LH 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.

H −1 Fact 11.5. π0 Mor(L C ) = Mor(C [W ]).

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 0 : ho C → ho D such that the diagram

C F D

0 ho C F ho D

Reuben Stern 27 Spring 2017 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 11.6. Let’s take the time to define Kan extensions.

Definition 11.7. Let F : C → E and K : C → D be functors. A left Kan extension of F along K is a functor LanK F : D → E , together with a natural transformation η : F ⇒ LanK F ◦ K satisfying the following universal property: given other pair (G : D → E , γ : F ⇒ G ◦ K), there is a unique natural transformation α : LanK F ⇒ G such that γ = α ◦ η. This can be summarized in the following diagrams:

C F E C F E η LanK F γ G = α K K K D D

Right Kan extensions are defined dually, with the natural transformation arrows reversed.

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.

11.2 Definitions, Examples, and Basic Properties Definition 11.8. Suppose we are given a diagram as follows, where the outer square commutes: A B

i p

X Y 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 Math 231br 11.2 Definitions, Examples, and Basic Properties

Definition 11.9. A model category C is a category with 3 distinguished wide subcate- gories10

∼ • 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 f · · and · ·

∼ ∼ · ·

Remark 11.10. Recall that a map f is a retract of g if there is a commuting diagram

idA A B A

f g f X Y X

idX

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 11.11. 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 11.12. We call the category W ∩fib the class of acyclic fibrations, and W ∩cof is the class of acyclic cofibrations. 10A wide subcategory is a subcategory that includes all objects and all identity morphisms.

Reuben Stern 29 Spring 2017 Math 231br 11.2 Definitions, Examples, and Basic Properties

Definition 11.13. 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 11.14. (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 ChR 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 = ChR 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 11.15.

cof = llp(W ∩ fib) fib = rlp(W ∩ cof) W ∩ cof = llp(fib) 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

r f pe X Y

Reuben Stern 30 Spring 2017 Math 231br 11.3 Homotopy Relations so we can get a lift r. Redraw this as

A A A

f i f

p X r B X

This is a retract diagram, so (MC3) gives that f is a cofibration.

Proposition 11.16. 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.

11.3 Homotopy Relations Fix a model category C .

Definition 11.17. A (good) cylinder object for A ∈ ob C is a factorization of the fold map ∇ : A t A → A as A t A cyl(A) ∼ A.

Two maps f, g : A → X are left homotopic (written f ∼` g) if there exists a cylinder object for A and a map cyl(A) −→H X such that the diagram

A f

A t A i0ti1 cyl(A) H X

A g commutes.

Lemma 11.18. With notation as above, if A is cofibrant, then i0 and i1 are cofibrations.

Reuben Stern 31 Spring 2017 Math 231br 11.3 Homotopy Relations

Proof. We have a factorization

A i0 cyl(A)

η0 A t A

Why is η0 a cofibration? The square

∅ A

η0 p A A t A is a pushout square, so η0 deserves a tail.

Proposition 11.19. If A is cofibrant, then ∼` is an equivalence relation on maps C (A, X).

Reuben Stern 32 Spring 2017 Math 231br February 17, 2017

12 February 17, 2017

12.1 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 ∼` g) if there exists a map cyl(A) −→H X such that the diagram f A

A t A cyl(A) H X

A g commutes. From this, it is clear that ∼` is symmetric and reflexive; the tough part is to show transitivity.

Proposition 12.1. Left homotopy is an equivalence relation.

Proof. Suppose f ∼` g via an object cyl(A) and a map H, and g ∼` h via an object cyl(A)0 and a map H0. The relevant diagram to chase is

A

i0 A ∼ cyl(A)

∼ ∼ i0 ∼ A 1 cyl(A)0 ∼ p 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 Math 231br 12.1 The Homotopy Category of a Model Category diagram commuting: f A cyl(A)

C X

A cyl(A) g which gives us the left homotopy relation.

` We thus define π`(A, X) to be the set of equivalence classes C (A, X)/ ∼.

Lemma 12.2. Suppose X is fibrant, and f ∼` g : A → X. Then if h : A0 → A is a map, we have fh ∼` gh.

Proof. Step 1: There exists a map H : cyl(A) → X witnessing f ∼` g. Factor H (by the axiom MC5) as A t A cyl(A) ∼ A

∼ ∼

cyl(A0)

Step 2: How do we know cyl(A0) interpolates f and h? Because X is fibrant, we have the diagram cyl(A) H X H ∼ e cyl(A0) ∗ which gives us a lift Hf. We can thus produce a “good cylinder object”. Step 3: Because cyl(A0) is a cylinder object for A0, we get the diagram

A0 t A0 hth A t A cyl(A) k ∼ cyl(A0) ∼ A0 h A

Reuben Stern 34 Spring 2017 Math 231br 12.1 The Homotopy Category of a Model Category which commutes, giving us a lift k. We thus have the diagram

A0 h A f

cyl(A0) k cyl(A) H X g

A0 h A giving H ◦ k as a left homotopy.

Definition 12.3. 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 12.4. Suppose f, g : A → X.

(a) If A is cofibrant, then f ∼` g ⇒ f ∼r g.

(b) If X is fibrant, then f ∼r g ⇒ f ∼` 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`(X,A) = πr(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 Math 231br 12.1 The Homotopy Category of a Model Category

Theorem 12.5. 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

f A ∼ X ∼ ∼ q p C

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 = idX .

Lemma 12.6. If A is cofibrant, and p : C → X is an acylic fibration, then

` ` p∗ : π (A, C) −→ π (A, X)

is a bijection.

` Assuming the lemma, we see that sp ∼ idC . 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

f A X ∼ p q C

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 ∼ idX . We have the diagram g q X A C He i0 ∼ p

cyl(X) H X.

Reuben Stern 36 Spring 2017 Math 231br 12.1 The Homotopy Category of a Model Category

Set s = Hif 1. Then ps = idX . We also have a homotopy inverse for q, say r. As pq = f, pqr = fr ⇒ p ∼ fr, so sp ∼ qgp ∼ qgfr ∼ qr ∼ idC .

Aside 12.7. 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 12.8. 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 12.9. 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 12.10. 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 Math 231br 12.2 Brief Introduction to Derived Functors

12.2 Brief Introduction to Derived Functors Definition 12.11. 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

γC γD

ho(C ) D LF

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 12.12. 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 12.13. (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 t idB : A t B → B as

ftid A t B B B p q ∼ C

Because A and B are cofibrant, the pushout square

∅ A

p B A t B tells us that the inclusions i0 : A → A t B and i1 : B → A t B are cofibrations (cofibrations are closed under pushouts). By the two out of three axiom (MC2), we see that both q ◦ i0

Reuben Stern 38 Spring 2017 Math 231br 12.2 Brief Introduction to Derived Functors

and q ◦ i1 are weak equivalences, and thus acyclic cofibrations of cofibrant objects. By assumption then, both F (q◦i0) and F (q◦i1) are weak equivalences. As F (p◦q◦i1) = F (idB) 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 ◦ i0) is a qeak equivalence. The upshot of this is that a left Quillen functor preserves weak equivalences between cofibrant objects.

Example 12.14. 1. Take the category of chain complexes, ChR. Fix some M ∈ ChR. We have an adjunction of functors

M ⊗R − a HomR(M, −).

L We get the derived functors M ⊗R − and RHomR(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 trans- formations give the data of a pseudo-2-functor between

ho ModelCat −−−−−→ Catad.

Reuben Stern 39 Spring 2017 Math 231br February 22, 2017

13 February 22, 2017

13.1 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 13.1. For all n-equivalences B,−→ Y , and commutative diagrams

B Y

ω|∂Dn ω Sn−1 Dn

13.2 Eilenberg-MacLane Spaces

Reuben Stern 40 Spring 2017 Math 231br February 24, 2017

14 February 24, 2017

Recall that yesterday we proved the following:

Lemma 14.1. There exist spaces K(A, n) with π∗K(A, n) = A if ∗ = n, and 0 otherwise.

' Lemma 14.2. If π∗nY = 0, then [X,Y ] −−→ [πnX, πnY ].

The idea of the second lemma is that X has no new cells of dimension less than n. From this we conclude

Corollary 14.3. The K(A, n) are unique up to homotopy.

A further corollary is:

Corollary 14.4. 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).

14.1 Brown Representability

Remark 14.5. The functor Spaces∗/(−,T ) describes a sheaf, i.e., the “pasting lemma” holds.

op Theorem 14.6. Brown. Take a functor F : hTop∗/ → Sets∗/ which satisfies W Q 1. Wedge axiom: F ( α Xα) = α F (Xα)

2. Sheaf condition (sometimes called the Mayer-Vietoris axiom): if X = A1 ∪A2,

and f1 ∈ F (A1) and f2 ∈ F (A2) such that f1|A1∩A2 = f2|A1∩A2 , then there exists

some f ∈ F (X) such that f1 = f|A1 and f2 = 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 7→ f ∗u is a natural bijection. Also, for τ : F ⇒ G between t∗ two such functors, there is a unique map t : YF → YG such that [X,YF ] −−→ [X,YG] is compatible.

Reuben Stern 41 Spring 2017 Math 231br 14.1 Brown Representability

Remark 14.7. This is sort of a converse to two claims: first, if you are given a functor, you can construct a representing object. Thus, the second part of the theorem is a consequence of Yoneda and the first part. Definition 14.8. An element u ∈ F (Y ) is n-universal if [Sq,Y ] → F (Sq) taking ω 7→ ω∗u is onto for all q ≤ n, and an isomorphism for all q < n.

Proof of Brown Representability. Suppose that un ∈ F (Y ) is n-universal. We want to ∗ correct un and Y to be (n + 1)-universal. Consider A = {α ∈ πnY : α un = 0} and L = F (Sn+1). Form W _ n α _ n+1 mapping cone 0 Sα −−−→ Y ∨ Sλ −−−−−−−→ Y . α λ Applying F , we have

W n  W n+1 0 F ( α Sα) F Y ∨ λ Sλ F (Y ) ∈

W ∗ W α α un (un, λ λ) ∃ un+1

Lemma 14.9. For Y a space with universal elementa u ∈ F (Y ), (X,A) a CW-pair, ∗ v ∈ F (X), g : A → Y a cellular map such that g u = v|A, there exists a cellular map eg ∗ X −→ Y extending g and classifying v, in the sense that v = ge (u). ai.e., n-universal for all n

Proof idea. Glue A × I into X and Y by attaching A0 to A in X, and A1 to g(A) in Y . Call this space T . We can split it into two halves: T1 and T2, where they overlap a little bit. For instance, we can have T1 be Y and the last 2/3rds of A × I, and T2 be X and the first 2/3rds. Note that u ∈ F (T1) and v ∈ F (T2). Gluing together, we get an element w ∈ F (T ). This is possible because v and u agree on the intersection of T1 and T2 by the assumption ∗ g u = v|A. We can thus extend T to a CW-pair (Y 0,T ) with universal element u0 restricting to w on T . There is a weak equivalence Y −−→j Y 0. Why? There is an inclusion of Y,−→ Y 0, and both Y and Y 0 have some universality condition. The square:

0 π∗Y π∗Y

=∼ =∼ F (S∗) F (S∗)

Reuben Stern 42 Spring 2017 Math 231br 14.1 Brown Representability commutes, so by Whitehead’s theorem, j is a homotopy equivalence. So j has a homotopy inverse j−1, and the composite

j−1 Y Y 0 X does the job of our wanted map ge.

Example 14.10. We like the spaces S1 and CP ∞, because they are K(Z, 1) and K(Z, 2), respectively. Suppose one wanted to build the functor that S1 represents “from scratch.” That is, start from a point, get a natural transformation [−, ∗] → [−,S1]. The Brown Representability proof says you should take a bouquet of 1-spheres, and W 1 ∞ consider [−, Z S ]. You can say a similar bunch of words for CP .

Finally, we can finish the proof:

Proof of Brown Representability, finished. To get surjectivity (of F (X) ←− [X,Y ]), take A = ∗ in the lemma. To get injectivity, set X0 = X × I, and set A0 = X × ∂I. Use A0 to support g1 and g2. Use the lemma, so g1 = g2.

Remark 14.11. Brown representability applied to Hfn(−,A) gives K(A, n). So Eilenberg- Maclane spaces are intimately rooted in simplicial homology.

Reuben Stern 43 Spring 2017 Math 231br February 27, 2017

15 February 27, 2017

Recall from last time: Brown Representability told us “if it looks like a sheaf, it is a sheaf.”

Aside 15.1. Recall that a sheaf has (i) the sheaf condition (i.e., Mayer-Vietoris axiom) and (ii) the wedge axiom. Furthermore, if you have a sequence of co/homology functors, we have hen+1(ΣX) ∼= hen(X), and the isomorphism is natural. ∗ There is thus associated to each co/homology theory he a sequence of spaces hn such that the following diagram commutes:

n ∼ he (X) = hTop∗/(X, hn) induced by hn→hn+1 ∼ = =∼ n+1 ∼ ∼ he (ΣX) = hTop∗/(ΣX, hn+1) = hTop∗/(X, Ωhn+1).

We can also read the results of Brown Representability backwards, and have the spaces

hn define a cohomology theory.

15.1 Spectra Goal 15.2. Use this presentation of cohomology functors by sequences of spaces to blend ho- motopy theory into the category of cohomology theories (with maps natural transformations of cohomology functors). More precisely, we want a category where

1. Cohomology theories “live” in this category as sungle objects.

2. There is a map from Spaces∗/ to this category in such a way that

[X, h] “computes” h0(X).

3. This functor in from Spaces∗/ is compatible with all of the connectivity-stabilized theorems from chapter 6 of Switzer.

Definition 15.3. A spectrum is a collection of CW-complexes {En}n together with maps (i.e., inclusions) ΣEn ,−→ En+1. Although we have not yet defined maps of spectra, the potential category Spectra receives a map from Spaces∗/ : a space X maps to the collection n ∞ {Σ X}n=0, with inclusions the identity maps. This is called the suspension spectrum of X, and is written Σ∞X.

Reuben Stern 44 Spring 2017 Math 231br 15.2 The Category of Spectra

Example 15.4. The suspension spectrum S = Σ∞S0 is called the stable sphere. It is sometimes written S0, to differentiate from Sn := Σ∞Sn.

15.2 The Category of Spectra We now work towards defining maps of spectra, so that they form a legitimate category Spectra.

Warning 15.5. Remember that we once calculated

0 1 2 3 π0S = {±1}, π1S = Z, π2S = Z, π3S = Z,... so if we were to define a map of spectra to be a compatible system of maps, what would the homotopy class of maps [S0, S0] be? Suppose we picked a map

0 fn 0 (S )n (S )n for each n, such that the diagram

0 fn 0 (S )n (S )n

0 Σfn−1 0 Σ(S )n−1 Σ(S )n−1 commutes. Because the linking (vertical) maps are surjective, fn is determined exactly by Σfn−1, so none of the fn≥1 matter. It follows that

0 0 [S , S ] = {±1}, which is the wrong answer.

We thus need to find a different definition of maps of spectra.

Definition 15.6. A subspectrum {Fn} of a spectrum {En} is a collection of subcomplexes Fn ⊆ En with compatible linking maps, induced by the linking maps for {En}.A cofinal m subspectrum is one for which any cell e ⊆ En eventually appears (after a finite number of ` ` linking map applications) in some Σ Fn ⊆ Σ En.

Example 15.7. A sort of n-truncation of the stable sphere is a cofinal subspectrum: we take (∗, ∗,..., ∗,Sn,Sn+1,... ) ⊆ (S0,S1,...,Sn−1,Sn,Sn+1,... ).

Definition 15.8. A morphism between two spectra E → G is a choice of cofinal sub- spectrum F ⊆ E and a compatible sequence of maps fn : Fn → Gn. Two morphisms are

Reuben Stern 45 Spring 2017 Math 231br 15.2 The Category of Spectra equivalent if they agree on a mutually cofinal subspectrum:

F 0 E

G

F 00 F

Remark 15.9. The map F,−→ E described by the inclusion of a cofinal subspectrum is equivalent to the identity map.

Definition 15.10. Two maps E G are homotopic if there is a common cofinal subspectrum and a witnessing level-wise homotopy

00 F ∧ I+ G.

Lemma 15.11. Spectra have wedge sums and cofiber sequences/mapping cones.

Corollary 15.12. As a corollary, we have a Whitehead theorem for spectra: set π E = [ n,E] = lim π (E ). If a map E → G induces an isomorphism on π , n S −→k→∞ n+k k ∗ then it is a homotopy equivalence.

Remark 15.13. We also need to account for n ∈ Z−, as seen by the following lemma.

Lemma 15.14. There is always a map (ΣEn)n −→ (En+1)n, which is an inclusion of a cofinal subspectrum, so an equivalence.

−1 Corollary 15.15. There is an inverse functor (Σ E)n = En−1.

Thus we have isomorphisms ΣSn ' Sn + 1, and can define S−1 = Σ−1S0.

Corollary 15.16. [Σ2Σ−2X, Σ2Σ−2Y ] is an abelian group, so [X,Y ] is too.

Reuben Stern 46 Spring 2017 Math 231br March 1, 2017

16 March 1, 2017

16.1 Co/homology Theories from Spectra Last time, we built this category Spectra such that the functor Σ∞ induces the diagram

∞ Spaces Σ Spectra

∞ hTop Σ hSpectra, and where Brown Representability gives a functor CohomThy −−−→B.R. hSpectra such that

Ee0(X) = hSpectra(Σ∞X,E), where E is the spectrum associated to the cohomology functor.

Question 16.1. Why do we just get cohomology theories? We’re missing half the picture here.

Definition 16.2. Given a spectrum E, we define its associated cohomology functors by

Een(X) = hSpectra(Σ∞X, ΣnE), and

Een(X) = πn(E ∧ X), where the spectrum E ∧ X is given by (E ∧ X)n = En ∧ X, and the suspension maps are the natural ones.

Claim 16.3. These are, in fact, co/homology theories.

∞ ∞ m m Lemma 16.4. [Σ X, Σ Y ] = limm→∞[Σ X, Σ Y ].

Remark 16.5. The assigment E (E , ΣE → E ) presents E as lim Σ−nΣ∞E , i.e., n n n+1 −→ n E is an ind-system of desuspensions of suspension spectra. Maps between ind-systems

Xα and Yβ are generally computed by

lim lim[X ,Y ], ←− −→ α β α β which looks like our business with cofinal subspectra.

We check the axioms for co/homology:

Reuben Stern 47 Spring 2017 Math 231br 16.1 Co/homology Theories from Spectra

1. (Suspension.)

Een+1(ΣX) = hSpectra(Σ∞ΣX, Σn+1E) = hSpectra(ΣΣ∞X, Σn+1E) = hSpectra(Σ∞X, ΣnE) n+1 n Een+1(ΣX) = [S ,E ∧ ΣX] = [S ,E ∧ X] = Een(X).

i 2. (Cofibrations.) If A ,−→ X −→ X ∪i CA is a coexact sequence, then

∞ ∞ ∞ ∞ ∞ Σ A −→ Σ X −→ Σ (X ∪i CA) = Σ X ∪i CΣ A

is coexact. Apply the functor E ∧ −, we have

∞ ∞ ∞ E ∧ Σ A E ∧ Σ X Σ (X ∪i CA) ∧ E

E ∧ A E ∧ X

checking that sequence is coexact is 100% a matter of unwinding the definitions.

3. (Wedge.)

! ! n _ ∗ ∞ _ n Ee Xα = hSpectra Σ Xα, Σ E α α ! ∗ _ ∞ n = hSpectra Σ Xα, Σ E α ∗ Y ∞ n Y n = hSpectra (Σ Xα, Σ E) = Ee (Xα). α α Apparently, all the equalities marked with a red ∗ are easy enough to believe.

Homology is considerably less clear. For instance, we have

! ! " # _ _ n _ Een Xα = πn E ∧ Xα = S ,E ∧ Xα α α∈A α∈A   " # ∼ = n, lim E ∧ _ X ←−−= lim n,E ∧ _ X S −→ α −→ S α S⊆A α∈S S finite α∈S Y = lim Ee (X ) −→ n α S α∈S

The isomorphism arrow uses compactness of Sn in a nontrivial way.

Reuben Stern 48 Spring 2017 Math 231br 16.1 Co/homology Theories from Spectra

Remark 16.6. The functors

B.R. CohomThy hSpectra ∗ (f−) do not give an equivalence of categories. It is almost an equivalence, but fails because of the existence of things called phantom maps of spectra: maps between spectra that cannot are not associated to natural transformations of the corresponding cohomology theories. However, we can get our hands on a lot of this “almost-equivalence” in one direction by Whitehead’s theorem, and on the other side by inventing spectral sequences. Remark 16.7. The cohomology of a union of spaces does not in general equal the limit of the cohomologies. Instead, we can invoke Milnor Sequences; see Homework #3. Remark 16.8. The map from CohomThy −→ hSpectra is secretly defined on objects at the level of spectra, without passing to the homotopy category. These spectra satisfy an adjunction:

Σhn hn+1 ⊥ hn Ωhn+1 The spectra are called Ω-spectra, and they satisfy

hSpectra(Σ∞X, h) = lim[ΣnX, h ] is a constant system. −→ n In general, one can convert an arbitrary spectrum into an Ω-spectrum as follows: note for example

hSpectra(Σ∞X, Σ∞Y ) = lim[ΣnX, ΣnY ] −→ = lim[X, ΩnΣnY ] −→ h i = X, lim ΩnΣnY , −→ and we set y = lim ΩnΣnY . More generally, lim ΩnΣn(ΣmY ) = y . 0 −→ −→ m If we apply this recipe to the spaces in an arbitrary spectrum, we recover a sequence of spaces in an equivalent Ω-spectrum:

E = lim ΩnE . m −→ m n

Remark 16.9. The functor − ⊗Z Z(p), where Z(p) is Z localized at the prime ideal (p), preserves exact sequences of abelian groups. So the assignment

∗ ∗ X 7→ E (X) 7→ E (X) ⊗Z Z(p) also sends cofiber sequences to long exact sequences, hence is represented by a spectrum

Reuben Stern 49 Spring 2017 Math 231br 16.1 Co/homology Theories from Spectra

E(p). The functor Hom(−, Q/Z) also does this, so

X 7→ E∗(X) 7→ Hom(E∗(X), Q/Z) is represented by a spectrum denoted IQ/Z(E).

Reuben Stern 50 Spring 2017 Math 231br March 3, 2017

17 March 3, 2017

17.1 The Hurewicz Theorem

Recall: the natural map Sn → K(Z, n) is an n-equivalence, so the induced map of spectra S0 → HZ is a 0-equivalence. This is equivalent to saying there is a fibration of spectra

fib F S0 HZ where π∗≤0F = 0

Theorem 17.1. (Hurewicz) If X is (n − 1)-connected (or n connective), there is an ∼ isomorphism πnX = Hn(X; Z).

Proof. Apply the functor − ∧ X to the fibration above to get

F ∧ X S0 ∧ X HZ ∧ X coexact, so homotopy induces a long exact sequence

··· π∗+1HZ ∧ X

∂∗ 0 π∗F ∧ X π∗S ∧ X π∗HZ ∧ X

∂∗

π∗−1F ∧ X ···

Which in turn induces an isomorphism (when ∗ = n)

0 ∼ πn(S ∧ X) = πn(HZ ∧ X). ∼ This is more commonly written πnX = Hn(X; Z).

17.2 Spectral Sequences Remark 17.2. Most (all) of our spaces have come to us as iterated gluing sequences:

X1 X2 X3 ··· X

A1 A2 A3

Reuben Stern 51 Spring 2017 Math 231br 17.2 Spectral Sequences

We want some sort of tool to compute H∗X. Ideally, this wouldn’t depend so much on H∗(Xn), but rather on H∗An. If we apply homology to the iterated gluing diagram, we get

H∗X1 H∗X2 H∗X3 ··· H∗Xn H∗Xn+1 ··· H∗X (−1) (−1) (−1) (−1) (−1)

H∗A1 H∗A2 H∗A3 ··· H∗An H∗An+1

For X = S∞ X , we have H (X) = lim H (X ). Hence, if σ ∈ H (X), there is some n n=1 n ∗ −→ ∗ n ∗ and σn ∈ H∗(Xn), such that σ is the image of σn in H∗(X). We can then push σn forward to an−1 ∈ H∗An−1:

σn σ ∈ ∈

··· H∗Xn−1 H∗Xn H∗Xn+1 ··· H∗X

an−1

H∗An−1 H∗An H∗An+1

Fact 17.3. The minimal n for which such a class σn appears has non-zero image an−1. From this, one can observe that each class σ ∈ H∗X has a unique place where it appears in the bottom row.

Now we want to sort out classes in H∗An−1 from those classes without lifts to H∗(X).

0 = σ0 σ1 ··· ∃σn−2 ∃σn−1 σn Q: ∃σn+1? ∈

∗ H∗X1 ··· H∗Xn−2 H∗Xn−1 H∗Xn H∗Xn+1 ··· H∗X

H∗An−2 H∗An−1 H∗An H∗An+1 d1 d1 d1

an−2 an−1 an

Pick an an ∈ H∗An. We want to know: is there some σn+1 ∈ H∗Xn+1 that hits an? The steps illustrated in the diagram are as follows:

Step 1: Push an forward to some σn ∈ H∗Xn. Send σn to some an−1 ∈ H∗An−1. If an−1 6= 0, then by exactness there cannot exist such a σn+1. If an−1 = 0, move onto step 2.

Step 2: Because an−1 = 0, σn ∈ ker of the map. By exactness, we can pull back to some σn−1 ∈ H∗Xn−1; the process continues.

Continue this process until either: (a) one of the σi has nonzero image ai−1, in which case

Reuben Stern 52 Spring 2017 Math 231br 17.2 Spectral Sequences

there cannot exist σn+1, or (b) we get to σ0 ∈ H∗(∗); conclude that this is zero, so there does exist a σn+1.

Definition 17.4. The differential d1 is the composite sending, e.g., an to an−1: push forward to σn, and push forward again. The r-th differential dr is the composite sending an to an−r. The claim is that each dr is well-defined up to (co)homology with respect to dr−1. This isn’t super difficult; you can work it out yourself.

r r These (co)homology groups, with respect to dr−1, are written En,∗. Each En,∗ is a r−1 subquotient of En,∗ .

Theorem 17.5. The group H∗X admits a filtration such that its n-th filtration quotient ∞ is En,∗, a sort of “limiting page” of the spectral sequence.

Remark 17.6. Things get complicated when you deal with a descending filtration, rather than an ascending one. You can see this by our argument with the complicated diagram: this relied on the sequence “bottoming out” eventually. Furthermore, things get complicated if you use cohomology instead of homology.

W Example 17.7. Filter a space X by skeleta, so that An = α Sα. Application of ordinary homology gives a spectral sequence with E1-page

! 1 _ M n E∗,n = E∗(An) = H∗ Sα = H∗±n(S ) α α We claim the isomorphism of the bottom two rows of the following diagram:

1 1 d1 : E∗,∗ E∗−1,∗−1

L L h∗+n h∗+n−2

Reuben Stern 53 Spring 2017 Math 231br March 6, 2017

18 March 6, 2017

18.1 Obstruction Theory

Reuben Stern 54 Spring 2017 Math 231br March 8, 2017

19 March 8, 2017

19.1 A Complicated Example

Reuben Stern 55 Spring 2017 Math 231br March 10, 2017

20 March 10, 2017

20.1 Smash Products

Reuben Stern 56 Spring 2017 Math 231br March 20, 2017

21 March 20, 2017

21.1 The Serre Spectral Sequence

Reuben Stern 57 Spring 2017 Math 231br March 22, 2017

22 March 22, 2017

22.1 More About the Serre Spectral Sequence 22.2 The Transgressive Differential Lemma 22.3 Multiplicative Structure

Reuben Stern 58 Spring 2017 Math 231br March 24, 2017

23 March 24, 2017

23.1 More Spectral Sequences 23.2 Multiplicative Extensions

∗ 23.3 Computing H (K(Z/2, 1); F2)

Reuben Stern 59 Spring 2017 Math 231br March 27, 2017

24 March 27, 2017

24.1 G-bundles and Fiber Bundles

Aside 24.1. A fiber bundle with fiber F , also called a locally trivial fibration, is a map p : E → B satisfying the following properties:

1. For any point b ∈ B, the fiber p−1(b) ∼= F .

2. The map p : E → B is surjective.

3. For every x ∈ B, there is an open neighborhood U ⊆ B containing x equipped −1 with a homeomorphism ϕU : p (U) → U × F such that the following diagram commutes: ϕ p−1(U) U U × F =∼

p pr1 U U

Definition 24.2. Let G be a topological group. A G-bundle is a fiber bundle E −−→p B with fiber G where G acts on E in such a way that

U ⊆ B, ϕU = p−1(U) ∼= G × U is G-equivariant. These are often called principal bundles. Remark 24.3. If G acts on another space F , then from a G-bundle E, one can form a

fiber bundle E ×G F → B with fiber F as follows:

E × F/(eg, f) ∼ (e, gf).

This is called the Borel construction.

Example 24.4. The unitary group U(n) acts on Cn. This gives an assignment U(n) bundles vector bundles. In fact, is a bijection. Explicitly, suppose given a U(n)-bundle E.

Lemma 24.5. The assigment X 7→ {isomorphism classes of G-bundles on X} is a functor satisfying the wedge axiom and Mayer-Vietoris, at least when X is a finite CW complex.

Proof. This should be evident. One can work it out if one is eager to.

Reuben Stern 60 Spring 2017 Math 231br 24.1 G-bundles and Fiber Bundles

Corollary 24.6. By Brown representability, there is thus a homotopy type BG repre- senting this functor.

Recipe 24.7. A G-bundle p : E → B gives a map ξ : B → BG. Picking a cohomology class ω ∈ H∗BG gives a class ξ∗ω ∈ H∗B. This choice of class ξ∗ω is natural, and we call them characteristic classes of G-bundles.

Lemma 24.8. Let p : E → B be a G-bundle, where E is n-connected. Then the classifying map ξ : B → BG is an n-equivalence; the natural transformation

[−,B] → [−,BG] ∼= {G-bundles on −}

is an isomorphism on complexes of dimension ≤ n.

Corollary 24.9. The universal bundle, classified by BG −→id BG has contractible total space, which is often denoted EG. ∼ Corollary 24.10. The homotopy groups of BG satisfy πn+1BG = πnG for all n.

Remark 24.11. There is a model of K(A, n), which is an actual topological group, so that BK(A, n) ∼= K(A, n + 1).

Example 24.12. Computing H∗BU(n). We have two fibrations: U(n) → ∗ → BU(n) and U(n − 1) → U(n) → S2n−1. First, we run the Serre Spectral Sequence on the second fibration: this gives that make ∗ ∼ V H U(2) = [e1, e2], the exterior algebra on two generators. Knowing this, we can run the spec- spectral sequence on the first fibration to compute H∗BU(2): one sees that tral se- quence ∗ ∼ H BU(2) = Z[c1, c2], dia- gram a polynomial algebra on two generators. ∗ ∼ V One can see by an inductive argument that in general, H U(n) = [e1, . . . , en], and ∗ ∼ H BU(n) = Z[c1, . . . , cn]. The classes ci are called Chern classes. When G is a finite (discrete) group, there is an especially useful model for BG called the bar construction. This requires some categorical definitions.

Definition 24.13. Let C be a category. The nerve of C , written NC , is the simplicial

Reuben Stern 61 Spring 2017 Math 231br 24.1 G-bundles and Fiber Bundles set defined by

NC0 = ob C

NC1 = mor C

NC2 = {pairs of composable maps • → • → • in C } . .

NCn = {strings of n composable arrows • → • → · · · → • in C }.

The face maps di : NCn → NCn−1 compose the i-th and (i + 1)-st arrows when 0 < i < n, and leaves out the first or last arrow if i = 0 or n, respectively. The degeneracy maps si : NCn → NCn+1 take a string

x0 → x1 → · · · → xi → · · · → xn

of n composable arrows and inserts the identity idxi at the i-th spot. Remark 24.14. The nerve is a functor N : Cat → sSet.

Example 24.15. There are two important examples we can construct. First, we define the homotopy quotient G//G to be the category with objects indexed by G, and 0 where there is a morphism xg → xh if and only if there exists an element g ∈ G such that gg0 = h. That is, morphisms correspond to multiplication in G. Similarly, we define ∗//G to be the category with one object ∗, and morphisms ∗ −→∗h for each h ∈ G, which compose in the expected ways.

Lemma 24.16. The category G//G is “contractible”: any map ∗ → G//G is fully faithful and essentially surjective.

Remark 24.17. The G-action on G//G is free.

Corollary 24.18. The map |N(G//G)| → |N(∗//G)| models EG → BG.

Remark 24.19. There is a “commutative diagram” where the vertical arrows are equiva- lences of categories: G//G ∗//G

nG-torsors witho marked point {G-torsors}

Reuben Stern 62 Spring 2017 Math 231br 24.1 G-bundles and Fiber Bundles

A G-torsor is a G-set for which the action is free and transitive. Intuitively, a G-torsor is “a copy of G with the identity forgotten.” For example, a C-torsor is just a complex line.

“Recall” that a map X −→ξ BG is the data of a G-bundle. We have that the correspon- dence taking a map Sing X → N(∗//G) to ξ is one to one, and bijective on homotopy classes.

Reuben Stern 63 Spring 2017 Math 231br Properties of Chern Classes

25 Properties of Chern Classes

Reuben Stern 64 Spring 2017 Math 231br The Steenrod Algebra and the Bar Spectral Sequence

26 The Steenrod Algebra and the Bar Spectral Sequence

Reuben Stern 65 Spring 2017 Math 231br April 10, 2017

27 April 10, 2017

27.1 The Steenrod Algebra, part II The key pieces of input for this lecture are the Serre spectral sequence, the transgression theorem (“proved” on the homework), and a strong stomach.

∗ ∞ ∼ Fact 27.1. We know from some lecture ago that H (RP ; F2) = F2[ι1], where ι1 is a ∗ ∞ generator in degree 2. The cohomology ring H (RP ; F2) is (clearly) isomorphic as a graded ring to ∞ [RP ; K(F2, ∗)], homotopy classes of maps from K(F2, 1) into K(F2, ∗). In turn, this is isomorphic to the 1 ∗ set of natural transformations H (−; F2) =⇒ H (−; F2). In particular, for each n, there is 1 n a unique natural transformation H (−; F2) ⇒ H (−; F2) corresponding to a chosen class n ∞ ω ∈ H (RP ; F2).

Example 27.2. Take for instance the case ∗ = 1. We are then considering natural transformations 1 1 H (−; F2) =⇒ H (−; F2).

The collection of all of these is isomorphic to the 1-graded bit of F2[ι1], and thus contains only

Reuben Stern 66 Spring 2017 Math 231br April 12, 2017

28 April 12, 2017

28.1 A Serre Spectral Sequence Trick 28.2 Serre Classes

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29 April 14, 2017

29.1 Serre’s Method

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30 April 17, 2017

30.1 The Adams Spectral Sequence 30.2 A Fun Computation Consider Eric’s favorite subalgebra of the Steenrod algebra: A(1) = hSq1, Sq2i. The goal is to compute ∗,∗ ExtA(1)(F2; F2). The subalgebra A(1) looks like

• • • •

• • • •

We take a free resolution of F2: in degree zero, we have •

• • • •

• • • •

The purple classes are all in the kernel, so survive to the degree 1 part:

• • •

• • • •

• • • •

• • • •

• • • •

• • • •

Reuben Stern 69 Spring 2017 Math 231br 30.2 A Fun Computation

• • • • • • • • •

• • • •

• • • •

• • • •

• • • •

• • • •

• • • •

Reuben Stern 70 Spring 2017 Math 231br Collected Homework Problems

A Collected Homework Problems

A.1 Problem Set 1

Problem A.1.1. Show that for the unreduced mapping cylinder Mf of a map f : X → Y , the retraction r : Mf → Y induces an isomorphism r∗ : π∗Mf → π∗Y .

Problem A.1.2. Show that if f : X → Y is a homotopy equivalence, then f∗ : π∗(X, x0) → π∗(Y, f(x0)) is an isomorphism for all choices of x0 ∈ X and ∗ ≥ 0.

Problem A.1.3. Let F : J → C be a functor. We say the cone of F is a functor x : J → C with a natural transformation x ⇒ F .A limit of F is an initial object in the category of cones.

1. Expand the definition of natural transformation and constant functor to reveal that a

cone is equivalent to the data of an object x ∈ C together with maps fj : x → F (j) for each object j ∈ J such that for any map g : j → j0 in the diagram, there is a commuting triangle x fj fj0

f(g) F (j) F (j0).

2. Now expand the definition of limit to see that a limit, expressed as an object ` together

with maps hj, has the property that any other cone point x and its maps fj factor uniquely through a map x → `.

3. Show that the product X × Y is the limit of the diagram with objects X and Y and no non-identity arrows.

4. Show that the equalizer E of a pair of functions X ⇒ Y of sets is indeed the limit of that diagram.

Problem A.1.4. A contravariant functor G : C → Sets is called representable when there exists an object Y and a natural isomorphism

G ==∼⇒ Sets(−,Y ).

1. From a morphism t : Y → Y 0 of representing objects, construct a natural transforma- 0 tion t∗ : G ⇒ G of the functors they represent.

2. From a natural transformation G ⇒ G0 of represented functors, construct a morphism Y → Y 0 of the representing objects.

3. Show also that your assignments respect composition of natural transformations and of morphisms.

Reuben Stern 71 Spring 2017 Math 231br A.2 Problem Set 2

4. Show that your assignments are mutual inverses, i.e., a natural transformation of representable functors is exactly the same information as a morphism of representing objects.

Problem A.1.5. Explain why the usual recipe for forming a group structure on π1(X, x0) does not apply to the relative homotopy object π1(I, ∂I). Problem A.1.6. Let p : E → B be a map and consider

Z = {(e, γ) ∈ E × BI : p(e) = γ(0)} ⊆ E × BI .

A path lifting function for p is a map λ : Z → EI with λ(e, γ)(0) = e and p ◦ λ(e, γ) = γ.

1. Show that p is a fibration if and only if there is a path lifting function λ for p.

2. Let p : E → B be a fibration with fiber F , and let Pp be the pathspace construction p described in class. Given a path lifting function λ : Z → EI for p, define maps

g : F → Pp, f 7→ (f, ω0) −1 h : Pp → F, (e, γ) 7→ [λ(e, γ )](1).

Show that g and h present the two halves of a homotopy equivalence.

A.2 Problem Set 2 Task A.2.1. Read Chapter 5 to see a “proper” definition of a CW-structure on a pre-existing space.

Task A.2.2. Skim through Chapter 6 and look at all the proofs we skipped. Try readng a few. Then try reading a few more. Move on to the rest of the problem set whenever you like.

Problem A.2.3. Show that if (X,A) is a relative CW-complex, then X/A is a CW-complex. Given CW-complexes X and Y , use this to concoct appropriate conditions so that X ∧ Y is a CW-complex.

Problem A.2.4. Suppose (X,A) is a relative CW-complex and p : E → B is a weak fibration. Show that for any map f : X → E and homotopies F : X × I → B and

H : A × I → E with F0 = p ◦ f, H0 = f|A, and p ◦ H = F |A×I , there is a homotopy G : X × I → E lifting F with G|A×I = H, G0 = f and p ◦ G = F . Diagrammatically, these conditions are summarized as

f∪H (X × {0}) ∪ (A × I) E G p

X × I F B.

Reuben Stern 72 Spring 2017 Math 231br A.3 Problem Set 3

n Problem A.2.5. Suppose X is obtained from A by attaching n-cells {eβ|β ∈ B}. Show ∼ W n that X/A = β∈B Sβ , and that the homeomorphism can be chosen so that the diagram

i n β W n  (S , ∗) β Sβ , ∗

p0 (Dn,Sn−1) =∼

fβ

p (X,A) (X/A, ∗)

n commutes, where fβ is the characteristic map of eβ. Problem A.2.6. Show that if f : X → Y is a cellular map of CW-complexes, then

Y ∪f CX is naturally a CW-complex. Problem A.2.7. Justify some of our ad hoc constructions from class by proving the following: let C be a category with finite products and a zero object and let F : C op → Groups be a group-valued functor. Show that if F is represented by an object Y via a natural transformation t : C (−,Y ) ⇒ F , then Y carries a group structure which causes C (−,Y ) to be group-valued and the comparison natural isomorphism t to respect the group structure.

A.3 Problem Set 3

Problem A.3.1. As in class, define S(p) to be the spectrum representing

∞ X 7→ hSpectra(Σ X, S) ⊗Z Z(p).

(a) Describe the homology functor associated to S(p).

(b) Demonstrate E(p) ' E ∧S(p) for any spectrum E. In particular, this gives S(p) ∧S(p) ' S(p).

n (c) Define πn,(p)E = [S(p),E]. Show that πn,(p)E(p) = πnE(p).

A.4 Problem Set 4 Problem A.4.1. Prove the transgressive differential lemma from class: let F −→i E → B be a fibration, and let B π C(i) δ ΣF be the naturally induced maps. Show that the following situations are equivalent:

(a) A class x ∈ HnB has d

Reuben Stern 73 Spring 2017 Math 231br A.4 Problem Set 4

(b) There is a class τ(x) ∈ HnC(i) with δ∗τx = y and π∗τx = x.

Problem A.4.2. The 2-adic Bockstein spectral sequence is the filtration spectral sequence arising from the diagram

∧ 2 2 Z2 ··· Z Z Z

··· Z/2 Z/2 Z/2.

Applying H∗(X; −) to this diagram of coefficients gives a spectral sequence of signature

∗,∗ M ∗ ∼ ∗ ∗ ∧ E1 = H (X; F2) = F2[w] ⊗ H (X; F2) =⇒ H (X; Z2 ), ∗

∗ where the E1-page consists of many duplicated copies of H (X; F2), which we can think of as tagged by monomials in w.

BSS k (a) Show that the differentials in this spectral sequence are “w-linear”, i.e., dr (w x) = k BSS w dr (x).

∗ ∧ BSS (b) Show that a torsion-free class x ∈ H (X; Z2 ) is in ker dr on all pages Er and never BSS in im dr . Demonstrate that this condition is equivalent to the corresponding class in the spectral sequence being w-torsion-free.

(c) More generally, show that the order of w-torsion of a class on the E∞ page of the spectral sequence is identical to the 2-primary torsion order of the corresponding ∗ ∧ cohomology class in H (X; Z2 ).

BSS 1 (d) Show that d1 in this spectral sequence is computed by the Steenrod square Sq .

∗ ∧ Problem A.4.3. Use this spectral sequence to make a calculation of H (K(Z/2, 2); Z2 ) ∗ from the calculation of H (K(Q/2, 2); F2) given in class. You will want to know the following even BSS mysterious formula: for any class x ∈ H (X; F2) where dr (x) is defined, we have

 1 |x| 1 Sq (x) · x + Sq Sq (x) for r = 2, dBSS = r BSS dr−1(x) · x for r > 2.

j p n Problem A.4.4. Let F E B be a fiber sequence, let u ∈ H (F ; F2) be a class n+1 that transgresses to τ(u) ∈ H (B; F2), and suppose that for some integer i ≥ 1 there is a BSS BSS ∗ Bockstein differential di v = τ(u). Show that di+1 p v is then defined and that

∗ BSS ∗ BSS j di+1 p (v) = d1 (u),

BSS where again d1 is the first Bockstein differential.

Reuben Stern 74 Spring 2017 Math 231br A.4 Problem Set 4

Problem A.4.5. In this problem, you will reinvent one of the main results of unstable ratio- nal homotopy. For a simply connected space X, we inductively define its rationalization to be a space Q ⊗ X under X as follows: given a Postnikov fibration

K(πnX, n) → X[0, n] → X[0, n), and the rationalization map X[0, n) → (Q⊗X)[0, n), we construct a corresponding Postnikov fibration for Q ⊗ X as the back face in

K(Q ⊗ πnX, n) K(Q ⊗ πnX, n)

K(πnX, n) K(πnX, n)

(Q ⊗ X)[0, n] ∗

X[0, n] ∗

(Q ⊗ X)[0, n) K(Q ⊗ πnX, n + 1)

X[0, n) K(πnX, n + 1)

Here the nodes X[0, n] and (Q ⊗ X)[0, n] are defined as the total spaces of the pullback fibrations, and the map between them is induced by the universal map of fibrations. We set Q ⊗ X to be the homotopy inverse limit

⊗ X = lim( ⊗ X)[0, n], Q n Q which has the factorization property

∼ π∗X Q ⊗ π∗X π∗(Q ⊗ X).

Now, justify the following claims:

Reuben Stern 75 Spring 2017 Math 231br A.4 Problem Set 4

1. The rational cohomology of rational Eilenberg-Mac Lane spaces is given by   [x ] if n is even, H∗(K( , n); ) = Q n Q Q 2 Q[xn]/xn if n is odd.

2. The cohomology H∗(X(n, ∞); Q) as well as its ring structure are completely deter- mined by the cohomology ring H∗(X[n, ∞); Q).

3. The map X → Q ⊗ X is an isomorphism on rational cohomology.

4. The Postnikov fibrations K(Q ⊗ πnX, n) → (Q ⊗ X)[0, n] → (Q ⊗ X)[0, n) give a model for C∗(X; Q) whose underlying graded-commutative algebra is free and which uses the minimal number of algebra generators11.

5. Any rational commutative differential-graded algebra A∗ with A0 = Q and A1 = 0 inductively receives a quasi-isomorphism from a Sullivan model.

6. There is a sequence of Postnikov sections X[0, n) → K(πnX, n + 1), hence a space X, whose Sullivan model is the one associated to A∗.

7. Given a Sullivan model for C∗(X; Q), its indecomposables compute the rational homotopy groups of X.

8. The rational homotopy groups of Sn, n > 1, are given by

 n n Σ Q if n is odd, Q ⊗ π∗S = ΣnQ ⊗ Σ2n−1Q if n is even.

Problem A.4.6. (a) The tensor product of line bundles induces a map

BU(1) × BU(1) ⊗ BU(1)

on the object BU(1) representing the functor X 7→ {iso-classes of line bundles on X}. Describe the behavior of this map in ordinary cohomology with Z coefficients. (b) In general, the tensor product of vector bundles induces a similar map

BU(n) × BU(m) ⊗ BU(nm).

Describe the behavior of this map in ordinary cohomology as well. 11Such a presentation of the rational cochain complex is called a Sullivan minimal model. It may please you to check that two such models are related by a chain homotopy equivalence.

Reuben Stern 76 Spring 2017 Math 231br A.5 Problem Set 5

Problem A.4.7. The dual Steenrod algebra is a Hopf algebra, meaning that it not only has a multiplication map but also a diagonal map ∆ : A∗ → A∗ ⊗ A∗ and an antipode map χ : A∗ → A∗. In class, we deduced a formula for ∆, and we showed that as an algebra the dual Steenrod algebra forms a polynomial ring. The antipode fits into the commutative diagram χ⊗1 A ⊗ A A ⊗ A ∆ µ

ε η A F2 A ∆ µ

1⊗χ A ⊗ A A ⊗ A

A.5 Problem Set 5 Problem A.5.1. Suppose you believe in complex Bott periodicity, so that the homotopy groups of BU(n) have the form πoddBU(n) = 0 and πevenBU(n) = Z in the range [0, 2n]. ∗ Set n = 3 and describe the action of the Steenrod algebra on H (BU(3); F2). Then try n = 4. Then n = 5. Stop once you get sick of the exercise.

Problem A.5.2. Return to the picture of the Adams spectral sequence computing π∗ko described in class. At a glance, it appears that there could be a potential differential r+1 drh1 = h0 . Without assuming Bott periodicity, argue why this differential cannot occur. (Hint: h0h1 = 0.)

Problem A.5.3. Compute the first several terms (until you get tired) of a free resolution of F2 as a module over the Steenrod algebra. Once you have the resolution, use it to compute Ext and compare your answer with the part of the Adams spectral sequence drawn in class.

Problem A.5.4. Let E(1) denote the exterior F2-algebra on two generators e1 and e3, of ∗,∗ degrees 1 and 3 respectively. Calculate ExtE(1)(F2, F2). Task A.5.5. Try to read Section 8 of Steve Wilson’s Brown-Peterson Homology: An Introduction and Sampler.

Problem A.5.6. Figure out both the statement and the proof of the 5-Lemma and the Snake Lemma in mod-C homological algebra. Solution. I won’t state and prove either lemmas in their greatest generality.

Lemma A.1 (Mod-C 5-Lemma). Suppose we have a commutative diagram

A1 A2 A3 A4 A5

α γ δ  θ

B1 B2 B3 B4 B5

Reuben Stern 77 Spring 2017 Math 231br A.5 Problem Set 5 such that the top and bottom rows are C -exact sequences: ker / im ∈ C . If α, γ, , and θ are C -isomorphisms, then δ is a C -isomorphism.

Proof. First, we show that ker δ ∈ C .

Lemma A.2 (Mod-C Snake Lemma). Suppose we have a commutative diagram

0 A B C 0

α β γ 0 A0 B0 C0 0 where the rows are C -short-exact. There is then a long C -exact sequence

0 ker α ker β ker γ δ coker α coker β coker γ 0.

Furthermore, this construction is natural: given a collection of maps f of commutative diagrams

0 A1 B1 C1 0 f f f α1 β1 γ1 0 A2 B2 C2 0

α2 β2 γ2 0 0 0 0 A1 B1 C1 0 f f f

0 0 0 0 A2 B2 C2 0, there is a commutative diagram of long exact sequences

δ1 0 ker α1 ker β1 ker γ1 coker α1 coker β1 coker γ1 0 f ∗ f ∗ f ∗ f ∗ f ∗ f ∗

δ2 0 ker α2 ker β2 ker γ2 coker α2 coker β2 coker γ2 0.

Reuben Stern 78 Spring 2017 Math 231br Notes from Switzer

B Notes from Switzer

This appendix exists to collect notes on proofs, definitions, examples, etc. that may have been shortchanged in lecture. All material is taken from [Swi02]. We begin with a more precise definition of a CW-structure on a preexisting space X:

Definition B.1. A CW-structure on a space X is a way of decomposing X into a CW-complex K satisfying C) K is closure-finite—each cell has only a finite number of immediate faces (other cells which intersect it nontrivially) and W) X has the weak topology induced by K—a subset S ⊂ X is closed if and only if n n S ∩ eα is closed in the cell eα for each n, α.

Lemma B.2. Simplicial approximation. Suppose X = A ∪ en is obtained from a space A by attaching a single n-cell, (K,L) is a finite simplicial pair, and f : (|K|, |L|) → (X,A) is a map of pairs. Then there exists a subdivision (K0,L0) of (K,L) and a map f 0 :(|K|, |L|) → (X,A) such that

i) f and f 0 agree on f −1(A) and f ' f 0 rel f −1(A);

0 0 n 0 n ii) for any simplex σ of K , if f (|σ|) meets e0 , then f (|σ|) is contained in e and f 0 is a linear map when restricted to |σ|.

Proof. The idea is to very finely subdivide K: since |K| is compact, the map g−1f, g the n −1 n n n characteristic map of e , is uniformly continuous on f e2 , where D2 = {x ∈ D : |x| ≤ −1 n 3/4}, for instance. In particular, there is some δ > 0 such that d(x, y) < δ in f e2 implies −1 −1 n that d(g f(x), g f(y)) < 1/4 in D2 . We can then subdivide K finely enough that no simplex of K0 has diameter more than δ.

Reuben Stern 79 Spring 2017 Math 231br REFERENCES

References

[Hat01] Allen Hatcher, Algebraic topology, 2001.

[Hov98] Mark Hovey, Model categories, vol. 63, American Mathematical Society, 1998.

[Koc96] Stephen O. Kochmann, Bordism, stable homotopy, and adams spectral sequences, vol. 7, American Mathematical Society, 1996.

[Swi02] Robert M. Switzer, Algebraic topology — homology and homotopy, Springer-Verlag Berlin Heidelberg, 2002.

Reuben Stern 80 Spring 2017 Index 2-adic Bockstein spectral sequence, 74 G-bundle, 60 G-equivariant, 21 acyclic cofibrations, 29 Gabriel-Zisman localization, 26 acyclic fibrations, 29 good cylinder object, 31 associated cohomology, 47 good path object, 35 bar construction, 61 group, 12 base change, 31 hammock localization, 26 bifibrant, 35 homotopic Borel construction, 60 maps of spectra, 46 category enriched over a category, 27 homotopy category, 37 characteristic classes of G-bundles, 61 homotopy groups, 15 Chern classes, 61 homotopy quotient, 62 closure-finite, 79 Hopf algebra, 77 cobase change, 31 ind-system, 47 coexact sequence, 16 injective model structure, 30 cofiber sequence, 16 cofibrant object, 30 left homotopic, 31 cofibrant replacement, 37 left Kan extension, 28 cofibrations, 29 left lifting property, 28 cofinal, 45 left Quillen functor, 38 cogroup, 13 locality,6 cone on a topological space, 16 locally trivial fibration, 60 CW-structure, 79 loop space, 12 delooping, 41 Mayer-Vietoris axiom, 41 differential, 53 Milnor Sequences, 49 model category, 29 equalizer,7 morphism, 45 equivalent morphisms of spectra, 46 n-connected, 19 Eric Peterson,4 n-universal, 42 exact sequence, 16 nerve, 61

fiber bundle, 60 Ω-spectrum, 49 fiber sequence, 16 fibrant object, 30 path space, 11, 18 fibrant replacement, 37 phantom maps, 49 fibrations, 29 principal bundles, 60 folding map, 14 product space,7 projective model structure, 30

81 Math 231br INDEX pseudo-2-functor, 39 rationalization, 75 reduced cone, 16 relative homotopy group, 19 representable functor, 71 retract, 29 right homotopic, 35 Right Kan extensions, 28 right lifting property, 28 right Quillen functor, 38 set of n-simplices, 27 sheaf,6, 44 sheaf condition,6 simplex category, 27 simplicial category, 27 simplicial object, 27 simplicial set, 27 smash product, 11 spectrum, 44 stable sphere, 45 subspectrum, 45 Sullivan minimal model, 76 suspension, 14 reduced, 14 suspension spectrum, 44 total left derived functor, 38 total right derived functor, 38 universal bundle, 61 weak equivalence, 19 weak equivalences, 29 weak factorization system, 29 weak topology, 79 wedge product, 10 wide subcategory, 29

Reuben Stern 82 Spring 2017