Three Degree Filtration. 7. 3.3. Exact Couples. 8. 3.4. General Filtrations. 9. 3.5. Misc. 9. 4. The Serre Spectral Sequ
RATIONAL HOMOTOPY THEORY JONATHAN CAMPBELL
Contents 1. Introduction 2. Building Spaces 2.1. Postnikov Towers 3. As Much as I Can Say About Spectral Sequences 3.1. Two Degree Filtration 3.2. Three Degree Filtration 3.3. Exact Couples 3.4. General Filtrations 3.5. Misc. 4. The Serre Spectral Sequence 5. Some Rational Computations 6. Serre Theory 7. Rational Homotopy Groups of Spheres 8. Rational Spaces and Localization 8.1. Rational H-spaces 9. DGAs 9.1. Basic Definitions 9.2. Sullivan Models 9.3. Minimal Sullivan Models 10. Appendix: Simplicial Sets 10.1. Geometric Realization and Model Structure 11. Appendix: Model Categories 11.1. The Small Object Argument 11.2. Cofibrantly Generated Model Categories References
1 2 2 5 6 7 8 9 9 10 11 12 14 15 17 18 18 19 21 23 25 27 31 34 35
1. Introduction Resources: Griffiths-Morgan, Felix-Halperin-Thomas. Should probably say something about prequisites. One should know
• Algebraic topology as in Hatcher (counting the homotopy theory chapter). • Basic spectral sequences There are also things it would be useful to know, but which I’ll review and/or create handouts on • simplicial sets / simplicial objects in a category C • model categories We’ll start off with Serre theory, which is a computational tool. While it is a computational tool, it gives us some indication that operating only with certain filters on our eyes can immensely simplify things. In particular, it hints that if we were to consider only the parts of spaces that rational (co)homology sees, life is a little bit easier. 1
This is where the philosophy of homotopy theory actually comes in. We think of homotopy theory as trying to classify all shapes up to the relation of “homotopy” which is our usual notion of being able to deform without tearing, etc. However, “modern” homotopy theory is really a rich theory of how to consider difference objects equivalent. For example, we can consider two objects equivalent if they are homotopy equivalent, or two chain complexes equivalent if they are quasi-isomorphic. There are even looser notions of equivalence, for example, two spaces are Q-equivalent if their rational homologies are equivalent. Using the tools of homotopy theory, we can examine what kind of theory we get if we allow ourselves such a notion of equivalence. In fact, what we can show is that if we restrict ourselves to only caring about rational equivalence, the category of topological spaces is completely algebraic. That is, it is equivalent to some category completely determined by algebraic data (some elaboration of a category of algebras). The goal in this course is to approach the problem of stating the most powerful version of rational homotopy theory (Quillen’s, as far as I know) by baby steps. (1) We’ll see how working rationally greatly simplifies many computations. In particular, computations with Eilenberg-MacLane spaces become easy, rather than a combinatorial nightmare producing Steenrod operations. (2) In fact, we’ll show that if we are computing rationally, homotopy and homology don’t differ all that much. This is achieved using Serre theory. (3) Once this is achieved, we’ll discuss the rationalization of a space, and look at some specific examples, namely rational H-spaces. We’ll see that H-spaces become purely algebraic. (4) I’ll review the construction of Postnikov towers, then note by earlier computations that these should simplify drastically if we consider them rationally. This leads to a discussion of DGAs and minimal DGAs. (5) We will discuss Q-differential forms. (6) We will discuss a version of the algebraic equivalence. (7) We’ll get our grubby hands on some actual spaces, and do some computations. (8) Finally, after the discussion of many algeraic gadgets, we’ll get to Quillen’s equivalence between model categories TopQ ↔ DGLAQ So, that’s is nominally the goal of the course. The goal is also to introduce a number of topics of independent interest. For example, some classical computations, simplicial sets, model categories, etc. GrVectQ O U proj
ChQ O 1 Top≥ Q
/ CDAGQ 6
Top≥1 O v Set∆
2. Building Spaces 2.1. Postnikov Towers. Since we’ll be building a lot of spaces, it will be useful to discuss how they actually get built. In particular, I’ll have to remind folks of Postnikov towers. Before I do that, I should remind folks of building blocks of Postnikov towers, which are called Eilenberg-MacLane spaces. 2
Definition. An Eilenberg-MacLane space for a group G is a space K ( G, n) such that ( πi K ( G, n) =
G 0
i=n otherwise
Remark. One should note that for n = 1 the group may be non-abelian, but for n > 1, the group G must be abelian, since it arising as the higher homotopy group of a topological space. Example. S1 is a K (Z, 1). CP∞ is a K (Z, 2). RP∞ is a K (Z/2, 1). One should maybe have an idea of why these exist. Proposition. Let G be a group. Then a K ( G, 1) exists. Proof Sketch. The idea is to create a fiber bundle G → EG → BG where EG is contractible. It will then follow that π1 ( BG ) = G, and higher homotopy groups vanish. Thus, BG is a K ( G, 1). For a full proof, see [?]. Proposition. Let G be an abelian group. Then K ( G, n) exists. Proof. G will have a presentation Z [r β ] → Z [ gα ] → G where r β are the relations and gα are the generators. We consider α Sn — a bouquet of spheres with one W sphere for each generator. Every relation gives us a map ϕ β Sn → α Sn . Using these as attaching maps, we form the pushout W
W β
W
/ Wα Sn
Sn
n +1 D β
/X
We claim now that X has the property that πi ( X ) = 0 for i < n. We have a long exact sequence πn+1 ( X, X (n) ) → πn ( X (n) ) → πn ( X ) → πn ( X, X (n) ) But, note that πn+1 ( X, X (n) ) = Z[r β ] and πn ( X (n) ) = Z[ gα ], πn ( X, X (n) ) = 0. So, the above exact sequence actually gives a presentation. Now, there are higher homotopy groups. But we go ahead and kill those off by inductively attaching cells. The following is (arguably) the most important property of Eilenberg-MacLane spaces, and it will be used repeatedly and without much mention below. Theorem. Let X be a space with the homotopy type of a CW-complex and let G be an abelian group. Then H n ( X; G ) = [ X; K ( G, n)]. That is, cohomology is represented by K ( G, n). We now note that we can build up topological spaces one homotopy group at a time by recourse to Eilenberg-MacLane spaces. 3
Theorem. Let X be a path-connected topological space. A Postnikov tower is a diagram
··· X G 3 > X2 / X1
X such that the following hold: ∼ =
• πi ( X ) − → πi ( Xn ) for i ≤ n. • πi ( Xn ) = 0 when i > n. • Xn → Xn−1 is a fibration with fiber K (πn ( X ), n). I won’t prove that these things exist. They do. You can look it up in [1] or any other book on algebraic topology. In fact, the provide successive approximations to a topological space in the following sense Proposition. Let X be a connected CW-complex. Then the natural map X → lim Xn is a weak homotopy equivalence. Given a random space X, it would be nice to sort of how to build the Postnikov tower. Suppose we’ve started at the bottom with a map X → K (π1 ( X ), 1) that induces an isomorphism on π1 ( X ). We’ve like to introduce a second space X2 such that we can produce a lift K ( π2 ( X ), 2)
/7 X2
X
/ K ( π1 ( X ), 1)
(the K (π2 ( X ), 2) is hanging around to indicate that it is the fiber of the right vertical map). That is, we’d like X2 to be a total space with fiber K (π2 ( X ), 2) and base K (π1 ( X ), 1). What we need are methods to produce fiber spaces like this. One tried and true method of producing fiber spaces is by classifying maps - in particular if we had a fibration sequence K (π2 ( X ), 2) → P → K (π2 ( X ), 3) where P is contractible, it would be nice if we could extend the above to a diagram K ( π2 ( X ), 2)
7/ X 2
/P
X
/ K ( π1 ( X ), 1)
/ K ( π2 ( X ), 3)
where the square on the left is now pullback. However, it is not always the case that we can do that. In order for this to be the case, the putative X2 → K (π1 ( X ), 1) has to be a special kind of fibration, which now define. 4
Definition. A fibration K (π, n) → E → B is said to be principal if it arises as a pullback from a path-loop fibration. That is, if there is a diagram as below where the lower square is pullback: K (π, n)
/ K (π, n)
E
/P
B
/ K (π, n + 1)
There is also a more general definition of principal: Definition. A fibration F → E → B is a principal fibration if there is a commutative diagram F
/E
/B
ΩB0
/ F0
/ E0
/ B0
such that the bottom row is a fibration sequence Remark. Roughly, a fibration sequence is principal if it can be extended to the right, and not just the left. We now have the following theorem, which I don’t think I will prove at the moment. It is pretty standard and can be found in [1] or [2]. Theorem. A connected CW-complex has a Postnikov tower of pricnipal fibrations if and only if the action of π1 ( X ) on πn ( X ) is trivial. Remark. Such a space is often called simple. What is the utility of this? Given that there is a Postnikov tower with principal fibrations, we have at each level of the postnikov tower a fibration sequence k
n X n +1 → X n − → K (πn+1 X, n + 2).
This means that Xn+1 is classified by an element
[ k n ] ∈ H n +2 ( X n , π n +1 X ). These k-invariants, as they are known, often prove useful — as all classifying maps do. Remark. A Look Ahead Why do we care about the above? Well, it’s a nice theoretical result of course. But our goal here is to produce nice “rational” approximations to a space. It will turn out that it is very easy to rationally approximate Eilenberg-MacLane spaces, and then we can bootstrap this to building up rationalizations. 3. As Much as I Can Say About Spectral Sequences SLOGAN: homology of the associated graded converges to the associated graded of homology. For later use, it will behoove us to carefully discuss spectral sequences and where they come from. I don’t know how much I’ll get to this in class. Suppose there is some topological space you’d like to compute the homology of. This should beW the case, given that you’re taking a topology class. If you’re lucky, the space splits up into parts X = i Xi and you know how to take the homology of those parts. You are never this lucky. Instead, what usually exists in nature is a filtration, i.e. some sequence of subspaces ∅ = X0 ⊂ X1 ⊂ X2 ⊂ · · · such that X = Xi . If there are only two steps in this filtration, i.e. X1 ⊂ X2 = X, you might hope to use a long exact sequence in homology to get at H∗ ( X ): S
· · · → Hi ( X1 ) → Hi ( X ) → Hi ( X, X1 ) → · · · . 5
You might again hope something like Hn ( X ) = Hn ( X1 ) ⊕ Hn ( X2 /X1 ). Of course, this is not true. What IS true is that there is an induced filtration on homology: p −1
p
Fn = Im( Hn ( X p ) → Hn ( X ))
Fn
p
⊂ Fn ⊂ · · ·
and that under right conditions we can compute the associated graded of H∗ ( X ): p −1
M p
gr F Hn ( X ) =
Fn /Fn
Of course, this doesn’t completely recover homology (there are extension problems), but it gets us closer. 3.1. Two Degree Filtration. Let’s look at an example more carefully. In fact, disgustingly carefully. Hn+1 ( X1 )
Hn+1 ( X2 )
Hn+1 ( X2 )
Hn+1 ( X2 )
/ Hn+1 ( X1 )
/0
/0
/0
/ Hn+1 ( X2 /X1 )
/ Hn ( X1 )
/ Hn ( X1 )
/0
/0
/ Hn ( X2 )
/ Hn ( X2 /X1 )
/ Hn−1 ( X1 )
/0
/ Hn ( X2 )
/0
/ Hn−1 ( X2 )
Let’s mess around with this a little bit. What we’ve decided is that we want to get at the associated graded of homology. That is, we want Im( H∗ ( X1 ) → H∗ ( X2 )) H∗ ( X2 )/(Im H∗ ( X1 ) → H∗ ( X2 )) We observe how these are contained in the diagram above. First, we have Im( Hn ( X1 ) → Hn ( X2 )) ∼ = Hn ( X1 )/ ker( Hn ( X1 ) → Hn ( X2 ) ∼ = Hn ( X1 )/ Im( Hn+1 ( X2 /X1 ) → Hn ( X1 )) And we also have Hn ( X2 )/(Im Hn ( X1 ) → Hn ( X2 )) ∼ = Hn ( X2 )/(ker Hn ( X2 ) → Hn ( X2 /X1 )) ∼ = Im( Hn ( X2 ) → Hn ( X2 /X1 ))
∼ = ker( Hn ( X2 /X1 ) → Hn−1 ( X1 )) If we make the definition 2 En,i = ker( Hn ( Xi , Xi−1 ) → Hn−1 ( Xi−1 , Xi−2 ))/ Im( Hn+1 ( Xi+1 , Xi ) → Hn ( Xi , Xi−1 ))
we have a diagram 0
6
5 En+2,2
/ Im(0 → Hn ( X1 ))
6 En,1
Im( Hn+1 ( X1 ) → Hn+1 ( X2 ))
50
/ Im( Hn ( X1 ) → Hn ( X2 ))
6 En,2
0
0
/ Im( Hn ( X2 ) → Hn ( X2 ))
En2 +1,1 2
2
/0
2
/0
where the colored arrows indicate exact sequences (the paths parallel to the colored arrows are exact as well). 6
3.2. Three Degree Filtration. We start with the diagram (which continues indefinitely in all directions): / Hn+1 ( X1 )
/0
/0
/0
/0
/ Hn+1 ( X2 /X1 )
/ Hn ( X1 )
/ Hn ( X1 )
/0
/0
/ Hn+1 ( X3 /X2 )
/ Hn ( X2 )
/ Hn ( X2 /X1 )
/ Hn−1 ( X1 )
/ Hn−1 ( X1 )
/0
/ Hn ( X3 )
/ Hn ( X3 /X2 )
/ Hn−1 ( X2 )
/ Hn−1 ( X2 /X1 )
/0
/ Hn ( X3 )
/0
/ Hn−1 ( X3 )
/ Hn−1 ( X3 /X2 )
Hn+1 ( X1 )
Hn+1 ( X2 )
Hn+1 ( X3 )
Hn+1 ( X3 )
Hn+1 ( X3 )
Now we turn the crank. For ease of notation, let Imna,b := Im( Hn ( Xa ) → Hn ( Xb )) 2 En,i = ker( Hn ( Xi , Xi−1 ) → Hn−1 ( Xi−1 , Xi−2 ))/ Im( Hn+1 ( Xi+1 , Xi ) → Hn ( Xi , Xi−1 ))
Then we have a diagram (where the colored arrows are exact sequences). E2
/ Im0,1
En2 +1,1
/0
E2
/ Im1,2
En2 +1,2
/ Im0,1 n =0
0
/ Im2,3
En2 +1,3
/ Im1,2 n
0
/ Im3,3
0. 12
Proof. We will obviously use induction and the fibration sequence K (π, n − 1) → ∗ → K (π, n). There are some reduction steps.
• It suffices to do n = 1 by the previous lemma • It suffices to do π = Z and π = Z/m by the Kunneth theorem (at least when C = FG or mb f Fp ) So, we do this for FG and F p . But, this is easy. A K (Z, 1) is S1 , and so the theorem holds. A K (Z/m, 1) is a lens space, so this holds too. Lemma. Let X be simply connected or π1 ( X ) acts trivially on πn ( X ), then πn ( X ) ∈ C ⇐⇒ Hn ( X; Z) ∈ C for all n > 0. Proof. The conditions would see to suggest that we build a Postnikov tower. If πi ( X ) ∈ C, then H∗ (K (πi ( X ), n)) ∈ C from the above. Then, by using induction and the Serre spectral sequence, we get Hn ( Xn ; Z) ∈ C. Before we go on, the following will be useful. Lemma. Let X be a topological space with π1 ( X ) acting trivially on πm ( X ) for all m. Let Xn denote the n Postnikov level of that space. Then ∼ =
Hn ( X ) − → Hn ( Xn ). Proof. We begin with the map X → Xn and turn it into a fibration, which we also call X → Xn . Take the fiber to obtain a fiber sequence X > n → X → Xn . Note that by the long exact sequence in homotopy, πi ( X ) = 0 for i ≤ n. Now, we apply the Serre Spectral sequence. The E2 -page is E2p,q = H p ( Xn ; Hq ( X >n )) and we note that Hq ( X >n ) is only non-trivial for q ≥ n + 1. It follows that E∞ p,n− p = 0 except when p = n and so Hn ( X ) = Hn ( Xn ). Theorem (Mod C Hurewicz). If X has πi ( X ) ∈ C for i < n then h : πn ( X ) → Hn ( X ) is an isomorphism mod C. The following corollaries are usually more useful than the theorem itself: Corollary. If for all i, πi ( X ) ∈ C, then h : π∗ ( X ) → H∗ ( X ) is an isomorphism mod C. e i ( X; Z) ⊗ Q = 0 for all i if and only if πi ( X ) ⊗ Q = 0 Corollary. For X a simply connected topological space, H for all i. Proof of Theorem. We assume for the moment that X is simply connected. Let { Xi } be the Postnikov tower of X. The maps πn ( X ) → Hn ( X ) and πn ( Xn ) → Hn ( Xn ) are the same, so we might as well work with the latter. We of course have the fibration K ( π n ( X ), n ) → X n → X n −1 and we can use our hammer of choice, the Serre spectral sequence. There is a 5 term exact sequence, coming from the Serre spectral sequence: Hn+1 ( Xn−1 )
d n +1
/ E∞ 0,n
/ Hn (K (πn ( X ), n))
/ Hn ( Xn )
The composition of maps in the middle is the inclusion of the map Hn (K (πn ( X ), n)) → Hn ( Xn ). 13
/ Hn ( Xn−1 )
Now, if we assume πi ( X ) ∈ C for i < n, then πn ( Xn−1 ) ∈ C and thus Hn ( Xn−1 ) ∈ C and Hn+1 ( Xn−1 ) ∈ C. Thus, the map Hn (K (πn ( X ), n)) → Hn ( Xn ) is an isomorphism mod C. We then consider the diagram π n ( K ( π n ( X ), n )
∼ =
∼ =
Hn (K (πn ( X ), n)
/ π n ( Xn ) / Hn ( Xn )
The left vertical map is an isomorphism by regular Hurewicz. The top horizontal map is an isomorphism by definition. We just showed the bottom horiztonal map is an isomorphism mod C. Thus, the right vertical map is an isomorphism mod C. 7. Rational Homotopy Groups of Spheres We now have enough information to prove an actual, big theorem in algebraic topology: we get some structure results on the homotopy groups of spheres. Theorem. πi (Sn ) are finite for i > n except for when n is even and i = 2n − 1. In this case π2n−1 (Sn ) = Z ⊕ finite. Proof. Consider the map Sn → K (Z, n) which represents the generator of πn (K (Z, n)) ∼ = Z. We can consider this map to be a fibration with fiber F. By the long exact sequence in homotopy for a fibration πi F ∼ = πi Sn in the range we care about (i > n). Further convert the map F → Sn into a fibration with fiber K (Z, n − 1): K (Z, n − 1) → F → Sn . This is amenable to attack by the Serre spectral sequence. Since K (Z, n − 1) have different homology groups when n is even or odd, we split into two cases. Case 1: n odd. In this case we know the E2 -page is E2p,q = H ∗ (Sn ; Q) ⊗ H ∗ (K (Z, n − 1); Q) ∼ = Λ( x ) ⊗ Q[y] and so it looks like (with d2 drawn in) 3n − 3
Qy3
2n − 2
Qy2
n−1
Qy
0
Q
0
Qxy3 &
Qxy2 & &
Qxy Qx n
Now, we know that F is (n − 1)-connected, so if we are to kill off any cohomology in dimension n − 1, we must have that d2 : Qy → Qx is an isomorphism. By the multiplicativity of the spectral sequence it is then e ∗ ( X; Q) = 0, and πi ( X ) ⊗ Q is 0 as well. Thus, πi (Sn ) is the case that ALL d2 ’s are isomorphisms. Thus, H finite for i > n. Case 2: n even. In this case the E2 -page is E2p,q = H ∗ (Sn ; Q) ⊗ H ∗ (K (Z, n − 1); Q) ∼ = Λ( x ) ⊗ Λ(y) 14
and so we have the E2 -page looking like Qy
Qxy "
Q
Qx
e ∗ ( X; Q) ∼ e (S2n−1 ; Q). By Hurewicz for C = FG, we have that πi (Sn ) is 0 for which means that H = H n n < i < 2n − 1 and π2n−1 (S ) = Z ⊕ finite. We now need to compute the higher homotopy groups. Let F → F
