RATIONAL HOMOTOPY THEORY JONATHAN CAMPBELL

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1 RATIONAL HOMOTOPY THEORY JONATHAN CAMPBELL Contents 1. Introduction 1 2. Building Spaces Postnikov Towers 2 3. As Much as I Can Say About Spectral Sequences Two Degree Filtration Three Degree Filtration Exact Couples General Filtrations Misc The Serre Spectral Sequence Some Rational Computations Serre Theory Rational Homotopy Groups of Spheres Rational Spaces and Localization Rational H-spaces DGAs Basic Definitions Sullivan Models Minimal Sullivan Models Appendix: Simplicial Sets Geometric Realization and Model Structure Appendix: Model Categories The Small Object Argument Cofibrantly Generated Model Categories 34 References 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

2 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 Top Q DGLA Q 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. GrVect Q U Ch proj Q Top 1 Q Top 1 CDAG Q 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

3 Definition. An Eilenberg-MacLane space for a group G is a space K(G, n) such that π i K(G, n) = { G i = n 0 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. S 1 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 π 1 (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[g α ] G where r β are the relations and g α are the generators. We consider α S n a bouquet of spheres with one sphere for each generator. Every relation gives us a map ϕ β S n α S n. Using these as attaching maps, we form the pushout β S n α S n β D n+1 X We claim now that X has the property that π i (X) = 0 for i < n. We have a long exact sequence π n+1 (X, X (n) ) π n (X (n) ) π n (X) π n (X, X (n) ) But, note that π n+1 (X, X (n) ) = Z[r β ] and π n (X (n) ) = Z[g α ], π n (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 That is, cohomology is represented by K(G, n). H n (X; G) = [X; 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

4 Theorem. Let X be a path-connected topological space. A Postnikov tower is a diagram X 3 X 2 X X 1 such that the following hold: π i (X) = πi (X n ) for i n. π i (X n ) = 0 when i > n. X n X n 1 is a fibration with fiber K(π n (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 is a weak homotopy equivalence. X lim X n 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(π 1 (X), 1) that induces an isomorphism on π 1 (X). We ve like to introduce a second space X 2 such that we can produce a lift K(π 2 (X), 2) X X 2 K(π 1 (X), 1) (the K(π 2 (X), 2) is hanging around to indicate that it is the fiber of the right vertical map). That is, we d like X 2 to be a total space with fiber K(π 2 (X), 2) and base K(π 1 (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(π 2 (X), 2) P K(π 2 (X), 3) where P is contractible, it would be nice if we could extend the above to a diagram K(π 2 (X), 2) X 2 P X K(π 1 (X), 1) K(π 2 (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 X 2 K(π 1 (X), 1) has to be a special kind of fibration, which now define. 4

5 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 There is also a more general definition of principal: B K(π, n + 1) Definition. A fibration F E B is a principal fibration if there is a commutative diagram such that the bottom row is a fibration sequence F E B ΩB F E B 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. standard and can be found in [1] or [2]. It is pretty Theorem. A connected CW-complex has a Postnikov tower of pricnipal fibrations if and only if the action of π 1 (X) on π n (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 X n+1 X n This means that X n+1 is classified by an element k n K(πn+1 X, n + 2). [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 be the case, given that you re taking a topology class. If you re lucky, the space splits up into parts X = i X i 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 = X 0 X 1 X 2 such that X = X i. If there are only two steps in this filtration, i.e. X 1 X 2 = X, you might hope to use a long exact sequence in homology to get at H (X): H i (X 1 ) H i (X) H i (X, X 1 ). 5

6 You might again hope something like H n (X) = H n (X 1 ) H n (X 2 /X 1 ). Of course, this is not true. What IS true is that there is an induced filtration on homology: F p n = Im(H n (X p ) H n (X)) F p 1 n F p n and that under right conditions we can compute the associated graded of H (X): gr F H n (X) = F p n /F p 1 n Of course, this doesn t completely recover homology (there are extension problems), but it gets us closer Two Degree Filtration. Let s look at an example more carefully. In fact, disgustingly carefully. H n+1 (X 1 ) H n+1 (X 2 ) H n+1 (X 2 ) H n+1 (X 1 ) 0 H n+1 (X 2 /X 1 ) H n (X 1 ) 0 H n (X 2 ) 0 0 H n (X 1 ) 0 H n (X 2 /X 1 ) H n 1 (X 1 ) H n+1 (X 2 ) 0 H n (X 2 ) 0 H n 1 (X 2 ) 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 (X 1 ) H (X 2 )) H (X 2 )/(Im H (X 1 ) H (X 2 )) We observe how these are contained in the diagram above. First, we have And we also have If we make the definition we have a diagram Im(H n (X 1 ) H n (X 2 )) = H n (X 1 )/ ker(h n (X 1 ) H n (X 2 ) = H n (X 1 )/ Im(H n+1 (X 2 /X 1 ) H n (X 1 )) H n (X 2 )/(Im H n (X 1 ) H n (X 2 )) = H n (X 2 )/(ker H n (X 2 ) H n (X 2 /X 1 )) = Im(H n (X 2 ) H n (X 2 /X 1 )) = ker(h n (X 2 /X 1 ) H n 1 (X 1 )) E 2 n,i = ker(h n(x i, X i 1 ) H n 1 (X i 1, X i 2 ))/ Im(H n+1 (X i+1, X i ) H n (X i, X i 1 )) E 2 n+1,1 0 E 2 n+2,2 Im(0 H n (X 1 )) E 2 n,1 0 Im(H n+1 (X 1 ) H n+1 (X 2 )) 0 Im(H n (X 1 ) H n (X 2 )) E 2 n, Im(H n (X 2 ) H n (X 2 )) where the colored arrows indicate exact sequences (the paths parallel to the colored arrows are exact as well). 6

7 3.2. Three Degree Filtration. We start with the diagram (which continues indefinitely in all directions): H n+1 (X 1 ) H n+1 (X 1 ) H n+1 (X 2 ) H n+1 (X 2 /X 1 ) H n (X 1 ) H n (X 1 ) 0 0 H n+1 (X 3 ) H n+1 (X 3 ) H n+1 (X 3 /X 2 ) H n (X 2 ) H n (X 2 /X 1 ) H n 1 (X 1 ) H n 1 (X 1 ) 0 H n (X 3 ) H n (X 3 /X 2 ) H n 1 (X 2 ) H n 1 (X 2 /X 1 ) H n+1 (X 3 ) 0 H n (X 3 ) 0 H n 1 (X 3 ) H n 1 (X 3 /X 2 ) Now we turn the crank. For ease of notation, let Im a,b n := Im(H n (X a ) H n (X b )) E 2 n,i = ker(h n(x i, X i 1 ) H n 1 (X i 1, X i 2 ))/ Im(H n+1 (X i+1, X i ) H n (X i, X i 1 )) Then we have a diagram (where the colored arrows are exact sequences). E 2 Im 0,1 n+1 E 2 n+1, E 2 Im 1,2 n+1 E 2 n+1,2 Im 0,1 n = 0 E 2 n,1 0 0 Im 2,3 n+1 E 2 n+1,3 Im 1,2 n E 2 n,2 0 0 Im 3,3 n+1 0 Im 2,3 n E 2 n,3 Im 1,2 n 1 0 Im 3,3 n+1 0 Im 3,3 n 0 Im 2,3 n 1 Note that we can prove these sequences are exact, just as we did above by messing around with long exact sequences. There is one exception, the sequence 0 Im 2,3 n+1 Im3,3 n+1 E2 n+1,3 Im1,2 n Im 2,3 n E 2 n,2 0 For fun, we prove this sequence is exact (we at least do some of the argument). We focus on the following portion of the diagram above: H n (X 1 ) i H n+1 (X 3 /X 2 ) k H n (X 2 ) j H n (X 2 /X 1 ) k H n 1 (X 1 ) i H n (X 3 ) First, let s prove the map Im 2,3 n En,2 2 is surjective. Let x Im2,3 n. We know there is an x such that ix = x. We then take jx. This is the map to En,2 2 (note that jx is in the kernel of H n (X 2 /X 1 ) H n 1 (X 1 )). 7

8 Now, suppose α ker(h n (X 2 /X 1 ) H n 1 (X 1 ))/ Im(H n+1 (X 3 /X 2 ) H n (X 2 /X 1 )) Thinking about it, this means α is in the image of j, i.e. there is α H n (X 2 ) such that jα = α. Then look at iα. This is in Im 2,3 n. This shows the map is a surjection. (There is a little bit of a lie in this argument; see if you can find it). Now, we prove the sequence is exact at Im(H n (X 2 ) H n (X 3 )). It is clear that the kernel of Im 2,3 n En,2 2 contains Im 1,2 n (why?). But we need to show that it s all of it. Let x Im(H n (X 2 ) H n (X 3 )) and suppose that ix = x and jx = 0 in En,2 2. This means that jx Im(H n+1 (X 3 /X 2 ) H n (X 2 /X 1 ). That is, there is a y such that jky = jx, i.e. j(ky x ) = 0. By exactness, ky x H n (X 2 ) lifts to an element in H n (X 1 ), call it x. Furthermore i(x ) = i(x ky) = ix = x since ik = 0 again by exactness. I ll stop proving exactness of this sequence here. The rest proceeds in substantially the same way. But! The fact that we ve verified exactness verifies that the diagram I drew above is actually useful. Now, we can continue! But first, note that from the above diagram we have E 2 n,1 = Im(H n(x 1 ) H n (X 2 )) En,2 2 = Im2,3 n / Im 1,3 n so that already at this stage we ve aleady done some computing of the associated graded. Define E 3 n+1,i = ker(e3 n,i E3 n 1,i 2 )/ Im(E3 n+1,i+2 E3 n,i ) We get a diagram 0 E 3 n+1, E 3 n+1,2 0 E 3 n,1 Im 1,3 n+1 E 3 n+1,3 0 E 3 n,2 Im 2,3 n+1 Im 3,3 n+1 0 Im 1,3 n 0 Im 2,3 n E 3 n, Im 3,3 n We can now verify that the requisite sequences are exact. But note that they are short exact! 3.3. Exact Couples. The general mess of the above calls for some more general organizational system. This is called an exact couple. Definition. An exact couple is a diagram of (possibly graded) abelian groups D k i j D E 8

9 with the diagram exact at each corner. Now, we define a map d : E E by d = jk. Note that d 2 = 0. We can then define new groups E = ker d/ Im d D = i(d) and new maps as follows. First, let i = i D. Let j (ix) = [jx] (check that this is well-defined!). Let k [y] = ky (again, check that this is well-defined!). These groups and maps give a derived couple (E, D, i, j, k ). Proposition. The derived couple is an exact couple. Proof. Diagram chasing exactly analogous to the above. This means that the process can be iterated over and over again. In fact, what we were doing was computing derived couples over and over again. We end up with a sequence (E r, D r, i (r), j (r), k (r) ) and we define d r = j (r) k (r). Now, a spectral sequence is a gadget that uses exact couples to compute (co)homology from filtrations. The degree of cohomology and degree of filtrations give us a bigrading on our abelian groups. That said, here is the definition of a spectral sequence. Definition. A spectral sequence is a sequence {E r,, d r } of (possibly graded of bigraded) groups such that (1) d r : E r E r satisfies d 2 r = 0 (2) E r+1 = ker d r / Im d r Remark. The collection of groups E r, is typically called the rth page of a spectral sequence. Remark. The differential d r has a degree. It s degree depends on a few things; for example, whether we are computing homology or cohomology. It depends on much more arbitrary choices, such as how we index things. However, it is common in homological spectral sequences for d r to have degree ( r, r 1) and common in cohomological spectral sequences for d r to have degree (r, 1 r). Example General Filtrations. Of course in this case, we have a buttload more information. We have exact sequences like that for pairs (X i, X i 1 ) and by direct summing the sequences we end up with a digram like this: H (X i ) H (X i ) This gives us an exact couple, H (X i, X i 1 ) 3.5. Misc. There are other useful tools we ll need from spectral sequences. For example, the following comparison theorem. Theorem. Let E r p,q and E r p,q be two homology spectral sequences. Suppose that there is a map E r p,q E r p,q with then Proof. ADD PROOF Various exact sequences will also come in handy. E p,q = E p,q E 2 0,q = E 2 0,q E 2 p,0 = E 2 p,0. Example. Suppose we have a homological spectral sequence converging to some H (X) with an E 2 -page such that E 2 p,q = 0 0 < q < n. 9

10 That is, there is a large gap between the zeroth row and the next. The first differential that is interesting is then d n+1 : E 2 n+1,0 E2 0,n. Since no other interesting differentials will hit the E 0,n-spot we have an exact sequence E 2 n+1,0 Now, we further know we have a filtration of H n (X): d n+1 E 2 0,n E 0,n 0 H n (X) = F n n F n 1 n F 0 n = 0 such that E p,n p = Fn p /Fn p 1. In particular, E0,n = F0 n. Since E1,n = 0 by assumption, we have F0 n = Fn 1 =. This continues until we hit E 0,n This gives an exact sequence E n,0 = Fn n /F n 1 n = F n n /E 0,n = H n(x)/e 0,n. 0 E 0,n H n(x) E n,0 0. Stitching these two exact sequences together we obtain E 2 n+1,0 d n+1 E 2 0,n E 0,n H n(x) E n, The Serre Spectral Sequence The Serre spectral sequence is a spectral sequence that will allow to organize information about fibrations. Given a fibration F E B it allows us to deduce information about H (E) from H (B), H (F). Theorem (Homological Serre Spectral Sequence). Given a fibration F E B with B path connected and π 1 (B) acting trivially on H (F, G), there is a homological spectral sequence {E r p,q, d r } with (1) d r : E r p,q E r p r,q+r 1, i.e. a differential of degree ( r, r 1) (2) E p,n p = Fn p /Fn p 1 where the F arise from some filtration of H (E; G) (3) E 2 p,q = H p (B; H q (F; G)). Remark. In many cases, the E 2 -page can be written as E 2 = H p (B) H q (F) by the Künneth theorem. Remark. The picture to keep in mind: d 4 d 3 d 2 There is also a cohomological version. This is perhaps more useful, since as wel ll note (but not prove), this comes equipped with a multiplicative structure that arises from the cup product in cohomology. Theorem (Cohomological Serre Spectral Sequence). Given a fibration F E B with π 1 (B) acting trivially on H (B; G), there is a cohomological spectral sequence {Er p,q, d r } with (1) d r : Er p,q E p+r,q r+1 with E p,q r+1 = ker d r/ Im d r (2) There is some filtration where E p,n p = Fp n /Fp 1 n (3) The E 2 -page can be computed as E p,q 2 = H p (B; H q (F; G)). This cohomology spectral sequence can be quite useful because of a multiplicative structure on it. 10

11 Theorem (Multiplicative properties of Serre Spectral Sequence). There is a multiplication E p,q r E s,t r with the following properties (1) d r is a derivation with respect to this product: (2) On the E 2 -page the product is the cup product E p+s,q+t r d r (ab) = (d r a)b + ( 1) p+q a(d r b) H p (B; H q (F; R)) H s (B; H t (F; R)) H p+q (B; H s+t (F; R)) (3) The products on H (E; R) restrict to products on the filtrations, and these agree with the product on E. Let s do some examples. Example. Let s do a classic example: Compute H (ΩS n ; Z). We use the fibration ΩS n PS n S n. We recall { H (S n Z = 0, n ; Z) = 0 otherwise. Now, the E 2 -page is So, let s draw a picture in the case n = 3 { E 2 p,q = H p (S n ; H q (ΩS n H q (ΩS n ; Z) p = 0, n )) = 0 otherwise H 3 (ΩS 3 ; Z) 0 0 H 3 (ΩS 3 ; Z) H 2 (ΩS 3 ; Z) 0 0 H 2 (ΩS 3 ; Z) H 1 (ΩS 3 ; Z) 0 0 H 1 (ΩS 3 ; Z) Z 0 0 Z The only differential that exists is d 3 as pictured. Furthermore, since the E page must be trivial, the map must be an isomorphism. We have H 2 (ΩS 3 ; Z) = H 4 (ΩS 3 ; Z) = H 6 (ΩS 3 ; Z) = 5. Some Rational Computations We ll use the Serre spectral sequence (mostly cohomological) to do some computations with coeffiecients in Q. The first few are perhaps the most important cases for us: The Eilenberg-MacLane spaces. As we ve seen in the construction of Postnikov towers, Eilenberg-MacLane spaces are the constituents of topological spaces (in a sense dual to cells). Thus, knowing rational cohomology will give us a hint to the rational cohomology of spaces in general. Theorem. We have where deg(x) = n. { H Q[x] (K(Z, n); Q) = Λ Q (x) n even n odd Proof. We first note that H (K(Z, 1); Q) = Λ Q (x) since S 1 is a K(Z, 1). How, we note that there is the path-space fibration K(Z, n 1) PK(Z, n) K(Z, n) and then the Serre spectral sequence for this fibration has E 2 term E p,q 2 = H p (K(Z, n); Q) H q (K(Z, n 1); Q). 11

12 We are assuming by induction that we have computed H (K(Z, n 1); Q). We have to work in cases. Case 1: n odd. In this case H (K(Z, n 1); Q) is a polynomial algebra on a generator degree n 1. First, we note that H (K(Z, n); Q) = 0 when < n and H n (K(Z, n); Q) = Qx. Let s do n = 3. The E 2 -page is pictured as: Qy Qx Qy Qy 0 0 Qx Qy Qx H 4 (K(Z, 3); Q) H 5 (K(Z, 3); Q) H 6 (K(Z, 3); Q) Now, since the E -page has to be zero, H 4 and H 5 both have to be zero, since there are no differentials that hit it. The differential d 2 doesn t exist for degree reasons, so E 2 = E 3. Now, the differential d 3 (pictured) can be non-trivial. Since we need E to vanish, it must be that dy = x. Then, this implies that d(y 2 ) = 2x y and thus this is an isomorphism (since we re working over the rationals). Since this must be an isomorphism, H 6 has to vanish. Thus x 2 = 0. We continue by induction to compute everything else. Proposition. There is a map K(Z, 1) K(Q, 1) induced by Z Q descends to an isomorphism on cohomology: H (K(Z, n); Q) = H (K(Q, n), Q) 6. Serre Theory Definition. Let C be one of the following FG - finitely generated abelian groups T P - torsion abelian groups whose elements have orders that are multiplies of primes drawn from P F P - the finite groups in T P. There are some obvious consequences of the definitions which will be useful. Lemma. We have (1) The classes C are closed under extension. That is, if A C, B C then if there is an exact sequence C C as well. (2) If A, B C then A B, Tor(A, B) C. Exercise. Prove the above statement. 0 A C B 0 We are going after a kind of Hurewicz theorem mod C. The following lemma will be useful for that Theorem. Let F X B be a fibration with π 1 (B) acting trivially the fiber. Assume F, X, B are path-connected. If two of F, X, B have H n C, then so does the third. Proof. Suppose H n (F), H n (B) C. We will figure out H n (X) from the spectral sequence. The E 2 page is E 2 p,q = H p (B, H q (F)) = H p (B; Z) H q (F; Z) Tor(H p 1 (B), H q (F)). Every element in this E 2 -page is in C by supposition. Since quotients of elements of C are in C, then E r p,q and thus E p,q C. So we have H n (X) C. There are two other cases, but these will be left as exercises to do on your own and look up. Lemma. π C, then H k (K(π, n); Z) C for all k, n > 0. 12

13 Proof. We will obviously use induction and the fibration sequence There are some reduction steps. K(π, n 1) K(π, n). 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 F p ) So, we do this for FG and F p. But, this is easy. A K(Z, 1) is S 1, 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 π 1 (X) acts trivially on π n (X), then for all n > 0. π n (X) C H n (X; Z) C Proof. The conditions would see to suggest that we build a Postnikov tower. If π i (X) C, then H (K(π i (X), n)) C from the above. Then, by using induction and the Serre spectral sequence, we get H n (X n ; Z) C. Before we go on, the following will be useful. Lemma. Let X be a topological space with π 1 (X) acting trivially on π m (X) for all m. Let X n denote the n Postnikov level of that space. Then H n (X) = Hn (X n ). Proof. We begin with the map X X n and turn it into a fibration, which we also call X X n. Take the fiber to obtain a fiber sequence X >n X X n. Note that by the long exact sequence in homotopy, π i (X) = 0 for i n. Now, we apply the Serre Spectral sequence. The E 2 -page is E 2 p,q = H p (X n ; H q (X >n )) and we note that H q (X >n ) is only non-trivial for q n + 1. It follows that E p,n p = 0 except when p = n and so H n (X) = H n (X n ). Theorem (Mod C Hurewicz). If X has π i (X) C for i < n then h : π n (X) H n (X) is an isomorphism mod C. The following corollaries are usually more useful than the theorem itself: Corollary. If for all i, π i (X) C, then h : π (X) H (X) is an isomorphism mod C. Corollary. For X a simply connected topological space, H i (X; Z) Q = 0 for all i if and only if π i (X) Q = 0 for all i. Proof of Theorem. We assume for the moment that X is simply connected. Let {X i } be the Postnikov tower of X. The maps π n (X) H n (X) and π n (X n ) H n (X n ) 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: H n+1 (X n 1 ) d n+1 H n (K(π n (X), n)) E 0,n H n (X n ) H n (X n 1 ) The composition of maps in the middle is the inclusion of the map H n (K(π n (X), n)) H n (X n ). 13

14 Now, if we assume π i (X) C for i < n, then π n (X n 1 ) C and thus H n (X n 1 ) C and H n+1 (X n 1 ) C. Thus, the map H n (K(π n (X), n)) H n (X n ) is an isomorphism mod C. We then consider the diagram = π n (K(π n (X), n) π n (X n ) = H n (K(π n (X), n) H n (X n ) 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. π i (S n ) are finite for i > n except for when n is even and i = 2n 1. In this case π 2n 1 (S n ) = Z finite. Proof. Consider the map S n K(Z, n) which represents the generator of π n (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 π i F = π i S n in the range we care about (i > n). Further convert the map F S n into a fibration with fiber K(Z, n 1): K(Z, n 1) F S n. This is amenable to attack by the Serre spectral sequence. groups when n is even or odd, we split into two cases. Case 1: n odd. In this case we know the E 2 -page is and so it looks like (with d 2 drawn in) E 2 p,q = H (S n ; Q) H (K(Z, n 1); Q) = Λ(x) Q[y] Since K(Z, n 1) have different homology 3n 3 Qy 3 2n 2 Qy 2 Qxy 3 Qxy 2 n 1 Qy Qxy 0 Q Qx 0 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 d 2 : Qy Qx is an isomorphism. By the multiplicativity of the spectral sequence it is then the case that ALL d 2 s are isomorphisms. Thus, H (X; Q) = 0, and π i (X) Q is 0 as well. Thus, π i (S n ) is finite for i > n. Case 2: n even. In this case the E 2 -page is E 2 p,q = H (S n ; Q) H (K(Z, n 1); Q) = Λ(x) Λ(y) 14

15 and so we have the E 2 -page looking like Qy Qxy Q which means that H (X; Q) = H(S 2n 1 ; Q). By Hurewicz for C = FG, we have that π i (S n ) is 0 for n < i < 2n 1 and π 2n 1 (S n ) = Z finite. We now need to compute the higher homotopy groups. Let F F <2n 1 be a map obtained by killing all homotopy groups i 2n 1 of F. We turn that map into a fibration to obtain a fiber sequence Qx F 2n 1 F F <2n 1. We note that all π i (F <2n 1 ) are finite (since we already know them for i < 2n 1), and thus H(F <2n 1 ; Q) = 0. The Serre spectral sequence then gives We now consider H (F 2n 1 ; Q) = H (F; Q) = H (S 2n 1 ; Q). F 2n 1 K(Z, 2n 1) which induces an isomorphism on π 2n 1 mod torsion. We use the argument as in the odd case to get that π i (F 2n 1 ) is finite for i > 2n 1. That is, we extend to a fibration sequence K(Z, 2n 2) F F 2n 1 K(Z, 2n 1). We have, as in our earlier case π i ( F) = π i (F 2n 1 ) for i > 2n 1, so it s enough to compute homotopy groups of F. How, we use the Serre spectral sequence. The E 2 -page is E 2 p,q = H p (F 2n 1 ; Q) H q (K(Z, 2n 2); Q) = H p (S 2n 1 ; Q) H q (K(Z, 2n 2); Q). We then argue as in the first case to show that H i ( F; Q) has to vanish. 8. Rational Spaces and Localization We construct rationalizations of spaces. That is, for a topological space, there is a functorial construction which assigns to X a new spaces X Q and a map X X Q such that π (X) Q π (X Q ) Q = π (X Q ) is an isomorphism. In fact, more is true about localization. Let s precisely state what a Q-local space is Definition. A Q-local space (or a Q-space) is a simply connected topologocal space X such that (1) π (X) is a Q-vector space (2) H (X; Z) is a Q-vector space. Note that this is implicitly a theorem too we have to prove that those two conditions are equivalent Proof. 1) = 2). We assume that π i (X) is a Q-vector space. To show that H i (X) is a Q-vector space, we use our favorite tool: Postnikov towers and the Serre spectral sequence. The induction can start at K(π 2 (X); Q) = X 2. Then we know that H (X 2 ; Z) is a Q-vector space if π 2 (X) is. We assume that we ve proved that H (X n 1 ) is a Q-vector space. We use the fibration K(π n (X), n) X n The E n -page of this spectral sequence is X n 1 E 2 p,q = H p (X n 1 ; Z) H q (K(π n (X), n); Z) then this gives that H (X n ; Z) must be a Q-vector space. 15

16 2) = 1) Suppose H (X; Z) is a Q-vector space. And suppose by induction that we know π (X n 1 ) are Q-vector spaces. Then H (X n 1 ; Z) is (by the above) and so is H (X, X n 1 ) (by the long exact sequence of a triple. But π n (X) = H n+1 (X, X n 1 ). So, all homotopy groups of X n are rational vector spaces. Say something about commuting with limits Definition. Let X, X Q be simply-connected CW-complexes. We call a map X X Q a localization if any of the three (equivalent!) conditions are satisfied: (1) The induced map is an isomorphism. (2) The map H (X; Q) H (X Q ; Q) π (X) Q π (X Q ) Q is an isomorphism (3) The map X X Q is universal in the following sense. If X Y Q is a map to a Q-space Y Q then this factors through X Q uniquely. Proposition. The above conditions are equivalent. Proof. I won t prove all of them (one of the equivalences requires an argument from obstruction theory, which I would prefer to ignore for the moment). 3) = 1) Let Y Q = K(Q, n). Then the fact that this factors uniquely means that [X, K(Q, n)] = [X Q, K(Q, n)] and thus H (X; Q) = H (X Q ; Q) by the representability theorem. 2) 1) In this case π (X) Q π (Y) Q will be an isomorphism if and ony if π (F) Q = 0 for all i. But, by the rational forms of Hurewicz, this will be the case if and only if H (F; Q) = 0. But by the Serre spectral sequence, this will imply that H (X; Q) = H (Y; Q). We would now like to construct localizations of spaces. That is, we would like to construct a map X X Q such that X Q is Q-local and the map induces an isomorphism on rational cohomology. First, we note that for K(G, n) with G abelian, this is easy. Proposition. The map K(G, n) K(G Q, n) induced by G G Q is a Q-localization. Proof. By definition. Using K(G, n) as building blocks, we can construct the localization of a space in general. Note that the above only makes sense when G is abelian and so for the moment we will neglect π 1 phenomena. Theorem (Q-localization of spaces). Given a (for now simply connected) CW-complex X, there is a localization X Q and a map X X Q. Proof. We construct this via inducation and Postnikov towers. Suppose we have constructed a localication l n 1 : X n 1 (X n 1 ) Q. We d like to construct (X n ) Q. Well, we know how to construct X n via the k-invariant. That is to say, there is a (pullback) commutative diagram X n X n 1 k n+1 K(π n (X), n + 1). We know exactly what we want the n-level of (X n ) Q to be we d like it to be K(π n (X) Q, n + 1). We can compose the k-invariant above with the localization map: X n 1 K(π n (X), n + 1) K(π n (X) Q, n + 1) 16

17 and this, by definitino of localization factors through (X n 1 ) Q. We now form the pullback square to define (X n ) Q (X n ) Q (X n 1 ) Q K(π n (X) Q, n + 1) and there is an induced map X n (X n ) Q. It is easy to check that this is the desired localization. Example (Rational Bott Periodicity). Chern classes produce maps (by representability) BU K(Q, 2k) k and the computation of H (BU; Q) show that that map becomes an isomorphism on rational cohomology. Thus, BU Q K(Q, 2k). k In particular, we note Ω 2 BU Q BU Q, which is rational Bott periodicity. The following should properly be an example, but it s interesting enough to state as a proposition. Proposition. S 2k+1 Q is a K(Q, 2k + 1). Also, there is a fibration so that K(Q, 4k 1) SQ 2k K(Q, 2k) π i (S 2k ) Q = { Q i = 2k, 4k 1 0 otherwise Proof. We know that S 2k+1 K(Q, 2k + 1) induces an isomorphism on rational cohomology (thus homology, thus homotopy). Or, we can observe that K(Q, 2k + 1) is a Moore space M(Q, 2k + 1), but M(Q, 2k + 1) = SQ 2k+1. For the rest, consider SQ 2k K(Q, 2k) which induces an isomorphism on H 2k (we can get such a map from the universal property of localization). Turn this into a fibration F SQ 2k K(Q, 2k) and use the Serre spectral sequence Q Qx Qx 2 Qx 3 0 2k 4k 6k 8.1. Rational H-spaces. 17

18 9. DGAs We now switch gears. We ve discussed how to build spaces, and how to do some computations. We ve discovered that if we are working with only rational coefficients, computations become much easier. There is, of course, a deeper reason for this. We will see later that when we put on our rational goggles, topological spaces are classified completely by certain types of DGAs. Furthermore, notions of maps and homotopies will extend to the category of DGAs. Since this is the case, it is important for us to understand the homotopy theory of DGAs Basic Definitions. Definition. A differential graded algebra is an algebra that is both graded, and equipped with a differential and the algebra multiplication and differential play well with the grading. More precisely, A is an algebra over a field k (in our case, Q, usually) with a grading A p such that A = A p p 0 and (1) A differential d : A A +1 with d 2 = 0 and d(ab) = da b + ( 1) p a db (2) The multiplication is graded: A p A q A p+q satisfies What are some examples? ab = ( 1) pq ba. Example. (1) Over R, differential forms under wedge product. (2) (t, dt). This is going to be the interval in much of what follows. This is kind of lame, not-too-clear notation for the algebra Let s just do a computation with this for fun: Q[x] Λ(y) dx = y, deg x = 1, deg y = 2. d(xy) = xdy + ( 1) 0 ydx = xd 2 x = 0 d(x n y) = x n dy + ( 1) n d(x n )y = ( 1) n n!(dx)y = 0 We can even compute the cohomology! And we will, for shits and giggles. We have H 0 (I) = ker d : I 0 I 1 = Q since the differential operating on the ground field is 0. Moving on, H 1 (I) = ker d/ Im d = 0 since dx = y and all the elements in degree 1 are of the form ax. H 2 (I) = Q y /Q y (3) The Koszul complex (4) H (X; Q) with a trivial differential (i.e. it s a graded vector space with a multiplication) (5) H (X; Z) with Bockstein. Example. An important non-example is C (X; Q). Example. There isn t really a notion of free DGA (as far as I know), but there are semi-free DGAs. These are DGAs whose underlying graded-commutative algebra is free. Free graded-commutative algebras are defined as follows. Given a graded vector space {V } the free graded-commutative algebra is F(V ) = k[v even ] k Λ(V odd ). 18

19 Definition. The cohomology of a DGA is defined how one would think it is. There is a differential d : A A +1 and we define H n (A ) = kerd : A n A n+1 / Im d : A n 1 A n Definition (Relative Cohomology). Let C, D be chain complexes (we forget structure from DGAs), and suppose f : C D is a map between chain complexes. We define a mapping cylinder, M f to be with differential d M : M n f M n+1 f We then make the definition Lemma. There is a log exact sequence Proof. We leave this as an exercise. given by M n f = C n D n 1 ( dc 0 f d C ) H (C, D) := H (M f ). H n (C ) H n (D ) H n+1 (C, D ) H n+1 (C ) 9.2. Sullivan Models. We would now like to do homotopy theory with DGAs. We could wave a wand and come up with a model category structure on CDGAs. And we will. But it s also useful to know how to work with them on an element-by-element level, so we ll develop the abstract and concrete in parallel (also as a good example of the abstractions). Recall we have the following model structure on CDGA Q Theorem. There is a model structure on CDGA Q such that The generating cofibrations, I, are the maps ΛS n 1 ΛD n The generating ayclic cofibrations, J, are the maps 0 ΛD n. The weak equivalences are quasi-isomorphisms. Furthermore, the cofibrations are retracts of I-cell complexes and the fibrations are (degree-wise) surjections. Now, this is a cofibrantly generated model category. This allows us to quickly identify the cofibrant objects in this category (or at least some of them). Corollary. The cofibrant objects in CDGA Q with the above model structure are exactly I-cells. A convenient property: Corollary. All objects in CDGA Q are fibrant. Definition. A Sullivan algebra is a cofibrant object in CDGA Q. Example. Let s see what this says. Suppose we ve built k 1 stages of a Sullivan algebra, and we call the algebra ΛV(k 1) at stage k 1. To build the next stage we build a pushout ΛS n ΛD n ΛV(k 1) ΛV(k) The map ΛS n ΛV(k 1) picks out an x V(k 1) and the pushout forces the condition that dx = y, where y corresponds to the element that D n maps to in ΛV(k). Remark. Look up the definition of a Sullivan algebra in Felix-Halperin-Thomas. The definition they have there is completely equivalent to this one, but this one is a bit more compactly stated. Remark. A lot of times below we ll suppress the notation of the derivation. 19

20 Definition. A Sullivan model for A CDGA Q is a sullivan algebra ΛV such that there is a quasiisomorphism ΛV A. Now, one can easily see from our knowledge of model categories that a Sullivan model exists this is because a Sullivan model is just cofibrant replacement! We now want to define homotopies. We need a quick auxilliary definition Definition. Λ(t, dt) = ΛD 1. Definition. Two morphisms f, g : A B of CDGA Q are homotopic if there is a morphism such that (Id ɛ 0 )H = f and (Id ɛ 1 )H = g. H : A B Λ(t, dt) Remark. Of course, it is easy to check that B Λ(t, dt) is a cylinder object for B and so the definition above corresponds with our usual definition of homotopy in a model category. We have a bunch of lifting properties for free from the model structure. They are usually a pain to prove because people insist on using algebraic structure. For some of the proofs we ll do that too, just to get a sense of how to do element-wise arguments in CDGA Q. Lemma (The Lifting Lemma). Suppose we are given a diagram ΛV where the right vertical arrow is a surjective quasi-isomorphism and ΛV is a sullivan algebra. Then there is an indicated lift. Proof 1. A surjective quasi-isomorphism is a trivial fibration. Since ΛV is cofibrant, the lift exists. Proof 2. ΛV can be inductively built, and so, as usual for cellular things, we build a lift inductively. Suppose we ve lifted ΛV(k 1). The next level is determined by a map d : V k Λ(k 1). Let V k = v α. Definition. A relative Sullivan algebra is a relative I-cell complex. Remark. As usual, this has an alternate definition. It is a CDGA Q of the form B ΛV where B is a subcochain algebra, 1 V = V and V is built as V(0) V(1) such that Lemma. B B ΛV is a cofibration. Proof. It s obviously a relative I-cell complex. d : V(0) B A B d : V(k) B ΛV(k 1) One would expect that we d be able to factor maps into these guys. This is, after, all exactly what the small object argument did for us. Let s first give a quick definition Definition. Let A CDGA Q. Let UA denote the underlying graded vector space E(A) = Λ(UA dua) d : UA = dua The literature of rational homotopy gives the following a special name (the surjective trick ) but it s really factorization in a model category. Theorem. Let A B be a morphism in CDGA Q as A A E(B) B where the first map is a relative Sullivan algebra and E(B) B is surjective. 20

21 Proof. Actually quite easy to see from definitions, but also follows from factorization. We continue to develop homotpy theory. Definition. Let [A, B] denote the set of homotopy classes of maps of CDGA Q s from A to B. Proposition. Let A C be a quasi-isomorphism and ΛV a Sullivan algebra. Then is a bijection. [ΛV, A] [ΛV, C] Proof. A and C are both fibrant objects, and ΛV is cofibrant. This then follows from a statement about model categories. This is one of those cases where doing an elemental analysis of it just isn t worth it. Lemma. Given a diagram B B ΛV L where the left vertical arrow is a relative Sullivan model and the right vertical arrow is a quasi-isomorphism there is a lifting L such that f L ψ. It is high time for an example. Example. Well, a classic non-example. Consider the algebra where ψ A f C A = Λ(a, bc) deg a, b, c = 1 da = bc db = ac dc = ab This is not a Sullivan algebra. Why? However, it can be approximated by a sullivan algebra. The whole algebra can written as 0 a, b, c ab, bc, ca abc 0 0 Computing homology is easy. ker d : A 1 A 2 = 0, so H 1 (A) = 0. ker d : A 2 A 3 = ab, bc, ac which is the whole image of d : A 1 A 2, so H 2 (A) = 0.Finally, d : A 3 A 4 is abc, which is not in the image of d : A 2 A 3. Thus H 3 (A) = Q abc. Define a CDGA Λ(x) with deg x = 3 and no differential. Then it is easy to check that there is a quasi-isomorphism f : Λ(x) A f (x) = abc Minimal Sullivan Models. Sullivan algebras are I-cell CDGAs, which should be thought of like cellular topological spaces. However, we all know from working with topological spaces that the concept of CW is also very important. The version of this for CDGAs is called a minimal Sullivan algebra. These, like CW complexes, will be easier to work with and have a sort of uniquness that Sullivan algebras lack. Definition. Let Λ + V denote the positive-degree elements of ΛV in positive degree. Definition. A Sullivan algbra is a minimal Sullivan if Im d Λ + V Λ + V. Lemma. Any CDGA of the form (ΛV, d) with V = V 2 such that Im d Λ + V Λ + V 21

22 Proof. It already satisfies the condition for minimality. Consider the degree k chunk of V, V k. We know d : V k V k+1, but also Im d Λ + V Λ + V. Looking at (Λ + V Λ + V) k+1 we see for degree reasons that it is inside ΛV k 1. Thus, d : V k ΛV k 1, which provides us with the filtration necessary for a Sullivan algebra. We know that every algebra has a Sullivan algebra replacement (since it is just cofibrant replacement). We can t use this technique to produce minimal Sullivan algebras. We ve got to do this by hand. We ll provide a general argument later, but for simply connected algebra, we really can do an element by element argument. Proposition. Suppose A CDGA Q such that H 0 (A) = Q and H 1 (A) = 0. Then there is a minimal Sullivan model (ΛV, d) for A. Proof. We, of course, proceed by induction. We start H 2 (A). Just pick ΛV 2 with no differential and a map f 2 : (ΛV 2, 0) A. We suppose that f k : (ΛV k, d) A has been constructed and we try to construct f k+1 : ΛV k+1 A. We also suppose by H i ( f k ) is an isomorphism for i k and injective for i = k + 1. We try to construct f k+1. Consider the long exact sequence H k+1 (ΛV k ) Hk+1 ( f k ) H k+1 (A) H k+1 (ΛV k, A) H k+2 (ΛV 2 ) Hk+2 ( f k ) H k+2 (A) We assume that H k+1 ( f k ) is injective. We d like to make it surjective as well, so we ll try to tack on some extra stuff to ΛV k to make this true. Similarly, we ll try to fix up H k+2 ( f k ) to be injective. We work on the first part. A failure of surjectivity is that H k+1 ( f k ) doesn t hit enought stuff. We pick a basis for the stuff it doesn t hit, i.e. H k+1 (A) = Im H k+1 ( f k ) i I Q a i where the a i is the basis and I is some indexing set. Also, a failure of injectivity means we have a kernel for H k+2 ( f k ). We pick a basis for this too: ker H k+2 ( f k ) = j J Q z j. So at least we have names for what s mucking us up. Now, we build these things into our algebra. Let where deg v, v = k + 1 and define V k+1 = v i, v j I,J dv i = 0 dv j = z j. This makes sure that v i represent elements of homology in degree k + 1 and makes sure that z j is exact. We let ΛV k+1 = ΛV k+1 ΛV k. Now, we extend the map to f k+1 : ΛV k+1 A by f k+1 (v i ) = a i f k+1 (v j ) = b j where f k z j = db j (this is just to make sure the differential works). Now we show that the inductive hypothesis holds. H k+1 ( f k+1 ) is surjectivey by construction. We now want both H k+1 ( f k ) and H k+2 ( f k+1 ) to be injective. Suppose f k+1 [α] = 0 where α is in degree k + 1. That is, we can write a representative where R is a product of lower order terms. α = λ i v i + λ j v j + R 22

23 Since α is a closed thing, dα = 0. Thus λ j z j + dr = 0. In homology, this becomes λ j [z j ] = 0, and because we chose z j to be a basis, each λ j = 0. So, now we know α = λ i v i + R. Apply f to get f (α) = λ i α i + f (R) Now, in homology this is λ i [α i ] + [ f (R)]. However, we know that [ f (R)] Im H k+1 ( f k ) and [α i ] is a basis for the rest. We thus have λ i = 0. We need to show that H k+2 ( f k+1 ) is injective now. But this is by construction. We ve made everything in ker H k+2 ( f k ) exact, and thus 0 in homology Sullivan and Minimal Models. We are going to claim that DGAs over Q provide a reasonable model for rational spaces. Spaces, as we know, can be built out of smaller bits in various ways. For example, we can build a cell complex or we can build a Postnikov tower (these are in a precise way dual to each other). In the same way, and perhaps it is even simpler to see, various algebra things are built out of smaller bits. Vector spaces have bases, algebras have generators and relations and these objects can be built up sequentially out of such things. Definition. A DGA A is minimal if (1) A is free as a graded-commutative algebra on generators in degree 2. (2) We also have d(a ) = A + A + A + = A k k>0 We now define how to build DGAs as we build Postnikov towers. Definition. Let A be a DGA. A Hirsch extension is an inclusion A A d Λ(V k ). The notation on the right means that we come equipped with a map d : V A k+1. We say two Hirsch extension are equivalent if...there is an obvious commutative diagram. Lemma. Two extensions A A d Λ(V) A A d Λ(V ) are equivalent iff there is an isomorphism ψ : V V and the diagram commutes: V V H k+1 (A) Proof. = direction. We assume that ϕ : A d Λ(V) = A d Λ(V ) is an isomorphism and that it furthermore extends the identity on A. Because of the condition of extending the identity, and the diagram commuting, it must be the case that ϕ(v) = a v + ψ(v) where a v is an element of A depending on v (linearly). Now, ϕ(dv) = d (ϕ(v)) = d (a v + ψ(v)) = d (a v ) + d(ψ(v)). This means that [dv] = [d ψ(v)] H k+1 (A ). 23

24 = direction If ψ : V = V then [dv] = [d ψ(v)] H k+1 (A ). This means dv d ψ(v) = da, a v A. Then, define ϕ(v) = a v + ψ(v) and this defines a map. The corollary of the proof is more important than the proof itself: Corollary. Equivalence classes of extensions are in 1-1 correspondence with maps d : V H k+1 (A ) which in turn is the same thing as a class in H k+1 (A ; V ). The following gives a characterization of minimality, and tells us why we introduced Hirsch extensions in the first place. Theorem. Let M be a DGA and M(n) M the subalgebra generated in degrees n. Then M is minimal if M(1) is the ground field, n M(n) = M and each inclusion M(n) M(n + 1) is a Hirsch extension with the new generator in degree n Forms on Simplicial Sets We ve studied rational spaces and dgas. We re going to try to link them together now. To a space, we ll want to assign a certain DGA. The issue, of course, is what DGA to assign to it. A first guess would be the chain complex C (X; k), but chain complexes are a somewhat poor proxy for our purposes. The issue with them is that there is only a quasiisomorphism between C (X Y; k) and C (X; k) k C (Y; k). If X were a manifold we d have a ready-made DGA: the complex of differential forms Ω (X; R) under wedge product. The idea is to create differential forms for more general types of spaces, e.g. simplicial sets. Of course, differential forms have their own theory which must be developed Differential Forms. We recall the following: Definition. Let C be a category. A simplicial object in C is a functor X : op C Remark. Simplicial rings arise in derived algebraic geometry, simplicial abelian groups pop up all over the place (e.g. higher Chow groups), simplicial simplicial sets (alias bisimplicial sets) are quite useful, etc. Definition. A simplicial commutative DGA is a functor A : op CDGA Q,. Remark. Note that a simplicial CDGA has two indices, and we will often denote one by A. The upper index will be the algebra grading and the lower index will be the simplicial grading. Remark. We think of A p as a simplicial set (by forgetting structure) We come to a very important definition Definition. Let A CDGA Q, and K Set. Define A p (K) as follows. As a set A p (K) = Hom Set (K, A p ). Addition, multiplication, and the differential are given by the second coordinate. We note that this construction is covariant in the algebra and contravariant in the simplicial set. Remark. This is slightly abstract. What is this saying? For every simplex σ K n (i.e. every map n K) we assign it an element x σ A p n. That is, Φ A p (K) is a map Φ(σ) = x σ. The map is additive, plays well with scalar multiplication and the differential. Remark. A further remark: How should we think of this in relation to differential forms on manifolds? Why doesn t the definition look the same? First, recall the definition of Ω p (M). We first define a bundle Λ p (T M) M and then define Ω p (M) to be a section of this bundle. A section, of course, is a continuously varying choice of p-form. That is, it s a continuously varying choice of some symbol that transforms correctly between charts. So what does it take to define a p-form? Really, a way to define it on a chart (local data), and then a way to move between charts (gluing). 24