CONSTRUCTION OF MAPS BY POSTNIKOV TOWERS
DISSERTATION
Presented in Partial Ful llment of the Requirements for the Degree Doctor of
Philosophy in the Graduate School of the Ohio State University
By
Chris Kennedy,
Graduate Program in Mathematics
The Ohio State University
2018
Dissertation Committee:
Jean-François R. Lafont, Advisor
James Fowler
John E. Harper c Copyright by Chris Kennedy
2018 ABSTRACT
Using Postnikov towers, we investigate the possible degrees of self-maps of various spaces, including SU(3), Sp(2), SU(4), and the principal Sp(1)-bundles over S7. This investigation requires determining the Postnikov invariants for these spaces, which involves a detailed exploration of the secondary and tertiary cohomology operations that relate their cohomology classes. We also present some conditions for nding and lifting H-maps in Postnikov towers, and conjecture what corresponding results about SU(n) and Sp(n) should look like.
ii For my parents
iii ACKNOWLEDGMENTS
This thesis would not have been possible without a lot of help. I am grateful to my advisor, Jean Lafont, for helping and encouraging me through the last several years, and dedicating countless hours to hearing me attempt to explain what I had read or done that week, even as my research strayed further and further from the original plan. I also thank Jim Fowler for giving me so much of his time to help unravel things, and the many helpful suggestions and clarifying remarks he made.
Among my peers, special thanks to Kyle Parsons, Jake Blomquist, Tom Dinitz, and Mike Steward for being great friends, and also for providing a supportive en- vironment for learning how to play board games about Renaissance-era commerce.
Jake deserves additional thanks for elding many questions about category and ho- motopy theory. Thanks also to the broader circle of math folks here, including but not limited to Laine Noble, Neal Edgren, Charles Baker, Mike Belfanti, Drew Meyer, and Stacy Kim (not math, but close enough), for helping me weather the storm of grad school.
Outside OSU, thank you to Phil Engel, who enthusiastically heard at least four very dierent research shpiels from me at various points, and to Lily Berger, who put
iv up with Phil and me talking about math more often than strictly necessary. Thanks to both for continuing to be great friends to me, as they have been for years.
I am indebted to the grant Algebraic topology and its applications, NSF-DMS
#1547357, which allowed me to focus on my research for a number of very productive semesters in the last couple of years.
Finally, to my parents, thank you so much for your unlimited, unbelievable sup- port and love through everything.
v VITA
2010 ...... B.Sc. in Chemistry, B.Sc in Physics, Massachusetts Institute of Technology
2011-Present ...... Graduate Teaching Associate, The Ohio State University
FIELDS OF STUDY
Major Field: Mathematics
vi TABLE OF CONTENTS
Abstract ...... ii
Dedication ...... iii
Acknowledgments ...... iv
Vita...... vi
CHAPTER PAGE
1 Introduction ...... 1
2 Postnikov Towers ...... 4
2.1 The Postnikov tower of a space ...... 4 2.2 Maps of Postnikov systems ...... 8 2.3 The Whitehead exact sequence ...... 12
3 H-Spaces ...... 18
3.1 H-spaces and algebraic structures ...... 18 3.2 Homotopy properties of H-maps ...... 21 3.3 Fibrations ...... 23 3.4 Postnikov stages of H-spaces ...... 26 3.5 Localization ...... 29
4 Higher Cohomology Operations ...... 33
4.1 Secondary operations ...... 33 4.2 Higher operations ...... 38
vii 5 Degree Sets ...... 43
5.1 Multi-degrees ...... 43 5.2 A criterion for degrees ...... 45 5.3 Self-maps of SU(3) ...... 48 5.4 Maps between principal SU(2)-bundles over S5 ...... 50 5.5 Self-maps of Sp(2) ...... 52 5.6 Maps between principal Sp(1)-bundles over S7 ...... 54
6 Self-maps of SU(4) ...... 60 6.1 Low dimensions ...... 60 6.2 Eliminating higher obstructions ...... 64
7 Future Directions ...... 67
7.1 General questions about H-spaces ...... 67 7.2 Problems about speci c H-spaces ...... 69
APPENDICES
A Index of Notation ...... 72
B The Leray-Serre Spectral Sequence ...... 74
B.1 Basic information ...... 74 B.2 The bration ...... 75 K(Z2, 4) → X4 → K(Z, 3) Bibliography ...... 78
viii CHAPTER 1
INTRODUCTION
Given two closed oriented n-manifolds M and N, one often wishes to know whether there is a continuous map f : M → N of nonzero degree; that is, such that f∗[M] 6=
0 ∈ Hn(N). An ideal answer would not only produce such a map, but also determine all possible such that there is a continuous with . One of k ∈ Z f f∗[M] = k[N] the foundational results of algebraic topology, that up to homotopy there is exactly one degree-d self-map of the n-sphere, f : Sn → Sn, is of this form. Other well- known results along these lines include the fact that there is a map f :Σg → Σg0 of positive degree between oriented surfaces if and only if g ≥ g0, and that an oriented n-manifold M always has maps of positive degree into Sn. Our goal is to construct maps of given degree between compact simple Lie groups of the same dimension, and to do the same for some related spaces that arise naturally, most of which are
H-spaces (a sort of Lie group up to homotopy).
The main tool we use is the Postnikov tower, rst introduced by Postnikov [16] in 1951. This object provides a way to reconstruct the homotopy type of a space
X from its homotopy groups and a set of Postnikov invariants kn, though there are some caveats (as detailed in, for example, [18]). By carefully matching up the
1 Postnikov invariants of two spaces X and X0, one can construct maps φ : X → X0 with desired characteristics. Using this technique, we prove the following:
Theorem 5.5.1. Let be the generators of the cohomology groups 3 x3, x7 H (Sp(2); Z) and 7 , respectively. There is a map such that ∗ H (Sp(2); Z) φ : Sp(2) → Sp(2) φ (x3) = ∗ t3x3 and φ (x7) = t7x7 if and only if t3 ≡ t7 (mod 12).
This is a result originally due to Maruyama-Oka (see [6]), but our technique, and in particular elucidation of the secondary cohomology operation relating x3 and x7, does not seem to have appeared in print to date. This can be extended to show:
7 Theorem 5.6.2. Let X,Y be principal Sp(1)-bundles over S , and let t3, t7 be de ned analogously to above. There is a map φ : X → Y realizing t3 and t7 if and only if certain congruences involving t3 and/or t7 are satis ed.
We defer the speci cs to 5.6. Finally, we use Postnikov towers and some tech- niques from the theory of H-spaces to prove:
Theorem 6.1.1. Let be the generator of i for . Then there xi H (SU(4); Z) i = 3, 5, 7 ∗ is a map φ : SU(4) → SU(4) such that φ (xi) = tixi for i = 3, 5, 7 if and only if t3 ≡ t5 (mod 2) and t3 ≡ t7 (mod 6).
The structure of this document is as follows. In Chapter 2, we review the notion of a Postnikov tower, and prove a series of lemmas that will be useful in constructing maps of given degree later on. Chapter 3 reviews H-spaces and H-maps, nishing with a criterion for an H-map of Postnikov stages 0 to be ex- φn−1 : Xn−1 → Xn−1 tended to an H-map of the succeeding stages, 0 . In Chapter 4, we φn : Xn → Xn 2 explore secondary and higher cohomology operations, which are necessary for ex- plicitly representing the Postnikov invariants of Sp(2), SU(4), and related spaces. Chapter 5 explains the multi-degree of a map, and after applying our techniques to some results of Lafont-Neofytidis [9] on SU(2)-bundles over S5, we prove Theorem 5.5.1 and Theorem 5.6.2. In Chapter 6, we go on to prove Theorem 6.1.1 using H- space machinery, and Chapter 7 concludes by listing some conjectures and questions that constitute a natural continuation of our program.
In what follows, all spaces are assumed to be 1-connected CW-complexes, and
(co)homology with unspeci ed coe cients has coe cients in Z unless otherwise noted. We will write ρm for the coe cient homomorphism on (co)homology induced by reduction , and for the generator of n (when is clear Z → Zm ιn H (K(G, n); G) G from context). In general, we will notate the Bockstein of the coe cient sequence
·n by to distinguish it from the Bockstein of the mod- Steenrod Z −→ Z → Zn βe β p algebra.
3 CHAPTER 2
POSTNIKOV TOWERS
In what follows, we will cast almost everything in the language of Postnikov towers, which provide a canonical homotopy decomposition for appropriately nice spaces.
To that end, in this section we introduce Postnikov towers and establish some key facts that will be useful in our study. Comprehensive references about Postnikov towers are di cult to nd, so we will cite a number of dierent ones; the main ones are, in increasing order of technicality, [4], [1], and [2].
2.1 The Postnikov tower of a space
As noted above, a Postnikov tower is a way to decompose a 1-connected CW-complex
X into homotopy-invariant pieces (we will assume from now on that X is such a space). It is de ned as follows:
De nition 2.1.1. If X is a 1-connected CW-complex, a Postnikov tower for X is a sequence of 1-connected CW-complexes
q3 q4 qn X2 o X3 o ··· o Xn−1 o Xn o ···
together with a set of Postnikov sections pn : X → Xn such that 4 1. For n ≥ 2, (pn)∗ : πk(X) → πk(Xn) is an isomorphism for k ≤ n, and
πk(Xn) = 0 for k > n; and
2. For n ≥ 3, qnpn is homotopic to pn−1.
We call X the total space of the tower X2 ← X3 ← · · · .
According to the de nition above, a Postnikov tower for X exists for all X we are considering (see [4]), but is not unique; it is, however, unique up to homotopy
(see below). We will usually be concerned with how to obtain Xn from Xn−1, and to that end, note that we can get a ber sequence up to homotopy:
in qn kn K(πn(X), n) / Xn / Xn−1 / K(πn(X), n + 1)
Thus, for each n, Xn can be viewed as the total space of a bration over Xn−1 with
ber K(πn(X), n), and classi ed by the map
n+1 kn ∈ H (Xn−1; πn(X)).
This object is called the nth k-invariant or Postnikov invariant for X. One way to recognize it is as the image of a transgression; in the Leray-Serre spectral sequence for the bration
K(πn(X), n) → Xn → Xn−1, the image τ(ιn) of the fundamental class of K(πn(X), n) is kn. Here are a few more facts about Postnikov stages and k-invariants (for proofs, see [1] 7.2):
• Each map pn is an (n + 1)-equivalence: an isomorphism on (co)homology in
n+1 dimension k ≤ n, surjective on Hn+1(−), and injective on H (−). 5 • Postnikov stages are homotopy invariant; that is, if X2 ← X3 ← · · · and 0 0 are both Postnikov towers for , then 0 for all . X2 ← X3 ← · · · X Xn ' Xn n Thus we may speak of the Postnikov tower for a space.
• Similarly, Postnikov towers are functorial: If f : X → Y is a map, there are
induced maps fn : Xn → Yn compatible with both Postnikov towers.
• Any stage Xn can be homotoped so that the restricted (cellular) Postnikov section (n+1) (n+1) is an inclusion with image (n+1). pn : X → Xn Xn
An illustrative example showing some of the strengths and weaknesses of Post- nikov towers is given by the complex Bott manifolds. One constructs a complex Bott manifold by setting ; then for , let P , the Mn M0 = {∗} 1 ≤ j ≤ n Mj = C (ξj ⊕ C) projectivization of the Whitney sum of and , where is a complex line bundle ξj C ξj over and is the trivial line bundle. This produces the -dimensional smooth Mj−1 C 2n manifold Mn. Its integral cohomology ring is generated by x1, . . . , xn with |xi| = 2 for all i, where xi is a pullback of c1(ξi) (the Chern class that classi es the line bundle
∗ ξi). Inductively, if Sj−1 = H (Mj−1), then the Leray-Hirsch theorem says that
j−1 ∗ 2 X Sj = H (Mj) = Sj−1[xj]/(xj = xj Aijxi) i=1
The resulting array of coe cients Aij thus describes the cohomology of Mn by listing the relations satis ed by the variables . Let us also note that, since P1 is x1, . . . , xn C 2 homeomorphic to S , Mn can be viewed as the total space of a ber bundle
2 S → Mn → Mn−1
6 which splits, as it has a section given by inclusion along the south pole of S2 (alternatively, corresponding to the trivial summand in P ), so the long C (ξn ⊕ C) exact sequence for homotopy splits as well, hence ∼ ∼ n (and more π2(Mn) = π3(Mn) = Z ∼ 2 n generally πi(Mn) = πi(S ) ). It turns out we can capture all of this data entirely in the rst nontrivial k-
∗ invariant for Mn. Fix generators x1, . . . , xn for H (Mn). The cohomology ring is j−1 ! ∗ 2 X H (Mn; Z) = Z[x1, . . . , xn]/ xj = xj Aijxi, 1 ≤ j ≤ n , i=1 and we will construct the third Postnikov stage X3. The rst nontrivial stage is , which is n ; the cohomology ring of this space is the free polynomial ring X2 K(Z , 2) with for each . To nd , we must identify 4 n ∼ Z[x1, . . . , xn] |xi| = 2 i X3 k3 ∈ H (X2; Z ) = n n, since n. To that end, consider the bration Z ⊗ Z π3(Mn) = Z
n K(Z , 3) → X3 → X2
p,q and the corresponding Leray-Serre spectral sequence E . Since X3 is a Postnikov
4 4 stage, we should have a monomorphism H (X3) → H (Mn), meaning all relations
4 4 in H (Mn) should pull back to H (X3). On the other hand, examining the di- agonal p,4−p shows that the only possible nonzero entry is 4,0 (in particular, E2 E2 4 ). Therefore, we must obtain all the relations as the image of 0,3 H (K(Z, 3)) = 0 E4 ∗ n under the transgression τ. Indexing the n copies of the generators of H (K(Z , 2); Z) by {xi,1}, {xi,2},..., {xi,n}, and recalling that the Postnikov invariant k3 is τ(ι3), we get
j 2 2 2 X 4 n n k3 = (x1,1, x2,2 − A12x1,2x2,2, . . . , xj,j − A`jx`,jxj,j,...) ∈ H (K(Z , 2); Z ) `=1 ∗ In other words, k3 is essentially a list of relations describing the ring H (Mn). 7 2.2 Maps of Postnikov systems
Since we will mostly want to use Postnikov towers to infer facts about the spaces they approximate, we need to be able to deal with Postnikov towers and the maps between them in the absence of a total space. To that end, one can build a Postnikov tower using just the data of the k-invariants and the homotopy groups:
De nition 2.2.1. A Postnikov system is a sequence of pairs, n , de ned Pn ((πi, ki))i=2 recursively such that
• The empty sequence (n = 1) is a Postnikov system,
• πn is a nitely generated abelian group,
n−1 is a Postnikov system, and • ((πi, ki))i=2
n+1 • kn ∈ H (Xn−1; πn).
To build the corresponding spaces, we begin with X1 = {∗}. Then for n ≥ 2,
Xn is the homotopy ber of the classifying map kn : Xn−1 → K(πn, n + 1), with the same ber sequence as before:
in qn kn K(πn(X), n) / Xn / Xn−1 / K(πn(X), n + 1)
This de nes the (homotopy types of) stages corresponding to the Postnikov system
((πi, ki)). These stages can be assembled into a total space recovering the notion of a Postnikov tower above.
Proposition 2.2.2. Let n for , and let be the Post- Pn = ((πi, ki))i=2 n = 2, 3,... Xn nikov stage corresponding to P for each n. If we set Xˆ = lim X , and X a CW n ←− n 8 ˆ approximation of X, then the sequence X2 ← X3 ← · · · and the induced maps
X → Xn together de ne a Postnikov tower as in 2.1.1.
ˆ ˆ Proof. Let η : X → X be a weak homotopy equivalence, and let φn : X → Xn be the ˆ projection maps induced from the limit de nition of X. Put pn = φn ◦ η : X → Xn; then we have a commutative diagram involving X and X2 ← X3 ← · · · as required by (2) of 2.1.1. The long exact sequence for homotopy shows that condition (1) is satis ed for Xˆ, and since η is a weak homotopy equivalence, condition (1) is also satis ed for X.
For two CW-complexes X and X0, we need to be able to build a putative map
φ : X → X0 stage-by-stage using the Postnikov towers for both spaces, and then at some point transform the map 0 into . To do so, we rst de ne a map φn : Xn → Xn φ between Postnikov systems in the following way, following [18].
De nition 2.2.3. For two Postnikov systems and 0 0 0 , a P = {πi, ki} P = {πi, ki} morphism of systems is a collection of group homomorphisms 0 with the fi : πi → πi following (recursive) coherence property: the collection {fi}i ∗ 0 ( ) φn−1(kn) = fn∗(kn) † We will often refer to φn−1 as the morphism of systems, rather than talking about {fi}. 9 Lemma 2.2.4. The above de nition makes sense; that is, if condition (†) is satis ed by a pair of systems n−1 and 0 0 n−1 for which there is a morphism {πi, ki}i=2 {πi, ki}i=2 φn−1 : 0 , then extends to a morphism 0 . Xn−1 → Xn−1 φn−1 φn : Xn → Xn Proof. Condition (†) is equivalent to the commutativity of the right-hand square in the diagram kn Xn / Xn−1 / K(πn, n + 1) φn φn−1 fn k0 0 / 0 n / 0 Xn Xn−1 K(πn, n + 1) since n+1 0 can be naturally identi ed with Hom 0 . As the H (K(πn, n + 1); πn) (πn, πn) right square commutes and and 0 are the homotopy bers of and 0 , respec- Xn Xn kn kn tively, there is an induced map 0 . φn : Xn → Xn This can be used to partially answer a question of Masuda-Suh (see [10], Problem 1) about complex Bott manifolds: if two 2n-dimensional Bott manifolds M and M 0 have isomorphic cohomology rings under an isomorphism ∗ 0 ∗ σ : H (M ; Z) → H (M; Z) is there a dieomorphism φ : M → M 0 such that φ∗ = σ? Assuming such an iso- n n morphism σ exists, one can see that it must be a linear transformation B : Z → Z between the bases 2 and 0 0 2 0 , and compat- {x1, . . . , xn} ⊂ H (M) {x1, . . . , xn} ⊂ H (M ) ibility of B with the matrix of relations Aij is precisely the matching condition for the k-invariants, ∗ 0 φ2(k3) = B∗(k3) 10 where n n is the induced map on Eilenberg-Maclane spaces B∗ : K(Z , 4) → K(Z , 4) from . Hence, existence of is equivalent to a map 0 (where now B σ φ3 : M3 → M3 M3 means the third Postnikov stage of M). One could then hope to continue extending 0 φ3 up the Postnikov tower to obtain a homotopy equivalence φ : M → M . There is a di culty, however. One can show that the Whitehead sequence (see next section) for M decomposes into short exact sequences k2j−1 H2j(M) / H2j(M2j−2) / π2j−1(M) and these sequences split, but not necessarily naturally. Extending a map φ2j−2 to φ2j−1, therefore, is equivalent to nding a map f2j−1 that makes the diagram below commute: b2j k2j−1 H2j(M) / H2j(M2j−2) / π2j−1(M) φ2j−2 f2j−1 b0 k0 0 2j / 0 2j−1 / 0 H2j(M ) H2j(M2j−2) π2j−1(M ) 2j 0 0 since the obstruction lies in H (M2j−2; π2j−1(M )) = Hom(H2j(M2j−2), π2j−1(M )). But as we will see later, this type of scenario, where one is essentially reduced to solving an extension problem, is very hard to do with Postnikov towers in the absence of extra structure (as we would have if, for example, φ2j−2 were an H-map of H- spaces). Finally, let us make a note of how many Postnikov stages must be computed to furnish a map of total spaces with the desired properties. 0 Proposition 2.2.5. Let X and X be n-dimensional CW-complexes. If φn−1 : 0 is a map of their respective Postnikov stages, it corresponds to a Xn−1 → Xn−1 11 map 0 for which ∗ i 0 i and ∗ i 0 i φ : X → X φn−1 : H (Xn−1) → H (Xn−1) φ : H (X ) → H (X) are the same for i ≤ n − 1. Proof. We may assume that Xn−1 has the same n-skeleton as X, and similarly for 0 and 0; we may also assume is cellular. The restriction of to (n) is Xn−1 X φn−1 φn−1 Xn−1 therefore the desired map φ, and induces the same maps on cohomology for i ≤ n−1 i i since H (X) can be naturally identi ed with H (Xn−1) for i ≤ n − 1 (and similarly for X0). In particular, to realize a map φ : X → X0 of a particular multi-degree (see 5.1), 0 one needs to construct φn−1 on the Postnikov towers for X and X . 2.3 The Whitehead exact sequence Computing the (co)homology of Postnikov stages of X is an elaborate way to relate ∗ the H (X) to π∗(X). One might ask, then, if there is a more general or higher version of the Hurewicz isomorphism hk : πk(X) → Hk(X) for a (k − 1)-connected space X. In fact, there is such a Hurewicz homomorphism hn for each n ≥ k, and those homomorphisms can be t into a long exact sequence that also involves the homology groups of Xn. Whitehead [23] called it a certain exact sequence: Proposition 2.3.1. Let X be a 1-connected CW-complex, and let X(n) be the n- skeleton of X. Then the following sequence is exact and natural: bn+1 jn hn ··· / Hn+1(X) / Γn(X) / πn(X) / Hn(X) / ··· , (n−1) (n) where Γn(X) = im(i∗ : πn(X ) → πn(X )). The above sequence is called the Whitehead sequence of X. 12 Proof. Whitehead's proof is unwieldy, mainly because he had to construct the notion of an exact couple from scratch, and it is hard to nd a simpler proof in the literature, so we provide one here. Recall that an exact couple is a diagram of double complexes f A•,• / A•,• b g h E•,• with maps f, g, h taking f : An,p → An,p+1, g : An,p → En,p, and h : En,p → An−1,p−1 such that the diagram above is exact at every corner. The long exact sequences for homotopy of the pairs (X(p),X(p−1)) provide the exact couple (p) (p) (p−1) An,p = πn(X ),En,p = πn(X ,X ). The derived couple A0 → A0 → E0 → A0 is the homology with respect to the dierential d = h ◦ g (which is the homotopy boundary operator). Pick out the exact sequence in the derived couple with p = n. There, the derived groups are 0 im (n−1) (n) An,n = (πn(X ) → πn(X )) 0 im (n) (n+1) An,n+1 = (πn(X ) → πn(X )) 0 (n+1) (n) (n) (n−1) En+1,n+1 = ker(πn+1(X ,X ) → πn(X ,X ))/ (n+2) (n+1) (n+1) (n) im(πn+2(X ,X ) → πn+1(X ,X )) and they t together into an exact sequence 0 0 0 0 · · · → En+1,n+1 → An,n → An,n+1 → En,n → · · · 13 We can identify these groups: the group 0 is, by cellular approximation, just An,n+1 (n) (n−1) πn(X). The pair (X ,X ) is (n − 1)-connected, so (n) (n−1) (n) (n−1) πn(X ,X ) ' Hn(X ,X ); applying this to the expression for E0, we nd exactly the formula for the homology of the cellular chain complex, which since X is a CW-complex is just the homology of . Hence 0 is . Putting these together with the de nition of X En+1,n+1 Hn+1(X) Γn(X) above, we get the promised exact sequence. The connection between the Whitehead sequence and Postnikov towers is pro- vided by the following two lemmas from Baues [2]: Lemma 2.3.2. Let X be a 1-connected CW-complex. Then Γn(X) is naturally iso- morphic to Hn+1(Xn−1). Moreover, if X has free homology and πi(X) is nite for all i ≥ N, then for any n > N the induced map p∗ : Hn(X) → Hn(Xn−1) is an isomorphism. Proof. Consider the map of Whitehead sequences induced by the Postnikov section p : X → Xn−1: jn hn b Hn+1(X) / Γn(X) / πn(X) / Hn(X) / Γn−1(X) p∗ 0 / Hn+1(Xn−1) / Γn(Xn−1) / 0 / Hn(Xn−1) / Γn−1(Xn−1) The vertical isomorphisms arise from identi cations (k) (k) for , and is Xn−1 = X k ≤ n p∗ surjective since p is an n-equivalence. The rst statement now follows from exactness of the bottom row. If X also satis es the second hypothesis, then πn(X) is nite and 14 Hn(X) is free, so hn is trivial and hence b is an injection. By commutativity of the nal square, that implies that p∗ is also injective, and hence an isomorphism. Let u : πn(X) → coker jn be the natural projection and n+1 n+1 u∗ : H (Xn−1; πn(X)) → H (Xn−1; coker jn) the induced homomorphism on cohomology; let ∆ be the map n+1 ∆ : Ext(Hn(Xn−1), coker jn) → H (Xn−1; coker jn) −1 from the universal coe cient theorem. De ne (kn)† as ∆ u∗(kn). Lemma 2.3.3. Let X be a simply-connected CW-complex, let bn, jn be maps in n+1 the Whitehead sequence as in 2.3.1, and let kn ∈ H (Xn−1; πn(X)) be a Post- nikov invariant. Then jn = µkn, the image of kn in Hom(Hn+1(Xn−1), πn(X)) in the universal coe cient theorem, and (kn)† = {Hn(X)}, the class of the extension represented by Ext(ker jn−1, coker jn). Proof. See [2], Thm. 2.5.10. Thus there is a natural relationship between the k-invariants for X and the White- head sequence for X. More precisely, and usefully for our purposes, kn can be identi- ed with jn if Ext(Hn(Xn−1, πn(X))) = 0, and kn can be identi ed with the extension πn(X) → Hn(X) → Hn(Xn−2) if Hn+1(Xn−1) = πn−1(X) = 0. The Whitehead sequence is useful both as a technical tool and as a framing device. As a technical tool, it can frequently assist in calculating Hn+1(Xn−1), a nec- n+1 essary component of H (Xn−1; πn(X)) if the latter is not available from a spectral 15 sequence. It is also used in proving the two lemmas below, as well as in establishing the connections between the multi-degree of a map φ (resulting from the induced multiplication on ), the maps 0 that characterize a morphism of H∗(X) fn : πn → πn Postnikov systems, and the maps ∗ that actually relate Postnikov stages. As a φn−1 framing device, the Whitehead sequence shows, at a glance, information about all the Postnikov stages of X, along with some information about the k-invariants, as in Lemma 2.3.3. With all this in mind, here is a simple but useful lemma for disposing of some nonzero k-invariants: Lemma 2.3.4. Let 0 be a map of Postnikov stages, and as- φn−1 : Xn−1 → Xn−1 sume that (i) and 0 are free, (ii) the maps Hn(Xn−1) Hn(Xn−1) bn+1 : Hn+1(X) → and 0 are both zero, and (iii) the Hurewicz maps Hn+1(Xn−1) bn+1 hn : πn(X) → and 0 0 0 are both zero. Then extends to a map Hn(X) hn : πn(X ) → Hn(X ) φn−1 0 . φn : Xn → Xn n+1 Proof. The k-invariants to extend φn−1 to φn are found in H (Xn−1; πn(X)) and the corresponding group for X0. According to the universal coe cient theorem, this group is n+1 ∼ H (Xn−1; πn(X)) = Hom(Hn+1(Xn−1), πn(X)) ⊕ Ext(Hn(Xn−1), πn(X)) but by assumption (i), the Ext term is zero. Hence the k-invariants are exactly the maps and 0 0 0 . Since j : Hn+1(Xn−1) → πn(X) j : Hn+1(Xn−1) → πn(X ) bn+1 = 0 0 and hn = 0, j is in fact an isomorphism, and similarly for j . By Lemma 2.2.4, φn 16 0 exists if a suitable map fn : πn(X) → πn(X ) exists, and in this case we can take 0 −1 fn = j φn−1j . 17 CHAPTER 3 H-SPACES We now restrict our attention to H-spaces. As a homotopy version of Lie groups, H- spaces are natural objects of study for algebraic topology, especially with homotopy- theoretic techniques. A great deal of work has been done on nite or mod-p nite H-spaces (that is, those whose homology or mod-p homology, respectively, is of nite dimension), and one can produce very stringent conditions for the cohomology rings and Steenrod algebras of nite H-spaces. We will need to be a bit more general, because while our end goal involves nite H-spaces (indeed, mostly nite-dimensional Lie groups), the Postnikov stages used to build them are not nite-dimensional. In this chapter we will lay out the algebraic and topological properties of H-spaces we will need later, and prove a lemma that lets us deal with k-invariants in dimensions above the generators'. 3.1 H-spaces and algebraic structures For a CW-complex X, an H-space structure consists of a multiplication µ : X ×X → X and a strict identity element e ∈ X such that the µ(x, e) = µ(e, x) = x. Since 18 all our spaces are 1-connected, we can stipulate that the identity is the basepoint whenever necessary. For the same reason, having a homotopy identity element (that is, such that x 7→ µ(x, e) and x 7→ µ(e, x) are both homotopic to the identity map) is equivalent to having a strict identity. The key property of H-spaces is that their homology and cohomology with eld coe cients possess dual Hopf algebra structures. Since many of the coe cient rings we will encounter are not elds, we must drop the word algebra. So, for our purposes, the relevant consideration is the existence of a product structure on H∗(X; R) for arbitrary rings R; this is induced by the homology cross product and H-space multiplication µ: µ∗ Hi(X; R) × Hj(X; R) / Hi+j(X × X; R) / Hi+j(X; R) Dually, there is also a coproduct on cohomology, but in the absence of a eld, we will be content to just use the notion of a primitive element, due to Kahn (in [7]): De nition 3.1.1. Let (X, µ) be an H-space and x ∈ Hn(X; R). Consider the natural splitting Hn(X × X; R) ∼= Hn(X ∨ X; R) ⊕ Hn(X ∧ X; R), where X ∧ X is the smash product (X × X)/(X ∨ X). Let p be projection onto the second term above. Then x is R-primitive if pµ∗(x) = 0. There are two plausible notions for a morphism of H-spaces that we will de ne, though one (H-maps) will almost always be the useful one. 19 De nition 3.1.2. If (X, µ) and (Y, ν) are H-spaces, then a continuous map f : X → Y is an H-map if the following diagram commutes up to homotopy: µ X × X / X f×f f / Y × Y ν Y If the diagram commutes, we call f an H-homomorphism. It follows that a composition of H-maps is also an H-map (similarly for H- homomorphisms), and that a continuous map homotopic to an H-map is also an H-map (the same is not true, of course, for H-homomorphisms). As one would hope, H-maps preserve products and primitivity: Proposition 3.1.3. If (X, µ) and (Y, ν) are H-spaces and f : X → Y is an H-map, then f induces a homomorphism of graded rings f∗ : H∗(X; R) → H∗(Y ; R) for any coe cient ring R, and if x ∈ H∗(X; R) is R-primitive, then so is f ∗(x). Proof. For the rst assertion, we need to verify the commutativity of both squares of the following diagram, where the coe cients are understood to be in R: × µ∗ Hi(X) × Hj(X) / Hi+j(X × X) / Hi+j(X) f∗×f∗ (f×f)∗ f∗ × µ∗ Hi(Y ) × Hj(Y ) / Hi+j(Y × Y ) / Hi+j(Y ) On the left, we have a square that commutes due to the naturality of the cross product on homology. On the right, the square commutes thanks to homology applied to the homotopy-commutative diagram for an H-map. Commutativity of the whole diagram now indeed says that for x, y ∈ H∗(X), we have f∗(µ(x, y)) = ν(f∗(x), f∗(y)) 20 as desired. The second assertion follows directly from the naturality of the splitting in De nition 3.1.1. 3.2 Homotopy properties of H-maps Since H-maps are de ned homotopically, it makes sense to search for an obstruction, cohomological or otherwise, to a map f : X → Y being an H-map between the H-space structures (X, µ) and (Y, ν). Here, we nd this obstruction and present a criterion for eliminating it in cases of interest. To do that, we will also investigate the properties of H-maps with respect to brations. If (X, µ) is an H-space and W is any space, then the set of (based) homotopy classes [W, X] has a multiplication induced by µ, sending a pair of maps f : W → X and g : W → X to f + g : W → X, f + g : w 7→ µ(f(w), g(w)). The set [W, X] has a unit, but is not in general associative or commutative, unless X is (respectively) homotopy associative or homotopy commutative. However, it is an algebraic loop, in that it has unique dierences: Lemma 3.2.1. If f, g ∈ [W, X], where (X, µ) is an H-space, then there exists a unique element D(f, g) ∈ [W, X] such that D(f, g) + g = f. Proof. See [24], Thm. 1.3.1. 21 The functor [−,X] has a convenient property with respect to products, namely that [W × Y,X] ts into a short exact sequence p∗ ∗ 0 / [W ∧ Y,X] / [W × Y,X] i / [W ∨ Y,X] / 0 where i : W ∨ Y → W × Y is inclusion and p : W × Y → W ∧ Y is the quotient map onto the smash product (see [24], Lemma 1.3.5). As a result, we can tell whether a map is an H-map by subtracting the two homotopic compositions of Def. 3.1.2: De nition 3.2.2. Let (X, µ) and (Y, ν) be H-spaces, and let f : X → Y be a map. Then the H-deviation of f is the element Df ∈ [X ∧ X,Y ] de ned so that Df ◦ p = D(f ◦ µ, ν ◦ (f × f)) One can verify that D(f ◦ µ, ν ◦ (f × f)) does in fact factor through X ∧ X, so that the above de nition makes sense, and that Df is trivial if and only if f is an H-map. This also shows that [X ∧ X,X] is in one-to-one correspondence with the set of H-space multiplications on X, since any multiplication µ can be modi ed to µ0 = λp + µ, i.e. µ0(x, x0) = µ(λ(x ∧ x0), µ(x, x0)) where λ ∈ [X ∧ X,X], and conversely all multiplications still restrict to the fold map on X ∨ X. Using all of these facts, Zabrodsky (Prop. 1.5.1(a) of [24]) provides the following useful criterion for potential H-maps: Lemma 3.2.3. Let (X, µ) and (Y, ν) be H-spaces, and let f : X → Y be a map. 22 Then Y admits another multiplication ν0 such that f :(X, µ) → (Y, ν0) is an H-map if and only if the H-deviation Df lies in the image of the map (f ∧ f)∗ :[Y ∧ Y,Y ] → [X ∧ X,Y ] Proof. Let pX : X × X → X ∧ X be the projection, and pY similarly for Y . Note ∗ that (f ∧ f)pX = pY (f × f). Now, suppose that Df ∈ im (f ∧ f) , which means there is a λ ∈ [Y ∧ Y,Y ] such that Df = λ(f ∧ f). By the de nition of Df we thus have fµ = Df pX + ν(f × f) = λ(f ∧ f)pX + ν(f × f) = λpY (f × f) + ν(f × f) = (λpY + ν)(f × f) 0 0 so using ν = λpY +ν, we have an H-map f :(X, µ) → (Y, ν ) as desired. Conversely, 0 0 if ν makes f an H-map, then we can write ν = λpY +ν for some λ ∈ [Y ∧Y,Y ] since the H-structures on Y correspond to elements of [Y ∧ Y,Y ]. Following the equations ∗ above in reverse then shows that Df ∈ im (f ∧ f) . 3.3 Fibrations As Postnikov stages are constructed as homotopy bers, we need to be able to pull H-space structures back. For technical convenience, we use slightly modi ed de nitions of path space and related concepts which are standard in the H-space literature (see e.g. [20]). For a space X with basepoint e, we de ne the free path space FX to be the space of pairs 23 (γ, r), where γ is a continuous map γ : [0, r] → X with r ∈ R. Addition of paths in FX is de ned in the natural way by concatenation, so that (γ, r) + (γ0, r0) is de ned when γ(r) = γ0(0) (in particular, the rst summand happens rst); note that this operation is strictly associative. When convenient, we will just write γ for (γ, r). Within FX, the subset PX consists of maps for which γ(0) = e; as a result the endpoint projection PX → X given by γ 7→ γ(r) is a bration, and we will de ne its ber to be ΩX. This last space (often called the Moore loop space) is homotopy equivalent to the usual loop space, but (as with FX) has the advantage of a strictly associative multiplication, as well as a strict identity given by the constant path of length 0. Since we no longer have just the unit interval as the domain for paths, a path γ : [0, r] → X will be said to have endpoint at time 1. When (X, µ) is an H-space, FX also has an H-space multiplication η de ned so that µ(γ(t), γ0(t)) if 0 ≤ t ≤ min(r, r0) η((γ, r), (γ0, r0))(t) = µ(γ(r), γ0(t)) if r ≤ t ≤ r0 and r ≤ r0 µ(γ(t), γ0(r0)) if r0 ≤ t ≤ r and r0 ≤ r The identity is 0e, which is the pair (0, 0). Using these constructions, we can prove the following two lemmas, the latter of which is due to Stashe (see [20], Thm. 2). Lemma 3.3.1. Let (X, µ) and (Y, ν) be H-spaces. If f : X → Y is an H-map, then there is an equivalent map f 0 : X0 → Y such that f 0 is an H-map, f 0 is a bration, and X0 is homotopy equivalent to X via an H-map g. 24 Proof. For the rst part, we turn f into a bration in the usual way by replacing X with the homotopy-equivalent space X0 = {(x, γ) ∈ X × FY : f(x) = γ(1)} so that the new map is f 0(x, γ) = γ(0). Since f is an H-map, there is a homotopy 0 0 0 0 ht : X × X → Y with h0(x, x ) = ν(f(x), f(x )) and h1(x, x ) = f(µ(x, x )), which can be taken to be stationary on X ∨ X. Denoting concatenation of paths in FY by addition, we have a multiplication µ0 on X0 given by 0 0 0 0 0 0 µ ((x, γ), (x , γ )) = (µ(x, x ), ν(γ, γ ) + ht(x, x )) 0 0 0 0 which is de ned since ν(γ, γ )(1) = h0(x, x ) and f(µ(x, x )) is by de nition h1(x, x ). It then remains to check that (e, 0e) is in fact a strict identity; but this follows from the fact that ht is stationary on X ∨ X. Lemma 3.3.2. Let (X, µ) and (Y, ν) be H-spaces and f : X → Y an H-map. Then the homotopy ber F of f has an H-space structure (F, η) such that q : F → X is an H-homomorphism. Proof. Adapted from [20]. Let F be {(x, γ)|f(x) = γ(1)} ⊂ X × PY, where PY is the space of based paths in Y . Since f is an H-map, we have a homotopy 0 0 0 0 ht : X × X → Y such that h0(x, x ) = ν(f(x), f(x )) and h1(x, x ) = f(µ(x, x )); we can also assume that ht(x, e) = ht(e, x) = f(x) for all t and x since our spaces 25 are 1-connected. Then there is an associated map G : X × X → Y I such that 0 0 G(x, x )(t) = ht(x, x ). To get an H-multiplication η on F , put η((x, γ), (x0, γ0)) = (µ(x, x0), ν(γ, γ0) + G(x, x0)), where ν(γ, γ0)(t) = ν(γ(t), γ0(t)) and addition means concatenation of paths in Y I . To see that this is well-de ned, note that ν(γ, γ0)(1) = ν(f(x), f(x0)) so the addition in the second coordinate is valid, and f(µ(x, x0)) = G(x, x0)(1) as required. With (e, 1e) as the identity, where 1e is the constant path at the identity in Y , (F, η) is an H-space. The map q : F → X is just q(x, γ) = x, and qη((x, γ), (x0, γ0)) = µ(q(x, γ), q(x0, γ0)) = µ(x, x0), so q is an H-homomorphism. 3.4 Postnikov stages of H-spaces As alluded to above, H-spaces are quite compatible with Postnikov towers, as shown by a theorem due to Kahn which we discuss in this section. We will then show that it is in some cases possible to construct an H-map stage by stage via Postnikov towers. Recall that for an H-space X, an element x ∈ Hn(X; R) is primitive if its pro- jection pµ∗(x) into Hn(X ∧ X; R) is trivial. Viewing x as a map X → K(R, n), n the H-deviation Dx of x is in [X ∧ X,K(R, n)] = H (X ∧ X; R), and in fact is the dierence between ∗ and ∗ ∗ , where is inclusion to the µ (x) i1(x) + i2(x) ij : X → X ∨ X jth factor. Hence x is primitive if and only if its classifying map is an H-map. 26 Applying this fact to the context of Postnikov towers, Kahn proves that primitive k-invariants are, in fact, exactly what is necessary to make X an H-space: Theorem 3.4.1. Let X be a 1-connected space with homotopy groups πn, n ≥ 2. Then X is an H-space if and only if it has a Postnikov tower in which each Postnikov invariant kn is πn-primitive. Furthermore, if X is an H-space, all its Postnikov stages Xn are H-spaces. Proof. The rst assertion as applied to spaces with only nitely many homotopy groups is Thm. 3.2 of [7], and can be extended to arbitrary X by an argument of Barratt (again see [7], nal remark and footnote 10). The second assertion is Corollary 3.1 of the same paper. Or, to put it another way, a Postnikov tower constructs an H-space if all the constituent maps kn and qn are H-maps. On the other hand, it is a notable omission from [7] that one can obtain necessary or su cient conditions for an H-map to be built from suitable Postnikov stages (given, e.g., [7], Thm. 2.2, which shows that the Postnikov towers of X and Y can be made to form a strictly commuting ladder given a map ). One would like to be able to say that, if 0 f : X → Y φn−1 : Xn−1 → Xn−1 is an H-map, and 0 are H-maps such that the matching condition of 2.2 holds, kn, kn then the map 0 is also an H-map. φn : Xn → Xn Without considering this in full generality, we can prove a criterion for ensuring the next stage of a map of Postnikov towers remains an H-map. We will use this extensively when discussing SU(4). 27 Lemma 3.4.2. Let and 0 be H-spaces, 0 an H-map, and put X X φn−1 : Xn−1 → Xn−1 and 0 0 so that 0 induces a map 0 . If πn = πn(X) πn = πn(X ) fn : πn → πn φn : Xn → Xn the induced map ∗ n 0 0 0 n 0 (φn ∧ φn) : H (Xn ∧ Xn; πn) → H (Xn ∧ Xn; πn) is surjective, then there is an H-structure on 0 such that is an H-map. Xn φn Proof. By Lemma 3.2.3, there will be an H-structure on 0 that makes an H-map Xn φn if the H-deviation is in the image of the map Dφn # 0 0 0 0 (φn ∧ φn) :[Xn ∧ Xn,Xn] → [Xn ∧ Xn,Xn] (where we have used # rather than ∗ to distinguish from the map on cohomology in the hypothesis). Let and 0 0 to get the diagram Kn = K(πn, n + 1) Kn = K(πn, n + 1) in qn kn ΩKn / Xn / Xn−1 / Kn Ωfn φn φn−1 fn i0 q0 k0 0 n / 0 n / 0 n / 0 ΩKn Xn Xn−1 Kn Since and 0 are H-spaces, and 0 are both H-maps, which implies by Lemma X X kn kn 3.3.2 that and 0 are also H-maps, so is also an H-map. Then using the qn qn φn−1qn ber sequence 0 0 0 , we get a commuting diagram of Puppe sequences ΩK → Xn → Xn−1 i0 q0 0 0 0 n / 0 0 0 n / 0 0 0 [Xn ∧ Xn, ΩK ] [Xn ∧ Xn,Xn] [Xn ∧ Xn,Xn−1] ∗ # (φn∧φn) (φn∧φn) i0 q0 0 n / 0 n / 0 [Xn ∧ Xn, ΩK ] [Xn ∧ Xn,Xn] [Xn ∧ Xn,Xn−1] Since 0 is an H-map, the image of under 0 is trivial, so φn−1qn : Xn → Xn−1 Dφn qn 28 is in the image of 0 . As the homotopy sets furthest left are exactly the coho- Dφn in mology groups of the hypothesis, and the left square above commutes, surjectivity of ∗ is enough to imply that im # as desired. (φn ∧ φn) Dφn ∈ (φn ∧ φn) 3.5 Localization The modern study of H-spaces is inextricably linked to the localization program propounded by Sullivan in the 1970s; as such, there are a number of topics related to H-spaces that should be stated with localization in mind, and some proofs when considering families of bundles later on that require it. Here we brie y tackle the process of localization, following Ranicki's edition of Sullivan's notes (see [21], Ch. 2). Let P the set of primes and S a (possibly empty) subset of P. Put a for all ZS = { b :(p, b) = 1 p ∈ S} (that is, −1 ). In the special case that , we will denote by ZS = (P\S) Z S = {p} ZS , and we will denote by (this is also , of course). We will say a space is Z(p) Z∅ Z(0) Q localized by means of a map into a space with suitable homotopy groups: De nition 3.5.1. Let X be a simply-connected CW-complex. Then XS is a space such that 1. is a -module; π∗(XS ) ZS 2. is a -module; H∗(XS ) ZS 29 3. there is a map fS : X → XS inducing localization maps on homotopy and homology; and 4. any map X → Y , where Y satis es the two conditions above, factors uniquely through XS . The map fS is the localization map for X with respect to S. The spaces and are de ned analogously to and , and we call X(p) X(0) Z(p) Z(0) X(0) the rationalization of X. One can show that a localization in the sense of the above de nition always exists and is unique up to homotopy, but we defer that statement until we can make it even stronger. In particular, we can specify (to some extent) the CW-structure on XS by using local CW-cells de ned in the following way. First we construct the local sphere d . Let be the poset in which SS (A, ) Y A = {n ∈ Z : n = pi, pi ∈ P \ S} and if and only if . Choose a sequence ∞ from that is co nal n m n|m {ak}k=1 A with respect to , and consider the mapping telescope Sd a1 / Sd a2 / Sd a3 / ··· in which each sphere is attached to the next via a map of degree ak. The sphere d includes into the limit d as the rst sphere in the sequence, and this inclusion S SS is the localization of Sd with respect to S, as can be seen by the induced map on homology. We can then build a local CW-complex by beginning with a point, and then glueing in local -cells (that is, a cone over d ) using maps of d into the (d + 1) SS SS 30 local d-skeleton. Using this procedure, Sullivan shows (in [21], Thm. 2.1 and Thm. 2.2) that Proposition 3.5.2. For any S ⊆ P and any 1-connected CW-complex X with one 0- cell and no 1-cells, there is a localization XS satisfying De nition 3.5.1 that is unique up to homotopy, and XS can be taken to have an isomorphic collection of cells to X (that is, if one attaches cells one by one, the intermediate spaces for XS are the localizations of the intermediate spaces for X). Furthermore, there is a localization map X → XS that is cellular in the same way. Localization can be extremely convenient because it is compatible with just about every other functor or construction in algebraic topology. We list the relevant facts without proof, xing a set of primes S: • Any space X with a Postnikov tower as de ned in 2 also has a local Postnikov tower (X2)S ← (X3)S ← · · · , such that all the k-invariants are localized. • Localization preserves brations and co brations (including products, wedges, and smash products). • If S ⊆ T ⊆ P, there is a canonical map XS → XT . • There is a natural map (ΩX)S → Ω(XS ) de ned by looping the original local- ization map fS . • If S and T are disjoint sets of primes such that S ∪ T = P, then X is the pullback of the diagram XS → X(0) ← XT , and in fact X is the ber product of all the X(p) over X(0). 31 • X is an H-space if and only if X(p) is an H-space for each p and all the rings , , are isomorphic. H∗(X(p); Q) p ∈ P In the context of H-spaces, we will usually want to be able to compare spaces X and Y using details about their p-local versions. To that end, we say that X and Y are equivalent mod S if XS and YS are homotopy equivalent, and we write X 'S Y (similarly, X and Y are equivalent mod p if X(p) ' Y(p), and they are rationally equivalent if X(0) ' Y(0)). A map f : X → Y , where X and Y are H-spaces, is said to be an H-map mod p if the induced map f(p) : X(p) → Y(p) is an H-map, and a map f : X → Y is a rational equivalence when f(0) : X(0) → Y(0) is a homotopy equivalence. Example 3.5.3. If X and Y are spaces such that maps f : X → Y have multi-degrees, then any map of nonzero (total) degree is a rational equivalence. Therefore, any map f of nonzero degree can be precisely reconstructed from the data f(p) for each p ∈ P, since all the maps f(p) automatically cohere on X(0). Example 3.5.4. (From [8], 9.2) If X is a nite H-space, then X is rationally equivalent to a product of spheres, r Y 2ni−1 X '(0) S i=1 since its Hopf algebra is the external algebra with H∗(X; Q) ΛQ(xi, . . . , xr) |xi| = 2ni − 1. There is also a mod p equivalence r Y 2ni−1 X '(p) S i=1 3 5 3 5 if and only if p ≥ max ni. For example, SU(3) 6'(2) S × S , but SU(3) '(p) S × S for p ≥ 3. 32 CHAPTER 4 HIGHER COHOMOLOGY OPERATIONS We will eventually construct maps by matching up k-invariants using Lemma 2.2.4, and we will need secondary and tertiary cohomology operations to express those k- invariants in a suitably functorial way. This also provides an opportunity to elucidate the close relationship between Postnikov towers and higher cohomology operations in some generality (enough, at least, for computations in low dimensions), a connection which is rarely made explicit in the literature. 4.1 Secondary operations Classically, primary cohomology operations arose from geometric considerations. The most familiar operations, the Steenrod squares Sqk (which will be used to relate el- ements of ∗ in the next chapter), were de ned axiomatically by Steen- H (SU(3); Z2) rod, and can also be viewed geometrically as part of the de nition of Stiefel-Whitney classes. But the most relevant de nition of Steenrod squares, for our purposes, is as stable operations represented by maps Sqi : K(Z2, n) → K(Z2, n + i) 33 which induce the familiar maps n n+i . In particular, the opera- H (X; Z2) → H (X; Z2) tions Sqi can be seen as elements in the groups n+i . More generally, H (K(Z2, n); Z2) a primary cohomology operation ξ : Hm(X; G) → Hn(X; G0), where G and G0 are abelian groups and 1 ≤ m ≤ n, can be identi ed with a map ξˆ : K(G, m) → K(G0, n) and therefore an element of n 0 . In particular, ˆ∗ 0 . We H (K(G, m); G ) ξ (ιn) = ξ(ιm) will say that ξ is of type (G, m, G0, n). So a primary cohomology operation is just a particular element of the cohomology ring of an Eilenberg-Maclane space. To avoid dealing with more notation, we will henceforth use the same name for a cohomology element y ∈ Hn(Y ; G) and its representing map y : Y → K(G, n). We will also con ate any stable primary operation θ with its loopings and deloopings ..., Ω−2θ, Ω−1θ, θ, Ωθ, Ω2θ, . . . so that, if θ is of type (G, m, G0, n), it can also be used in a context where an operation of type (G, m + i, G0, n + i) would be required. If one wanted to generalize the notion of primary cohomology operations, one way to do so would be to broaden the class of cohomology rings to draw from. Since an Eilenberg-Maclane space is the rst step of a Postnikov tower, and a map between Eilenberg-Maclane spaces is the rst k-invariant of such a tower, we could try to construct higher cohomology operations as cohomology elements of higher stages of Postnikov towers. These operations would then incorporate data about the groups 34 πj involved in the tower, as well as all of the k-invariants that dictate how the stages are built. Let us recast the Postnikov tower to make it convenient to talk about higher operations. If X is a space whose rst nonzero homotopy groups are π`, πm, and πn in dimensions `, m, and n, respectively (1 < ` < m < n), then the rst three nontrivial stages of its Postnikov tower look like: in / K(πn, n) Xn Wkn qn im kn K(πm, m) / Xm / K(πn, n + 1) qm km K(π`, `) X` / K(πm, m + 1) where is the homotopy ber of and is, by de nition, the homotopy ber Wkn kn Xm of km. For a CW-complex Y , the primary cohomology operation km applied to an ` m+1 element y ∈ H (Y ; π`) yields km(y) ∈ H (Y ; πm) by composing the maps km and y. A secondary operation arises in the following manner. Assume that, in the Serre long exact sequence for the bration whose total space is Xm, we have τ(ιm) = θ(ι`), where θ is a stable primary cohomology operation of type (π`, `, πm, m+1) (of course, τ(ιm) is km, by de nition, so we are assuming km takes a particular form). If ψ is a stable primary operation of type (πm, m, πn, n + 1) for which ψ ◦ θ = 0, then ψ(ιm) is transgressive and τ(ψ(ιm)) = 0, which means there is an element λn+1 ∈ n+1 such that ∗ . If can be uniquely speci ed (perhaps H (Xm; πn) im(λn+1) = ψ(ιm) λn+1 by imposing other conditions), it represents a secondary operation Θ corresponding to the relation ψ ◦θ = 0. This is crucial for our study of Sp(2) and SU(4) later, since 35 various Postnikov invariants for those spaces can be most conveniently described in this way. In order to compute Θ, we use the diagram above, which displays the universal example for , in which ∗ . More generally, for a CW-complex Θ Θ(qmι`) = λn+1 Y ` and a class y ∈ H (Y ; π`), the map y : Y → X` will have a lift y : Y → Xm if θ(y) = km ◦ y = 0, and we put Θ(y) = kn ◦ y. Hence, Θ takes elements of a subgroup ` n+1 of H (Y ; π`) to H (Y ; πn). We will denote the domain of Θ by ` Def(Θ,Y ) = {y ∈ H (Y ; π`): θ(y) = 0}. There is also an indeterminacy associated to Θ based on the fact that y may not be unique (even up to homotopy). The indeterminacy is given by m n+1 Ind(Θ,Y ) = ψH (Y ; πm) ⊆ H (Y ; πn), n+1 and strictly speaking the codomain of Θ is the coset space H (Y ; πn)/Ind(Θ,Y ). As an example, let us consider X = Sp(2): Example 4.1.1. The rst three nontrivial stages of its Postnikov tower are i5 k7 K(Z2, 5) / X5 / K(Z, 8) q5 i4 k5 K(Z2, 4) / X4 / K(Z2, 6) q4 i3 k4 K(Z, 3) X3 / K(Z2, 5) Let be the reduction of 3 , and let be the generator of ρ2ι3 Z2 ι3 ∈ H (X3; Z) ι4 4 . Since is nontrivial, is transgressive in the Leray-Serre spectral H (K(Z2, 4); Z2) k4 ι4 36 2 sequence for the bration whose total space is X4, so τ(ι4) = Sq ρ2ι3 (see Appendix B for full details on this spectral sequence). Therefore in , 6 ∼ , X4 H (X4; Z2) = Z2 generated by an element such that ∗ Sq2 . This element speci es a λ6 i4(λ6) = ι4 2 2 secondary cohomology operation Σ corresponding to the relation Sq ◦ Sq ρ2 = 0 in dimension 3, and is characterized by the fact that ∗ . In general, applied Σ(q4ι3) = λ6 to a CW-complex Y , we have Def 3 Sq2 Ind Sq2 4 (Σ,Y ) = {y ∈ H (Y ; Z): ρ2y = 0}, (Σ,Y ) = H (Y ; Z2) In particular, Def(Σ, Sp(2)) = H3(Sp(2)) and Ind(Σ, Sp(2)) = 0, so Σ takes exact values in 6 , which is trivial, so vanishes on 3 . H (Sp(2); Z2) Σ H (Sp(2)) A good example of a secondary operation modulo 3 occurs for X = Sp(3): Example 4.1.2. Let us try to compute the generator of 11 ∼ . Thinking H (X10; Z3) = Z3 ∼ 3-locally, there are only two nonzero homotopy groups to worry about, π3(Sp(3)) = ∼ . The bration is nontrivial, so , the π7(Sp(3)) = Z K(Z, 7) → X7 → K(Z, 3) ρ3ι7 generator of 7 , must transgress to 1 (where 1 is 1 ). There H (K(Z, 7); Z3) βP ι3 P P ρ3 is an Adem relation P 1βP 1 = βP 2 + P 2β in the mod 3 Steenrod algebra, so in dimension 11, we have 1 1 1 1 τ(P ι7) = P τ(ρ3ι7) = P βP ι3 = 0 2 1 since both P ρ3ι3 and βρ3ι3 are zero. One also nds that d8(ι3ι7) = ρ3ι3βP ι3, so 11 is generated by an element such that ∗ 1 . This element H (X10; Z3) θ11 i7θ11 = P ι7 37 1 de nes a secondary operation Λ based on the relation P 1 ◦ βP = 0 in dimension 3, which can act on the cohomology of a CW-complex Y with Def 3 1 Ind 1 7 (Λ,Y ) = {y ∈ H (Y ; Z): βP y = 0}, (Λ,Y ) = P H (Y ; Z3) and taking values in 11 . In particular, is de ned on all of 3 and H (Y ; Z3) Λ H (Sp(3)) takes exact values in 11 . Unlike before, the latter fact is not completely H (Sp(3); Z3) obvious since 7 is nontrivial, but its generator is 1 , so H (Sp(3); Z3) ρ3x7 P ρ3x3 1 1 1 2 P ρ3x7 = P P ρ3x3 = 2P ρ3x3 = 0 and hence Ind(Λ, Sp(3)) = 0. 4.2 Higher operations We will also need to express obstructions in terms of higher cohomology operations, and for those, it is convenient to follow the more general setup of Harper [3]. There, a secondary operation arises from a pair of maps θ ψ K0 −→ K1 −→ K2 such that ψ ◦ θ is nullhomotopic, where the Ki are all 1-connected and K2 is an H-space. As we have seen, one could have the Ki as Eilenberg-Maclane spaces (or products thereof) and the maps as primary cohomology operations. Let Wθ be the homotopy ber of θ. Then there is a map ψe : Wθ → ΩK2 that lifts ψ, and this map depends on the choice of nullhomotopy for ψ ◦ θ. If y : Y → K0 is a map such 38 that θ ◦ y is nullhomotopic (in the prior setup, a cohomology class annihilated by θ), then there is also a lift y : Y → Wθ making the following diagram commute: ψ W e / ΩK > θ 2 y qθ y θ Y / K0 / K1 This speci es a secondary operation Θ with domain Def(Θ,Y ) = {y : Y → K0, θ ◦ y ' ∗} ⊆ [Y,K0] and taking values in [Y, ΩK2], and Θ(y) = ψe ◦ y. Since this depends on the choices of y and ψe, there is an indeterminacy Ind(Θ,Y ) associated with Θ, and it is the subgroup im Ωψ∗ ⊆ [Y, ΩK2], where Ωψ∗ :[Y, ΩK1] → [Y, ΩK2] is the map induced by looping ψ : K1 → K2. Hence the value of Θ(y) is, strictly speaking, a coset in [Y, ΩK2], or alternatively an element in [Y, ΩK2]/Ind(Θ,Y ). The portion of the diagram not including Y is the universal example of Θ, and Θ(qθ) = ψe. We de ne higher operations in terms of syzygies, which yield, in some sense, generalized Postnikov pieces. A syzygy is a sequence θ0 θ1 θn−1 K0 / K1 / ··· / Kn j−1 such that Kj is j-connected (and hence Ω Kj is 1-connected) and each successive pair of maps θj ◦ θj−1 is nullhomotopic. Inductively, a syzygy is said to be admissible if it is of length 2 or there is a nullhomotopy for θ1 ◦ θ0 such that the lift θe1 ts into an admissible syzygy Ωθ θe1 / Ωθ2 / Ωθ3 / n−1/ Wθ0 ΩK2 ΩK3 ··· ΩKn 39 j−1 As a result, we can build a tower out of the homotopy bers Wj of the maps Ω^θj, for j ≥ 1: n−1 n^−2 n−1 Ω θn−1 Ω θn−1 n−1 Ω Kn−1 / Wn−2 / Ω Kn qn−2 n−2 n^−3 n−2 Ω θn−2 Ω θn−2 n−2 Ω Kn−2 / Wn−3 / Ω Kn−1 qn−3 . . q1 Ωθ1 / θe1 / ΩK1 Wθ0 ΩK2 q0 θ0 K0 / K1 Note that each stair (that is, a horizontal map, a vertical map, and another hori- zontal map) is a ber sequence up to homotopy. Applying this to Sp(2) and using the diagram above, we have a syzygy de ning Σ (as before) and a tertiary operation Π in the following way (these operations were also eectively constructed by [22] as part of the Postnikov tower for BSp(2)). First, the syzygy is 2 2 2 Sq ρ2 Sq ρ2,4Sq K(Z, 3) / K(Z2, 5) / K(Z2, 7) / K(Z4, 9) where ρ2,4 is a coe cient homomorphism de ned below. Using the same spectral sequence as before (see also B.2), we compute that 3 2 1 2 τ(Sq ι4) = 0, τ(Sq Sq ι4) = 0, d4(ι4ρ2ι3) = ρ2ι3Sq ρ2ι3 so that 7 ∼ , generated by Sq1 and an element for which H (X4; Z2) = Z2 ⊕ Z2 λ6 µ7 40 ∗ Sq2Sq1 . We also have 8 ∼ , generated by Sq1 , which is equal i4µ7 = ι4 H (X4; Z2) = Z2 µ7 2 1 2 1 2 2 to Sq λ6 since Sq Sq Sq ι4 = Sq Sq ι4. At the next stage, let us try to determine the generator of 7 ∼ . For H (X5; Z4) = Z4 this, let k k k k ρ4,2 : H (Y ; Z4) → H (Y ; Z2), ρ2,4 : H (Y ; Z2) → H (Y ; Z4) denote the reduction and inclusion, respectively, induced by the coe cient sequence (with the connecting homomorphism, of course, being the Bockstein Z2 → Z4 → Z2 Sq1). Using this sequence, we can see that 7 ∼ generated by H (X4; Z4) = Z2 ⊕Z2 ρ2,4µ7 1 and an element ω7 such that ρ4,2ω7 = Sq λ6. Since the bration is nontrivial, the Leray-Serre transgres- K(Z2, 5) → X5 → X4 sion satis es . If we use coe cients, then 6 is generated τ(ι5) = λ6 Z4 H (K(Z2, 5); Z4) 1 by an element ν6 for which ρ4,2ν6 = Sq ι5, and hence 1 ρ4,2τ(ν6) = τ(ρ4,2ν6) = Sq λ6 which means that τ(ν6) = ω7. In the next dimension, 2 2 2 1 τ(ρ2,4Sq ι5) = ρ2,4τ(Sq ι5) = ρ2,4Sq λ6 = ρ2,4Sq µ7 = 0 and therefore 7 ts into a short exact sequence H (X5; Z4) i∗ 7 7 5 7 H (X4; Z4)/(ω7) / H (X5; Z4) / H (K(Z2, 5); Z4) The spectral sequence leaves us with the extension problem of the middle group being or , but we know from the Whitehead sequence (see proof of Theorem Z2 ⊕ Z2 Z4 5.5.1) that it is . So let be the generator of 7 (of course, is one as Z4 ζ7 H (X5; Z4) −ζ7 41 2 well, but the choice of sign makes no dierence), and note that ρ2,4Sq ι5 generates the rightmost group. Then ∗ Sq2 , which means that we have de ned a i5ζ7 = ρ2,4 ι5 2 tertiary cohomology operation Π based on the relation ρ2,4Sq ◦ Σ = 0 in dimension 3, and in the universal example we have ∗ ∗ . More generally, we have X5 Π(q5q4ι3) = ζ7 Def 3 Ind Sq2 5 (Π,Y ) = {y ∈ H (Y ; Z) : Σ(y) = 0}, (Π,Y ) = ρ2,4 H (Y ; Z2) and in particular Def(Π, Sp(2)) = H3(Sp(2)) and Ind(Π, Sp(2)) = 0, so the genera- 3 7 tors x3 ∈ H (Sp(2)) and x7 ∈ H (Sp(2)) are related by Π(x3) = ρ4x7. 42 CHAPTER 5 DEGREE SETS 5.1 Multi-degrees Now we turn brie y to the general question of which degrees of maps are possible between various spaces. For now, we will explain the terminology and state a criterion by which one can establish necessary and su cient conditions for maps of given multi-degrees to exist. De nition 5.1.1. Let X be a 1-connected manifold with free integral homology and cohomology, generated by classes in distinct dimensions ( ). xdi di 1 ≤ i ≤ r We call a manifold with those properties a multi-degree manifold (or MDM). For a self-map , we say that has -degree if ∗ . When f : X → X f di tdi f (xdi ) = tdi xdi the meaning is clear, we will say that has multi-degree . Similarly, f (td1 , . . . , tdr ) for a map f : X → X0 of multi-degree manifolds such that H∗(X) and H∗(X0) are isomorphic as rings, f has d -degree t if f ∗(x0 ) = t x . i di di di di We will always use the notational convention above (namely, for each |xdi | = di of the cohomology generators ) when discussing multi-degree manifolds. xdi Since an MDM X has free homology, we can view the di-degrees as multiplication 43 by on a generator of instead when convenient. Note that if is the tdi Hdi (X) h fundamental coclass of , we can compute the degree of in terms of the using X f tdi 's expression as a product of ; if is an H-space as well, the same holds for h xdi X homology. In the case where and 0 are dierent manifolds, each is only well- X X tdi de ned up to sign, since there is not necessarily an intrinsic choice of generator for Hdi (X) or Hdi (X0). We will note this when necessary in discussing maps between dierent manifolds. When X is an H-space with multiplication µ and also an MDM with r generators, we can make the multi-degree into a homomorphism. In particular, as in 3.2, we can put an algebraic loop structure on [X,X] where the operation is pointwise multiplication in the target, f, g f + g, (f + g)(x) = µ(f(x), g(x)). r With this structure in place, we can de ne the multidegree as a map tˆ:[X,X] → Z that sends a map to its multidegree . As observed in [6], this map is f (td1 , . . . , tdr ) a homomorphism: Proposition 5.1.2. If X is an H-space multi-degree manifold with r generators, the r multi-degree tˆ:[X,X] → Z is a homomorphism. Proof. Let f, g ∈ [X,X], and consider the map f + g, which can be decomposed as (f,g) µ X / X × X / X ∗ ∗ Pick x ∈ H (X). Since X has free homology, a class x ∈ H (X) is Z-primitive if ∗ and only if the corresponding class x ∈ H (X; Q) is primitive in the sense of a Hopf 44 ∗ algebra (which matches our notion of Q-primitive). Since H (X; Q) is commutative and associative and is a manifold, its dual is also commutative and X H∗(X; Q) associative, so both algebras are primitively generated (see [8] 1.5, Corollary C(i)). ∗ Hence H (X) is generated by Z-primitive elements. Since our de nition of multi- degree only depends on the action on the cohomology generators, we may therefore assume that x is primitive. Now we can compute (f, g)∗µ∗(x): (f, g)∗µ∗(x) = (f, g)∗(x ⊗ 1 + 1 ⊗ x) = f ∗(x) + g∗(x) which shows that f + g has degree tˆ(f) + tˆ(g). One can also do this for the loop [W, X] if W has isomorphic cohomology to X, but only as a map into r (sending a degree to its reduction mod 2), due to the Z2 tdi aforementioned sign ambiguity of a map between dierent manifolds. 5.2 A criterion for degrees Before stating our criterion, let us brie y return to an important detail about higher cohomology operations. As with primary operations, secondary operations need not be additive. The Pontryagin square, which maps 2m to 4m , H (−; Z2k ) H (−; Z2k+1 ) is such an example for primary operations, while the operation Φ of [22] is one for secondary operations. However, secondary (and higher) operations do satisfy a semi- additivity property that guarantees that the loop of a secondary operation is additive; in particular, the operation Π above is ΩΦ for adek-Vanºurí's operation Φ. In fact we will use an even stronger de nition: 45 De nition 5.2.1. A primary, secondary, or higher cohomology operation Θ is said to be a linear operation on a space X if 1. Θ(x + y) = Θ(x) + Θ(y) for all x, y ∈ Def(Θ,X), and 2. Ind(Θ,X) = 0. So all stable primary operations are linear, while the operations Σ and Π are linear on Sp(2) and additive on every space. But Σ is not a linear operation on every space, since for example has the nonzero element Y = K(Z, 3) × K(Z2, 3) Sq2Sq1 0 Ind , where 0 is a generator of 3 that is outside of im . ι3 ∈ (Σ,Y ) ι3 H (Y ; Z2) ρ2 With this de nition in mind, we are ready for a lemma that will be used ex- tensively in the next sections. Despite the unwieldy hypotheses, the situations for torsion-free H-spaces are so similar that this lemma will always apply: Lemma 5.2.2. Let be a multi-degree manifold with generators , , X {xdi } 1 ≤ i ≤ r and x a dimension ` = di < dj = n, for some i, j ≤ r. Assume that 1. there is a map φn−1 : Xn−1 → Xn−1; 2. the Whitehead sequence for X contains an exact fragment 0 hn bn 0 ··· / πn(X) / Hn(X) / Hn(Xn−2) / ··· where ∼ ; Hn(X) = Z n+1 3. H (Xn−1; πn(X)) is naturally isomorphic to Ext(Hn(Xn−2), πn(X)); 4. there is a cohomology class ` such that ∗ ; χ` ∈ H (Xn−1; π`(X)) pn−1(χ`) = x` 46 5. ∼ ; and π`(X) = Z 6. there is a linear cohomology operation Θ such that kn = Θ(χ`). Then φn−1 may be extended to φn if and only if the degree tn satis es tn ≡ t` (mod m), where m is the smallest positive integer such that mΘ(χ`) = 0. Proof. To avoid confusion (since we will need to distinguish between πn of the source and the target), we will call the target X0; every fact deduced about X obviously also 0 holds for X . Since pn−1 : X → Xn−1 is an isomorphism on cohomology in dimension , ∗ 0 . On the other hand, by (2), ∼ and is injective, so ` φn−1(χ`) = t`χ` Hn(X) = Z hn is either or 0. If then there is no obstruction to extending , πn(X) Z πn(X) = 0 φn−1 so we may assume ∼ and is multiplication by for some , and πn(X) = Z hn c c ∈ Z hence that ∼ n+1 ∼ Hn(Xn−2) = H (Xn−1; πn(X)) = Zc by (3) and (4). Therefore, in the diagram hn bn πn(X) / Hn(X) / Hn(Xn−2) fn tn (φn−1)∗ h0 b0 0 n / 0 n / 0 πn(X ) Hn(X ) Hn(Xn−2) we must have that fn is multiplication by tn, and (φn−1)∗ is multiplication by tn (mod c). We then apply those facts to the following diagram obtained from (5) and (6) illustrating both the operation and the matching condition ∗ 0 : Θ fn∗(kn) = φn−1(kn) t t ` ` / ` 0 o ` ` 0 0 H (Xn−1; π`(X)) H (Xn−1; π`(X )) H (Xn−1; π`(X )) Θ Θ Θ n+1 tn / n+1 0 o tn n+1 0 0 H (Xn−1; πn(X)) H (Xn−1; πn(X )) ∗ H (Xn−1; πn(X )) fn∗ φn−1 47 On the bottom row, multiplication by tn is taken to be modulo c. The map φn−1 extends to φn if the above diagram can be made to commute by an appropriate choice of fn, which is precisely a choice of tn. But the condition for commutativity of each square is 0 , which implies that the diagram commutes and the map t`Θ(χ`) = tnΘ(χ`) extends if and only if t` ≡ tn (mod m), where m is the order of Θ(χ`) (which divides but may not necessarily equal c). A couple of remarks about applicability are in order. Remark 5.2.3. The lemma applies to the lower stages of self-maps of SU(3), SU(4), SU(5), Sp(2), Sp(3), and numerous other manifolds obtainable as principal SU(n)- or Sp(n)-bundles over spheres. However, one begins to encounter di culties when generators and products of generators occur in the same dimension. For example, in , we have 15 ∼ 2 (generated by and ), contradict- SU(8) H (SU(8)) = Z x15 x3x5x7 55 ing condition (3). In the same vein, in Sp(8), H (Sp(8)) is generated by x55 and x3x7x11x15x19. Remark 5.2.4. We will apply logic very similar to that of this lemma in a few cases where X 6= X0, but some of those cases are not completely compatible with the lemma above since the condition for commutativity of the diagram turns out to be a condition on only one of t` or tn. 5.3 Self-maps of SU(3) The simplest application of the lemma in the previous section is to SU(3), a case which has been studied by Mimura-Oshima [12], Püttmann [17], and Lafont-Neofytidis 48 [9]. With the machinery we have built up, the conclusion is nearly immediate, in a more precise form than in prior work: Theorem 5.3.1. For integers t3, t5, there is a self-map f : SU(3) → SU(3) of degree (t3, t5) if and only if t3 ≡ t5 (mod 2). Proof. To realize a particular pair , we just need to produce 0 re- (t3, t5) φ7 : X7 → X7 alizing that pair, where X = X0 = SU(3). Hence, the last k-invariant to consider lies 8 7 in H (X6; π7(X)); but since π7(X) = 0, it actually su ces to stop at H (X5; π6(X)). We can avoid even this step, since any map φ5 extends to φ6 by Lemma 2.3.4. So we 6 need H (X4; π5). Recall that for n ≤ 7, if n = 3, 5 Z if πn(SU(3)) = Z6 n = 6 0 otherwise Let . Since , we can construct a map 0 by simply t3 ∈ Z X3 = K(Z, 3) φ3 : X3 → X3 choosing the cohomology class 3 . Since , this is also a t3ι3 ∈ H (X3) = Z π4(X) = 0 map 0 . For the next stage, we need a bration φ4 : X4 → X4 K(Z, 5) → X5 → X4 (and similarly for 0). The relevant cohomology groups 6 and 6 0 0 X H (X4; π5) H (X4; π5) are isomorphic to with generators Sq2 and Sq2 0 respectively, where Sq2 is Z2 βe (ι3) βe (ι3) Sq2 and is the Bockstein homomorphism from the coe cient sequence ρ2 βe Z → Z → . The space is not , so and 0 are nontrivial. Therefore Z2 X5 K(Z, 3) × K(Z, 5) k5 k5 Sq2 and 0 Sq2 0 . k5 = βe (ι3) k5 = βe (ι3) 2 Now apply Lemma 5.2.2 with n = 5, χ3 = ι3, and Θ = βeSq (which is stable and primary, hence linear), and the conclusion follows. 49 In particular, this illustrates that the obstruction to other mapsthe nonzero Steenrod square between 3 and 5 is really the only phe- H (SU(3); Z2) H (SU(3); Z2) nomenon involved. 5.4 Maps between principal SU(2)-bundles over S5 We can fairly easily use the techniques of the previous section to compute (again verifying and sharpening [9]) the possible multi-degrees of maps among the principal -bundles over 5. These bundles are classi ed by 5 3 ∼ , SU(2) S [S ,BSU(2)] = π4(S ) = Z2 so the only two are S3 × S5 and SU(3). Theorem 5.4.1. Let . Then there is a map t3, t5 ∈ Z 1. φ : SU(3) → SU(3) of degree (t3, t5) if and only if t3 ≡ t5 (mod 2); 3 5 2. φ : SU(3) → S × S of degree (t3, t5) if and only if t3 ≡ 0 (mod 2); 3 5 3. φ : S × S → SU(3) of degree (t3, t5) if and only if t5 ≡ 0 (mod 2); and 3 5 3 5 4. φ : S × S → S × S of degree (t3, t5). Note that, in general, degrees of maps between dierent manifolds are only well- de ned up to sign, but since everything here is mod 2, no such concern arises. Proof. We have already proved (1) in Theorem 5.3.1, and (4) is clear. So we have only two cases to do: Case 1: φ : SU(3) → S3 × S5. Let X = SU(3) and X0 = S3 × S5. As in the previous section, the only Postnikov invariants that matter are 6 8 k5 ∈ H (X4; π5), k8 ∈ H (X6; π7) 50 as well as 0 and 0 , since the rest of the Postnikov invariants match up by Lemma k5 k8 6 2 0 2.3.4. Recall that k5 ∈ H (X4; π5) is βeSq ι3. For X , Lemma 2.3.2 implies that 0 , so 6 0 0 ∼ (since 0 3 5 ∼ ), and since H5(X4) = 0 H (X4; π5) = Z2 π5 = π5(S ) ⊕ π5(S ) = Z2 ⊕ Z up to this point 0 is , the Postnikov invariant is 0 0 , where is the X4 (Sp(2))4 k5 = Σ(ι3) Σ secondary operation of Example 4.1.1. The induced map ∗ 6 0 0 6 0 φ4 : H (X4; π5) → H (X4; π5) is therefore zero. On the other hand, since h5 : π5 → H5(X) is multiplication by 2 and 0 0 0 takes to , the map 0 is , h5 : π5 → H5(X ) (a, b) ∈ Z2 ⊕ Z b ∈ Z f5 : π5 → π5 2t5 meaning 6 6 0 f5∗ : H (X4; π5) → H (X4; π5) is also zero, since the groups above are and , respectively. We therefore Z2 Z2 ⊕ Z2 have the diagram 3 t3 / 3 0 o t3 3 0 0 H (X4; π3) H (X4; π3) H (X4; π3) 2 2 βeSq βeSq Σ 6 0 / 6 0 o 0 6 0 0 H (X4; π5) H (X4; π5) H (X4; π5) which can be made to commute if and only if t3 ≡ 0 (mod 2). We still need to deal with k8, but this is less complicated. Since 2|t3, the 0 map H8(X) → H8(X ) is also multiplication by an even number, which annihi- lates 0 ∼ . One can then set 0 to be zero to satisfy the π7(X ) = Z2 ⊕ Z2 f7 : π7 → π7 condition ∗ 0 . φ6(k7) = f7∗(k7) Case 2: φ : S3 × S5 → SU(3). Let X = S3 × S5 and X0 = SU(3). Noting that 0 and hence 0 , we can always take 0 to be zero to π7(X ) = 0 k8 = 0 f7 : π7(X) → π7(X ) 51 satisfy the matching condition ∗ 0 . So the only invariants to match are φ6(k8) = f7∗(k8) and 0 . But the group which is the target of both and ∗ is 6 0 , k5 k5 f5∗ φ4 H (X4; π5) = 0 so the matching condition is satis ed as long as f5 is de ned. Looking again at the Whitehead sequences of X and X0, we have the square (0,id) π5 / H5(X) 1 2 t5 t5 0 ·2 / 0 π5 H5(X ) in which all the groups are except for , which is . In particular, is Z π5 Z2 ⊕ Z f5 de ned as long as t5 ≡ 0 (mod 2), and this is also the condition that allows φ to be de ned. Remark 5.4.2. It is again possible to come up with the congruences of Theorem 5.4.1 as necessary conditions using only the Steenrod algebras of SU(3) and S3 × S5, though in this case one could argue that something a little more subtle is going on because of the nontrivial secondary operation Σ that appears in the Postnikov tower (though, notably, it is again trivial on S3 × S5 itself). 5.5 Self-maps of Sp(2) Building on our use of Sp(2) as an example in 4, we now prove that the tertiary operation Π, as well as a primary operation modulo 3, are the only obstructions to creating self-maps: Theorem 5.5.1. For integers t3, t7, there is a self-map of Sp(2) with multi-degree (t3, t7) if and only if t3 ≡ t7 (mod 12). 52 Proof. Recall that ∗ (in particular, is an MDM), and H (Sp(2)) = ΛZ(x3, x7) Sp(2) that the homotopy groups of Sp(2) are n 1 2 3 4 5 6 7 8 9 πn(Sp(2)) 0 0 ZZ2 Z2 0 Z 0 0 0 (see, for example, [13]). Let X = Sp(2) = X . Beginning with a map φ3 = t3ι3 : , we can extend to a map 0 by the triviality of K(Z, 3) → K(Z, 3) φ5 : X5 → X5 several homology groups and Lemma 2.3.4; since π6(X) = 0, this is equivalently a map 0 . From the other direction, if we had a map , it would also φ6 : X6 → X6 φ7 be a map 0 since and are trivial, and this would be enough to φ9 : X9 → X9 π8 π9 get a map φ : X → X0. So the only obstruction to be dealt with is in the group 8 0 H (X6; π7(X )). First we identify this group. The Hurewicz map h7 : π7(X) → H7(X) is, by the 3 ∼ 7 relative Hurewicz theorem, precisely the map π7(Sp(2)) → π7(Sp(2),S ) = π7(S ) in the long exact sequence for the bration S3 → Sp(2) → S7. Since Sp(2) is the principal 3-bundle associated to the generator of 3 ∼ , it follows that S π7(BS ) = Z12 h7 is multiplication by 12. Since H8(X) = 0 and h7 is injective, H8(X6) = 0, so 8 0 Ext H (X6; π7(X )) = (H7(X6), π7(X)) = Z12 since is also (by the diagram in Lemma 2.3.2). Another application of H7(X6) Z12 universal coe cients shows that 7 , and combining this fact with the H (X6; Z) = 0 identi cation above, we see that the Bockstein 7 8 is an βe : H (X6; Z12) → H (X6; Z) isomorphism. Next we nd the generators for this group. Ignoring 2-torsion, X6 and X3 are the same, so in particular 7 7 , which is generated by 1 . H (X6; Z3) = H (K(Z, 3); Z3) P ι3 53 Additionally, we saw in 4.2 that 7 is generated by , which is ∗ ∗ . H (X6; Z4) ζ7 Π(q5q4ι3) Therefore the generator and Postnikov invariant we need is 1 k7 = βe(P + Π)(χ3) where ∗ ∗ and we have been casual about the inclusions . In χ3 = q5q4ι3 Z3, Z4 → Z12 1 Sp(2), and therefore in X6, Π is a linear operation, as is P , so Lemma 5.2.2 now applies with m = 12 to show that φ6 extends to φ7, and therefore to φ9 and a self-map of Sp(2), if and only if t3 ≡ t7 (mod 12). This result is known (see [6]), using a cell-by-cell technique (that is, φ is con- structed through the skeleton rather than the Postnikov tower). Remark 5.5.2. There is a bit of danger hidden in our expression for k7. First, it is only actually well-de ned up to sign, since Postnikov stages are invariant up to homotopy under automorphisms of , and in this case has the sign change. πn π7 = Z But worse, if we compared X7 with the space Y7 constructed using Y 1 k7 = βe(P − Π)(χ3), they are not related by an automorphism of Z, and so are not homotopy equivalent. The 10-skeleton of the space Y7 would be the space E5 of the next section. 5.6 Maps between principal Sp(1)-bundles over S7 We are now in a position to generalize the results of 5.5 to the rest of the principal Sp(1)-bundles over the 7-sphere. These bundles are classi ed by elements of 7 3 ∼ [S , BSp(1)] = π6(S ) = Z12 54 and corresponds to the generator . We will denote the space corre- Sp(2) ω ∈ Z12 3 7 sponding to kω by Ek, so that Sp(2) is E1 and S × S is E0. These spaces have been studied in a number of contexts. The Hilton-Roitberg criminal (see [5], as well as [8] 10-2), the space E5, is the most well-known, and it has the property that 3 3 E5 × S is homotopy equivalent to Sp(2) × S , but E5 is not homotopy equivalent to Sp(2). This is an example of two non-homotopic nite-dimensional spaces of the same genus; that is, we have E5 '(p) Sp(2) for all primes p. For the rest of the Ek, 7 note that Ek and E−k are homotopic, since there is an involution on S available; but they are otherwise distinct (again see [8]). We will consider maps among the Ek that are H-spaces, which are the ones corresponding to k = 0, 1, 3, 4, 5. Our job will be made easier by considering them as being similar to Sp(2) or S3 × S7 either 2-locally or 3-locally. Speci cally, we have 3 7 Lemma 5.6.1. Localized at 2, E3 '(2) Sp(2) and E4 '(2) S × S . Localized at 3, 3 7 E3 '(3) S × S and E4 '(3) Sp(2). Proof. Each space is characterized by (up to the sign of ); so if Ek kω ∈ Z12 k k1 ≡ ±k2 in , then . Using the fact that and 3 7, (Z12)(p) Ek1 '(p) Ek2 E1 = Sp(2) E0 = S × S the result follows. As a nal note before stating the result, recall that multi-degree for maps between distinct spaces is only de ned up to sign. We will prove: 55 Theorem 5.6.2. Let . Then there is a map of multi-degree t3, t7 ∈ Z φ : Ek → Ek0 (t3, t7) if and only if k = 0 k0 = 1 k0 = 3 k0 = 4 k0 = 5 k = 0 − t7 ≡ 0 (12) t7 ≡ 0 (4) t7 ≡ 0 (3) t7 ≡ 0 (12) t3 ≡ 0 (3) t3 ≡ ±t7 (3) t3 ≡ ±t7 (3) k = 1 t3 ≡ 0 (12) t3 ≡ t7 (12) t3 ≡ ±t7 (4) t3 ≡ 0 (4) t3 ≡ ∓t7 (4) t7 ≡ 0 (3) t3 ≡ 0 (3) t7 ≡ 0 (3) k = 3 t3 ≡ 0 (4) t3 ≡ t7 (4) t3 ≡ ±t7 (4) t7 ≡ 0 (4) t3 ≡ ±t7 (4) t3 ≡ ±t7 (3) t3 ≡ 0 (3) t3 ≡ ±t7 (3) k = 4 t3 ≡ 0 (3) t3 ≡ t7 (3) t7 ≡ 0 (4) t7 ≡ 0 (4) t7 ≡ 0 (4) t3 ≡ ±t7 (3) t3 ≡ 0 (3) t3 ≡ ±t7 (3) k = 5 t3 ≡ 0 (12) t3 ≡ t7 (12) t3 ≡ ∓t7 (4) t3 ≡ ±t7 (4) t3 ≡ 0 (4) where − means there is no restriction. Proof. Let us reduce the number of cases to deal with. First, the rows and columns for E1 and E5 involving other spaces are the same; for example, maps E1 → E3 have the same restrictions as E5 → E3 since, localized at any single prime, E1 and E5 actually are homotopy equivalent. Second, as before, we only need to actually compute the four cells in the top left corner (E0 → E0, E0 → E1, E1 → E0, and E1 → E1), and the rest of the cells involving E0,E3, or E4 follow by using the p-local equivalences for E3 and E4 listed in Lemma 5.6.1. Next, we have already computed the possible maps E1 → E1 in Theorem 5.5.1, and it is clear that any multi-degree (t3, t7) can be achieved for a map E0 → E0. So we just have three cases to do. Case 1: φ : Sp(2) → S3×S7. Let X = Sp(2) and X0 = S3×S7. By Lemma 2.3.4, 56 8 10 we only need to look at the k-invariants k7 ∈ H (X6; π7(X)) and k9 ∈ H (X8; π9(X)) (and similarly for 0 0 and 0). k7, k9 X Recall the diagram from Lemma 5.2.2, which laid out conditions for matching k-invariants. In the k7 case, we are looking at 3 t3 / 3 o t3 3 0 H (X6; Z) H (X6; Z) H (X6; Z) Θ Θ Ξ (12t ,0) 8 7 / 8 o 0 8 0 H (X6; Z) H (X6; Z ⊕ Z2) H (X6; Z ⊕ Z2) 1 where Θ is either βeP or βeΠ depending on whether we are examining the situation 3- locally or 2-locally, respectively, and Ξ is a cohomology operation from the Postnikov 3 tower of S . Note the (12t7, 0) in the bottom row, which comes from the fact that the map 0 must satisfy f7 : π7 → π7 0 h7f7 = t7h7 from the commuting square involving π7 and H7(X) in the Whitehead sequence, and h7 is multiplication by 12. Now, inspecting the diagram above, we see that it can be completed if and only if t3 ≡ 0 (mod 12). The other -invariants to match are and 0 . As 0 ∼ is annihilated k k9 k9 π9(X ) = Z3 ⊕Z2 by 12, and the induced map 0 is multiplication by , which is H10(X8) → H10(X8) t3t7 divisible by 12, the obstruction 0 in k9(φ8)∗ 10 0 0 H (X8; π9(X )) = Hom(H10(X8), π9(X )) 0 must vanish. Hence φ9 and therefore φ : X → X exists. Case 2: φ : S3 × S7 → Sp(2). Let X = S3 × S7 and X0 = Sp(2). Once again we only need to look at 8 and 10 in addition to 0 k7 ∈ H (X6; π7(X)) k9 ∈ H (X8; π9(X)) k7 57 and 0 ; but in this case since 0 , 0 , so there cannot be an obstruction. k9 π9(X ) = 0 k9 = 0 Thus the only relevant Postnikov invariant is k7. Looking again at the diagram from the previous part, but this time with Θ on the right and 0 in the middle, and the coe cient groups rearranged on the bottom row, we see that the diagram can be made to commute as long as f7 exists; as in the SU(3) case, this can only happen as long as t7 ≡ 0 (mod 12). As this is the only nontrivial invariant to match, this is enough to provide a map φ9 and hence φ as desired. The rest of the table can be deduced from these results. As an example, let us 0 compute the box with k = 3 and k = 1. Here, we are interested in maps E3 → E1. Localized at 2, this is equivalent to asking about maps E1 → E1, for which the (2-local) criterion reads t3 ≡ t7 (mod 4). Localized at 3, we are asking about maps E0 → E1, where the (3-local) criterion is instead t7 ≡ 0 (mod 3). 0 Case 3: φ : Sp(2) → E5 and φ : E5 → Sp(2). Let X = Sp(2) and X = E5. As in the proof of Theorem 5.5.1, the only obstruction to be considered is k7. Eectively, the k-invariants for X and X0 are 1 0 1 0 k7 = βe(P + Π)(χ3), k7 = βe(P − Π)(χ3) so we nd that the usual diagram will commute if and only if 1 1 0 t3(P + Π)(χ3) = t7(P − Π)(χ3) This is possible if t3 ≡ t7 (mod 4) and t3 ≡ −t7 (mod 3). But due to sign ambi- guity, this translates to the pair of congruences in the table; this also justi es our arbitrary choice that Sp(2) would be the sum and E5 would be the dierence of their 58 0 components, since the same procedure for X = E5 and X = Sp(2) yields the same result. 59 CHAPTER 6 SELF-MAPS OF SU(4) In this chapter we apply the same techniques as last chapter to a nal group, SU(4). While we will still primarily be identifying k-invariants in terms of cohomology op- erations, in this chapter we will need sharper data on the maps φn. This is required because the dimension of SU(4) (15) is su ciently larger than the dimension of the highest generator (7) that the highest obstructions to extension do not yield congru- ence conditions as our method in the previous chapter always did (more succinctly, this happens because H∗(SU(4)) has 3 generators instead of 2). 6.1 Low dimensions In this section and the next, we will prove the following: Theorem 6.1.1. Let . There is a map of multidegree t3, t5, t7 ∈ Z φ : SU(4) → SU(4) (t3, t5, t7) if and only if t3 ≡ t5 (mod 2) and t3 ≡ t7 (mod 6). Self-maps of SU(4) are not as well-studied as those of H-spaces with fewer cells; as one example, there is the work of Oshima-Yagita© [15] on the nilpotency of the group of self-maps [G, G] for various Lie groups G. 60 The congruence hypothesis is the pair of relations given above. The proof will be accomplished in the following steps (with X = X0 = SU(4)): 1. Show that there is a map 0 if and only if the congruence hypothesis φ7 : X7 → X7 is satis ed; it remains then to show that φ7 can be extended to φ14 if the congruence hypothesis is satis ed. 2. If necessary, modify (t3, t5, t7) so that t3, t5, and t7 are all units modulo 30. 3. Verify that φ8 exists and can be taken to be an H-map. 4. Extend φ8 to φ10. 5. Show that φ10 can be taken to be an H-map. 6. Extend φ10 to φ14. Before beginning the process above, let us record the essential data about SU(4). Analogously to SU(3), SU(4) is the total space of a principal ber bundle SU(3) → SU(4) → S7 classi ed by a map in 7 , and that map is the generator [S ,BSU(3)] = π6(SU(3)) = Z6 of . Recall that is 2-connected, and the rest of its homotopy groups are Z6 SU(4) n 3 4 5 6 7 8 9 10 11 12 13 14 πn(SU(4)) Z 0 Z 0 ZZ24 Z2 Z120 ⊕ Z2 Z4 Z60 Z4 Z1680 ⊕ Z2 (see [13]). The cohomology of SU(4) is ∗ H (SU(4)) = ΛZ(x3, x5, x7), 61 and in particular free, so the nonzero homology groups occur in dimensions 0, 3, 5, 7, 8, 10, 12, and 15 and SU(4) is an MDM. Step 1: Construct φ7. First, construction of φ5 occurs exactly as in the case of SU(3), so we get a condition (necessary and su cient for building φ5) that t3 ≡ t5 8 (mod 2). Then we must extend φ5 to φ7, the k-invariant for which lies in H (X6; π7). As in the case of SU(3) and Sp(2), applying the relative Hurewicz theorem to the pair (SU(4),SU(3)) and inspecting the long exact sequence for the bration SU(3) → 7 SU(4) → S , we see that the Hurewicz map h7 : π7 → H7(X) in the Whitehead 7 sequence is exactly the map π7(SU(4)) → π7(S ) in the long exact sequence; hence the map π7 → H7(X) is multiplication by 6 and (since π6 = 0) H7(X5) = H7(X6) = . This also implies that the map is zero, which combined with the Z6 H8(X6) → π7 facts that and is nite implies that . Putting all of these H8(X) = Z π8 H8(X6) = Z facts together, we have 8 Hom Ext H (X6; π7) = (H8(X6), Z) ⊕ (H7(X5), Z) = Z ⊕ Z6 where the splitting is natural since the Hom portion is free. Furthermore, since 7 , the long exact sequence for the coe cients implies that H (X6; Z) = 0 Z → Z → Z6 the group 7 must be , and therefore that the Bockstein H (X6; Z6) Z6 7 8 βe : H (X6; Z6) → H (X6; Z) identi es it with the of the latter group. Z6 Now we need to identify the generators. We rst identify the piece, where Z2 2 2 again for brevity, we write Sq for Sq ρ2. Consider the Leray-Serre spectral sequence for the bration , with coe cients. Since the bration K(Z, 5) → X5 → K(Z, 3) Z 62 is nontrivial, the transgression takes to the generator of 6 , which is ι5 H (K(Z, 3)) 2 βeSq ι3. So we compute that 2 2 2 2 τ(ι5) = βSq ι3 τ(Sq ι5) = Sq βeSq ι3 = 0 2 2 τ(βeSq ι5) = 0 d6(ι3ι5) = ι3βeSq ι3 Then 7 is generated by an element for which ∗ Sq2 , which de nes H (X6; Z2) λ7 i5λ7 = ι5 a secondary operation Φ based on the relation 2 2 Sq ◦ βeSq = 0 in dimension 3. In general, for a CW-complex Y , we have 3 2 2 5 Def(Φ,Y ) = {x ∈ H (Y ; Z): βeSq x = 0}, Ind(Φ,Y ) = Sq H (Y ; Z) and in particular Φ is a linear cohomology operation for Y = SU(4) (since it is the looping of an operation on ), taking values in 7 . It follows that BSU(4) H (SU(4); Z2) generates the portion of 8 . βe(λ7) Z2 H (X6; Z) The piece is much more straightforward. The -invariant attaching Z3 k K(Z, 5) to is trivial mod 3, so we just get 7 ∼ , generated by ∗ 1 . K(Z, 3) H (X5; Z3) = Z3 q5P ρ3ι3 Now, the k-invariant for constructing X7 is not trivial mod 3, and the generator for the piece of 8 is ∗ , which is not primitive. But , as a -invariant Z H (X5; Z) 2(q5ι3)ι5 k7 k for an H-space, must be primitive, so it follows that ∗ 1 k7 = βe(λ7 + q5P ρ3ι3) Since this has order 6, applications of Lemma 5.2.2 modulo 2 and 3 show that φ5 can be extended to φ7 if and only if t3 ≡ t7 (mod 6). 63 6.2 Eliminating higher obstructions We continue the plan laid out in the previous section. Step 2: Modify t3, t5, t7. By Lemma 5.1.2, the algebraic loop [SU(4),SU(4)] 3 has a homomorphism to Z given by the multidegree; since the identity (1, 1, 1) is clearly a valid map of SU(4) and by assumption t3 ≡ t5 ≡ t7 (mod 2), we may change (t3, t5, t7) to (t3 + 15, t5 + 15, t7 + 15) if necessary to ensure the ti are all units mod 2 without changing their class modulo 3 or 5. Similarly, since t3 ≡ t7 (mod 3), we may add a multiple of (10, 10, 10) to make the ti all units mod 3, and a multiple of (6, 6, 6) to do it mod 5. Step 3: Ensure φ8 is an H-map. First we must check that φ3, φ5, and φ7 are H-maps. The map is a group homomorphism of , so is certainly an φ3 K(Z, 3) H-map. For the latter two, the sets in which their H-deviations lie are surjected onto by the groups 5 7 H (X5 ∧ X5; Z) = 0,H (X7 ∧ X7; Z) = 0 so φ5 and φ7 are both H-maps (independent of the modi cation from Step 2). Next, φ7 can be extended to φ8 by Lemma 2.3.4. Finally, by Lemma 3.2.3, φ8 can be modi ed to be an H-map if ∗ 8 0 0 8 (φ8 ∧ φ8) : H (X8 ∧ X8; Z24) → H (X8 ∧ X8; Z24) is a surjection. A simple calculation shows that the groups on each side are 8 ∼ 3 5 5 3 H (X8 ∧ X8; Z24) = H (X8; Z24) ⊗ H (X8; Z24) ⊕ H (X8; Z24) ⊗ H (X8; Z24) ∗ and similarly with primes. Since t3 and t5 are units mod 2 and 3, (φ8 ∧ φ8) is 64 multiplication by a unit of , hence a surjection. So can be taken to be Z24 ⊕ Z24 φ8 an H-map, possibly after modifying the H-structure of 0 . X8 10 Step 4: Extend φ8 to φ10. For this, we must rst determine H (X8; π9). In the Whitehead sequence, H10(X8) sits in the middle of the short exact sequence Z = H10(X) → H10(X8) → π9 = Z2 so is either or . The -invariant group 10 is therefore either Z Z ⊕ Z2 k H (X8; π9) Z2 or . But 10 must contain a nonzero multiple of (a combination of Z2 ⊕ Z2 H (X8; π9) pullbacks of) ι3ι7, which is not primitive, whereas the k-invariant is primitive (and nonzero), so the group is , generated by and a version of . It is then Z2 ⊕ Z2 k9 ι3ι7 clear that the matching condition ∗ 0 is satis ed since is an H-map, φ8(k9) = f9∗(k9) φ8 and hence must take a primitive element to a primitive element. Once φ9 has been constructed, φ10 follows immediately by Lemma 2.3.4. Step 5: Ensure φ10 is an H-map. This is very similar to the procedure for φ8. First, note that φ9 is an H-map since 9 H (X9 ∧ X9; Z2) = 0 ∗ so there is no H-deviation. For φ10, we examine the map (φ10 ∧ φ10) , which involves the groups 10 ∼ 3 7 7 3 H (X10 ∧ X10; I) = H (X10; I) ⊗ H (X10; I) ⊕ H (X10; I) ⊗ H (X10; I) and similarly with 0 , where the coe cient group is . As X10 I = π10(X) = Z120 ⊕ Z2 before, we have already stipulated that t3 and t7 are units mod 2, 3, and 5, so that 65 ∗ (φ10 ∧ φ10) is multiplication by a unit and hence a surjection, meaning φ10 can be an H-map. Step 6: Extend φ10 to φ14. Extending φ10 to φ11 is the same argument as Step 4, but with instead of , the other parts carrying through mutatis π11 = Z4 π9 = Z2 mutandis. Extending to φ12 is an application of Lemma 2.3.4, and φ12 can be taken to be an H-map by logic analogous to step 5 (again using the fact that t3, t5, t7 are all units mod 60). Construction of φ13 again follows by Lemma 2.3.4, and it can be made an H-map since 13 H (X13 ∧ X13; π13) = 0 and so construction of φ14 is again analogous to step 4. This completes the proof for SU(4). Remark 6.2.1. If we wished, we could also have ensured that t3, t5, t7 were units modulo 7, in which case we could make φ14 an H-map. However, this does not imply that the nal map φ : SU(4) → SU(4) is an H-map, because we have ignored the H-structure when collapsing the higher cells of X14 to get X. Indeed, one would need to do an analysis similar to the above with X = SU(4) ∧ SU(4) and X0 = SU(4) in order to construct an H-map on SU(4) itself. See also [6], [14], and [12] for more discussion of existence or non-existence of nontrivial self-H-maps. 66 CHAPTER 7 FUTURE DIRECTIONS In this nal section, we will brie y state some conjectures and directions in which one could hope to proceed with the work described to this point. Some of the problems posed in this chapter have known answers, but have not (to our knowledge) been solved using the approach of Postnikov towers. Others are theoretically open in the sense that they are unrecorded, but are perhaps known as folklore. Others still may be genuinely unknown. We will highlight the problems that fall de nitively into the rst category, but we cannot be sure about the latter two. 7.1 General questions about H-spaces Though there exist a number of reasonably thorough treatments of H-spaces (e.g. [8], [24], and [19]), it is still not hard to generate simple questions that seem not to have obvious answers. One of the technical issues we have encountered is that of lifting an H-map up through a Postnikov tower. While Lemma 3.2.3 can be useful, as we have seen 67 in building the tower for SU(4), one would like a more general statement like the following: Conjecture 7.1.1. Let 0 be 1-connected H-spaces, and 0 Postnikov X,X Xn−1 Xn−1 stages for and 0, and 0 an H-map with chosen so that X X φn−1 : Xn−1 → Xn−1 fn∗ ∗ 0 . Then the induced map 0 is also an H-map. fn∗(kn) = φn−1(kn) φn : Xn → Xn As remarked in 3.4, this would serve as a nice complement to Kahn's results [7] that translate an H-map φ : X → Y into H-maps φn : Xn → Yn. One could also hope for a much more general statement: Conjecture 7.1.2. Let A1,B1,A2,B2 be 1-connected H-spaces, and f1 : A1 → B1, f2 : A2 → B2, α : A1 → A2, and β : B1 → B2 H-maps so that β ◦ f1 = f2 ◦ α. Let Fi be the ber of fi for i = 1, 2. Then the induced map φ : F1 → F2: f1 F1 / A1 / B1 φ α β f2 F2 / A2 / B2 is also an H-map. In the self-map vein of ideas, we have seen that all the obstructions to self-maps of given degrees have been in the form of primary or higher cohomology operations. Let us brie y make a de nition to handle nite H-spaces that are not necessarily free of homological torsion: ∗ De nition 7.1.3. Let X be a 1-connected nite H-space such that H (X; Q) = , and let di be elements such that for all , and ΛQ(yd1 , . . . , ydr ) xdi ∈ H (X) ρpxdi 6= 0 p 68 , where is induced by the coe cient homomorphism . Then ρ(0)xdi 6= 0 ρ(0) Z → Q the -degree of a self-map is the integer such that ∗ . di φ : X → X tdi φ (xdi ) = tdi xdi One can check that, despite the ambiguity in choosing the , the multi-degree xdi is still well-de ned. We pose the following two questions: Question 7.1.4. Does there exist a nite H-space X for which the obstruction to a self-map of given multi-degree is not a primary or higher cohomology operation? Question 7.1.5. Does there exist a nite H-space X for which the multi-degrees of self-maps are not de ned by a nite set of congruences among the (td1 , . . . , tdr ) ? tdi It would seem that a negative answer to the rst question would also imply a negative answer to the second. 7.2 Problems about speci c H-spaces Given the spaces we have worked with to this point, there are some natural candi- dates for spaces to apply our methods to in the future, some of which already have well-known conditions for self-maps, and others which do not. The most obvious candidates would be the next spaces in the SU and Sp series: Conjecture 7.2.1. There is a self-map of SU(5) of multi-degree (t3, t5, t7, t9) if and only if t3 ≡ t5 (mod 2), t3 ≡ t7 (mod 6), and t3 ≡ t9 (mod 24). Conjecture 7.2.2. There is a self-map of Sp(3) of multi-degree (t3, t7, t11) if and only if t3 ≡ t7 (mod 12) and t3 ≡ t11 (mod 120). 69 These are based on the fact that ∼ and ∼ , which π8(SU(4)) = Z24 π10(Sp(2)) = Z120 imply that the next relevant k-invariants for X = SU(5) and Y = Sp(3) would be 10 9 12 11 k9 ∈ H (X8; π9) = βHe (X8; Z24), k11 ∈ H (Y10; π11) = βHe (Y10; Z120) implying that congruences mod 24 and mod 120 would come into play. More gener- ally, given the facts that if n is even ∼ Z(2n+1)! πn(SU(n)) = Zn!, π4n+2(Sp(n)) = if is odd Z2(2n+1)! n one might guess that Conjecture 7.2.3. There is a self-map of SU(n) of multi-degree (t3, t5, . . . , t2n−1) if and only if t3 ≡ t2k+1 (mod k!) for 2 ≤ k ≤ n − 1. Conjecture 7.2.4. There is a self-map of Sp(n) of multi-degree (t3, t7, . . . , t4n−1) if and only if t3 ≡ t4k+3 (mod (2k + 1)!) when k is even and t3 ≡ t4k+3 (mod 2(2k + 1)!) when k is odd, for 1 ≤ k ≤ n − 1. A natural next step might be the groups Spin(n), the double covers of the special orthogonal groups SO(n). But the cohomology ring and Steenrod algebra of Spin(n) are not nearly as simple to describe as they are for SU(n) or Sp(n), so this seems di cult. It should be noted, however, that through sporadic isomorphisms we have already covered the cases Spin(n) for n ≤ 6. Going in another direction, one might try the exceptional Lie groups. The small- 6 est, G2, which arises as a principal SU(3)-bundle over S , is only 14-dimensional and has rational cohomology generators in dimensions 3 and 11, though unlike our prior examples, its homology is not free. It is known (see [14]) that 70 Theorem 7.2.5. There is a self-map of G2 of multi-degree (t3, t11) if and only if t3 ≡ t11 (mod 30). It would be interesting to try to prove this using the Postnikov tower approach. One potential complication is that the Hurewicz map π11(G2) → H11(G2) is known to have cokernel , and this is the group in which the nal relevant -invariant Z120 k would arise. So this would be a case where the congruence relation does not match the order of the k-invariant group, as it has every time until now. Finally, we should remark that self-H-maps have been studied as well for a number of groups; Oka's study of G2 [14] includes the result that G2 has no nontrivial self- H-maps. Maruyama-Oka [6] reached a similar conclusion for the spaces Ek obtained 3 7 as S -bundles over S , with respect to any multiplication on the Ek (not just the canonical one). It would be interesting to try to use our methods to prove that the various spaces we have studied do not admit any nontrivial self-H-maps, especially since our technique of 6.2 provably does not produce self-H-maps of the total spaces (see Remark 6.2.1). 71 Appendix A INDEX OF NOTATION Let X be a 1-connected pointed topological space. Symbol Description Xn nth Postnikov stage of X qn Postnikov tower map Xn → Xn−1 pn Postnikov section X → Xn K(G, n) Eilenberg-Maclane space with πn = G and πk = 0 if 1 ≤ k 6= n τ Leray-Serre spectral sequence transgression kn Postnikov invariant mapping Xn−1 → K(πn, n + 1) ιn Fundamental class of K(G, n) X(n) n-skeleton of X ΩX (Moore) loop space of X SU(n) Unitary n × n matrices of determinant 1 Sp(n) Compact symplectic group of n × n quaternionic unitary matrices 72 X ∨ Y Coproduct of pointed spaces X,Y : (X, x0) t (Y, y0)/(x0 ∼ y0) X ∧ Y Smash product of pointed spaces X,Y : (X × Y )/(X ∨ Y ) [X,Y ] Set of pointed homotopy classes of maps X → Y (X, µ) H-space X with multiplication µ : X × X → X Reduction of coe cients mod , ∗ ∗ ρm m H (X; Z) → H (X; Zm) Sqi Steenrod square mapping k k+i H (X; Z2) → H (X; Z2) Sq2 The composition Sq2 k k+2 ρ2 : H (X; Z) → H (X; Z2) i Steenrod power operation, i k k+2(p−1)i P P : H (X; Zp) → H (X; Zp) 1 The composition 1 k k+2(p−1) P P ρp : H (X; Z) → H (X; Zp) Bockstein associated with coe cient sequence β Zp → Zp2 → Zp Bockstein associated with coe cient sequence ·n βe Z −→ Z → Zn 73 Appendix B THE LERAY-SERRE SPECTRAL SEQUENCE B.1 Basic information Here we brie y list some important properties of the Leray-Serre spectral sequence, and give a detailed example of calculating the cohomology of a Postnikov stage, based on the bration that occurs in the construction of K(Z2, 4) → X4 → X3 Sp(2). For an in-depth construction of the spectral sequence and its properties, see McCleary [11]. As usual, all spaces are 1-connected CW-complexes. Recall that the Leray-Serre spectral sequence is a bigraded object p,q equipped En with pages En and dierentials dn which are de ned so that p,q p+n,q−n+1 dn : En → En and the next page En+1 is obtained from En by the rule p,q p,q p+n,q−n+1 im p−n,q+n−1 p,q En+1 = ker(dn : En → En )/ (dn : En → En ) The spectral sequence computes the cohomology of the total space of a bration using the cohomology of the ber and base space as inputs: 74 Theorem B.1.1. Let F → E → B be a bration, and G a nitely-generated abelian group. There is a cohomology spectral sequence p,q such that En p q ∼ p,q p+q H (B; H (F ; G)) = E2 =⇒ H (E; G), that is, the pages converge to groups p,q that yield p+q via ltrations. En E∞ H (E; G) The important facts for us are: • The cup product is compatible with the bigrading, and the product on En induces one on En+1. • The dierential dn is a derivation with respect to cup product. The dierential 0,n−1 n,0 is called the transgression, notated . • dn : En → En τ • The transgression is natural, and in particular commutes with Steenrod oper- ations. B.2 The bration K(Z2, 4) → X4 → K(Z, 3) Now we demonstrate some computations for the spectral sequence associated to , which is the rst nontrivial stage of the Postnikov tower K(Z2, 4) → X4 → K(Z, 3) for X = S3 or X = Sp(2). 75 Cohomology SS for K(Z2, 4) → X4 → K(Z, 3) 3 1 Sq Sq ι4 8 2 ι4 2 1 Sq Sq ι4 7 3 Sq ι4 2 ) Sq 2 6 ι4 Z 4); , 2 1 1 5 Sq ι4 ι3Sq ι4 Z ( K ( ∗ H 4 ι4 ι3ι4 . . k4 Sq4Sq2ι ι Sq2 2 Sq2 3 0 3 ι3 ι3 ι3 ι3 3 ι3 0 ··· 3 4 5 6 7 8 9 ∗ H (K(Z, 3)) We work with coe cients, and for easier notation, we will, in this section Z2 only, use for the generator of 3 (instead of ). Since the bration ι3 H (K(Z, 3); Z2) ρ2ι3 2 is nontrivial, τ(ι4) = Sq ι3. Along with Adem relations, this is the only piece of information we need. The Adem relations we use include: Sq1Sq2 = Sq3 Sq2Sq2 = Sq3Sq1 Sq3Sq2 = 0 Sq2Sq3 = Sq4Sq1 + Sq5 Sq3Sq3 = Sq5Sq1 k as well as the the fact that Sq ι3 = 0 if k = 1 or if k ≥ 4. 76 2 τ(ι4) = Sq ι3 Sq1 Sq1Sq2 Sq3 2 τ( ι4) = ι3 = ι3 = ι3 2 2 2 3 1 τ(Sq ι4) = Sq Sq ι3 = Sq Sq ι3 = 0 3 3 2 τ(Sq ι4) = Sq Sq ι3 = 0 2 1 2 3 4 1 5 τ(Sq Sq ι4) = Sq Sq ι3 = Sq Sq ι3 + Sq ι3 = 0 2 Sq4 Sq4Sq2 τ(ι4) = τ( ι4) = ι3 3 1 3 1 2 5 1 τ(Sq Sq ι4) = Sq Sq Sq ι3 = Sq Sq ι3 = 0 2 τ(ι3ι4) = τ(ι3)ι4 + ι3τ(ι4) = ι3Sq ι3 Sq1 Sq1 3 τ(ι3 ι4) = ι3τ( ι4) = ι3 There are thus elements 6 and 7 such that λ6 ∈ H (X4; Z2) µ7 ∈ H (X4; Z2) ∗ Sq2 ∗ Sq2Sq1 i4λ6 = ι4, i4µ7 = ι4. Additionally, ∗ Sq1 Sq3 ∗ Sq1 Sq3Sq1 i4( λ6) = ι4, i4( µ7) = ι4 thereby accounting for all the boxed elements (that is, those that survive to X4 in dimension ≤ 8). 77 BIBLIOGRAPHY [1] Martin Arkowitz. 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