Quaternionic Projective Space (Lecture 34)

Transcription

1 Quaternionic Projective Sace (Lecture 34) July 11, 2008 The three-shere S 3 can be identified with SU(2), and therefore has the structure of a toological grou. In this lecture, we will address the question of how canonical this structure is. In the category of toological grous, the grou structure on S 3 is unique u to isomorhism. However, the urely homotoy-theoretic situation is not quite so nice: there exist uncountably many airwise inequivalent grou structures on saces which are homotoy equivalent to S 3 (we will return to this oint at the end of the lecture). However, the situation is much simler in -adic homotoy theory, where is a fixed rime. In this case, we again have a unique grou structure on (the -adic comletion) of the homotoy tye of S 3. We will sketch a roof of this result when is odd, following the ideas of Dwyer, Miller and Wilkerson. We begin by formulating the roblem more recisely. In homotoy theory, giving a grou structure on a homotoy tye G is equivalent to realizing G as the loo sace of a ointed sace X. In this case, we have a fiber sequence G X. If G = S 3, then we can use the Serre sectral sequence to comute the (mod ) cohomology ring of X: H (X) F [t], where t lies in degree 4 (and transgresses to the fundamental class of G = S 3 ). Moreover, we have the same icture in the -rofinite category. We can now state the main result: Theorem 1 (Dwyer, Miller, Wilkerson). Let X be a -rofinite sace such that H (X) F [t], where t lies in degree 4. Then X is equivalent to the -rofinite comletion BSU(2) of the classifying sace of the grou SU(2) (in other words, infinite dimensional quaternionic rojective sace). The first ste is to describe the cohomology H (X) as a reresentation of the mod- Steenrod algebra A. To simlify the exosition, we will consider only the case 2. We therefore begin with a few recollections on the structure of A : For any sace X (or any -rofinite sace), the algebra A acts on the cohomology ring H (X; F ). The algebra A is generated the Bockstein oerator β of degree 1, together with oerations P i of degree 2i( 1), for i > 0. We have a Cartan formula P n (xy) = P n n=n +n (x)p n (y), and a similar formula for β (which involves a sign). Here we agree by convention that P 0 = id. If x H 2i (X; F ), then P i (x) = x and P j (x) = 0 for j > i (instability). We have P 1 P 1 = 2P 2 (this is a secial case of the Adem relations, which we will not write out in full). Lemma 2. Let X be as in the statement of Theorem 1. Then there exists an isomorhism α : H (X) F [t] such that the action of A on H (X) F [t] is determined by the Cartan formula, together with the relations βt = 0 1

2 2t +1 2 if i = 1 P i t = t if i = 2 0 otherwise. Proof. The formula βt = 0 is obvious, since H 5 (X) 0. The exressions P i t vanishes for i > 2 by instability, and P 2 t = t. We have P 1 (t) = ct +1 2 for some constant c F ; the only nontrivial oint is to comute c. For this, we observe 2t = 2P 2 (t) = P 1 P 1 (t) = cp 1 t +1 2 = c t so that c 2 = 4 +1 = 4. This has solutions c = ±2. However, if c = 2 then we can adjust the isomorhism α via the substitution t λt, where λ F is not a quadratic residue, to obtain an isomorhism with the desired roerty. Corollary 3. There exists an isomorhism α : H (X) H (BSU(2)) of unstable A -algebras. We now make a few remarks about the structure of the grou SU(2). We have injective grou homomorhisms Z/Z S 1 SU(2). These induce mas of classifying saces hence we get mas on cohomology BZ/Z BS 1 BSU(2), H (BZ/Z) H (BS 1 ) H (BSU(2)). A simle comutation shows that each of these mas is injective, and we can identify the above with the sequence F [u, ɛ] F [u] F [t]. Here t u 2, where u has degree 2, and ɛ has degree 1 in H (BZ/Z) (and therefore squares to zero). Lemma 4. There exists a ma β : BZ/Z X such that the diagram commutes. H (X) α H (BZ/Z) H (BSU(2)) 2

3 Proof. We have π 0 Ma(BZ/Z, X) Hom(H (X BZ/Z ), F ) Hom(T H (X), F ) Hom(H (X), H (BZ/Z)) Here the Hom-sets on the right hand side are comuted in the category of unstable A -algebras. In other words, any ma of A -algebras from H (X) to H (BZ/Z) is necessarily induced by a ma of -rofinite saces BZ/Z to X (which is then uniquely determined u to homotoy). Let Y be the connected comonent of the maing sace X BZ/Z containing the ma β. We then have isomorhisms H (Y ) H (X BZ/Z ) H 0 (X BZ/Z ) F T H (X) (T H (X)) 0 F. Consequently, the cohomology ring H (Y ) deends only on H (X). Let us temorarily assume that X = BSU(2) and that β is the ma induced by the grou homomorhism Z/Z SU(2). The loo sace ΩY can be identified with the sace of homotoies from β to itself, which is a sace of sections of a certain fibration E BZ/Z with fiver SU(2). This fibration corresonds to an action of Z/Z on SU(2), which is simly induced by the action of Z/Z by conjugation. We therefore may therefore identify ΩY with the homotoy fixed set (SU(2) ) hz/z. Using the -rofinite Sullivan conjecture, this can be identified with the -rofinite comletion of the actual fixed set SU(2) Z/Z, which is simly the centralizer of Z/Z in SU(2). A simle calculation shows that this centralizer coincides with the circle grou S 1 SU(2). It follows that ΩY (S 1 ). Using the Serre sectral sequence, we conclude that H (Y ) is isomorhic to F [u], where u lies in degree 2. Moreover, the translation action of BZ/Z on itself determines a ma BZ/Z Y, which (after scaling u if necessary) is given on cohomology by the canonical inclusion F [u] F [u, ɛ]. We now return to the general case. Since H (Y ) deends only on H (X), we conclude that H F [u] in general. Evaluation at the base oint of BZ/Z induces a ma e : Y X. Moreover, the comosition BZ/Z Y e X can be identified with the ma β. It follows that the above sequence induces, on cohomology, the mas F [u, ɛ] F [u] F [t]. Consider the ma from X BZ/Z to itself, given by comosition with the ma Z/Z 1 Z/Z. This ma induces the identify on H 4 (BZ/Z), and therefore induces the identity ma on Hom(H (X), H (BZ/Z)) π 0 X BZ/Z. It therefore induces an involution on Y, which we will denote by i. We have a commutative diagram BZ/Z Y 1 BZ/Z Y, i 3

4 which gives a commutative diagram of cohomology grous F [u, ɛ] F [u] F [u, ɛ] F [u] Since the left vertical ma carries u to u, the right vertical ma does as well. Let Y hz/2z denote the homotoy coinvariants of the involution on Y. Then the canonical ma Y Y hz/2z induces an isomorhism H (Y hz/2z ) H (Y ) Z/2Z F [u 2 ]. The base oint of BZ/Z is invariant under the ma given by multilication by ( 1), so the evaluation ma e : Y X is invariant under the action of i. Consequently, we obtain a factorization Y e e Y hz/2z. X This induces a commutative diagram of cohomology grous F [u] F [t] F [u 2 ]. We conclude that e induces an isomorhism on cohomology, and therefore a homotoy equivalence of - rofinite saces Y hz/2z X. We now identify the -rofinite sace Y. Since the cohomology of Y lies entirely in even degrees, we can choose a comatible family of cohomology classes u i H 2 (Y ; Z/ i Z) lifting u. These cohomology classes determine a ma of -rofinite saces Y lim K(Z/ k, 2), which we can identify with a ma Y (BS 1 ). This ma induces an isomorhism on cohomology, and is therefore an equivalence of -rofinite saces. We may therefore identify Y with the (-rofinite) Eilenberg- MacLane sace K(Z, 2). Now consider the involution i on Y. We claim that the homotoy fixed set Y hz/2z is nonemty: this follows from the vanishing of the cohomology grou H 3 (BZ/2Z; Z ) (since is different from 2). We may therefore assume without loss of generality that Y contains a oint fixed by the involution i. In this case, i can be regarded as a ointed ma from the Eilenberg-MacLane sace K(Z, 2) to itself, which is given by a grou homomorhism h : Z Z. Since h has order 2, we deduce that h is given by the formula h(z) = λz, where λ = ±1. Since i carries u H 2 (Y ) to u, we deduce that λ = 1. We have therefore roven: Theorem 5. Let X be as in Theorem 1 and an odd rime. Then there is an equivalence of -rofinite saces X K(Z, 2) hz/2z, where the grou Z/2Z acts on Z by the sign involution. 4

5 In articular, there is only one ossibility for the homotoy tye of X. Theorem 1 follows. Let us now consider the same roblem in the non--rofinite world. Let X be a simly connected sace such that H (X; Z) H (BSU(2); Z) Z[t], where t lies in degree 4 (this is equivalent to the assertion that the loo sace ΩX is homotoy equivalent to a three shere S 3, by the Serre sectral sequence). We have a homotoy ullback diagram X X X Q ( X ) Q. Using Theorem 1 (and its analogue in the case = 2), we deduce that for each rime we have a homotoy equivalence X BSU(2). A much easier argument shows that X Q K(Q, 4) BSU(2) Q. We can therefore rewrite the above homotoy ullback diagram as X BSU(2) BSU(2) φ Q ( BSU(2) ) Q. However, this does not imly that X BSU(2), because the ma φ has not been determined. The domain of φ can be identified with an Eilenberg-MacLane sace K(Q, 4), and the codomain of φ with an Eilenberg-MacLane sace K(( Z ) Q, 4), so that φ is determined u to homotoy by secifying an element η ( Z ) Q. Every invertible element η ( Z ) Q gives rise to a sace X which is a delooing of the shere S 3. Not all of these choices are distinct (as an exercise, you can try to figure out when two choices of η give homotoy equivalent delooings), but this mixing construction nevertheless yields uncountably many grou structures on the homotoy tye S 3. 5