Positive curvature and torus symmetry in small dimensions, I -- Dimensions 10, 12, 14, and 16
aa r X i v : . [ m a t h . DG ] A ug POSITIVE CURVATURE AND TORUS SYMMETRY IN SMALLDIMENSIONS, I – DIMENSIONS 10, 12, 14, AND 16
MANUEL AMANN AND LEE KENNARD
Abstract.
This is the first part of a series of papers where we compute Euler characteris-tics, signatures, elliptic genera, and a number of other invariants of smooth manifolds thatadmit Riemannian metrics with positive sectional curvature and large torus symmetry. Inthe first part, the focus is on even-dimensional manifolds in dimensions up to 16. Many of thecalculations are sharp and they require less symmetry than previous classifications. Whenrestricted to certain classes of manifolds that admit non-negative curvature, these resultsimply diffeomorphism classifications. Also studied is a closely related family of manifoldscalled positively elliptic manifolds, and we prove the Halperin conjecture in this context fordimensions up to 16 or Euler characteristics up to 16.
Introduction
In this article, we prove topological obstructions to the existence of Riemannian metricswith positive sectional curvature and large symmetry. This is part of a well establishedresearch program was initiated by K. Grove in the 1990s and has led to the proofs of a largenumber of topological obstructions, the construction of many interesting examples, and thedevelopment of new tools for studying Riemannian manifolds with curvature bounds andsymmetry. For some recent surveys, we refer the reader to Grove [Gro09], Wilking [Wil07],and Ziller [Zil07, Zil14].In this article, we focus on torus symmetry, as it is perhaps the most understood. Threefundamental results in this area are due to Grove and Searle [GS94], Fang and Rong [FR05],and Wilking [Wil03]. They prove, respectively, equivariant diffeomorphism, homeomorphism,and homotopy and cohomology classifications for positively curved manifolds with torussymmetry, where the rank of the torus action is bounded from below by a constant that onlydepends on the dimension of the manifold.While these results provide strong classifications that hold in arbitrary dimensions, they donot always reduce to the best known results in small dimensions. For example, in dimensions 2and 3, positive sectional curvature by itself already implies that the manifold is diffeormorphicto a sphere. This follows from the Gauss–Bonnet theorem, the classification of surfaces, andHamilton’s work on the Ricci flow. Furthermore, in dimension 4, the best result is due toHsiang–Kleiner and Grove–Wilking (see [HK89, GW14], cf. [GGR15, GR15, PP03]), while indimension 5, the best result is due to Rong and Galaz-Garcia–Searle (see [Ron02, GGS14],cf. [Goz15, Sim]). In dimensions 6 and 7, the above results are the best known, however thereremains a large gap in our understanding due to the vast number of known positively curvedexamples (see [Zil07, Dea11, GVZ11, PW]).
Mathematics Subject Classification.
Key words and phrases. positive sectional curvature, torus symmetry, Euler characteristic, elliptic genus,biquotient, Halperin conjecture.
MANUEL AMANN AND LEE KENNARD
This article focuses on even dimensions 8 through 16. Our philosophy is motivated by Des-sai [Des11], which considers positively curved manifolds with torus symmetry in dimension8. The result of Fang and Rong implies that a closed, simply connected, positively curved 8–manifold with T symmetry is homeomorphic to S , C P , or H P , i.e., to one of the manifoldsknown to admit a positively curved metric. Dessai studies precisely this problem, except thathe only assumes T symmetry, and he computes a number of topological invariants, includingthe Euler characteristic χ ( M ) and signature σ ( M ). These computations provide obstructionsto the existence of positively curved Riemannian metrics on 8–manifolds with T symmetry. Theorem (Dessai, 2011) . If M is a closed, simply connected Riemannian manifold withpositive sectional curvature and T symmetry, then one of the following occurs: • χ ( M ) = 2 , σ ( M ) = 0 , and, if M is spin, the elliptic genus vanishes. • χ ( M ) = 3 , σ ( M ) = ± , and, if M is spin, the elliptic genus is constant. • χ ( M ) = 5 , σ ( M ) = ± , and M is not spin. Dessai also restricts his results to well studied classes of non-negatively curved manifolds(e.g., biquotients and certain cohomogeneity one manifolds), and for such manifolds hisresults imply stronger classification (e.g., up to diffeomorphism).The main results of this article extend this work of Dessai into dimensions 10, 12, 14, and16. For example, we prove the following (see Theorem 5.1 for a more detailed statement).
Theorem (Theorem 5.1) . If M is a closed, simply connected Riemannian manifold withpositive sectional curvature and T symmetry, then one of the following occurs: • M is homeomorphic to S . • χ ( M ) = χ ( C P ) and H i ( M ; Z ) ∼ = H i ( C P ; Z ) for i ≤ . We remark that the sphere and complex projective space are the only manifolds knownto admit positive curvature in dimension 10. There is a similar story in dimension 14.
Theorem (Theorem 7.1) . If M is a closed, simply connected, positively curved Riemannianmanifold with T symmetry, then one of the following occurs: • χ ( M ) = 2 and M is –connected. • H ∗ ( M ; Z ) ∼ = H ∗ ( C P ; Z ) , and the cohomology is generated by some z ∈ H ( M ; Z ) and x ∈ H ( M ; Z ) subject to the relation z = mx for some integer m . In dimensions divisible by four, the quaternionic projective spaces arise as examples. Inaddition, dimension 16 is home to the Cayley plane. The sphere, the Cayley plane, H P ,and C P admit positively curved metrics with T symmetry. The next result shows thatany closed, simply connected 16–manifold with positive curvature and T symmetry has thesame Euler characteristic as one of these spaces (2, 3, 5, or 9, respectively). It also providesa sharp calculation of the signature and elliptic genus under these assumptions. Theorem (Theorem 8.1) . If M is a closed, simply connected Riemannian manifold withpositive sectional curvature and T symmetry, then one of the following occurs: • χ ( M ) = 2 and σ ( M ) = 0 . • χ ( M ) = 3 , σ ( M ) = ± , and M is –connected. • χ ( M ) = 5 , σ ( M ) = ± , and H i ( M ; Z ) = 0 for all i . • χ ( M ) = 9 , σ ( M ) = ± , H i ( M ; Z ) ∼ = H i ( C P ; Z ) for i ≤ , and M is not spin.In any case, if M is spin, then the elliptic genus is constant. OSITIVE CURVATURE AND TORUS SYMMETRY, I 3
We now discuss dimension 12. The rank one symmetric spaces S , C P , and H P admitpositively curved metrics with T symmetry, and they are the only spaces up to tangentialhomotopy that admit such metrics by Wilking [Wil03, Theorem 2]. If one requires only T symmetry, there is no classification, and there is one known additional example, the Wallachmanifold W = Sp(3) / Sp(1) .The Euler characteristics of these manifolds are 2, 7, 4, and 6, respectively, and the absolutevalues of their signatures are 0 or 1, according to the parity of χ ( M ). Moreover, the ellipticgenus is constant for each of these manifolds, and C P is not spin while the other three arespin. The next theorem provides a partial recovery of all of these properties. To state it, wedenote by C (6) the maximum Euler characteristic achieved by a closed, simply connected,positively curved 6–manifold with T symmetry. Note that C (6) < ∞ by Gromov’s Bettinumber estimate (see [Gro81]). In fact, it is not difficult to see that C (6) ∈ { , , . . . , } (see Lemma 6.1). Theorem (Theorem 6.2) . Let M be a closed, simply connected, positively curved Rie-mannian manifold with T symmetry. Either χ ( M ) ∈ { , , , . . . , C (6) } or M is not spinand ≤ χ ( M ) ≤ C (6) . Moreover,(1) if C (6) = 6 , then χ ( M ) ∈ { , , } or M is not spin and χ ( M ) ∈ { , , } .(2) if M is rationally elliptic, then χ ( M ) ∈ { , , , , , , , } .Regarding the signature and elliptic genus, the following hold:(1) If χ ( M ) ≤ , then | σ ( M ) | ∈ { , } according to the parity of χ ( M ) .(2) If M is spin, then the elliptic genus is constant. In light of the known examples in dimension 12, this result is sharp in the spin case underthe additional assumption that C (6) = 6, which coincides with the Euler characteristics ofthe Wallach manifold SU(3) /T and the Eschenburg biquotient SU(3) //T . No compact 6–manifold with larger Euler characteristic is known to admit a positively curved metric with T symmetry.In each theorem above, if the torus T r is replaced by T r +1 , there are stronger classificationsdue to Fang–Rong [FR05] and Wilking [Wil03]. In particular, the Cayley plane and theWallach 12–manifold are excluded. If instead the T r is replaced by T r − , then little is knownbeyond the positivity of the Euler characteristic and the vanishing of some higher ˆ A genera(see Rong–Su [RS05] and Dessai [Des05, Des07]).For the proofs of these results, we build upon on a great deal of previous work. One majorsource of ideas is Wilking [Wil03], where the following are developed:(1) the connectedness lemma and the resulting periodicity in cohomology,(2) techniques for studying the fixed-point sets of involutions,(3) spherical recognition theorems, and d(4) the classification of maximal rank, smooth torus actions on H P n .In the first three of these cases, refinements have been made or consequences have beendeduced. In particular, we rely on the refinements of (1) and (2) in our previous work (see[Ken13, AK14, AK]). In this article, we add to this a simple but useful result related to(2) which we call the containment lemma (see Lemma 2.1). Also useful for our purposes isLemma 3.3, which is proven here using (3).A second important source of ideas and motivation is Grove–Searle [GS94]. First, we applyin a crucial way the equivariant diffeomorphism classification of positively curved manifoldswith maximal symmetry rank. We apply this to prove Lemma 6.7, which is crucial to the MANUEL AMANN AND LEE KENNARD proof of our result in dimension 12. In addition, we prove a partial generalization of theirclassification of fixed-point homogeneous circle actions, which we call the codimension twolemma (see Lemma 4.1).A third important source of ideas, and the main motivation for this article, is Dessai[Des11]. There is a fair amount of work required to understand the global picture of thefixed-point set data. Our basic strategy is the same: When there are not enough fixed-pointsets of small codimension to classify the homotopy type using the connectedness lemma, oneobtains isotropy rigidity at fixed-point sets of the torus action. For the reader hoping to getjust a passing idea of how this strategy works, we recommend the proof of the dimension14 case, as it involves the smallest number of special cases and other hiccups. For the moreinterested reader, the proof of the dimension 12 case is by far the most involved but also,we believe, the most interesting. In particular, there is a significant amount of combinatorialanalysis required in this dimension that comes out of the isotropy rigidity.Fourth, regarding the elliptic genus calculations, see Section 9. These calculations aremotivated by a question of Dessai [Des05, Des07] and work of Weisskopf [Wei].All manifolds studied in this article are shown to have positive Euler characteristic. In ad-dition, they are positively curved, so a conjecture of Bott–Grove–Halperin suggests that theyare rationally elliptic (see Grove [Gro02]). Putting these properties together, it is suspectedthat the class of manifolds studied in this paper is closely related to the class of positivelyrationally elliptic , or F , spaces. Motivated by this, we compute the homotopy groups of F spaces of formal dimension at most 16 (see Section 10). The resulting tables are used tostudy the four questions we discuss next.First, we specialize the results above to rationally elliptic spaces and derive rational ho-motopy classifications in dimensions 10, 14, and 16. We also provide a partial classificationof this kind in dimension 12. For this, the Euler characteristic calculations are helpful butnot sufficient. One must also apply the conclusions above regarding the product structure incohomology. See Section 12.Second, we specialize further to biquotients, a large class of manifolds that contains all ho-mogeneous spaces and that provides a source of numerous examples of manifolds admittingpositive sectional curvature, as well as weaker notions such as positive curvature almost ev-erywhere (see, for example, [Zil07], DeVito [DeV14, DDRW14, DeVa], Kerin [Ker11, Ker12],Kerr–Tapp [KT14], and Wilking [Wil02]). In our context, our results in dimensions 10, 14,and 16 imply diffeomorphism classifications when restricted to the case of biquotients. Corollary (Theorem 13.1) . Let M n be a closed, simply connected biquotient that indepen-dently admits a positively curved Riemannian metric with T r symmetry. • If n = 10 and r ≥ , then M is diffeomorphic to S , C P , S ˜ × H P , SO(7) / (SO(5) × SO(2)) , or ∆SO(2) \ SO(7) / SO(5) . • If n = 14 and r ≥ , then M is diffeomorphic to S , C P , S ˜ × H P , SO(9) / (SO(7) × SO(2)) , or ∆SO(2) \ SO(9) / SO(7) . • If n = 16 and r ≥ , then M is diffeomorphic to S , C P , H P , or CaP .Here S ˜ × H P m denotes one of the two diffeomorphism types of total spaces of H P m –bundlesover S whose structure group reduces to circle acting linearly. This corollary is proved in two steps. First, we apply the rational ellipticity of biquotientsto derive the rational homotopy type. We then apply a diffeomorphism classification ofKapovitch and Ziller [KZ04], together with a recent generalization due to DeVito [DeVb],
OSITIVE CURVATURE AND TORUS SYMMETRY, I 5 to conclude the result, up to a small number of other possibilities. To exclude these, wereturn to what our calculations imply about the integral cohomology, and this is sufficientto complete the classification.We remark that the 16-dimensional case of this corollary provides a diffeomorphism char-acterization of the Cayley plane among biquotients admitting positively curved metrics with T symmetry. To our knowledge, previous results along these lines have either had too stronga symmetry assumption to allow the Cayley plane or too weak an assumption to detect it.Third, we specialize even further in dimension 12 to the class of symmetric spaces, andhere we obtain a diffeomorphism classification (see Theorem 13.3).Finally, we study the conjecture of Halperin that any fibration E → B of simply connectedspaces where the fiber F is an F space has the property that H ∗ ( E ; Q ) ∼ = H ∗ ( B ; Q ) ⊗ H ∗ ( F ; Q ) as H ∗ ( B ; Q )–modules. Theorem (Theorem 11.6) . The Halperin conjecture holds for F spaces of formal dimensionat most or Euler characteristic at most . For further discussion of F spaces and Halperin’s conjecture, cases in which this conjectureis known to hold, and the proof of this theorem, see Sections 10 and 11. Acknowledgements.
The second author was supported by National Science FoundationGrant DMS-1404670/DMS-1622541. The authors would also like to acknowledge the supportof NSF Grant DMS-1440140 while they were in residence at the MSRI in Spring 2016.
Contents
Introduction 1Acknowledgements 51. Preliminaries 52. Containment lemma 73. Recognition theorems for spheres 84. Low codimension lemmas 95. Dimension 10 126. Dimension 12 157. Dimension 14 228. Dimension 16 249. Elliptic genus calculation 2810. Low dimensional positively elliptic spaces 2911. The Halperin conjecture 3412. Positive curvature and rational ellipticity 3913. Biquotients 42References 451.
Preliminaries
There are a large number of relatively old results from the theory of transformation groupsand newer tools developed over the past two decades that have grown out of work on theGrove program. We attempt to efficiently summarize those which we use.
MANUEL AMANN AND LEE KENNARD
Let M be an even-dimensional, closed Riemannian manifold with positive sectional curva-ture, and let T denote a torus that acts isometrically on M . A theorem of Berger states thatthe fixed-point set M T = { x ∈ M | g ( x ) = x for all g ∈ T } of the torus is non-empty. Ingeneral, for g ∈ T , each component N of the fixed-point set M g = { x ∈ M | g ( x ) = x } is aclosed, even-dimensional, totally geodesic (hence positively curved) embedded submanifoldon which T acts, possibly ineffectively. Applying Berger’s theorem to the induced T –actionon N , we see that every fixed-point component of every isometry in T contains a fixed pointof T . We use this fact frequently.Another issue is proving that components N ⊆ M g are orientable. If N is a fixed-pointcomponent of some subgroup H ⊆ T not equal to Z , then N inherits orientability from M . Also, if N has dimension greater than dim M , then Wilking’s connectedness lemma(see below) implies that N is simply connected. In some cases, other arguments are used.For example, if N ⊆ M Z is a component and Z ⊆ S ⊆ T , and if Q ⊆ N is a fixed-pointcomponent of S inside N such that dim( Q ) ≥ dim( N ), then Q is orientable, and hencesimply connected, and so N is simply connected, and hence orientable, by the connectednesslemma.Next, we recall some results from Smith theory. The first is due to Conner and Kobayashi(see [Con57, Kob58]): The Euler characteristic of M and its fixed-point set M T coincide. Thesame holds for the signature, i.e., σ ( M ) = σ ( M T ) (see [Des11, Theorem 2.4]). For this latterresult, one has to take care how one assigns orientations to the components of M T , howeverfor our purposes this will not matter since we will always show | σ ( M ) | ≤ | σ ( M T ) | ≤ P | σ ( F ) | ≤
1, where the sum runs over components F ⊆ M T . Another result dueto Conner is that sum of the odd Betti numbers of M T is at most that of M , and likewisefor the even Betti numbers.Regarding the Euler characteristic, we use frequently the following inclusion-exclusionproperty. If M T is contained in the union N ∪ N where N and N admit induced T –actions, then M T = ( N ∪ N ) T . Applying the property above together with the Mayer–Vietoris sequence, we conclude χ ( M ) = χ ( N ) + χ ( N ) − χ ( N ∩ N ) . There are obvious extensions of this formula in the case of three or more submanifolds.To close this section on preliminaries, we discuss a collection of results related to Wilking’sconnectedness lemma. We recall these at many points in the paper, so for ease of referencewithin this paper, we give them names.
Theorem 1.1 (Frankel, [Fra61]) . Let M n be a closed Riemannian manifold with positivesectional curvature. If N , N ⊆ M are closed, totally geodesic, embedded submanifolds suchthat cod( N ) + cod( N ) ≤ n , then N and N non-trivially intersect. Throughout this paper, cod( N ) denotes the codimension of a submanifold N ⊆ M . Theorem 1.2 (Connectedness lemma, [Wil03]) . Let M n be a closed Riemannian manifoldwith positive sectional curvature.(1) If N n − k → M is a closed, totally geodesic, embedded submanifold, then the inclusionis ( n − k + 1) –connected.(2) If N n − k → M is as above, and if N is a fixed point component of an isometric actionby a Lie group G , then the inclusion is ( n − k + 1 + δ ) –connected, where δ is thedimension of the principal orbits of G . OSITIVE CURVATURE AND TORUS SYMMETRY, I 7 (3) If N n − k i i → M are closed, totally geodesic, embedded submanifolds with k ≤ k , thenthe inclusion N ∩ N → N is ( n − k − k ) –connected. Note that, for our purposes, if M is a closed, positively curved manifold and if H actsisometrically on M , then every component of M H is a closed, embedded, totally geodesicsubmanifold.When the codimensions in the connectedness lemma are sufficiently small, there are strongcohomological consequences. The following is a corollary of the connectedness lemma togetherwith [Wil03, Lemma 2.2]. Corollary 1.3 (Wilking’s periodicity corollary) . Let M n be a closed, simply connected,positively curved Riemannian manifold.(1) If N n − k → M is a closed, totally geodesic, embedded submanifold, it follows that H k − ≤∗≤ n − k +1 ( M ; Z ) is k –periodic.(2) If N n − k → M is as above, and if N is a fixed point component of an isometricaction by a Lie group G , then H k − − δ ≤∗≤ n − k +1+ δ ( M ; Z ) is k –periodic, where δ is thedimension of the principal orbits of G .(3) If N n − k i i → M are closed, totally geodesic, embedded submanifolds with k ≤ k , andif the intersection N ∩ N is transverse, then H ∗ ( N ; Z ) is k –periodic. Throughout this paper, we say that H m ≤∗≤ n − m ( M ; Z ) is k –periodic if there exists x ∈ H k ( M ; Z ) such that the multiplication map H i ( M ; Z ) → H i + k ( M ; Z ) induced by x is asurjection from H m ( M ; Z ), an injection into H n − m ( M ; Z ), and an isomorphism everywherein between. When m = 0, we say simply that H ∗ ( M ; Z ) is k –periodic.In [Ken13], refinements to this corollary are proved by applying Steenrod powers to k –periodic cohomology rings. For this paper, we only require the following result: Lemma 1.4 (Classification of 4–periodic cohomology) . Let M n be a simply connected closedmanifold of dimension n ≥ such that H ∗ ( M ; Z ) is –periodic.(1) If n ≡ , then M is a cohomology S n , C P n , or H P n .(2) If n ≡ , then either M is a mod cohomology sphere or M is a homology C P n with cohomology generated by some z ∈ H ( M ; Z ) and x ∈ H ( M ; Z ) such that z = mx for some m ∈ Z .Proof. If n ≡ n ≡ M is a mod 2 cohomology S n , C P n , or S × H P n − (see [Ken13, Section 6]). In particular, H ( M ; Z ) = H ( M ; Z ) =0, and so H ( M ; Z ) is isomorphic to Z k +1 for some integer k or to Z by the universalcoefficients theorem and the definition of periodicity. Applying the definition of periodicityand Poincar´e duality, these two cases correspond to and imply the two possibilities claimedin the lemma. (cid:3) Containment lemma
The first result applies Frankel’s theorem to provide a sufficient condition for the fixed-point set M T of the torus action to be contained in a small number of fixed-point componentsof involutions. We will use this lemma many times, so we will call it the containment lemma . MANUEL AMANN AND LEE KENNARD
Lemma 2.1 (Containment lemma) . Suppose M n is a closed, positively curved Riemannianmanifold, and assume T is a torus acting isometrically on M . Fix x ∈ M T . Let H ⊆ T denote a subgroup isomorphic to Z r . Let δ = 4 if n ≡ and M is spin, and otherwiselet δ = 2 if n is even and δ = 1 if n is odd. If cod (cid:0) M Hx (cid:1) + cod (cid:0) M Hy (cid:1) < (cid:18) r − r − (cid:19) ( n + δ ) , then y ∈ S ι ∈ H \{ } M ιx . Note that every component of M H has codimension at most n . Hence, the containmentlemma implies that, if cod (cid:0) M Hx (cid:1) < n + 2 δ − n + δ r − , then M T ⊆ S M ιx , where the union runs over non-trivial elements ι ∈ H . For example, when M has even dimension, this assumption is satisfied in each of the following two cases: • H ∼ = Z and cod (cid:0) M Hx (cid:1) ≤ n + 2. • H ∼ = Z and cod (cid:0) M Hx (cid:1) ≤ n + 3.We will use this consequence frequently. We remark that, if we take r = 1 in this statement,this is just a restatement of Frankel’s theorem for the case of components of the fixed-pointset of some copy of Z in T . Proof.
Let y ∈ M T and ι ∈ H . If y M ιx then the components M ιx and M ιy are disjoint, soFrankel’s theorem implies cod (cid:0) M ιy (cid:1) + cod ( M ιx ) > n. If, moreover, M is spin and n ≡ n , hence it is at least n + 4. In any case, the left-hand side is at least n + δ ,where δ is defined as in the theorem. The lemma follows by summing these inequalities over ι ∈ H and using the fact that P ι ∈ H cod M ιx = 2 r − cod (cid:0) M Hx (cid:1) since H is some copy of Z r inside T (see Borel [Bor60]). (cid:3) Recognition theorems for spheres
It is a basic result from Smith theory that a smooth action by Z p on a mod p homologysphere has fixed-point set a mod p homology sphere (see Bredon [Bre72, Chapter III, Theorem5.1]). Wilking proved two results which can be viewed as partial converses to this theorem.The first provides a sufficient condition for recognizing when a manifold is homeomorphic toa sphere (see [Wil03, Theorem 4.1]). Theorem 3.1 (Recognition theorem, even-dimensional case) . Let M m be a compact ( m − k ) –connected manifold. Suppose that T k − acts smoothly and effectively on M with non-emptyfixed-point set. Assume that every σ ∈ T k − of prime order p has the property that itsfixed-point set is either empty or a mod p homology sphere. Then M is homeomorphic to asphere. The second provides a partial recognition result for mod p homology spheres (see [Wil03,Theorem 5.1]): OSITIVE CURVATURE AND TORUS SYMMETRY, I 9
Theorem 3.2.
Let M n be a closed, simply connected, positively curved Riemannian man-ifold, and let p be a prime. If Z p acts isometrically on M with connected fixed-point set ofcodimension k , then H i ( M ; Z p ) = 0 for k ≤ i ≤ n − k + 1 . If, moreover, M Z p is fixed by acircle acting isometrically on M , then H k − ( M ; Z p ) = 0 and H n − k +2 ( M ; Z p ) = 0 as well.Proof. The first statement is just [Wil03, Theorem 5.1], which is based on the fact that M Z p → M is ( n − k + 1)–connected according to the connecteness lemma. When M Z p is fixed by a circle as in the second assertion, the connectedness lemma implies that theinclusion M Z p → M is ( n − k + 2)–connected, and the proof of [Wil03, Theorem 5.1] implythat H k − ( M ; Z p ) = 0 and H n − k +2 ( M ; Z p ) = 0 as well. (cid:3) Using Theorem 3.2, we prove the following, which can be viewed as another recognitiontheorem for the sphere.
Lemma 3.3.
Let M n be a closed, simply connected, positively curved Riemannian manifoldwith n = 2 m ≥ . Assume there exists a circle S acting isometrically on M such that M S has a component N of codimension four. If b ( M ) = 0 and χ ( M ) = χ ( N ) , then M ishomeomorphic to a sphere. Note, in particular, that b ( M ) = 0 and χ ( M ) = χ ( N ) = 2 if M is a rational homologysphere. Proof.
Let Z p ⊆ S . If M Z p has codimension two, then the result follows from Wilking’speriodicity corollary. Suppose that M Z p has codimension four. By Frankel’s theorem, N isthe only component of codimension four and all others are points or positively curved 2–manifolds. Since χ ( M ) = χ (cid:0) M Z p (cid:1) , it follows that M Z p is connected. By Theorem 3.2 itfollows that H i ( M ; Z p ) = 0 for all 3 ≤ i ≤ n −
6. For n ≥
12, it follows immediately fromPoincar´e duality and the assumption that b ( M ) = 0 that M is an integral homology sphere.For n = 10, the same holds by a similar argument using the additional observation that b ( M ) = χ ( M ) − (cid:3) Low codimension lemmas
It follows immediately from Wilking’s periodicity corollary that a positively curved, ori-ented, closed manifold of odd dimension contains a codimension two, totally geodesic sub-manifold only if it is homeomorphic to S n . It is an open question whether an analogousresult holds in the even dimensional case. By Grove and Searle’s result, if the codimensiontwo submanifold is fixed by an isometric circle action, then M is in fact diffeomorphic to S n or C P n/ . For our purposes, we will use the following, different partial result in this direction. Lemma 4.1 (Codimension two lemma) . Let M n be a closed, simply connected, positivelycurved Riemannian manifold with T symmetry. If some involution ι ∈ T has fixed-pointset of codimension two, then M is homeomorphic to S n or C P n/ .Proof. Let N ⊆ M ι be a component of codimension two. By Wilking’s periodicity corollary,it follows that M is an cohomology S n or C P n/ if b ( M ) ≤
1. In the first of these cases, thehomeorphism classification follows by Perelman’s resolution of the Poincar´e conjecture. Inthe second case, the homeomorphism classification follows from Lemma 3.6 in Fang–Rong[FR05]. Hence it suffices to show that b ( M ) ≤
1. We may assume that n ≥
8, since otherwisethis holds by the theorem of Hsiang and Kleiner and the connectedness lemma.
Suppose first that an involution ι ′ ∈ T \ h ι i exists whose fixed-point set contains acomponent N ′ has codimension at most n − . The intersection is transverse, so Wilking’speriodicity corollary implies that N ′ is 2–periodic. By the connectedness lemma, b ( M ) ≤ b ( N ′ ) ≤
1, as required.Suppose now that cod( M ι ′ ) ≥ n for all involutions ι ′ ∈ T \ h ι i . Consider the isotropyrepresentation at any point x ∈ M T \ N . The codimensions of M ι ′ x and M ιι ′ x sum to n and hence equal n . Fixing any such involution ι ′ and denoting its component of maximaldimension by N ′ , it follows by Frankel’s theorem that M T ⊆ N ∪ N ′ . Applying the inclusion-exclusion property to this containment, we have χ ( M ) − χ ( N ) = χ ( N ′ ) − χ ( N ∩ N ′ ) . By Wilking’s periodicity corollary, Parts (1) and (3), the left-hand side equals b ( M ), theright-hand side equals b ( N ′ ), and b ( N ′ ) ≤
1. This completes the proof. (cid:3)
Using the codimension two lemma, we prove a similar lemma for the case of codimensionfour fixed-point sets.
Lemma 4.2 (Codimension four lemma, part 1) . Let M n be a closed, simply connected,positively curved Riemannian manifold with even dimension n ≥ and T r symmetry with r ≥ . Assume there exists x ∈ M and involutions ι , ι ∈ T r such that cod ( M ι x ) = 4 and cod ( M ι x ) < n . One of the following occurs:(1) M is a cohomology S n , C P n , or H P n , or M is a homology C P m +1 with cohomologygenerated by some z ∈ H ( M ; Z ) and x ∈ H ( M ; Z ) such that z = mx for some m ∈ Z .(2) n = 4 m + 2 , M ∼ Z S n , and T r acts almost effectively on M ι x .(3) n = 4 m + 2 , cod( M ι x ) = n − and T r acts almost effectively on M ι x or M ι x .Proof. Let N ⊆ M ι denote the component of codimension four, and let N ⊆ M ι denotethe component of codimension k ≤ n −
1. Also let N ⊆ M ι ι be the component containing N ∩ N , which is connected by the connectedness lemma. We may assume that N hasmaximal dimension among all such choices, so it admits T symmetry. As a result, N has 4–periodic cohomology by the connectedness lemma (if N ∩ N is transverse) or thecodimension two lemma (if not).First assume k ≤ n −
2. We claim that b ( M ) = b ( M ). Since N has 4–periodic coho-mology, H ( M ; Z ) = 0 and H ( N ; Z ) = 0 by Lemma 1.4. Since the inclusion N → M is 5–connected, H ( M ; Z ) = 0 and H ( M ; Z ) = 0 as well. By the periodicity corollary, H i +1 ( M ; Z ) = 0 for all i and(4.1) χ ( M ) − χ ( N ) = b ( M ) + b ( M ) . By the connectedness lemma, N , N , and the common 2–fold intersection M h ι ,ι i x of anythree of submanifolds N , N , and N are 4–periodic with common values of b and b .Moreover, the M T r is contained in N ∪ N ∪ N by the containment lemma, so the inclusion-exclusion property of Euler characteristics implies(4.2) χ ( M ) − χ ( N ) = b ( M ) + b ( M ) . Indeed, if N ∩ N is transverse, then N = N ∩ N by the connectedness lemma and χ ( N ) − χ ( N ∩ N ) = b ( N ) + b ( N ) by 4–periodicity. Similarly, if N ∩ N is not transverse,then N , N , and N ∩ N are actually 2–periodic and the right-hand side equals b ( N ) + OSITIVE CURVATURE AND TORUS SYMMETRY, I 11 b ( N ) = b ( N ) + b ( N ). In both cases, the equality holds by the connectedness lemma.Comparing Equations 4.1 and 4.2, we conclude the claim that b ( M ) = b ( M ). Given thisclaim, it follows from Lemma 1.4, the connectedness lemma, the naturality of cup products,and the periodicity corollary that Conclusion (1) holds or that M is a mod 2 homologysphere. Moreover, we may refine the latter conclusion as follows: • If some circle in T r fixes N , then M is homeomorphic to S n by Lemma 3.3 andConclusion (1) holds. • If n ≡ H ( M ; Z ) ∼ = H ( M ; Z ), which vanishes since it injects into H ( N ; Z ) = 0. Hence H ( M ; Z ) is torsion-free in this case, N and hence M are integral homology spheres,and Conclusion (1) holds. • If neither of these cases occurs, then T r acts almost effectively on N , the dimension n ≡ k = n −
1. Note that n ≡ k is even. Moreover,assume that both dk( N i ) ≥ N has4–periodic cohomology and the inclusion N → M is 4–connected. We may assume furtherthat b ( M ) ≥
1. Indeed, if b ( M ) = 0, then the periodicity corollary implies b ( M ) = 0 andhence that M is a rational homology sphere. By Lemma 3.3, M is homeomorphic to S n andConclusion (1) holds.Lemma 1.4 implies that N is a cohomology C P n +24 or that n ≡ H ∗ ( N ; Z )is generated by some z ∈ H ( N ; Z ) and x ∈ H ( N ; Z ) such that z = mx for some m ∈ Z .Since N → M is 4–connected, it follows by Poincar´e duality that M is a cohomology C P n ,so we may assume the cohomology of N is as in the latter case. We have that H ( M ; Z ) = Z , H ( M ; Z ) = 0, and H ( M ; Z ) is the image of H ( N ; Z ) under the map induced by inclusion.In particular, the induced injection H ( M ; Z ) → H ( N ; Z ) is either an isomorphism or thezero map. If it is an isomorphism, the naturality of cup products implies that H ∗ ( M ; Z )is generated by elements in degree two and four as is the case for N and Conclusion (1)holds. If, instead, it is the zero map, then H ( M ; Z ) = 0 and hence M ∼ Z S × S n − and N ∼ Z S × S n − by the periodicity corollary and the connectedness lemma. In particular, χ ( M ) = χ ( N ), so M ι is connected by Frankel’s theorem. By the connectedness lemmaagain, the action of ι on N has connected fixed point set ( N ) ι = M ι ∩ N = N ∩ N .If dim( N ) ≥
12, this implies H ( N ; Z ) = 0 by Lemma 3.2, a contradiction, so we mayassume dim( N ) ≤
10, which is equivalent to n ≤
18. Since n ≡ n ≥
12 byassumption, we have that n = 18 and dim( N ) = 10. Note that χ ( M ) = 4 and χ ( N ) = 6,so there exists a component M ι y ⊆ M ι with negative Euler characteristic. By Frankel’stheorem, dim( M ι y ) ≤
6. But χ ( M ι y ) > M ι y has dimension 0, 2, or 4, or if M ι y admits anisometric circle action, so we may assume dim( M ι y ) = 6 and that T r fixes M ι y . Since r ≥ ι ∈ T r \ h ι , ι i such that M ιy strictly contains M ι y . By replacing M ιy by M ι ιy , if necessary, we may assume cod( M ιy ) ≤ cod( M ι y ) = 6. But now M ι x and M ιy intersect by Frankel, and we may proceed as in the case where cod( M ι ) ≤ n −
2. Altogetherone concludes that this case does not actually occur. (cid:3)
In the case where there is no involution ι as in Lemma 4.2, we need a companion lemma. Lemma 4.3 (Codimension four lemma, part 2) . Let M n be a closed, simply connected,positively curved Riemannian manifold with even dimension and T r symmetry such that r ≥ . Assume there exists an involution ι ∈ T r such that cod ( M ι ) = 4 and that everyother involution ι satisfies cod ( M ι ) ≥ n . Either(1) M ι is connected, or(2) r divides n , and every involution ι ∈ T r \ h ι i has cod ( M ι ) = n .Proof of codimension four lemma, part 2. Suppose that M ι is not connected. Using Berger’stheorem, choose y ∈ ( M ι ) T such that y does not lie in the component of M ι of codimensionfour. Frankel’s theorem implies cod (cid:0) M ι y (cid:1) ≥ n −
2. Under the map Z r → Z n/ induced bythe isotropy representation of T r at y , ι (1 , , , . . . , , ∗ ) , where the asterisk is 0 or 1 according to whether cod (cid:0) M ιy (cid:1) is n − n . In the first case,since r ≥
3, we may choose ι ∈ Z r \ h ι i whose image has a 0 in the last entry. It thenfollows that M ι y or M ι ι y has codimension less than n , a contradiction. In the second case,we have that cod (cid:0) M ιy (cid:1) + cod (cid:0) M ιι y (cid:1) = n for all ι ∈ Z r \ h ι i . Under the assumption that cod (cid:0) M ιy (cid:1) ≥ n for all ι ∈ Z r \ h ι i , it followsthat every such codimension equals n . It is not hard to see that there exists a decomposition Z n = V ⊕ . . . ⊕ V r − such that • for all ι ∈ Z r , the projection onto V i of image of ι has all 0s or all 1s, and • The V i have equal dimensions.In particular, n is divisible by 2 r − , which proves that 2 r divides n . (cid:3) Dimension 10
The only two compact, simply connected, smooth 10–manifolds known to admit positivesectional curvature are S and C P . Equipped with the standard metrics, each of thesespaces admits T symmetry. We partially recover a classification of these two spaces underthe assumption of T symmetry. Theorem 5.1.
Let M be a closed, simply connected Riemannian manifold with positivesectional curvature and T symmetry. One of the following occurs: • M is homeomorphic to S . • χ ( M ) = χ ( C P ) , H i ( M ; Z ) = H i ( C P ; Z ) for i ≤ , and H ( M ; Z ) is generated byan element of the form x y with x ∈ H ( M ; Z ) and y ∈ H ( M ; Z ) . • H ∗ ( M ; Z ) ∼ = H ∗ ( C P ; Z ) , and the cohomology is generated by some z ∈ H ( M ; Z ) and x ∈ H ( M ; Z ) subject to the relation z = mx for some m ∈ Z .In any case, H ( M ; Z ) is or Z , H ( M ) = 0 , and χ ( M ) = 2 + 4 b ( M ) ∈ { , } . For context, we remark that, if M is as in the theorem but has T symmetry, work of Fangand Rong implies that M is homeomorphic to S or C P (see [FR05]). On the other hand, itfollows from Rong–Su [RS05] that a T action on M as in the theorem is sufficient to show χ ( M ) ≥ M is rationally elliptic or admits a biquotient structure (see Theorems12.1 and 13.1).We spend the rest of this section on the proof. Denote the torus by T . If some involution ι ∈ T has fixed-point set M ι of codimension two, then M is homeomorphic to S or C P OSITIVE CURVATURE AND TORUS SYMMETRY, I 13 by Lemma 4.1. We assume therefore that cod( M ι ) ≥ ι ∈ T .In this situation, there exist two involutions with fixed point sets of codimension four, so weare in the setting of the codimension four lemma, part 1. However, the arguments there donot suffice in dimension 10. Fortunately, four is large enough relative to the dimension of M in this case to force isotropy rigidity. The main consequence of this rigidity is the following: Lemma 5.2 (Containment lemma for dimension 10) . Assume no involution ι ∈ T has fixed-point set of codimension two. There exist x ∈ M T and independent involutions ι , ι ∈ T such that both cod ( M ι i x ) = 4 and M T ⊆ M ι x ∪ M ι x ∪ M ι ι x . Proof.
Consider the map Z → Z induced by the isotropy representation T → SO( T x M ) atsome fixed point x ∈ M T . We claim that one of the following possibilities occurs. • (Case 1) There exists x ∈ M T and independent involutions ι , ι ∈ T such thatcod ( M ι x ) = cod ( M ι x ) = 4 and M ι x ∩ M ι x is not transverse. • (Case 2) At every x ∈ M T , there exist involutions ι , ι ∈ Z such thatcod ( M ιx ) = ι ∈ { ι , ι } ι
6∈ h ι , ι i ι = ι ι Indeed, suppose Case 1 does not occur, and let x ∈ M T . It is not possible for the image ofevery ι ∈ Z to have weight at least three, so we may choose ι ∈ Z with cod ( M ι x ) = 4.Next, we may choose ι ∈ Z \h ι i such that M ι x ∩ M ι x is transverse. Supposing for a momentthat cod ( M ι x ) = 6, it follows that some ι ∈ Z \ h ι , ι i exists such that M ιx has codimensionfour and intersects M ι x non-transversely. This is a contradiction since we assumed Case 1does not occur. It follows that cod ( M ι x ) = 4. Finally, using again the fact that Case 1 doesnot occur, it follows that every ι ∈ Z \ h ι , ι i has cod ( M ιx ) = 6. This concludes the proofof the claim.If Case 1 occurs, then the lemma follows immediately from the containment lemma. Wesuppose now that Case 2 occurs and proceed by contradiction. If z ∈ M T does not lie in M ι x ∪ M ι x , then cod ( M ι z ) > M ι z ) > (cid:3) We keep the notation N = M ι x and N = M ι x throughout the rest of the proof. Theproof of Theorem 5.1 is finished below in the following two lemmas. Lemma 5.3. If N ∩ N is not transverse, or if N or N admits T symmetry, then • M is homeomorphic to S , or • χ ( M ) = χ ( C P ) , H i ( M ; Z ) = H i ( C P ; Z ) for i ≤ , and H ( M ; Z ) is generated byan element of the form x y with x ∈ H ( M ; Z ) and y ∈ H ( M ; Z ) .Proof. We first calculate H ( M ; Z ), H ( M ; Z ), and χ ( M ). Suppose first that N and N intersect transversely, and assume without loss of generality that N is admits T symmetry.It follows from Grove and Searle’s diffeomorphism classification that N is diffeomorphic to S or C P . By either applying the same reasoning to N or by using the connectedness lemma,it follows that H ( M ; Z ) ∼ = H ( N ; Z ) = 0 and H ( N ; Z ) ∼ = H ( M ; Z ) ∼ = H ( N ; Z ), whichis isomorphic to 0 or Z . Note that N ∩ N → N i → M is 2–connected by the connectednesslemma, so M ι ι x = N ∩ N = S . In particular, M T ⊆ N ∪ N , so χ ( M ) = 2+4 b ( M ) ∈ { , } by the inclusion-exclusion formula for Euler characteristics. Suppose now that N and N do not intersect transversely. By the connectedness lemma,all two-fold intersections of N , N , and N = M ι ι x coincide and equal M h ι ,ι i x . By thecodimension two lemma, all four of these submanifolds have 2–periodic integral cohomology,which means their third homology groups vanish, while their second homology groups coin-cide and equal 0 or Z . By the connectedness lemma, H ( M ; Z ) = 0 and H ( M ; Z ) ∈ { , Z } .Since N ∪ N ∪ N contains M T , the inclusion-exclusion formula for Euler characteristicsimplies χ ( M ) = 2 + 4 b ( M ) ∈ { , } .This completes the calculation of H ( M ; Z ), H ( M ; Z ), and χ ( M ). We now complete theproof. Suppose first that χ ( M ) = 6. Let N be a six-dimensional submanifold of M homotopyequivalent to C P such that the inclusion N → M is 3–connected. By the naturality of cupproducts, it follows that the third power of a generator x ∈ H ( M ; Z ) is non-zero and not anon-trivial multiple. By Poincar´e duality, there exists y ∈ H ( M ; Z ) such that x y generates H ( M ; Z ).Suppose now that χ ( M ) = 2. By the calculations above, it follows that M is 3–connected.Moreover, we claim that the T –action on M has the property that every Z p ∈ T has fixed-point set equal to a mod p homology sphere. Indeed, since Z p ⊆ T , we have χ (cid:0) M Z p (cid:1) = χ ( M ) = 2, so it suffices to show that every component P ⊆ M Z p has vanishing odd Bettinumbers. This clearly holds if dim( P ) ≤ P ) = 8 by Wilk-ing’s periodicity corollary. If dim( P ) = 6, it follows by Grove and Searle’s diffeomorphismclassification if P is not fixed by a circle in T and by the connectedness lemma if it is.The homeomorphism classification now follows from Wilking’s spherical recognition theorem(Theorem 3.1). (cid:3) Lemma 5.4. If N and N are fixed by circles in T and intersect transversely, then • M is homeomorphic to S . • H ∗ ( M ; Z ) ∼ = H ∗ ( C P ; Z ) , and the cohomology is generated by H ( M ; Z ) and H ( M ; Z ) .Proof. Since N and N intersect transversely, the connectedness lemma implies that theinclusions N ∩ N → N i → M are 2–connected. In particular, N ∩ N is diffeomorphic to S and equals the fixed point component M ι ι x . In particular, M T ⊆ N ∪ N by Lemma5.2. Note also that H ( M ; Z ) is either finite or isomorphic to Z since the natural map H ( N ∩ N ; Z ) → H ( M ; Z ) is surjective.First suppose H ( M ; Z ) is finite. By the periodicity corollary, the Betti numbers of M satisfy b ( M ) = b ( M ) = 0 and b ( M ) = b ( M ) = 0. By Poincar´e duality, the Eulercharacteristic of M is even and equals 2 − b ( M ) − b ( M ). But χ ( M ) >
0, so b ( M ) = b ( M ) = 0. Hence M is a rational sphere, and N is as well by the connectedness lemma.Hence M is homeomorphic to S by Lemma 3.3.Now suppose that H ( M ; Z ) ∼ = Z . Since each N j is fixed by a circle in T , the connectednesslemma implies that N j → M is 4–connected. Applying the inclusion-exclusion formula forthe Euler characteristic, we conclude that χ ( M ) = χ ( N ∪ N ) = χ ( N ) + χ ( N ) − χ ( N ∩ N ) = 6 − b ( M ) . Comparing with the alternating sum of Betti numbers formula for χ ( M ), we conclude that2 = 2 b ( M ) − b ( M ). The periodicity corollary and Poincar´e duality imply that b ( M ) ≤ b ( M ) is even, so this equality implies that b ( M ) = 1 and b ( M ) = 0. ApplyingPoincar´e duality and the periodicity corollary again, we conclude that H ( M ; Z ) = 0 andthat x ∈ H ( M ; Z ) and z ∈ H ( M ; Z ) exist such that xz generates H ( M ; Z ), that x z OSITIVE CURVATURE AND TORUS SYMMETRY, I 15 generates H ( M ; Z ), and hence that x generates H ( M ; Z ). Finally, it follows as in theproof of Lemma 1.4 that H ( M ; Z ) = 0, so H ( M ; Z ) = 0 and M is as in the secondconclusion of the lemma. (cid:3) Dimension 12
Let C (6) denote the maximum Euler characteristic achieved by a closed, simply con-nected 6–manifold that admits a Riemannian metric with positive sectional curvature and T symmetry. Note that C (6) < ∞ by Gromov’s Betti number estimate. In fact, we havethe following: Lemma 6.1.
The maximum Euler characteristic C (6) of a closed, simply connected, posi-tively curved manifold with T symmetry satisfies ≤ C (6) ≤ .Proof. The Wallach flag M = SU(3) /T admits a metric with both positive curvature andan isometric T action, so C (6) ≥
6. For the upper bound, let M be a manifold as in thetheorem. Suppose for a moment that some g ∈ T has cod ( M g ) = 2. Let N ⊆ M g denotea four-dimensional component. By the result of Hsiang and Kleiner, N has b ( N ) ≤
1. Bythe connectedness lemma and Wilking’s periodicity corollary, it follows that M is homotopyequivalent to S or C P .Suppose therefore that every g ∈ T has fixed-point set of codimension at least four. Inthis case, the fixed-point set of T is made up of isolated fixed points. Moreover, this propertyimplies that the isotropy representations of T at fixed points are such that each fixed pointprojects to an extremal point in M/T . Since this space is a four-dimensional Alexandrovspace, the number of extremal points is strictly less than 2 (see Lebedeva [Leb15]). Since,on the other hand, the number equals χ ( M ), which is even, the proof is complete. (cid:3) The main result in dimension 12 is presented in terms of the constant C (6). Theorem 6.2.
Let M be a closed, simply connected, positively curved manifold with T symmetry. Either χ ( M ) ∈ { , , . . . , C (6) } or M is not spin and ≤ χ ( M ) ≤ C (6) .Moreover, the following hold.(1) if C (6) = 6 , then χ ( M ) ∈ { , , , , , } .(2) if M is rationally elliptic, then χ ( M ) ∈ { , , , , , , , } .In any case, the following hold for the signature and elliptic genus.(1) if χ ( M ) ≤ , then | σ ( M ) | ∈ { , } according to the parity of χ ( M ) .(2) if M is spin, then the elliptic genus is constant. Recall that 2, 4, 6, and 7 are realized as Euler characteristics of positively curved manifoldswith T symmetry, namely, S , H P , the Wallach flag W , and C P . As for the possibilitiesof χ ( M ) >
7, we note that there are many examples of non-negatively curved manifolds M with isometric T –actions such that χ ( M ) ∈ { , , , } and σ ( M ) ∈ { , } . Indeed, onefinds such examples among compact symmetric spaces of rank two (e.g., the GrassmannianSO(8) / SO(2) × SO(6) or products of rank one spaces such as S − m × C P m or C P × H P )or among certain connected sums of rank one symmetric spaces (e.g., C P H P ) endowedwith Cheeger metrics (see [Che73]).The proof of Theorem 6.2 takes the rest of the section. The bulk of the proof is containedin a sequence of lemmas that together prove the Euler characteristic calculation claimed in Theorem 6.2. The signature calculation is then proved at the end of the section, and theelliptic genus calculation is proved in Section 9.We assume for the rest of the section that M is a 12–dimensional, compact, simply con-nected Riemannian manifold with positive curvature and an effective, isometric T action. Lemma 6.3.
If there exists an involution with fixed-point set of codimension two, or if thereexist two involutions whose fixed-point sets have codimension four, then M has the integralcohomology of S , C P , and H P . This follows immediately from the codimension two and part 1 of the codimension fourlemma (Lemmas 4.1 and 4.2). The next case we consider is the following:
Lemma 6.4.
If the codimension of the fixed-point set is four for one involution but at leastsix for every other, then M is homeomorphic to S .Proof. Note that N = M ι is connected by the codimension four lemma, part 2 (see Lemma4.3), so H i ( M ; Z ) = 0 for 4 ≤ i ≤ N ) = 1, then thistheorem implies H ( M ; Z ) = 0 and H ( M ; Z ) = 0 as well. Since M ι is connected, wehave χ ( N ) = χ ( M ) = 2 + 2 b ( M ). But N admits an isometric T action, so Dessai’s Eulercharacteristic calculation in dimension 8 implies that χ ( N ) ∈ { , , } . Hence b ( M ) = 0and M is a rational sphere. It now follows that M is homeomorphic to S by Lemma 3.3.We may assume therefore that dk( N ) = 0. By Fang and Rong’s homeomorphism classi-fication, N is homeomorphic to S , C P , or H P . By the connectedness lemma, N → M is5–connected, so N is homeomorphic to S and M is 5–connected. It follows that H ( M ; Z )is torsion-free with rank b ( M ) = χ ( M ) − χ ( N ) − M is againhomeomorphic to S . (cid:3) Before continuing with the proof, we remark that it might be surprising that the Eulercharacteristic of C P does not appear in Lemma 6.4. The example below shows that, whilethere exist T –actions on C P that realize the above isotropy data at one fixed point, theyneed not globally realize the isotropy data at all fixed points. Example . Denote points in C P as equivalence classes [ z , z , . . . , z ] where z j ∈ C suchthat P | z j | = 1. Define the actions of three circles on C P by the following three mapsS → PU(7): w diag(1 , w, w, , , , w diag(1 , w, , w, w, , w diag(1 , , , w, , w, w )Note that, at the point x = [1 , , , , , , x of the fixed-point set is four for one involution but is greater than four forall of the others. Also note, however, that the product of the involutions in the first andthird circle factors of T also has a codimension-four fixed-point set, so actually this actiondoes not satisfy the hypotheses of Lemma 6.4.To complete the proof of the Euler characteristic calculation claimed in Theorem 6.2, weneed to consider the case where every involution in the torus acts with fixed-point set ofcodimension at least six.The key aspect of this case is the rigidity of the maps Z → Z induced by the isotropyrepresentations at fixed points of the torus action. More specifically, at each fixed point x , OSITIVE CURVATURE AND TORUS SYMMETRY, I 17 there exists a choice of basis for the tangent space at x and a choice of ρ, σ, τ ∈ Z such thatthe map Z → Z induced by the isotropy representation at x takes the form ρ (1 , , , , , σ (0 , , , , , τ (0 , , , , , M ιx ) = (cid:26) ι ∈ { ρ, σ, τ, ρστ } ι ∈ { ρσ, ρτ, στ } In particular, every x ∈ M T is an isolated fixed point, and we can associate to it a subgroupisomorphic to Z inside Z , the complement of which has the property that every member ι has cod ( M ιx ) = 6. We call these complements “clubs”. In particular, the club C ( x ) at x consists of the four members ρ , σ , τ , and their three-fold product ρστ . In fact, it is animportant property that the product of any three elements of a club is in that club. We callthis the “triple product property” of clubs.We analyze how these clubs overlap at distinct fixed points x, y ∈ M T . One possibility isthat the clubs at x and y coincide. By Frankel’s theorem, this implies that M ιx = M ιy for all ι ∈ C ( x ) = C ( y ). As it turns out, there is only one other possibility, namely, that the clubs C ( x ) and C ( y ) intersect in exactly two members. Indeed, • C ( x ) ∩ C ( y ) contains at least one member, since each club consists of four of theseven non-trivial elements of Z , • if C ( x ) ∩ C ( y ) contains exactly one member, ι , then the product of the three elementsof C ( x ) \ C ( y ) is both equal to ι (by the triple product property) but not in C ( y )(since clubs are complements of subgroups), a contradiction, and • if C ( x ) ∩ C ( y ) contains at least three members, then the three-fold product equalsboth the fourth member of C ( x ) and that of C ( y ) by the triple product propertyapplied to both clubs, hence these clubs coincide in this case.Next, we analyze how clubs at three distinct fixed points x, y, z ∈ M T might over lap.There are two possibilities (up to relabeling the involutions in Z ). The first is(Type I) C ( x ) = { ρ, στ, σ, ρτ } C ( y ) = { ρ, στ, τ, ρσ } C ( z ) = { σ, ρτ, τ, ρσ } The second is (Type II), in which C ( x ) and C ( y ) are exactly the same as above, and C ( z )contains ρστ as well as exactly one involution from each of the sets C ( x ) ∩ C ( y ), C ( x ) \ C ( y ),and C ( y ) \ C ( x ). Note however these choices are neither unique nor arbitrary since C ( z )satisfies the triple product property. We omit the proof, as it follows simply from furtheranalysis using the triple product property of clubs.Using this club analysis, we claim the following. Lemma 6.6 (Club analysis) . One of the following occurs:(1) There exists ι ∈ Z whose fixed-point set is connected, has dimension six, and admitsan effective, isometric T –action.(2) There exist three clubs with Type I intersection data, and M is not spin.Proof. It might happen that some involution ι is in every club. In this case, Frankel’s theoremimplies that M ι has dimension six and is connected. Moreover, by the rigidity of the isotropy representation, M ι is fixed by at most a circle, so in this case the first possibility of theconclusion occurs.We claim such an involution ι exists in each of the following three cases:(1) There exists exactly one club.(2) There exist exactly two clubs.(3) There exist at least three clubs, and every subset of three has intersection data ofType II.Indeed, in each of the first two cases, the club analysis above implies that some involutionis in all clubs. In the last case, we verify this as follows. Suppose C ( x ) = { ι , ι , ι , ι } and C ( y ) = { ι , ι , ι , ι } are two of the clubs. We assume that no involution is in every club andproceed by contradiction. Since the intersection data of every club with C ( x ) and C ( y ) is ofType II, there exist clubs C ( z ) and C ( w ) such that one contains ι but not ι and vice versafor the other club. Next our assumption implies that the clubs C ( y ), C ( z ), and C ( w ) haveintersection data of Type II. A general property of such triples of clubs is that their unioncontains all seven of the non-trivial involutions in Z . In particular, C ( z ) ∪ C ( w ) containsboth ι and ι . By applying the same line of reasoning to the triple of clubs C ( x ), C ( z ), and C ( w ), we conclude that C ( z ) ∪ C ( w ) contains ι and ι . Altogether we have that C ( z ) ∪ C ( w )contains all seven non-trivial involutions in Z . This contradicts the fact that any two clubseither coincide or intersect in exactly two elements. This completes the proof of the claim.Assuming the claim now, there exist at least three (distinct) clubs, and among these thereexist three with intersection data of Type I. We see immediately in this case that M isnot spin. Indeed, fixed-point sets of involutions on spin manifolds have all components ofcodimension congruent to c modulo four, for some c ∈ { , } . In our setting, no involutionis in every club, so any member ι ∈ C ( x ) has fixed point components of codimension 6 and8. (cid:3) We require one more lemma whose proof relies on further club analysis, together with theequivariant diffeomorphism classification of Grove and Searle.
Lemma 6.7.
Assume M is as in Theorem 6.2, and that every involution in the torus T hasfixed-point set of codimension at least six. Suppose ι ∈ T is an involution and P ⊆ M ι is acomponent of dimension four. • If P is S , then the two fixed-points in P T = P ∩ M T have the same club. • If P is C P , then the three fixed points in P T represent three distinct clubs.Proof. Fix ι ∈ Z and a four-dimensional component P ⊆ M ι . By the rigidity of the isotropymaps, the induced T –action on P has one-dimensional kernel that does not contain anyother involution. Let T denote the two-dimensional torus equal to the quotient of T by thekernel of this induced action. The induced action of T on P is equivariant to a linear actionon S or C P .Applying isotropy rigidity again, it follows that, for all p ∈ P T = P ∩ M T , there exist ι and ι such that the isotropy map at p takes the form ι (1 , , , , , ι ( ∗ , ∗ , ∗ , ∗ , , ι ( ∗ , ∗ , ∗ , ∗ , , OSITIVE CURVATURE AND TORUS SYMMETRY, I 19
By the rigidity of the isotropy at p , it follows that the club at p is given by C ( p ) = { ι , ιι , ι , ιι } . To complete the proof, first suppose P is a sphere, and denote the fixed points in P ∩ M T by p and p . Since the T –action on P is equivariant to a linear action on S , it follows that P τp and P τp have the same dimension for all involutions τ in T . In particular, the ι and ι from the previous paragraph are the same for p and p , and hence the clubs coincide atthese points.Now suppose P is C P . We use the fact that the T –action on P is equivariant to a linearaction. In particular, each involution ι ∈ T has the property that P ι consists of a copy of S together with an isolated point. In particular, if p and p are two points in P T with the sameclub, and if ι and ι are the involutions as above such that C ( p ) = C ( p ) = { ι , ιι , ι , ιι } , then the third point, p ∈ P T , is an isolated fixed point of the actions of ι and ι on P . Butthen the product ι ι acts on T p P as the identity and hence fixes P . This contradicts therigidity of the isotropy maps. (cid:3) We now complete the proof of the Euler characteristic calculation claimed in Theorem 6.2.We do this is three steps (see Lemmas 6.8, 6.9, and 6.10).
Lemma 6.8.
If every involution has fixed-point set of codimension at least six, then one ofthe following occurs:(1) χ ( M ) ∈ { , , , . . . , C (6) } , or(2) ≤ χ ( M ) ≤ C (6) and M is not spin.Proof. If there exists ι ∈ Z with connected fixed-point set of dimension six, then we have χ ( M ) = χ ( M ι ) ∈ { , , , . . . , C (6) } by Lemma 6.1. Suppose then, as in Lemma 6.6, thatdistinct clubs, C ( x ), C ( y ), and C ( z ), exist and have Type I intersection data. We recall thenotation from the proof of that lemma. Furthermore, we denote the non-trivial involutionsby ι = ρστ , ι = ρ , ι = σ , ι = τ , and ι i = ι ι i − for 2 i ∈ { , , } . In particular, C ( x ) ∩ C ( y ) = { ι , ι } , C ( x ) ∩ C ( z ) = { ι , ι } , C ( y ) ∩ C ( z ) = { ι , ι } , and ι is not in any of these clubs. Let N , . . . , N denote the (unique, by Frankel) componentsof dimension six of the fixed-point sets of ι , . . . , ι , respectively. If ι is in some club, let N denote the six-dimensional component of its fixed-point set; otherwise, let N denote theempty set.Let X ⊆ M T denote the set of fixed points whose club equals C ( x ). Define Y and Z similarly, and let W denote the fixed points of T whose club is distinct from C ( x ), C ( y ),and C ( z ). Note the following facts: • X , Y , Z , and W partition the fixed-point set M T . • N i − ∩ W and N i ∩ W partition W for all i ∈ { , , } . Indeed, if w ∈ N i , then ι i ∈ C ( w ) and hence ι i − = ι ι i C ( w ). Moreover, if w N i ∪ N i − , then ι i and ι i − are not in C ( w ), and hence ι = ι i − ι i C ( w ), a contradiction. • Each w ∈ W lies in exactly one or all three of N , N or N . Indeed, this follows fromthe previous fact together with the triple product property and the fact that ι ι ι isthe identity.We denote χ ( X ) by | X | and similarly for the orders of Y , Z , and W . Observe that w ∈ W if and only if ι ∈ C ( w ), hence W = ( N ) T and | W | = χ ( N ) . (6.1) Next, note that X ∪ Y = ( N ∩ N ) T and that, similarly, X ∪ Z = ( N ∩ N ) T and Y ∪ Z =( N ∩ N ) T . Hence | X | + | Y | = χ ( N ∩ N ) , (6.2) | X | + | Z | = χ ( N ∩ N ) , (6.3) | Y | + | Z | = χ ( N ∩ N ) . (6.4)Next, note that X ∪ Y ∪ W = ( N ∪ N ) T and that we have similar statements for N ∪ N and N ∪ N . Hence | X | + | Y | + | W | = χ ( N ) + χ ( N ) − χ ( N ∩ N ) , (6.5) | X | + | Z | + | W | = χ ( N ) + χ ( N ) − χ ( N ∩ N ) , (6.6) | Y | + | Z | + | W | = χ ( N ) + χ ( N ) − χ ( N ∩ N ) . (6.7)Adding together Equations (6.1)–(6.7), we conclude4 χ ( M ) = X j =0 χ ( N i ) . (6.8)We first use this equation to bound χ ( M ) from below. Since no ι j is in every club, every M ι j is composed of the six-manifold N j together with a disjoint union of 4–manifolds, eachof which has Euler characteristic at least two. Hence χ ( M ) ≥ χ ( N j ) + 2 for all 1 ≤ j ≤
6. Inaddition, M ι contains N as well as X , Y , and Z , so χ ( M ) ≥ χ ( N ) + 3. Hence Equation(6.8) implies 4 χ ( M ) ≤ ( χ ( M ) −
3) + X j =1 ( χ ( M ) − , which implies that χ ( M ) ≥
5. In fact, χ ( M ) = 5 would imply that χ ( N j ) ≤ ≤ j ≤
6. But χ ( N j ) is both positive and even, so χ ( N j ) = 2 for all 1 ≤ j ≤
6. Since χ ( N ) ≤ χ ( M ) − χ ( M ) ≤ χ ( M ) = 6 also cannot occur. Indeed, suppose χ ( M ) = 6. Then χ ( N ) ≤
3, so χ ( N ) = 2.But now χ ( M ι \ N ) = 4, a contradiction to Lemma 6.7, which implies that these four pointsmust come in two pairs such that, in each pair, the two clubs are the same. But these fourpoints together represent three clubs (their union is X ∪ Y ∪ Z ), a contradiction.To complete the proof of the lemma, it suffices to prove χ ( M ) ≤ C (6). By the rigidity ofthe isotropy maps, N i admits an effective, isometric T –action and hence has χ ( N i ) ≤ C (6)for all 1 ≤ i ≤
6. The same estimate on χ ( N ) holds, even if it is empty. The upper boundnow follows from Equation 6.8. (cid:3) Lemma 6.9.
Let M be as in Lemma 6.8. If C (6) = 6 , then χ ( M ) ≤ .Proof. We keep the notation from above, but now we assume χ ( N i ) ≤ ≤ i ≤ χ ( M ) ≤
42, and hence χ ( M ) ≤ χ ( M ) = 10. Equation (6.8) implies χ ( N ) ≥
4. Moreover, if χ ( N ) = 6, thenwe have | X | = | Y | = 1 and | Z | = 2 without loss of generality, and one can show that x and y can be replaced by suitable choices of w and w in W so that the three clubs C ( z ), C ( w ),and C ( w ) are distinct, have intersection data of Type I, and have the property that theinvolution not in C ( z ) ∪ C ( w ) ∪ C ( w ) has maximal component of Euler characteristic four.In other words, we may assume without loss of generality that χ ( N ) = 4. By Equation (6.8), OSITIVE CURVATURE AND TORUS SYMMETRY, I 21 χ ( N j ) = 6 for all 1 ≤ j ≤
6. By Equations (6.3)–(6.7), it follows that | X | = | Y | = | Z | = 2.Write W = { w , w , w , w } . Up to relabeling the w i , we have that N T = X ∪ Y ∪ { w , w } . Now every w ∈ W lies in exactly one or three of N , N , and N . Without loss of generality,this implies that N T = X ∪ Z ∪ { w , w } ,N T = Y ∪ Z ∪ { w , w } . In particular, ι is in the club C ( w ) but not C ( w ), so C ( w ) = C ( w ).On the other hand, consider the fixed-point set M ι . One component is N , which hasEuler characteristic six. The others are closed, oriented, positively curved 4–manifolds. ByLemma 6.7, we have that M ι = N ∪ P ∪ Q where P and Q are diffeomorphic to S , andwhere the two fixed-points in P have equal clubs and likewise for the fixed points of Q . Inparticular, the four points in ( P ∪ Q ) T represent only two clubs. But ( P ∪ Q ) T contains thetwo points in Z , as well as w and w . These points represent three distinct clubs, so we havethe desired contradiction. (cid:3) To complete the proof of the Euler characteristic calculation claimed in Theorem 6.2, itsuffices to prove the following.
Lemma 6.10.
Let M be as in Lemma 6.8. If M is rationally elliptic, then χ ( M ) ≤ or χ ( M ) = 12 .Proof. We keep the notation from the proof of Lemma 6.8. Consider one of the submanifolds N i in Equation 6.8. If N i admits an effective, isometric T action, then χ ( N i ) ∈ { , } byGrove and Searle’s classification. Otherwise, N i is a fixed point component of a circle actionon M . Since M is rationally elliptic, it follows that N i is rationally elliptic. Since N i is asimply connected, closed manifold, it follows that χ ( N i ) ≤ χ ( S × S × S ) = 8. Since thisestimate holds for all 0 ≤ i ≤
6, Equation 6.8 implies that χ ( M ) ≤ (cid:3) This completes the proof of the Euler characteristic calculation. For the elliptic genus, seeSection 9. It suffices to compute the signature.
Proof of Theorem 6.2, signature calculation.
First, if M is as in Lemmas 6.3 or 6.4, then M in an integral cohomology S , C P , or H P . In each of these cases, it follows that | σ ( M ) | is0 or 1, according to the parity of χ ( M ).Second, if M is as in the first possibility of Lemma 6.6, then there exists and involution ι such that M ι is a closed, connected, simply connected 6–manifold. Hence σ ( M ) = σ ( M ι ) = 0.Note that this is consistent with the fact that χ ( M ) is even in this case by the proof of Lemma6.8.We may therefore assume M is as in the second possibility of Lemma 6.6. In particular, M is not spin, no fixed-point set M ι i is connected, and 7 ≤ χ ( M ) ≤ C (6). Since C (6) ≤ ≤ χ ( M ) ≤ We cannot calculate the signature in all cases, so we further assume χ ( M ) ≤
13. Theproof of Lemma 6.8 (in particular, Equation 6.8) implies χ ( M ) − χ ( N i ) ∈ { , , , } forsome 0 ≤ i ≤
6, where the N i are the 6–dimensional fixed-point components of the ι i as inthe proof of Lemma 6.8. In particular, we have the following possibilities: • M ι i = N i ∪ S , and so σ ( M ) = σ ( S ) = 0, • M ι i = N i ∪ C P , and so | σ ( M ) | = | σ ( C P ) | = 1, • M ι i = N i ∪ S ∪ S , and so σ ( M ) = 0, or • M ι i = N i ∪ S ∪ C P , and so | σ ( M ) | = 1.Hence the signature is 0 or ± χ ( M ). (cid:3) We remark that an extension of this argument shows that | σ ( M ) | = 1 if there exists some N i with χ ( M ) − χ ( N i ) = 7. Using this together with the fact that | σ ( M ) | ≡ χ ( M ) mod 2,one can further compute that | σ ( M ) | = 1 if χ ( M ) ∈ { , , } .7. Dimension 14
The only simply connected, smooth, closed manifolds of dimension 14 known to admitpositive sectional curvature are S and C P . The following result provides a sharp calculationof the Euler characteristic and the second and third homology groups of a positively curved14–manifold in the presence of T symmetry. Theorem 7.1. If M is a closed, simply connected, positively curved Riemannian manifoldwith T symmetry, then one of the following occurs. • M is –connected and χ ( M ) = 2 . • H ∗ ( M ; Z ) ∼ = H ∗ ( C P ; Z ) , and H ∗ ( M ; Z ) is generated by some z ∈ H ( M ; Z ) and x ∈ H ( M ; Z ) subject to the relation z = mx for some m ∈ Z . Note that, if T is replaced by T in this statement, then M is tangentially homotopyequivalent to S or C P (see [Wil03, DW04]). We proceed to the proof. Lemma 7.2.
If a non-trivial involution ι ∈ T exists such that cod ( M ι ) ≤ , then one ofthe following occurs: • M is homeomorphic to S . • H ∗ ( M ; Z ) ∼ = H ∗ ( C P ; Z ) , and H ∗ ( M ; Z ) is generated by some z ∈ H ( M ; Z ) and x ∈ H ( M ; Z ) subject to the relation z = mx for some m ∈ Z . As the proof indicates, a weak version of this lemma is an easy consequence of the codi-mension four lemma. For the stronger conclusion, we require our calculation in dimension10 (see Theorem 5.1).
Proof.
Let M ι x be a fixed point component of an involution of maximum dimension. By thecodimension two lemma, we may assume cod( M ι x ) = 4. By maximality, M ι x has T (or T )symmetry. Theorem 5.1 implies that b ( M ι x ) = 1 or M ι x is homeomorphic to S . In the lattercase, the connectedness lemma implies M is homeomorphic to S . In the former case, if M ι x has T symmetry, then it is a cohomology C P by Wilking’s homotopy classification, and M is a cohomology C P by the connectedness lemma. We assume now that H ( M ; Z ) ∼ = Z andthat M ι x is fixed by a circle in T .Consider the map Z → Z induced by the isotropy representation of T at x . There exists ι ∈ Z \ h ι i such that cod ( M ι x ) ≤
6, and the codimension four lemma, part 1, applies. The
OSITIVE CURVATURE AND TORUS SYMMETRY, I 23 lemma follows if cod( M ι x ) ≤
4, so we may assume cod( M ι x ) = 6. Returning to the isotropymap Z → Z , we see that we may choose ι so that, in addition, M ι x and M ι x intersectnon-transversely. From the previous paragraph, we have without loss of generality that M ι x has T symmetry and hence is diffeomorphic to C P by the diffeomorphism classification ofGrove and Searle. Since the intersection of M ι x and M ι x is not transverse, the intersectionof any two of M ι x , M ι x , and M ι ι x equals the 6–dimensional manifold M h ι ,ι i x . By the con-nectedness lemma, M ι x , M ι ι x , and M h ι ,ι i x are cohomology complex projective spaces. Bythe containment lemma, M T ⊆ M ι x ∪ M ι x ∪ M ι ι x , so the inclusion-exclusion property of Euler characteristics implies χ ( M ) − χ ( M ι x ) = 5 + 5 − . On the other hand, the periodicity corollary together with the fact that M ι x is fixed by acircle in T imply that χ ( M ) − χ ( M ι x ) = − b ( M ) + 2 b ( M ) − b ( M ) ≤ b ( M ) = 2 . Combining the previous two computations, we conclude that M has the Betti numbers of C P . Applying the periodicity corollary again, we conclude that Z ∼ = H ( M ; Z ) ∼ = H ( M ; Z )and hence that H ( M ; Z ) = 0. Recalling that H ( M ; Z ) = 0, it follows that H ∗ ( M ; Z ) ∼ = H ∗ ( C P ; Z ). Finally, the connectedness lemma applied to the inclusion M ι x → M andPoincar´e duality imply that H ∗ ( M ; Z ) ∼ = H ∗ ( C P ; Z ). (cid:3) To complete the proof of the theorem, it suffices to prove the following:
Lemma 7.3.
If the fixed-point set of every involution has codimension at least , then M is -connected and has Euler characteristic two. The proof in this case is similar to, but not as hard as, the corresponding lemma indimension 12. As there, the key aspect is the rigidity of the maps Z → Z induced by theisotropy representations at fixed points x ∈ M T . Proof.
Let T denote a torus of rank four acting effectively and isometrically on M . Since everynon-trivial involution ι ∈ T has cod ( M ι ) ≥
6, we have rigidity in the isotropy representation Z → Z ⊆ SO( T x M ). Indeed, for every x ∈ M T , there exists a basis of T x M and a generatingset of involutions ι , . . . , ι ∈ Z such that the map Z → Z can be represented as follows: ι (1 , , , , , , ,ι (1 , , , , , , ,ι (0 , , , , , , ,ι (1 , , , , , , . In particular, at every x ∈ M T , there exist seven distinct involutions whose fixed-pointcomponent containing x has codimension six, seven with codimension eight, and exactly onewith codimension 14.As a consequence of this rigidity and Frankel’s theorem, if ι and ι are distinct involutionssuch that N = M ι x and N = M ι x have codimension six, then N ∪ N contains M T . Bythe connectedness lemma, N ∩ N is one–connected and b ( N ∩ N ) ≥ b ( N i ) = b ( M ) for i ∈ { , } . By the rigidity of the isotropy representation, N and N have T symmetry and N ∩ N has T symmetry. By the classifications of Grove–Searle and Fang–Rong, all three submanifolds are homotopy spheres or complex or quaternionic projective spaces. By theinclusion-exclusion property of Euler characteristics, we obtain the estimate χ ( M ) = χ ( N ) + χ ( N ) − χ ( N ∩ N ) ≤ . In fact, χ ( M ) ≡ b ( M ) ≡ χ ( M ) ∈ { , , } .In particular, the number of fixed points of T is less than the number of involutions ι ∈ T with cod ( M ιx ) = 6. At most one of these involutions can have cod (cid:0) M ιy (cid:1) = 14 at any onefixed point y ∈ M T , so there exists some involution ι with cod (cid:0) M ιy (cid:1) <
14 for all y ∈ M T . Bythe rigidity of the isotropy representation and Frankel’s theorem, M ι is connected. ApplyingFang and Rong’s classification again, we conclude that M ι either is homeomorphic to S orhas odd Euler characteristic. Since χ ( M ) = χ ( M ι ), we conclude that M ι is homeomorphicto S , χ ( M ) = 2, and M is 3–connected. (cid:3) Dimension 16
The four known, simply connected, compact, positively curved examples in dimension 16are the rank one symmetric spaces, S , C P , H P , and C P , and each of these admits apositively curved metric with T symmetry. Theorem 8.1. If M is a closed, simply connected Riemannian manifold with positivesectional curvature and T symmetry, then one of the following occurs: • χ ( M ) = 2 and σ ( M ) = 0 . • χ ( M ) = 3 , σ ( M ) = ± , and M is –connected. • χ ( M ) = 5 , σ ( M ) = ± , H i ( M ; Z ) = 0 for all i , and b ( M ) = 1 + 2 b ( M ) . • χ ( M ) = 9 , σ ( M ) = ± , H ( M ; Z ) = Z , H ( M ; Z ) = 0 , H ( M ; Z ) = Z , and H ( M ; Z ) is generated by an element of the form x y with x ∈ H ( M ; Z ) and y ∈ H ( M ; Z ) . Moreover, M is not spin.If moreover M is spin, then the elliptic genus is constant. The proof of the last claim is in Section 9. The proof of the first claim takes the rest ofthe section and is contained in Lemmas 8.2, 8.4, and 8.5.Let M be as Theorem 8.1, let T denote a torus of rank four acting effectively and isomet-rically on M , and let Z ⊆ T denote the subgroup of involutions. Lemma 8.2.
If there exists ι ∈ Z with cod ( M ι ) ∈ { , } , then(1) M is homeomorphic to S ,(2) M is homotopy equivalent to C P , or(3) χ ( M ) = 5 , σ ( M ) = ± , H i ( M ; Z ) = 0 for all i , and b ( M ) = 1 + 2 b ( M ) . The proof is a bit involved, and we would like to illustrate by example some of the structurewe recover in the most difficult case of the proof.
Example . Denote points in H P as equivalence classes [ q , q , . . . , q ] where q s ∈ H suchthat P | q s | = 1. Define the actions of four circles on H P by the following three maps OSITIVE CURVATURE AND TORUS SYMMETRY, I 25 S → Sp(5): w diag( w, , , , w diag( w, w, w, w, w ) w diag(1 , , w, , w ) w diag(1 , , , w, w )The involution ι in the first circle fixes a component H P of codimension four. The involution ι in the second circle acts trivially, so really we consider the action by the quotient of thesecond circle by {± } . The involution in this circle is then represented by ( i, i, i, i, i ), and itsfixed point set is a C P . Notice that all fixed points of the T action are isolated, and thatthey come in two types. If ι , . . . , ι denote the involutions in the four circles above, then atthe point z = [1 , , , ,
0] one hascod( M ιz ) = (cid:26) ι
6∈ h ι i
16 if ι = ι while at any of the other fixed points y , one hascod( M ιy ) = ι = ι ι ι C ( y )12 if ι ∈ ι C ( y ) \ { ι } for some subgroup C ( y ) isomorphic to Z depending on y . For example, C ( y ) = h ι , ι i at y = [0 , , , , M ιy is C P for ι ∈ ι C ( y ) ∪ ι ι C ( y )and is H P for non-trivial ι ∈ C ( y ). Finally, note that the fixed point set of the T action iscontained in the two-fold union of a large number of choices of eight-dimensional fixed pointcomponents of involutions. We recover this combinatorial and topological fixed point datain one case of the proof of Lemma 8.2, however we are unable to fully recover the topologyof M . Proof of Lemma 8.2.
By the codimension two lemma, we may assume that cod ( M ι ) = 4.Choose x ∈ M T such that cod ( M ι x ) = 4. By the codimension four lemmas, we may assumethat every other non-trivial involution ι ∈ T has cod ( M ι ) ≥ M ιz ) = 8 forany z ∈ M T \ M ι x and ι ∈ Z \ h ι i .Let y ∈ M T ∩ M ι x . The isotropy map at y is also rigid in the sense that there exist arank-two subgroup C ( y ) ⊆ Z and an involution ι ( y ) C ( y ) such that cod (cid:0) M ιy (cid:1) = 8 foreach of the 11 non-trivial involutions ι not in the coset ι C ( y ). Moreover, if ι is one of these11 involutions, the intersection M ι x ∩ M ιy is transverse if and only if ι ∈ C ( y ).Fix ι to be any choice of ι ( x ), and fix any choice of distinct ι , ι ∈ C ( x ). From thisisotropy rigidity and Frankel’s theorem, all of the following hold:(1) cod ( M ιx ) = 8 and M T ⊆ M ι x ∪ M ιx for all ι ∈ { ι , ι , ι , ι ι } .(2) M ι x ∩ M ι x ⊆ M ι x is 4–connected and has codimension two.(3) M ι x ∩ M ιx ⊆ M ιx is 4–connected and has codimension four for ι ∈ { ι , ι , ι ι } .(4) M T ⊆ M ι x ∪ M ι x ∪ M ι ι x .Set ǫ = χ ( M ) − χ ( M ι x ). By the first claim above, ǫ = χ ( M ι x ) − χ ( M ι x ∩ M ι x ). By thesecond claim and the codimension two lemma, M ι x and M ι x ∩ M ι x are cohomology spheresor complex projective spaces. In either case, the difference in their Euler characteristics is ǫ ∈ { , } , and χ ( M ι x ) = 2 + 3 ǫ . Note that the fixed points of T are isolated in this case, so χ ( M ) ≥ χ ( M ι x ) ≥ ǫ. We prove now that the opposite inequality holds. Fix for a moment ι ∈ { ι , ι , ι ι } . Bythe first claim, ǫ = χ ( M ιx ) − χ ( M ι x ∩ M ιx ). By the third claim, M ιx has 4–periodic integralcohomology and second Betti number equal to that of M ι x ∩ M ιx . In particular, since ǫ ∈ { , } ,we have χ ( M ιx ) = 2+ ǫ for all ι ∈ { ι , ι , ι ι } . Now consider any two-fold intersection M ιx ∩ M ι ′ x of these three submanifolds. Since the fixed points of T are isolated, the Euler characteristicof M ιx ∩ M ι ′ x is at least that of its component M h ι,ι ′ i x . This component is a closed, oriented,positively curved 4–manifold, so its Euler characteristic is at least two. In addition, this Eulercharacteristic is at least that of the three-fold intersection M ι x ∩ M ι x ∩ M ι ι x . Putting theseestimates together, the fourth claim above and the inclusion-exclusion property of Eulercharacteristics implies χ ( M ) ≤ ǫ ) − − − ǫ. Since the opposite inequality also holds, we conclude χ ( M ) = 2 + 3 ǫ ∈ { , } .For the signature, recall that M ι x has 2–periodic cohomology and is either S or C P ,according to whether ǫ is 0 or 1. In either case χ ( M ) = χ ( M ι x ), so M ι is connected sincethe fixed points of T are isolated. In particular, σ ( M ) = σ ( M ι x ), which is either 0 or ± χ ( M ) is 2 or 5, respectively.We conclude the rest of the lemma by considering cases. Let N = M ι x and recall thatdk( N ) denotes the dimension of the kernel of the induced T –action on N . • If dk( N ) = 0, Wilking’s homotopy classification implies that N is a cohomologysphere or quaternionic projective space. By the connectedness lemma, M is as well. • If dk( N ) = 1 and χ ( M ) = 2, then χ ( N ) = 2 as well. Since the fixed points of T are isolated, M ι is connected and hence H i ( M ; Z ) = 0 for 3 ≤ i ≤ n −
3. Since χ ( M ) = 2 + 2 b ( M ), we have b ( M ) = 0 and hence that M is homeomorphic to S by Lemma 3.3. • If dk( N ) = 1 and χ ( M ) = 5, the proof shows that M ι x is homeomorphic to H P . ByWilking’s maximal smooth symmetry rank classification for a integral quaternionicprojective spaces (see [Wil03, Theorem 3]), M ι x is fixed by a circle in T . In particular, M ι x → M is 2–connected, so H ( M ; Z ) = 0. Wilking’s periodicity corollary nowimplies that H i ( M ; Z ) = 0 and H ( M ; Z ) ⊇ Z . (cid:3) We now consider the possibility that the minimal codimension of a fixed-point set of aninvolution in T is six. Lemma 8.4.
Suppose min cod ( M ι ) = 6 , where the minimum runs over involutions in T .One of the following occurs: • χ ( M ) = 2 , σ ( M ) = 0 , and M is –connected. • χ ( M ) = 9 , σ ( M ) = ± , H ( M ; Z ) = Z , H ( M ; Z ) = 0 , H ( M ; Z ) = Z , H ( M ; Z ) =0 , and H ( M ; Z ) is generated by an element of the form x y with x ∈ H ( M ; Z ) and y ∈ H ( M ; Z ) , and M is not spin.Proof. Choose x ∈ M T such that N = M ι x has codimension six. The isotropy at x impliesthat some ι ∈ Z \ h ι i exists such that M ι x has codimension six. Moreover, if N and N intersect transversely, there exists ι ∈ Z \ h ι , ι i such that M ι x has codimension six and OSITIVE CURVATURE AND TORUS SYMMETRY, I 27 intersects N non-transversely. We may assume therefore that N = M ι x has codimensionsix and intersects N non-transversely.By the containment lemma, M T ⊆ N ∪ N ∪ N , where N = M ι ι x . By the connectedness lemma, all two-fold intersections of N , N , and N are connected and hence equal N ∩ N = M h ι ,ι i x . Since the intersection N ∩ N is nottransverse, N ∩ N ⊆ N has codimension two. Since no involution ι has cod ( M ι ) < N has T symmetry, so N and N ∩ N are both cohomology spheres or both cohomologycomplex projective spaces. We consider two cases:(1) Suppose N ∩ N is homeomorphic to S . Since N ∩ N → N i is 4–connected for i ∈ { , } , each N i is a 4–connected 10–manifold. Since N i is positively curved andhas T symmetry, χ ( N i ) >
0, which implies that N i is homeomorphic to S . By theinclusion-exclusion property of the Euler characteristic, χ ( M ) = 2. Moreover, since N → M is 5–connected, M is 5–connected as well. For the signature, note that σ ( M ) = σ ( M ι ) = σ ( N ) = 0.(2) Suppose N ∩ N is a cohomology C P . Since N ∩ N → N i is 4–connected for i ∈ { , } , it follows by arguments similar to those in the dimension 10 result that each N i is a cohomology C P . It follows that χ ( M ) = 9, H ( M ; Z ) = Z , H ( M ; Z ) = 0, H ( M ; Z ) = Z , H ( M ; Z ) = 0, and some x ∈ H ( M ; Z ) satisfies the property that x is not a multiple. In particular, H ( M ; Z ) is generated by an element of the form x y with y ∈ H ( M ; Z ).For the signature, note that σ ( M ) = ± σ ( M ) = σ ( M ι ) unless M ι consists of N together with three isolated points y j . Inthis latter case, no y j ∈ N , as that would imply that cod( M ι y j ) = 8, a contradictionto Frankel’s theorem since cod ( M ι ) = 6. It follows that each y j ∈ N and hence thateach y j lies in a 6-dimensional component of M ι ι . Since χ ( N ) = 6, the fixed-pointset of ι ι is comprised of N together with some number of six-dimensional compo-nents whose Euler characteristic is at least, and hence sums to, three. This contradictsthe fact that closed, oriented six-manifolds have even Euler characteristics.Finally, if M is spin, then M ι has, in addition to N , components of dimensiontwo. These two-dimensional components are oriented, so they are diffeomorphic tospheres. This is a contradiction since χ ( M ) − χ ( N ) = 3. (cid:3) To complete the proof of Theorem 8.1, we prove the following:
Lemma 8.5 (Rigid isotropy lemma for n = 16) . If cod ( M ι ) ≥ for every non-trivialinvolution ι ∈ T , then one of the following occurs: • χ ( M ) = 2 and σ ( M ) = 0 . • χ ( M ) = 3 , σ ( M ) = ± , and M is –connected.Proof. The isotropy representation is rigid in this case. Indeed, for each x ∈ M T , there existsan involution ι such that cod ( M ι x ) = 16 and cod ( M ιx ) = 8 for every other non-trivialinvolutions ι .Fix x ∈ M T and choose distinct, non-trivial involutions ι , ι ∈ T such that each M ι i x hascodimension eight. By the rigidity of the isotropy, each N i has T symmetry, so χ ( N i ) ≤ χ ( C P ) = 5 by Fang and Rong’s homeomorphism classification in dimension eight. ByFrankel’s theorem, M T ⊆ M ι x ∪ M ι x , so we have χ ( M ) ≤ < ι with cod ( M ιx ) = 8, and since at most one of these hascod (cid:0) M ιy (cid:1) = 8 at any other fixed point y ∈ M T , not every involution gets a turn at havinga 0–dimensional fixed point component. By Frankel’s theorem, there exists an involution ι ∈ T such that M ι is connected and has dimension eight. By the rigidity of the isotropyrepresentation, M ι has T symmetry and hence is homeomorphic to S , H P , or C P by theFang–Rong classification. We consider these three cases separately.(1) M ι = S . The Euler characteristic and signature of M and M ι are the same, so thecalculation χ ( M ) = 2 and σ ( M ) = 0 follows immediately. This is the first possibleconclusion of Lemma 8.5.(2) M ι = H P . As in the previous case, we immediately conclude χ ( M ) = 3 and σ ( M ) = ±
1. Moreover, we conclude by Wilking’s maximal smooth symmetry rank bound for H P that there is a circle in T fixing M ι . The connectedness lemma implies that M ι → M is 2–connected, so we have the additional conclusion in this case that π ( M ) = 0.(3) M ι = C P . We claim this case cannot occur. Indeed, choose another non-trivialinvolution τ , and consider the induced action of τ on M ι . Since M ι is homeomorphicto C P , the fixed point set ( M ι ) τ is comprised of exactly two components, and thesetwo components are integral complex projective spaces whose dimensions add todim( C P ) − χ ( M ι ) = χ (( M ι ) τ ). Inparticular, one of the components of ( M ι ) τ has dimension two or six. This contradictsthe rigidity of the isotropy representation, which implies that every component of M h ι,τ i has dimension zero or four.This concludes the proof of Lemma 8.5 and hence of Theorem 8.1. (cid:3) Elliptic genus calculation
We present a unified proof of the elliptic genus claims in dimensions 12 and 16.
Theorem 9.1.
Let M m be a closed, simply connected, spin manifold admitting positivesectional curvature and T m symmetry. If m ≤ , the elliptic genus is constant. Note that, if T m is replaced by T m +1 , then this claim holds by Wilking [Wil03, Theorem2]. Indeed, restricting his result to the spin case, such a manifold is homeomorphic to S m or H P m , and so the elliptic genus is constant by Novikov’s theorem. Proof.
Since M is spin, every involution ι ∈ T m has the property that there exists c ∈ { , } such that cod ( N ) ≡ c mod 4 for all components N ⊆ M ι . If c = 2 for some involution, theaction of that involution is of odd type and it follows that the elliptic genus vanishes by aresult of Hirzebruch–Slodowy (see corollary on [HS90, page 317]). We assume therefore thatevery component of the fixed-point set of every involution has codimension divisible by four.If there exists an involution ι ∈ T m for which cod ( M ι ) ≥ dim( M ), then the ellipticgenus is constant by another result of Hirzebruch–Slodowy (see the above-cited corollary).In particular, we are done if m ≤
2. Moreover, we are done m ∈ { , } and there existsan involution ι ∈ T m and a component N ⊆ M ι satisfying cod ( N ) = 8. Indeed, Frankel’stheorem implies that any other such component has codimension at least 8, which is at least OSITIVE CURVATURE AND TORUS SYMMETRY, I 29 half of the dimension of M . Assume therefore that no component of a fixed-point set of aninvolution has codimension eight.Next, suppose there exist two involutions with fixed point components, N and N , ofcodimension four. If N ∩ N is transverse, then the product of these involutions has fixed-point set of codimension eight, a contradiction. If, on the other hand, N ∩ N is not transverse,then the codimension two lemma implies that N is homotopy equivalent to a sphere orcomplex projective space, and the codimension four lemma implies that M is an integralsphere, complex projective space, or quaternionic projective space. Since N → M is 5–connected and M is spin, it follows that M is homeomorphic to S m and hence that theelliptic genus is constant.Assume therefore that at most one involution has codimension four fixed-point set. Choose Z m − ⊆ Z m such that every ι ∈ Z m − has cod ( M ι ) ≥
12. Summing these codimensions atsome fixed point x ∈ M T , we have (cid:0) m − − (cid:1) (12) ≤ X ι ∈ Z m − cod ( M ιx ) = 2 m − cod (cid:16) M Z m − x (cid:17) ≤ m m. Since m ∈ { , } , this is a contradiction, so the proof is complete. (cid:3) Low dimensional positively elliptic spaces
A simply connected, rationally elliptic topological space with positive Euler characteristicis called an F , or positively elliptic , space. These spaces have a (formal) dimension andsatisfy Poincar´e duality, and they admit pure minimal models (see [FHT01]). Specifically,they admit minimal models (Λ V, d ) with generators x , . . . , x k ∈ V of even degree andgenerators y , . . . , y k ∈ V of odd degree such that each d x i = 0 and each d y i is a homogeneouspolynomial in the x , . . . , x k . We classify all possible tuples(deg x , . . . , deg x k , deg y , . . . , deg y k )of degrees for F spaces of dimension up to 16.In dimensions up to eight, this follows from previous work (see [PP03, Pav02, Her]). Theorem 10.1 (Paternain–Petean, Pavlov, Herrmann) . If M is an F space of formaldimension , , , or , then M admits a pure model whose tuples of homotopy generatordegrees satisfy one of the following: • dim M = 2 and the tuple of degrees is (2 , , and M ≃ Q S • dim M = 4 and the tuple of degrees is (2 , , (4 , , or (2 , , , , and M is rationallyhomotopy equivalent to S , C P , S × S , C P C P , or C P − C P . • dim M = 6 and the tuple of degrees appears in Table 10.1. • dim M = 8 and the tuple of degrees appears in Table 10.2. Table 10.1: Dimension 6deg π ∗ ( M ) ⊗ Q χ ( M )(6 ,
11) 2(2 , , , ,
7) 4(2 , , ,
5) 6(2 , , , , ,
3) 8
Table 10.2: Dimension 8deg π ∗ ( M ) ⊗ Q χ ( M )(8 ,
15) 2(4 ,
11) 3(2 , , , , , ,
7) 4(2 ,
9) 5(2 , , ,
7) 6(2 , , , , , , , ,
7) 8(2 , , ,
5) 9(2 , , , , ,
5) 12(2 , , , , , , ,
3) 16In dimensions six and eight, Herrmann [Her] proves a partial classification of the rationalhomotopy types in dimensions, but we will not need this here. We proceed to the computationin dimensions 10, 12, 14, and 16. Let M be an F -space. Theorem 10.2. If M is an F space of formal dimension , , , or , then M admitsa pure model whose tuples of homotopy generator degrees appear in one of the Tables 10.3,10.4, 10.5, or 10.6. Moreover, each tuple that appears in this table is realized by such a space.Proof. Let (Λ
V, d ) be a pure model of M with k generators x i of even degree 2 a i and k generators y i of odd degree 2 b i −
1. Following [FHT01, Section 32], we may choose such x i and y i such that the following hold: • ≤ a ≤ . . . ≤ a k and 2 ≤ b ≤ . . . ≤ b k . • b i ≥ a i for all 1 ≤ i ≤ k . • P ki =1 ( b i − a i ) = dim M .Note in particular that dim( M ) ≥ P a i ≥ k , so in each dimension there are finitelymany possible values for k and for the a i . It follows that there are only finitely many possiblevalues for the b i as well. We lead a computer based search using Mathematica to enumerateall possible tuples (2 a , . . . , a k , b − , . . . , b k −
1) where k ranges from one up to half thedimension of M . Moreover, we compute in each case the Euler characteristic, which satisfiesthe following formula: χ ( M ) = k Y i =1 deg y i + 1deg x i = k Y i =1 b i a i . Of course, we can rule out any tuples for which the Euler characteristic is non-integral.However, there is a single criterion called the “arithmetic condition” due to Friedlander andHalperin that characterizes precisely which tuples of degrees are realized by F spaces (see[FHT01, Proposition 32.9]). In practice, it is not too difficult to check whether any giventuple of degrees can be realized using the integrality of the Euler characteristic together withdirect arguments.Let us demonstrate such a direct argument in one exemplary case, in which we ex-clude the existence of such a model. Suppose that homotopy groups are given degrees(2 , , , , , V = h x , x , x , y , y , y i with deg x = 2 , deg x = 4 , deg x = 4,deg y = 5 , deg y = 7 , deg y = 9. Let us see that there is no differential d on Λ V making H (Λ V, d) finite dimensional. In fact, for degree reasons we see that d y ∈ x · h x , x , x i and OSITIVE CURVATURE AND TORUS SYMMETRY, I 31 d y ∈ x · h x , x x , x , x i . Thus these two relations taken together cannot reduce the Krulldimension of Q [ x , x , x ] by two.For all the configurations in the tables it is easy to construct minimal Sullivan models.(Again, one can alternatively check that the arithmetic condition of Friedlander and Halperinholds.) Indeed, nearly all of them can be realized by products of spaces with singly-generatedcohomology algebra. Let us also provide the arguments in a few cases where we need slightlymore complicated realizing models:The first configuration which is not realizable as a non-trivial rational product is given by(4 , , , h x , x , y , y i , d)with deg x = 4 , deg x = 6, deg y = 9 , deg y = 11 and d x = d x = 0, d y = x x , d y = x + x .The next case is (2 , , , , , , , ,
13) is realized by(Λ h x , x , y , y i , d)with deg x = 4 , deg x = 6, deg y = 11 , deg y = 13 and d x = d x = 0, d y = x + x , d y = x x .The next case is (2 , , , , , C P . In the case (2 , , , , , , , , , , , , , , , , , , ,
11) can be realized by(Λ h x , x , x , y , y , y i , d)with deg x = deg x = 4, deg x = 6, deg y = 7 , deg y = 9 , deg y = 11 and d x = d x =d x = 0, d y = x , d y = x x , d y = x + x . (cid:3) Table 10.3: Dimension 10deg π ∗ ( M ) ⊗ Q χ ( M )(10 ,
19) 2(2 , , , , , ,
11) 4(2 , , , , , , ,
11) 6(2 , , , , , , , , , , , , ,
7) 8(2 , , ,
9) 10(2 , , , , , , , ,
7) 12(2 , , , , , , , , , , , ,
7) 16(2 , , , , ,
5) 18(2 , , , , , , ,
5) 24(2 , , , , , , , , ,
3) 32Table 10.4: Dimension 12deg π ∗ ( M ) ⊗ Q χ ( M )(12 ,
23) 2 (6 ,
17) 3(4 , , , , , , , , , ,
11) 4(4 , , ,
11) 5(2 , , , , , ,
11) 6(2 ,
13) 7(2 , , , , , , , , , , , , , , , , , ,
7) 8(2 , , ,
11) 9(2 , , ,
9) 10(2 , , , , , , , , , , , , , , , , , ,
7) 12(2 , , ,
9) 15(2 , , , , , , , , , , , , , , , , , , , , , ,
7) 16(2 , , , , ,
7) 18(2 , , , , ,
9) 20(2 , , , , , , , , , , , ,
7) 24(2 , , , , ,
5) 27(2 , , , , , , , , , , , , , , , ,
7) 32(2 , , , , , , ,
5) 36(2 , , , , , , , , ,
5) 48(2 , , , , , , , , , , ,
3) 64Table 10.5: Dimension 14deg π ∗ ( M ) ⊗ Q χ ( M )(14 ,
27) 2(2 , , , , , , , , ,
15) 4(2 , , , , , , , , ,
11) 6(2 , , , , , , , , , , , , , , , , , , , , , , , , , , ,
11) 8(2 , , , , , , , ,
11) 10(2 , , , , , , , , , , , , , , , , , ,
11) 12(2 , , ,
13) 14(2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,
7) 16(2 , , , , , , , , , , , , ,
11) 18(2 , , , , , , , ,
9) 20(2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,
7) 24(2 , , , , ,
9) 30(2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,
7) 32(2 , , , , , , , , , , , ,
7) 36(2 , , , , , , ,
9) 40(2 , , , , , , , , , , , , , , , ,
7) 48(2 , , , , , , ,
5) 54(2 , , , , , , , , , , , , , , , , , , , ,
7) 64(2 , , , , , , , , ,
5) 72
OSITIVE CURVATURE AND TORUS SYMMETRY, I 33 (2 , , , , , , , , , , ,
5) 96(2 , , , , , , , , , , , , ,
3) 128Table 10.6: Dimension 16deg π ∗ ( M ) ⊗ Q χ ( M )(16 ,
31) 2(8 ,
23) 3(2 , , , , , , , , , , , ,
15) 4(4 ,
19) 5(2 , , , , , , , , ,
15) 6(4 , , ,
13) 7(2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,
11) 8(2 , , , , , , ,
11) 9(2 , , , , , , , ,
11) 10(2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,
11) 12(2 , , ,
13) 14(2 , , , , , , , ,
11) 15(2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,
7) 16(2 , , , , , , , , , ,
11) 18(2 , , , , , , , , , , , , , , , , ,
11) 20(2 , , ,
13) 21(2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,
7) 24(2 , , ,
9) 25(2 , , , , ,
11) 27(2 , , , , ,
13) 28(2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,
7) 32(2 , , , , , , , , , , , , , , , , , , , , , , , , , ,
7) 36(2 , , , , , , , , , , , ,
9) 40(2 , , , , ,
9) 45(2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,
7) 48(2 , , , , , , ,
7) 54(2 , , , , , , ,
9) 60(2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,
7) 64(2 , , , , , , , , , , , , , , , ,
7) 72(2 , , , , , , , , ,
9) 80 (2 , , , , , , ,
5) 81(2 , , , , , , , , , , , , , , , , , , , ,
7) 96(2 , , , , , , , , ,
5) 108(2 , , , , , , , , , , , , , , , , , , , , , , , ,
7) 128(2 , , , , , , , , , , ,
5) 144(2 , , , , , , , , , , , , ,
5) 192(2 , , , , , , , , , , , , , , ,
3) 25611.
The Halperin conjecture
In this section we will show that F -spaces of dimension at most 16 satisfy the Halperinconjecture; and so do F -spaces with Euler characteristic at most 16. Recall that the Halperinconjecture states that any fibration with an F -space as a fiber should yield a Leray–Serrespectral sequence degenerating at the E -term.Keeping the notation from the previous section, let M be an F space of dimensiondim M ≤
16, and let (Λ( x , . . . , x k , y , . . . , y k ) , d) be a pure model satisfying the followingproperties: • The degrees deg( x i ) are increasing, and likewise for the degrees deg( y i ). • deg(d y i ) ≥ x i ) for all i . • each d x i = 0 and each d y i is a homogeneous polynomial in x , . . . , x k .Recall that the Halperin conjecture is known in the following cases: • k ≤
3, i.e., if the cohomology algebra of M is generated by at most three elements(see Lupton [Lup90]). • deg( x ) = . . . = deg( x k ), i.e., if all cohomology generators have the same degree (seeLemma 11.1 below). • If the model is the total space splits as a rational fibration whose base and fibersatisfy the Halperin conjecture, then it too satisfies the Halperin conjecture (seeMarkl [Mar90]).We use the following characterization of the Halperin conjecture due to Meier [Mei82].If the rational cohomology algebra H ∗ ( M ; Q ) does not admit a derivation of degree d < M . Using this, the following is a well known and easyconsequence: Lemma 11.1.
Let δ be a derivation of negative degree on H ∗ ( M ; Q ) , where M is an F space. If x ∈ H i ( M ; Q ) for some i > such that δ ( x ) ∈ H ( M ; Q ) , then δ ( x ) = 0 . Inparticular, if H ∗ ( M ; Q ) is generated by elements of the same degree, then δ = 0 . In other words, any image of δ landing in H ( M ; Q ) is zero. Proof.
Suppose x a is the maximal nonzero power of x . Since δ ( x ) is a non-zero element of H ( M ; Z ), we have δ ( x ) x a = 0. However δ is a derivation, so it follows that δ ( x a +1 ) = 0.This is a contradiction.To prove the last statement, note that δ = 0 if δ is zero on all generators. By the assumptionon the degrees of the generators, these images either land in H ( M ; Q ) or a zero group. Eitherway, these images are zero, so the proof is complete. (cid:3) The proof of the Halperin conjecture for Euler characterics up to 16 now follows easily, sowe prove it first.
OSITIVE CURVATURE AND TORUS SYMMETRY, I 35
Theorem 11.2.
The Halperin conjecture holds for F -spaces M with χ ( M ) ≤ .Proof. Recall from the previous section that the Euler characteristic of M is given by theformula χ ( M ) = Q deg(d y i ) / deg( x i ). Since each factor in the product is at least 2, we have χ ( M ) ≥ k where k is the number of cohomology generators, as above. By assumption, either k ≤ k = 4 and deg(d y i ) = 2 deg( x i ) for all i . In the first case, the conjecture holds byLupton’s result above. Suppose therefore that k = 4 and deg(d y i ) = 2 deg( x i ) for all i .Let δ be a degree d derivation on H ∗ ( M ; Q ) for some d <
0. If all of the x i have the samedegree, then Lemma 11.1 implies that δ = 0. If this is not the case, then there exists someinteger l such that deg( x ) = . . . = deg( x l ) < deg( x l +1 ) ≤ . . . ≤ deg( x ) . Since deg(d y i ) = 2 deg( x i ) for all i , it follows for degree reasons that d y i ∈ Λ( x , . . . , x l )for all i ≤ l . In particular, the model splits as a rational fibration over the subalgebraΛ( x , . . . , x l , y , . . . , y l ). Since both the base and fiber have fewer than four generators, weconclude from Markl’s result that the Halperin conjecture holds for M as well. (cid:3) We proceed to the proof of the Halperin conjecture for F spaces of dimension up to 16.We require two more observations that will facilitate the arguments. Lemma 11.3.
Suppose M is an F space and that its pure model (Λ V, d) is chosen as above.If δ is a derivation of negative degree on H ∗ ( M ; Q ) such that δ ( x i ) = 0 for k − of the k generators x i , then δ = 0 .Proof. For this proof only, we reorder the x i so that δ ( x i ) = 0 for all i ≥
2. We proceed bycontradiction, i.e. we assume that δ ( x ) = 0. Then, by Poincar´e duality, there is an element x ∈ H ∗ ( M ; Q ) such that δ ( x ) x generates the top cohomology group. Moreover we maychoose x to be a monomial in the x i . Write x = x l x ′ where x ′ is a monomial in x , . . . , x k .It follows that δ ( x l +11 ) x ′ generates top cohomology. But x l +11 x ′ = 0 for degree reasons and δ ( x ′ ) = 0 by assumption, so we compute that0 = δ ( x l +11 x ′ ) − x l +11 δ ( x ′ ) = δ ( x l +11 ) x ′ = 0 , a contradiction. (cid:3) Lemma 11.4.
Suppose M is an F space and that its pure model (Λ V, d) is chosen as above.If there exists l < k such that deg(d y l ) < deg( x )+deg( x l +1 ) , then (Λ V, d) splits as a rationalfibration with base algebra generated by x , . . . , x l , y , . . . , y l .Proof. This lemma is proved by the observation that under the above assumptions, theregular sequence d y , . . . , d y l lies in Λ h x , . . . , x l i , so splitting follows. (cid:3) With these preparations, we proceed to the proof of the Halperin conjecture for F spaceswith dimension at most 16. As a warm-up, we provide here a short proof in the case wherethere are at most three cohomology generators. Of course, this already follows by Lupton’stheorem, but we include it here for the convenience of the reader. Lemma 11.5.
The Halperin conjecture holds for F spaces M such that M has dimensionat most and H ∗ ( M ; Q ) has at most three generators.Proof. If there are at most two generators of the cohomology algebra, the proof follows imme-diately from Lemmas 11.1 and 11.3. Suppose there are exactly three cohomology generators.
We denote by (Λ h x , x , x , y , y , y i , d) a corresponding minimal model with d x = d x =d x = 0. Denote by δ a non-trivial derivation of negative degree on its cohomology algebra.Let the x i and the y i be ordered by degree. We want to show that it has to be trivial andassume the contrary. Denote the cohomology classes represented by x i by [ x i ]. We make thefollowing observations: • δ ([ x ]) = 0 by Lemma 11.1. • δ ([ x ]) = 0 and δ ([ x ]) = 0 without loss of generality by Lemma 11.3. • deg( x ) < deg( x ) by degree reasons and Lemma 11.1. • deg( x ) < deg( x ). Indeed, since δ maps H deg( x ) ( M ; Q ) linearly into a zero- or one-dimensional cohomology group, the property deg( x ) = deg( x ) would imply that,up to a change of basis, δ ([ x ]) = 0, a contradiction. • deg(d y ) ≥ deg( x ) + deg( x ) without loss of generality by Lemma 11.4. In fact,deg(d y ) ≥ deg( x ) + deg( x ) since otherwise d y ∈ Λ( x , x ). If d y ∈ Λ( x ), themodel splits as a rational fibration. If, on the other hand, d y Λ( x ), then applying δ to [d y ] = 0 in cohomology yields a relation among [ x ] and [ x ], and this is acontradiction since d y induces the relation of smallest degree. • deg(d y ) ≥ deg( x ) + deg( x ) without loss of generality by Lemma 11.4.Recall also that deg(d y i ) ≥ x i ) for all 1 ≤ i ≤
3. With these estimates in hand, thedimension formula, 16 ≥ n = X i =1 (deg(d y i ) − deg( x i )) , implies that the tuple of degrees is (2 , , , , , y ) ≥ deg( x ) + deg( x ), we may assume that d y contains a non-zero term involving x . By degree reasons, up to a scaling of y , we mayassume d y = x x + p ( x , x ). By a similar argument, we may assume that d y also hasa non-zero x x term, and hence that d y = x x + q ( x , x ) after scaling y . Finally, byreplacing y by y − y , we have that d y ∈ Λ( x , x ). Again, we obtain a contradiction ifd y has a non-zero term involving x , so we actually have d y ∈ Λ( x ) and hence that themodel of M splits as a rational fibration. (cid:3) Theorem 11.6.
The Halperin conjecture holds for F spaces of dimension up to .Proof. The proof proceeds by stepping through Tables 10.1 to 10.6. Let k denote the numberof cohomology generators.First, any model with k ≤ k − k − x ) < deg( x ) = . . . = deg( x k ), then δ maps H deg( x ) ( M ; Q ) linearly into a zero- or one-dimensional space. By a change of basis,we can arrange that δ ([ x ]) = . . . = δ ([ x k − ]) = 0 and hence that δ = 0. Finally, if somedeg(d y i ) < deg( x )+deg( x i +1 ) as in Lemma 11.4, then the model splits as a rational fibrationwhere the base and fiber are F spaces of smaller dimension. By induction over the dimension,this implies by Markl’s theorem that M satisfies the Halperin conjecture in this case.Taking into account these observations, one can either scan the tables or apply the dimen-sion formula to show that we only need to provide special arguments for the following fivecases: OSITIVE CURVATURE AND TORUS SYMMETRY, I 37 • (2 , , , , , , ,
7) and dim( M ) = 14. • (2 , , , , , , ,
7) and dim( M ) = 16. • (2 , , , , , , ,
7) and dim( M ) = 16. • (2 , , , , , , ,
11) and dim( M ) = 16. • (2 , , , , , , , , ,
7) and dim( M ) = 16.We consider these cases one at a time, proving in each case either that there is no non-trivial derivation of negative degree or that the model for M splits as a rational fibrationand apply Markl’s theorem.(2 , , , , , , , y = p ( x , x ) + q d y = p ′ ( x , x ) + q ′ where p, p ′ ∈ Λ h x , x i and where q and q ′ lie in the ideal ( x , x ) ⊆ Λ h x , x , x , x i gener-ated by x and x . Suppose for a moment that p is a rational multiple of p ′ . Up to a changeof basis, we may assume that d y = q ∈ ( x , x ). We derive a contradiction as follows. Bythe finite-dimensionality of H ∗ ( M ; Q ), some power [ x ] m is zero. At the level of the model,we have that some α i ∈ Λ h x , x , x , x i exist such that x m = d X i =1 α i y i ! = X i =1 α i d y i = X i =1 α i d y i + α ( p ′ + q ′ ) . Since d y i ∈ ( x , x ) for i ≤ q ′ ∈ ( x , x ), it follows that p ′ is a rational multipleof x . On the other hand, we can apply the same argument using a sufficiently large powerof x to conclude that p ′ is a rational multiple of x . This is a contradiction, hence we mayassume that p and p ′ are not multiples of each other.Let δ be a non-trivial derivation on H ∗ ( M ; Q ) of negative degree. By Lemma 11.1, we mayassume that δ ([ x ]) = δ ([ x ]) = 0 and that δ has degree −
2. Moreover, in view of Lemma11.4, we may assume that δ ([ x ]) and δ ([ x ]) are linearly independent, since otherwise achange of basis would yield δ ([ x ]) = 0 and hence that δ = 0. By a further change of basis,we may assume that δ ([ x ]) = [ x ] and δ ([ x ]) = [ x ].Note, in particular, that δ maps [ q ] , [ q ′ ] ∈ H ∗ ( M ; Q ) into the subalgebra generated by [ x ]and [ x ]. Hence δ maps [ q ] and [ q ′ ] to zero in H ( M ; Q ). Moreover, if we write p = ax + bx x + cx , for a, b, c ∈ Q , then we have that δ ([ p ]) = 2 a [ x ] + 2 b [ x ][ x ] + 2 c [ x ] , and similarly for [ p ′ ]. Since the triple ( a, b, c ) and the corresponding triple for p ′ are notmultiples of each other, we have constructed two independent relations in degree four. Thiscontradicts the fact that there is only one homotopy generator in degree three, so the proofis complete.(2 , , , , , , , δ ([ x ]) = [ x ], δ ([ x ]) =[ x ], and δ ([ x i ]) = 0 for i ∈ { , } . Moreover, it follows as in the last case that there is somepair y i and y j such that d y i and d y j are linearly independent modulo the ideal ( x , x ). As in the previous case, this implies the existence of two linearly independent relations in degreefour and hence provides a contradiction to degree considerations.(2 , , , , , , , δ ([ x ]) = [ x ], δ ([ x ]) =[ x ], and δ ([ x i ]) = 0 for i ∈ { , } . The proof can be handled similarly to the previous twocases, but there is a shortcut here. Without restriction, we may assume that that d y has x and possibly x as non-trivial summands, up to multiples. We compute that δ (d y ) hassummands x and possibly x , up to non-trivial multiples. Since there is no relation in degree3, we have a contradiction.(2 , , , , , , , δ ([ x ]) = δ ([ x ]) = 0 and that δ has degree two. In view of Lemma 11.3, we may assume further that δ ([ x ]) ∈ h [ x ] , [ x ] i and δ ([ x ]) ∈ h [ x ] , [ x ] , [ x ][ x ] , [ x ] i are non-zero. Up to an isomorphism of the minimalmodel, we may assume that either δ ([ x ]) = [ x ] or δ ([ x ]) ∈ h [ x ] , [ x ][ x ] , [ x ] i . Further,we may assume δ ([ x ]) = [ x ].Using the finite-dimensionality of H ∗ ( M ; Q ), it follows that x appears as a non-trivialsummand of d y . Up to scaling x , we may expressd y = x + k x + l ( x , x ) x x + p, where k ∈ Q , l ( x , x ) is a linear function of x and x , and p ∈ ker( δ ). We break the proofinto two cases.First, we suppose that δ ([ x ]) = [ x ]. For this we apply δ to the relation in cohomologyinduced by y . Since δ ([ x ] ) = 0, we conclude 0 = δ ([ x ] ) = 6[ x ] . This implies that d y is a multiple of x and hence that the model splits as a rational fibration over Λ( x , y ). ByMarkl’s result, the Halperin conjecture follows.Second, we suppose that δ ([ x ]) ∈ h [ x ] , [ x ][ x ] , [ x ] i . For this case, we apply δ to therelation induced by d y . This time, δ ([ x ] ) = 0, so we conclude that 0 = δ ( k [ x ] ) =6 k [ x ] . We consider now two subcases, according to whether k = 0 or [ x ] = 0.First suppose k = 0. Using the finite-dimensionality of H ∗ ( M ; Q ), the fact that x doesnot appear in d y implies that x must appear as a summand in d y . Up to scaling y , wemay write d y = x + l ′ ( x , x ) x + p ′ , where l ′ ( x , x ) is a linear function of x and x and p ′ ∈ ker( δ ). In fact, δ also sends [ x ] tozero since we are assuming δ ([ x ]) is a polynomial in [ x ] and [ x ]. Hence applying δ to theinduced relation in cohomology yields 0 = 2[ x ] . As before, this implies a rational fibrationsplitting, and hence that the Halperin conjecture holds.Finally, suppose that [ x ] = 0. This implies that x = m ( x , x )d y + c d y for some linear function m ( x , x ) of x and x and some c ∈ Q . Now, for degree considera-tions, we may express d y = l ′′ ( x , x ) x + p ′′ , where l ′′ ( x , x ) is a linear function of x and x and p ′′ ∈ Λ h x , x i ⊆ ker( δ ). Moreover,unless the model of M splits as a rational fibration over the subalgebra (Λ h x , x , y , y i , d),we may assume that l ′′ ( x , x ) = 0. Applying δ to the induced relation in cohomology yields0 = l ′′ ([ x ] , [ x ])[ x ]. Since the only relation in this degree is that induced by y , we conclude OSITIVE CURVATURE AND TORUS SYMMETRY, I 39 from this that d y ∈ Λ h x , x i is divisible by x . But now the relation above for x implies thatd y is also divisible by x . Looking again at the expression for d y , we conclude that l ′′ ( x , x )is a (non-trivial) multiple of x . Since l ′′ ([ x ] , [ x ])[ x ] = 0, we conclude that [ x ] = 0 andhence that, again, the model for M splits as a rational fibration over Λ h x , y i . This provesthe Halperin conjecture for this tuple of homotopy generator degrees.(2 , , , , , , , , , δ ([ x i ]) = 0 for i ∈ { , , } and that δ ([ x ]) =[ x ] and δ ([ x ]) = [ x ]. The proof here is similar to the first few exceptional cases, where theideal ( x , x ) is replaced by the ideal ( x , x , x ).This concludes the proof of Halperin’s conjecture in the five special cases above. Altogether,this completes the proof of the conjecture in dimensions up to 16. (cid:3) Positive curvature and rational ellipticity
We now combine the information of the previous sections in order to classify rationallyelliptic Riemannian manifolds of dimension at most 16 with positive sectional curvature andtorus symmetry. The additional assumption of rational ellipticity does not add to our under-standing in dimensions two, four, and six since the existing theorems do not see improvementupon adding the assumption of rational ellipticity. Indeed, each of these results show thatthe manifold is rational elliptic. Starting in dimension eight, however, Dessai’s result is onewhere the conclusion (an Euler characteristic, signature, and elliptic genus calculation) isimproved to a rational homotopy classification by adding the assumption of rational ellip-ticity (see [Des11, Theorem 1.2]). We have corresponding rational homotopy classificationsin dimensions 10, 14, and 16, and a partial result along these lines in dimension 12.
Theorem 12.1. If M is a closed, simply connected, rationally elliptic Riemannian mani-fold with positive curvature and T symmetry, then M is rationally homotopy equivalent to S , C P , or S × H P .Proof. We may assume that M is not homeomorphic to S . By our classification in dimension10 (Theorem 5.1), χ ( M ) = 6, H ( M ; Q ) = Q , and H ( M ; Q ) is generated by a product ofelements of degree two and four. Rational ellipticity implies H i +1 ( M ; Q ) = 0 for all i , andit follows that H ∗ ( M ; Q ) ∼ = H ∗ ( C P ; Q ). Fix generators z ∈ H ( M ; Q ) and x ∈ H ( M ; Q ).By Theorem 5.1, in the case when cohomology is generated by x and z , the manifold M has the rational type of a complex projective space (corresponding to m = 0) or S × H P (when m = 0) using intrinsic formality of these spaces.In the remaining case, the element x does not vanish and generates H ( M ; Q ). Hence, byPoincar´e duality, M ≃ Q C P . (cid:3) Theorem 12.2.
Let M be a closed, simply connected, rationally elliptic Riemannian man-ifold with positive curvature and T symmetry. If χ ( M ) ∈ { , , } , then one of the followingoccurs: • If χ ( M ) = 2 , then M ≃ Q S . • If χ ( M ) = 4 , then M ≃ Q H P , M ≃ Q S × S , M ≃ Q S × S , or the rational type of M comes out of a -parameter family tensoring to the real homotopy type of S × S . • If χ ( M ) = 7 , then M ≃ Q C P . If χ ( M )
6∈ { , , } , then χ ( M ) ∈ { , , , , } and correspondingly M has the Bettinumbers and homotopy Betti numbers of one of the following spaces: • S × C P or S × H P . • S × C P , S × S × S , S × S × S , or S × S × S . • C P × H P . • S × C P . • S × C P , S × S × C P , C P × S × S , or S × S × H P .Remark . Theorem 6.2 further computes the signature in this setting, but this additionalinformation does not exclude any of the manifolds listed in the conclusion of this theorem.For example, in the case of rational homotopy generators in degrees 2, 6, 7, and 11, weconstruct one class of possible minimal models by(Λ h x, y, x ′ , y ′ i , d)with deg x = 2 , deg y = 6 , deg x ′ = 7 , deg y ′ = 11, d x = d y = 0, d x ′ = xy , d y ′ = x + y . Itsintersection form is represented by (cid:18) − (cid:19) For the configuration (2 , , , , ,
11) we may use S × S × S with the intersection matrixbeing the same with respect to the basis ([ S ] + [ S ][ S ]) , ([ S ] − [ S ][ S ]).Similarly, when χ ( M ) ∈ { , } , we obtain spaces which can be realized as manifolds bya product containing a sphere factor, i.e. by boundaries. Since the signature is a bordisminvariant, these manifolds have vanishing signatures. Proof. If χ ( M ) = 2 or χ ( M ) = 7, Table 10.4 implies that M has the rational homotopygroups of S or C P . By formality, the rational homotopy classification follows in this case.Suppose next that χ ( M ) = 4. Table 10.4 implies that M has the rational homotopy groupsof one of four types. We consider these cases one at a time. Listed first in each case is thesequence of degrees of homotopy generators. • (4 , M has the rational homotopy groups of H P and hence the samerational homotopy type. • (2 , , , M has therational homotopy type of S × S . • (4 , , , S × S . • (6 , , , M has a minimal model of the form (Λ h u, v, u ′ , v ′ i , d) withdeg u = deg v = 6, deg u ′ = deg v ′ = 11, d u = d v = 0. Since the signature vanishes,we obtain that the intersection form over the reals is represented by the matrix (cid:18) − (cid:19) This relates to the minimal model over R by determining the differentials d u ′ = u + v and d v ′ = uv up to isomorphism. Let us now investigate how many rational homotopytypes tensor to this real homotopy type. Since the product of the two elements u, v OSITIVE CURVATURE AND TORUS SYMMETRY, I 41 is exact over R if and only if it is exact over Q , we see that all the rational types ofthis space are given by(Λ h u, v, u ′ , v ′ i , d) , deg u = deg v = 6 , deg u ′ = deg v ′ = 11 , d u = d v = 0d u ′ = u + kv , d v ′ = uv, k ∈ Q and k > Q . It is easy to see thatfor k = 1, this is the model for S × S .We proceed to the proof in the second case, where χ ( M )
6∈ { , , } . Here our Eulercharacteristic calculation (Theorem 6.2) implies that χ ( M ) ∈ { , , , , } . In particular, M is an F space and hence admits a pure model. Recall from [FHT01, p. 446] that theBetti numbers of a pure space only depend on the rational homotopy groups, i.e. pure spaceswith the same rational homotopy groups have the same Betti numbers. As a result, we onlyneed to compute the dimensions of the rational homotopy groups. This information can beimmediately deduced from Table 10.4. (cid:3) Next we come to the rational homotopy classifications in dimensions 14 and 16.
Theorem 12.4. If M is a closed, simply connected, rationally elliptic Riemannian mani-fold with positive curvature and T symmetry, then M is rationally homotopy equivalent to S , C P , or S × H P .Proof. This follows directly from Theorem 7.1, since χ ( M ) = 2 for a rationally ellipticspace M n implies M n ≃ Q S n and since the rational homotopy types of C P and S × H P (corresponding to m = 0 and m = 0 in the conclusion of Theorem 7.1, respectively) areintrinsically formal. (cid:3) Theorem 12.5. If M is a closed, simply connected, rationally elliptic Riemannian mani-fold with positive curvature and T symmetry, then M is rationally homotopy equivalent toa compact, rank one symmetric space, i.e., to S , C P , H P , or CaP .Proof. From Theorem 8.1 we derive that χ ( M ) ∈ { , , , } . In the first three cases, Table10.6 implies that M has singly generated rational cohomology with generator in degree 16,8, or 4, and so the result follows by formality. Suppose χ ( M ) = 9. By Theorem 8.1 again,the fifth power of a generator x ∈ H ( M ; Q ) is non-zero. Since M is an F space, theodd Betti numbers are zero, and by this fact the even Betti numbers are equal to one. ByPoincar´e duality, M is a rational cohomology C P and hence a rational homotopy C P byformality. (cid:3) In [AK15], the authors proved the Wilhelm conjecture for spaces with rational cohomologyalgebra generated by one element. We have the following consequence.
Corollary 12.6.
The Wilhelm conjecture holds for closed, simply connected, rationally el-liptic manifolds of dimension that admit a Riemannian metric with positive sectionalcurvature and T symmetry. Biquotients
In this section, we combine our Euler characteristic and other calcultions with work ofKapovitch–Ziller, Totaro, and DeVito to provide diffeomorphism classifications for biquo-tients in dimensions 10, 14, and 16 that admit positively curved metrics with torus symme-try. In particular, we obtain a characterization of the Cayley plane in this context. We alsoconsider dimension 12 and restrict further the case of symmetric spaces to obtain a partialdiffeomorphism classification.The main additional ingredient is the classification of biquotients whose rational cohomol-ogy is four–periodic in the sense of [Ken13, Definition 1.1]. This class includes those withsingly generated cohomology, and this case was classified in Kapovitch and Ziller [KZ04].In even dimensions, there is one more possible rational cohomology ring, namely, that of S × H P m . Here we require recent work of DeVito [DeVb].To be clear, we state explicitly that in the subsequent theorem the biquotient structuredoes not have to be related to the positively curved metric. Theorem 13.1.
Let M n be a closed, simply connected biquotient that independently admitsa positively curved Riemannian metric with T r symmetry. • If n = 10 and r ≥ , then M is diffeomorphic to S , C P , S ˜ × H P , N , or N . • If n = 14 and r ≥ , then M is diffeomorphic to S , C P , S ˜ × H P , N , or N . • If n = 16 and r ≥ , then M is diffeomorphic to S , C P , H P , or CaP .Here S ˜ × H P m denotes one of the two diffeomorphism types of total spaces of H P m –bundlesover S whose structure group reduces to a circle acting linearly, and N k − and N k − denotethe free circle quotients SO(2 k +1) / (SO(2 k − × SO(2)) and ∆SO(2) \ SO(2 k +1) / SO(2 k − of the unit tangent bundle of S k described in [KZ04] .Proof. First suppose M has the rational homotopy type of S × H P or S × H P . FromDeVito’s classification we deduce that M is diffeomorphic to one of two bundles over S or with fiber H P or [ G / SO(4)] × S . For M we obtain that M is one of the tworespective H P -bundles over S . Now, in dimension ten, we know that H ( M ; Z ) = Z , whereas H ([ G / SO(4)] × S ) = Z ⊕ Z .Next suppose that the rational homotopy type is not as above. The calculations abovefor rationally elliptic spaces imply that M has the rational homotopy type of a compact,rank one symmetric space (see Theorems 12.1, 12.4, and 12.5). In particular, the rationalcohomology of M is generated by one element, so we can apply the classification of Kapovitchand Ziller (see [KZ04, Theorem 0.1]). In these dimensions, either the conclusion of Theorem13.1 holds, or M is diffeomorphic to one of the two rational complex projective spaces indimensions 10 or 14. These spaces are N and N from the conclusion of the theorem. (cid:3) A similarly strong result in dimension 12 seems out of reach. We make a few general re-marks on the structure of a 12–dimensional biquotient admitting positive sectional curvatureand T symmetry. We then restrict further to the case of symmetric spaces and provide adiffeomorphism classification in this case.Suppose that M is a closed, simply connected Riemannian manifold with positive sec-tional curvature and T symmetry. Assume moreover that M admits a possibly independentbiquotient structure, i.e., suppose that M is diffeomorphic to a quotient G (cid:12) H of G by atwo-sided action by a subgroup H ⊆ G . By the results of Kapovitch–Ziller [KZ04] or Totaro[Tot02], we may replace G and H , if necessary, so that G is a connected, simply connected OSITIVE CURVATURE AND TORUS SYMMETRY, I 43
Lie group and H is a product of a torus T r with a connected, simply connected Lie group.In other words, G = Q ki =1 G i and H = T r × Q li =1 H i , where G i and H i are compact, one–connected, simple Lie groups. Since χ ( M ) > G and H agree, so l = k − r . Moreover, an extension of the arguments in [KZ04, Section 1] or [Tot02,Lemma 3.3 and Corollary 4.6] show that G has at most three simple factors, i.e., k ≤ G i to the oddrational homotopy groups of M . This work implies that, if G i has the rational homotopytype of Q j S n j − , then the largest value of n j appearing is at most 2 n = 24. In particular,there exist finitely many such G i . Finally we recall that the quotient map G → M is afibration with fiber H , so the associated long exact homotopy sequence relates the rationalhomotopy groups of G and H (which are both concentrated in odd degrees) with those of M . In particular, this implies that r = b ( M ) = dim π ( M ) ⊗ Q ≤ Theorem 13.2.
Let M be a closed, simply connected Riemannian manifold with positivesectional curvature and T symmetry. Assume M admits a possibly independent biquotientstructure. Then there exist k ∈ { , , } and r ∈ { , } and compact, one–connected, simpleLie groups G i and H i such that π j ( G i ) ⊗ Q = π j ( H i ) ⊗ Q = 0 for j ≥ and such that M isdiffeomorphic to a biquotient of the form G (cid:12) H , where G = Q ki =1 G i and H = T r × Q k − ri =1 H i . Using the last statement together with the classification in Table 10.4 of the rationalhomotopy groups of M (with dimension 12 and Euler characteristic 2, 4, 6, 7, 8, 9, 10, or12), one could analyze the possible groups G and H appearing in the conclusion of thistheorem. Indeed, the theorem together with the classification of compact, simply connected,simple Lie groups implies that there are finitely many choices of G . Given G and M , therational homotopy groups of H follow from the long exact homotopy sequence associated tothe fibration G → M , so there are only finitely many possibilities for H . We could thereforeenumerate the finite number of possibilities for G and H , however we do not pursue thishere.Instead, we restrict ourselves further to the class of symmetric spaces. These are homo-geneous spaces and therefore biquotients, so everything above carries over. However, weproceed by a different route to get a classification in this case. Theorem 13.3.
Let M be a compact, simply connected symmetric space. Assume M ad-mits a possibly independent Riemannian metric with positive sectional curvature and T symmetry. One of the following occurs: • χ ( M ) = 2 , M is spin, and M is S . • χ ( M ) = 4 , M is spin, and M is H P or a product of the form S k × S − k . • χ ( M ) = 6 , M is spin, and M is S × H P or S × (G / SO(4)) . • χ ( M ) = 6 , M is not spin, and M is SO(7) / SO(3) × SO(4) or S × C P . • χ ( M ) = 7 , M is not spin, and M is C P . • χ ( M ) = 8 , M is spin, and M is SO(8) / SO(2) × SO(6) , Sp(3) / U(3) , S × C P , or aproduct of the form S k × S l × S − k − l . • χ ( M ) = 8 , M is not spin, and M is S × (SO(5) / SO(2) × SO(3)) . • χ ( M ) = 9 , M is not spin, and M is C P × H P or C P × (G / SO(4)) . • χ ( M ) = 10 , M is not spin, and M is S × C P . • χ ( M ) = 12 , M is not spin, and M is S × (SO(7) / SO(2) × SO(5)) or a product ofthe form S k × S − k × C P . We note that the signature in each case is 0 or ±
1, according to the parity of M . In addition,the elliptic genus is constant for all of these manifolds since they are homogeneous (seeHirzebruch–Slodowy [HS90]). These remarks are consistent with the calculation in Theorem6.2 of the Euler characteristic, signature, and elliptic genus.We also remark that Theorem 13.3 could be improved if the constant C (6) defined in theintroduction is shown to be six. Indeed, if C (6) = 6, our Euler characteristic calculation indimension 12 implies that χ ( M ) = 12, χ ( M ) = 10, and that χ ( M ) = 8 only if M is not spin. Proof.
Let M be a closed, simply connected symmetric space that admits a (possiblyindependent) Riemannian metric with positive curvature and T symmetry. By Theorem6.2, either χ ( M ) ∈ { , , , } or M is not spin and χ ( M ) ∈ { , , , } . Note in the firstcase, the theorem does not claim that M is spin, however we will see for symmetric spaces, χ ( M ) ∈ { , } only if M is spin.By the classification of symmetric spaces, M = Q ki =1 M i where each M i is a compact,simply connected, irreducible symmetric space. Since dim( M ) = 12, there are a finite num-ber of possibilities to check. In fact, the classification also implies that each M i has Eulercharacteristic zero or Euler characteristic at least two. Since χ ( M ) >
0, we conclude thateach χ ( M i ) ≥ M = M , i.e., where M is irreducible. For this case, wesimply enumerate all compact, simply connected, irreducible symmetric spaces of dimension12. We first eliminate those spaces with Euler characteristic zero (i.e., any space G/H where G and H have unequal rank). Second, we eliminate the rank two complex GrassmannianU(5) / U(2) × U(3), which has Euler characteristic 10 and is not spin, because it has signature ± / U(4) from the ta-ble, since it is diffeomorphic to SO(8) / SO(2) × SO(6) by way of low-dimensional, accidentalisomorphisms (see Ziller [Zil, Section 6.3]). The result is Table 13.1.
Table 13.1.
Irreducible symmetric spaces M with 2 ≤ χ ( M ) ≤
12 and | σ ( M ) | ≤ M χ ( M ) spin? | σ ( M ) | S H P / SO(3) × SO(4) 6 no 0 C P / SO(2) × SO(6), Sp(3) / U(3) 8 yes 0Next consider the case where M = M × M or M = M × M × M where each M i isirreducible. In these cases, 2 ≤ dim( M i ) ≤
10 and 2 ≤ χ ( M i ) ≤ χ ( M ) ≤
12. Using Ziller [Zil, Section 6.3] again,we eliminate redundancy in our list by observing the following accidental isomorphisms:
OSITIVE CURVATURE AND TORUS SYMMETRY, I 45 (1) SO(4) / SO(2) × SO(2) is diffeomorphic to S × S , and hence it is not irreducible.(2) Sp(2) / U(2) is diffeomorphic to C P .(3) SO(6) / U(3) is diffeomorphic to C P .We also record which spaces are spin using the classification of Cahen and Gutt mentionedabove. The list of possible irreducible factors is presented in Table 13.2. Table 13.2.
Irreducible symmetric spaces M n with 2 ≤ n ≤
10 and 2 ≤ χ ( M ) ≤ M dim( M ) χ ( M ) spin? S n n C P H P , G / SO(4) 8 3 yes C P / SO(2) × SO(3) 6 4 no C P / SO(2) × SO(4), U(4) / U(2) × U(2) 8 6 yesSO(7) / SO(2) × SO(5) 10 6 no C P
10 6 yesWith Table 13.2 complete, it is not difficult to enumerate the possible products M × M and M × M × M that have dimension 12 and Euler characteristic satisfying χ ( M ) ∈{ , , , } or χ ( M ) ∈ { , , , } , where the latter case only occurs if M is not spin. Hereit is useful to recall that the product of manifolds is spin if and only if each factor is spin.For example, we can exclude the products of the form S × N where N is one of the threespin manifolds in Table 13.2 with Euler characteristic six. (cid:3) References [AK] M. Amann and L. Kennard. On a generalized conjecture of Hopf with symmetry.
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Manuel AmannFakult¨at f¨ur MathematikInstitut f¨ur Algebra und GeometrieKarlsruher Institut f¨ur TechnologieEnglerstraße 2, Karlsruhe76131 KarlsruheGermany [email protected] http://topology.math.kit.edu/21_54.php
Lee KennardDepartment of MathematicsUniversity of OklahomaNorman, OK 73019-3103USA [email protected]@math.ou.edu