Moduli spaces of Ricci positive metrics in dimension five

We use the $\eta$ invariants of spin$^c$ Dirac operators to distinguish connected components of moduli spaces of Riemannian metrics with positive Ricci curvature. We then find infinitely many non-diffeomorphic five dimensional manifolds for which these moduli spaces each have infinitely many components. The manifolds are total spaces of principal $S^1$ bundles over $\#^a\mathbb{C}P^2\#^b\overline{\mathbb{C}P^2}$ and the metrics are lifted from Ricci positive metrics on the bases. Along the way we classify 5-manifolds with fundamental group $\mathbb{Z}_2$ admitting free $S^1$ actions with simply connected quotients.

The conditions on the first Chern class in Theorem A are equivalent to the statement that 1 .M 5 / D Z 2 , M 5 is nonspin, and the universal cover of M 5 is spin.M 5 can be constructed by taking five-dimensional homotopy real projective spaces, removing tubular neighborhoods of generators of the fundamental group, and gluing along the boundaries of the tubular neighborhoods.By the classification of Smale [39] and Barden [4], the universal cover z M 5 is diffeomorphic to # aCb 1 S 3 S 2 .But we do not know an explicit description of the deck group action by Z 2 on z M 5 .
Our second theorem identifies conditions under which M 5 admits one, and infinitely many, free S 1 actions.
As an application, we will show that the manifolds in Theorem A admit infinitely many free S 1 actions.We construct the metrics used in Theorem A by lifting metrics from the quotients of M 5 by those actions.Here b 2 .M / is the second Betti number of M.
Theorem B Let M 5 be a 5-manifold with 1 D Z 2 .Then M admits a free S 1 action with a simply connected quotient if and only if M is orientable, H 2 .M; Z/ is torsion-free and 1 .M / acts trivially on 2 .M /.Furthermore, if b 2 .M / D 0, then M is diffeomorphic to RP 5 .If b 2 .M / > 0 and M admits a free S 1 action with simply connected quotient B 4 , then M admits infinitely many inequivalent free S 1 actions with quotients diffeomorphic to B 4 .
Note that here B 4 can be any simply connected 4-manifold, and need not be one of the manifolds of Theorem A. Theorem 1.11 provides greater detail about the correspondence between a 5-manifold M 5 and the set Q.M / of possible quotients B 4 D M 5 =S 1 .Given M 5 satisfying the hypotheses of Theorem B, we give conditions on the cohomology ring of a 4-manifold B 4 which are necessary and sufficient for B to be in Q.M /.In particular, any smooth manifold homeomorphic to a manifold in Q.M / is in Q.M /.
In Corollary 1.12 we see that for any such M, Q.M / contains either # c S 2 S 2 or # a CP 2 # b CP 2 for some a; b; c 2 Z.Those manifolds admit metrics with positive Ricci curvature, which can be lifted to M. Thus we have: Corollary Let M be a 5-manifold with 1 .M / D Z 2 admitting a free S 1 action with a simply connected quotient.Then M admits a metric with positive Ricci curvature.
Furthermore, it follows from Theorem 1.11 that given a simply connected 4-manifold B 4 , the set of diffeomorphism types of total spaces M 5 with 1 .M 5 / D Z 2 of S 1 bundles over B 4 depends only on the cohomology ring of B 4 .In particular, Theorem A would describe the same set of 5-manifolds if we replaced # a CP 2 # b CP 2 with one of the manifolds homeomorphic to it.
We first review previous work with methods and results relevant to Theorem A. In [30] Kreck and Stolz invented a moduli space invariant s.M; g/ 2 Q for a metric g of positive scalar curvature on a closed spin manifold M. The metric is based on the Á spectral invariant of the Dirac operator defined in Atiyah, Patodi and Singer [1].If s.M; g 1 / ¤ s.M; g 2 / then g 1 and g 2 represent elements in different path components of M scal>0 .Kreck and Stolz use the invariant to prove that for M 4kC3 with a unique spin structure and vanishing rational Pontryagin classes M scal>0 .M / is either empty or has infinitely many components.
Since a path of Riemannian metrics which maintains positive Ricci curvature maintains positive scalar curvature as well, the s invariant can detect connected components of M Ric>0 .Kreck and Stolz calculated s for the Einstein metrics on S 1 bundles N 7 k;l over CP 1 CP 2 described by Wang and Ziller [41].Kreck and Stolz showed, using the diffeomorphism classification in [28], that when k is even and gcd.k; l/ D 1, N k;l is diffeomorphic to infinitely many manifolds in the same family.As the s invariant takes infinitely many values on those metrics, the authors concluded that M Ric>0 .N k;l / has infinitely many components.Similar results have since been proved for S 1 bundles over CP 1 CP 2n with n 1; see Dessai, Klaus and Tuschmann [15].
Wraith showed that for a homotopy sphere 4k 1 bounding a parallelizable manifold, M Ric>0 ./ has infinitely many components.The procedure known as plumbing with disc bundles over spheres produces infinitely many parallelizable manifolds with boundaries diffeomorphic to .Wraith [43] constructed metrics of positive Ricci curvature on each boundary, and calculated the s invariant of each metric in [45].
Dessai [13] and the author [21] used the s invariant to find several infinite families of seven-dimensional sphere bundles M 7 such that M Ric>0 .M / and M sec 0 .M / have infinitely many path components.Grove and Ziller [22; 24] constructed metrics of nonnegative sectional curvature on the manifolds in those families, and the diffeomorphism classifications in Crowley and Escher [12] and Escher and Ziller [18] show that each manifold is diffeomorphic to infinitely many other members of the family.
More recently, Dessai and González-Álvaro [14] showed that if M 5 is one of the four closed manifolds homotopy equivalent to RP 5 then M sec 0 .M / and M Ric>0 .M / have infinitely many path components.López de Medrano [32] showed that each such M 5 admits infinitely many descriptions as a quotient of a Brieskorn variety, and Grove and Ziller [23] showed the each quotient admits a metric of nonnegative sectional curvature.Dessai and González-Álvaro calculated the relative Á invariant for those metrics to distinguish the path components.Wermelinger [42] extended their method to prove the same conclusion for five Z 2 quotients of S 2 S 3 .
We now outline the proof of Theorem A. We use Theorem B to show that each manifold M 5 in Theorem A admits infinitely many inequivalent free S 1 actions with quotient B 4 D # a CP 2 # b CP 2 .We modify a result of Perelman [34] to show that B admits a metric of positive Ricci curvature.That metric can be lifted to a metric of positive Ricci curvature on M by Gilkey, Park and Tuschmann [20].The lifted metrics depend on the S 1 action, and we get infinitely many distinct metrics on M.
We show that in dimensions 4k C 1, the Á invariant of a certain spin c Dirac operator constructed for a positive Ricci curvature metric g depends only on the connected component of the class of g in M Ric>0 .
To complete the proof we calculate Á for each metric on M and show that it obtains infinitely many values.This is the most intricate part of our proof.
The standard method for calculating the Á invariant of a spin Dirac operator on a manifold M with positive scalar curvature is to extend the metric over a manifold W with @W D M so that the extension has positive scalar curvature as well.When M is not spin but spin c , both the metric and a unitary connection on the complex line bundle associated to the spin c structure must be extended.The desired condition then involves the curvatures of both metric and connection.In their work, Dessai and González-Álvaro passed to the universal cover to find a suitable W over which the connection could be extended to a flat connection.They use equivariant Á invariants on the cover to compute the Á invariant on the quotient.
In this paper we work directly on M and use a manifold with boundary W over which the connection cannot be extended to a flat connection, but the curvature of the extension can be explicitly controlled.To be specific, we extend the metric and connection on M to a metric h and connection r on the disc bundle W D M S 1 D 2 associated to the S 1 bundle.We then use the Atiyah-Patodi-Singer index theorem [1] to obtain a formula for Á in terms of the index of the spin c Dirac operator on W and topological data on W. The index will vanish as long as scal.h/> 2jF r j h ; where F r is the curvature form of the connection r.We accomplish the extension for a general class of S 1 -invariant metrics of positive scalar curvature.This is more general than we need but may be of independent interest.In fact we construct h and r such that scal.h/> `jF r j h ; where `is a positive integer such that the first Chern class of the S 1 bundle is `times the canonical class of a spin c structure on the quotient.
Sha and Yang [38] constructed metrics of positive Ricci curvature on the 4-manifolds # a b CP 2 # b S 2 S 2 with a > b.Those manifolds are diffeomorphic to # a CP 2 # b CP 2 , so a manifold M satisfying the hypotheses of Theorem A also admits a free S 1 action with quotient # a b CP 2 # b S 2 S 2 .One can lift the Sha-Yang metric to M, and there is no reason to expect that the resulting metric lies in the same component as the metric lifted from # a CP 2 # b CP 2 in the proof of Theorem A. We will see, however, that the computation of the Á invariant involves only the cohomology ring of the quotient, and we cannot distinguish any new components in this way.
In [37] Sha and Yang also found metrics of positive Ricci curvature on # b S 2 S 2 .One might expect our methods to yield a similar result in this case.The 5-manifolds, however, would be spin, and the Á invariant of the spin Dirac operator in dimension 4k C 1 vanishes, even when twisted with certain complex line bundles; see Botvinnik and Gilkey [7].
We now discuss Theorem B. In [26], Hambleton and Su find a complete diffeomorphism classification of 5-manifolds M with 1 .M / D Z 2 when M is orientable, H 2 .M; Z/ is torsion-free, and 1 .M / acts trivially on 2 .M /.They apply the classification to investigate the diffeomorphism type of the total space of an S 1 bundle over a simply connected 4-manifold.When the total space is nonspin but has a spin universal cover, as is the case in Theorem A, they can only restrict the diffeomorphism type to two possibilities.Furthermore, an error is present in that calculation, which we correct in Lemma 1.7.
To prove Theorem B, we use the data of a principal S 1 bundle, namely the base and the first Chern class, to compute the diffeomorphism invariants used by Hambleton and Su for the total space.One, the second Betti number, is calculated easily.When the total space is nonspin but has a spin universal cover, we show how the other invariant can be computed by applying a map from to the base.While a two-fold ambiguity remains in determining which diffeomorphism type corresponds to a specific first Chern class, we are nonetheless able to determine which pairs of invariants are achieved, and achieved infinitely many times, by bundles over a given 4-manifold.
The paper is organized as follows.In Section 1 we examine S 1 actions on 5-manifolds with 1 D Z 2 and prove Theorem B. In Section 2 we discuss the Á invariant of a spin c Dirac operator and show that it can be used to detect connected components of the moduli space in the context of Theorem A. In Section 3 we compute Á in the case of certain .4nC1/-manifoldsadmitting free S 1 actions and prove Theorem A. In Section 4 we construct the metrics and connections used in the computations of Section 3. We will take the data of a principal S 1 bundle, namely the base and the first Chern class, and identify the diffeomorphism type of the total space.In particular, we will identify when the total space satisfies the hypotheses of Theorem 1.2, and then compute b 2 and OEP .That computation combined with the classification of Type I and II total spaces in [26, Theorems 6.5 and 6.8] finishes the proof of Theorem 1.11, which in turn implies Theorem B.
A straightforward computation using the long exact homotopy and Gysin sequences proves the following; see for instance [ The condition w 2 .TB/ D d mod 2 implies the existence of a spin c structure on B. We call d the canonical class of that spin c structure.On a simply connected manifold a spin c structure is uniquely determined by its canonical class.Thus in the Type III case, given a simply connected spin c 4-manifold B 4 with primitive canonical class d , we want to know the diffeomorphism type of the total space M 5 of the S 1 bundle over B 4 with first Chern class 2d.Since b 2 .M / is determined by Lemma 1.3, it remains to find the pin C cobordism class of a characteristic submanifold P 4 M 5 .In fact, the spin c structure on B 4 will naturally induce a pin C structure on P 4 .
To see this let W M !B be the bundle map and let !B be a complex line bundle with first Chern class d; then d is the unique nontrivial torsion element of H 2 .M; Z/.Let !M be the unique nontrivial real line bundle over M. As in the proof that a characteristic submanifold of M will admit a pin C structure -see [ In the next lemma, we will see that ˇis a spin c cobordism invariant whenever it is defined.Lemma 1.6 Let B 1 and B 2 be spin c manifolds with primitive canonical classes d 1 and d 2 , respectively.Then: Proof Part (a) follows immediately since the total space of the relevant bundle and the characteristic submanifold of that total space will be disjoint unions.
To prove part (b), let W be a simply connected spin c cobordism between B 1 and B 2 with canonical class d.Then dj B i D d i for each i D 1; 2, and d must be a primitive class.Let W N !B be the principal S 1 bundle over W with first Chern class 2d.By Lemma 1.3, 1 .N / D Z 2 .We have that @N D 1 .B 1 / q 1 .B 2 / and M i D 1 .B i / !B i is the principal S 1 bundle with first Chern class 2d i .
Let f W N !RP N be a classifying map for the universal cover of N which is transverse to RP N 1 .By Lemma 1.3, 1 .N / is generated by any S 1 orbit, so 1 .M i / ! 1 .N / is an isomorphism, and f j M i is a classifying map for the universal cover of M i .Thus / is a cobordism between P 1 and P 2 .The argument before Lemma 1.5 proves that the spin c structure on W induces a pin C structure on f 1 .RP N 1 /.That pin C structure restricts to the pin C structures induced on P i by the spin c structures on B i .To see this one must simply note that the nontrivial real line bundle over N restricts to the nontrivial real line bundle over M i .We conclude that ˇ.B 1 ; We now see that ˇdefines a map between the spin c and pin C cobordism groups.The four-dimensional spin c cobordism group The isomorphism takes a spin c manifold B with canonical class d to the characteristic numbers hd 2 ; OEBi and 1 8 .hd 2 ; OEBi sign B/: Here sign.B/ is the signature, and the second integer is the index of the spin c Dirac operator, which we denote by ind.B; d/.See [3; 40] for details.To construct generators of Spin c 4 let x 2 H .CP 2 ; Z/ be the generator which is the first Chern class of the Hopf bundle.Give X D CP 2 the spin c structure with canonical class x and represent .1;0/ and .9;1/ under the isomorphism with Z 2 and form a minimal generating set of Spin  This lemma corrects a mistake in the statement of [26,Theorem 6.7].Our argument uses ideas from the proof in [26] as well as corrections suggested to the author by Yang Su.
Proof We will see that ˇ.X; x/ D 1 and ˇ.Y; d Y / D 5 or 13.The lemma then follows since ˇis a homomorphism and The principal S 1 bundle RP 5 !CP 2 which is a Z 2 quotient of the Hopf bundle has first Chern class 2x.
Since RP 4 is a characteristic submanifold of RP 5 , it follows that The second calculation is more involved.We use the notation OEz 0 ; z 1 ; z 2 2 CP 2 and OEz 0 ; z 1 ; z 2 ˙2 RP 5 for the respective images of the point .z0 ; CP 2 be a classifying map for z which is transverse to CP 1 CP 2 and has a regular value OE1; 0; 0 2 CP 1 .Then Since the fundamental groups of M and RP 5 are generated by S 1 orbits (see Lemma 1.3), the homomorphism f W 1 .M / ! 1 .RP 5 / is an isomorphism and f is a classifying map for the double cover z M !M .Thus if we show that f is transverse to RP 4 RP 5 , we can conclude that P D f 1 .RP 4 / is a characteristic submanifold of M .Then given the correct pin C structure on To see that f is transverse to RP 4 D fOEz 0 ; z 1 ; r ˙2 RP 5 j r 2 Rg note that at points in 1 .CP 2 nCP 1 /, RP 4 is transverse to the S 1 orbits, which are contained in the image of the equivariant map f .At points in 1 .CP 1 /, we associate the horizontal space of the S 1 action with T CP 2 .By assumption on g, f is transverse to T CP 1 , and For later, we also note that f is transverse to RP 2 D fOEz 0 ; r; 0 2 RP 5 j r 2 Rg since T CP 1 T RP 2 except at OE1; 0; 0, which is a regular value of f by assumption on g.
There is a short exact sequence where is given by taking the cobordism class of a submanifold dual to w 2 1 ; see [26, page 172] and [27, page 217] for details.Thus Pin 2 is isomorphic to Z 8 with generator OERP 2 .We now compute .OEP / D 5, which restricts the possible values of ˇ.Y; d Y / D 5 or 13, as desired.
We need to find a submanifold of P dual to w 2 1 .TP /.Denote by N RP 4 the normal bundle of RP 4 in RP 5 and by NP the normal bundle of P in M. Then f N RP 4 D NP .Since RP 5 and M are orientable, Since OE1; 0; 0 is a regular point of g, OE1; 0; 0 ˙is a regular point of f , and the degree of f is the same as the degree of f j † .
The degree of f is the same as the degree of g.The degree of g is given by Thus the mod 2 degree of Let U be a tubular neighborhood of the S 1 orbit of OE1; 0; 0 ˙and V D RP 2 nU.Since OE1; 0; 0 is a regular value of g we can choose U to be made up of regular values of f .Then The S 1 orbit of OE1; 0; 0 ˙is a nontrivial loop in RP 2 , and U \ RP 2 is a tubular neighborhood of that loop, diffeomorphic to RP 2 nD 2 (the Möbius band).The local inverses to f j f 1 .U / are equivariant embeddings of the oriented tubular neighborhood U and are all isotopic.It follows that the 9 embedding of RP 2 nD 2 making up f 1 .U \ RP 2 / are all isotopic.Thus the process by which TM induces a pin C structure on P , which in turn induces a pin structure on †, will induce the same pin structure on each of the 9 copies of RP 2 nD 2 .
Since .RP 2 / D CP 1 and .U / \ CP 1 is diffeomorphic to a disc D 2 around OE1; 0; 0 made up of regular values of g, g 1 ..U /\CP 1 / is 9 copies of D 2 and .V / D CP 1 nD 2 .j RP 2 is injective away from the orbit of OE1; 0; 0 ˙, and thus is injective on V.It follows that maps f 1 .V / injectively onto g 1 ..V //.Thus f 1 .V / is diffeomorphic to g 1 .CP 2 / with 9 discs removed while f 1 .U \ RP 2 / is 9 copies of RP 2 nD 2 .In other words, and the nine summands of RP 2 all have the same pin structure.Pin 2 is generated by OERP 2 , and so it remains to compute the value of OEg 1 .CP 1 /.
Let D g 1 .CP 2 /.We will use a general method to define a pin structure called r on and compute OE 2 Pin 2 with this structure.We will then show that r is the correct pin structure to use, that is, r is compatible under (1.9) with the pin structure used to identify OE † with .OEP /, which we will call r .
Consider a simply connected spin c 4-manifold B with canonical class d and the complex line bundle with c 1 ./ D d.Let N B be a smooth submanifold dual to d.Then j N is isomorphic to the normal bundle of N .The spin c structure on B is equivalent to a spin structure, called s, on TB ˚ .Restricted to N , this is a spin structure on T N ˚2 .The transition functions for 2 admit a canonical lift from SO.4/ to Spin.4/; simply multiply two copies of any lift for the transition functions of , and the sign ambiguities cancel.Note that the identity lifts to the identity in this way.Using this lift, s induces a spin structure s N on N .
The spin cobordism class of N depends only on the spin c cobordism class of B. To see this, note that the dual to the canonical class of a spin c cobordism will be a spin cobordism between the two relevant submanifolds.Thus we have a homomorphism Thus on .O/ the transition functions for can be chosen to be the identity and the transition functions for .T Y ˚ /j can be chosen to be t ij .The spin structure s gives a lift of t ij to z t ij in Spin.2/.Since the canonical lift of the transition functions for 2 will also be the identity, z t ij is also the lift given by s and r .Furthermore, using (1.4), t ij ı are transition functions for E on O.By definition, s E gives the lift z t ij ı .Using (1.10), t ij ı are transition functions for both Ej O and T †, compatible by picking trivial transition functions for 5 det.T †/.The canonical lift of the transition functions for 5 det.T †/ will also be trivial, and the lift given by r will simply be the inclusion of z t ij ı into Pin .2/.Thus r D r on O.This completes the proof of Lemma 1.7.
We can now prove Theorem B. In fact, we prove the following more detailed theorem, which includes the statement of Theorem B.Here we use the notation of Hambleton and Su, where # S 1 is gluing along the boundary of a tubular neighborhood of a generator of 1 .The X.q/ for q D 1; 3; 5; 7 are the four closed manifolds homotopy equivalent to RP 5 , with X.1/ D RP 5 , and the X.q/ for q D 0; 2; 4; 6; 8 are constructed from pairs of homotopy RP 5 's using the operation # S 1 .The labeling is such that a characteristic submanifold P X.q/ has class q 2 Pin C 4 =˙D f0; : : : ; 8g.See the discussion before [26,Theorem 3.7] for details.
Theorem 1.11 Let M be a 5-manifold with 1 D Z 2 .Let P M be a characteristic submanifold.
(1) M admits a free S 1 action with a simply connected quotient if and only if M is orientable, H 2 .M; Z/ is torsion-free, and 1 .M / acts trivially on 2 .M /.Furthermore if b 2 .M / D 0 then M is diffeomorphic to RP 5 .X.q/ # S 1 .CP2 S 1 / with q D 0; 4 B is nonspin, b 2 D 3 and jsign Bj D 1.
Table 1 (2) Suppose M 5 satisfies the conditions in (1).Let Q.M / be the set of quotients of M by free S 1 actions.Table 1 gives necessary and sufficient conditions for a 4-manifold to be in Q.M /.
S is a set of four exceptional 5-manifolds of Type I described in the final two rows.If b 2 .M / > 0 then for each B 2 Q.M /, M admits infinitely many inequivalent S 1 actions with quotients diffeomorphic to B.
Thus given M 5 satisfying the hypotheses of (1) and matching the description of one of the rows in the left column, a 4-manifold B 4 is diffeomorphic to a quotient of M 5 by a free S 1 action if and only if it satisfies the conditions given in the corresponding row of the right column.
Proof We prove (2) first.Let M be an orientable and some d i must be even.Since d is primitive, some d i must be odd.One easily checks that under these conditions, hd 2 ; OEBi ¤ 0; 4 mod 8.So sign.B/ D ˙1.
If M D X.q/# S 1 .S 2 RP 3 / with q D 0; To prove the converse, suppose M is an orientable 5-manifold with 1 .M / D Z 2 acting trivially on 2 .M / and H 2 .M; Z/ torsion-free.Let P M be a characteristic submanifold.Since RP 5 admits a free S 1 action induced by the Hopf action we assume b 2 .M / > 0. We must show the set Q.M / described in Table 1 We note that the final paragraph of the proof above in fact shows the following, which we will make use of later.
Corollary 1.12 Let M be a 5-manifold with 1 D Z 2 admitting a free S 1 action with a simply connected quotient.Then M admits a free S 1 action with quotient diffeomorphic to either # c S 2 S 2 or # a CP 2 # b CP 2 for some a; b; c 2 Z.
Combining Theorem 1.11 with [26, Theorem 3.7], we can characterize the manifolds satisfying Theorem A.
Corollary 1.13 Let M 5 be a 5-manifold.The following are equivalent: (1) M 5 is Type III and admits a free S 1 action with a simply connected quotient.

Remark 1.14
By the discussion preceding [26,Theorem 3.7] the manifolds described in Corollary 1.13 can also be constructed by applying # S 1 to the homotopy RP 5 's.For instance, It is shown in [14] that M Ric>0 also has infinitely many components for the homotopy RP 5 's X.3/; X.5/ and X.7/.
If we fix a nonspin simply connected 4-manifold B 4 , then a Type III total space of a principal S 1 bundle over B will be diffeomorphic to X.q/# S 1 .#k .S 2 S 2 / S 1 /.Using Table 1 we see that q must satisfy q D ˙sign B mod 4. It follows that there are 2, 3 or 4 choices of q, and the same number of diffeomorphism types of Type III total spaces, if sign.B/ is 2, 0 or ˙1 mod 4, respectively.The value of q can be determined, up to two possibilities, using Lemma 1.7.The set of diffeomorphism types of Type I total spaces is more complicated, but can be computed using Theorem 1.11 and [26,Theorem 3.7].If B 4 is a simply connected spin 4-manifold, there exists a unique diffeomorphism type of total spaces with Using a result of Gilkey, Park and Tuschmann, we can lift metrics from the quotients described by Corollary 1.12 to prove the following: Corollary 1.15 Let M be a 5-manifold with 1 .M / D Z 2 admitting a free S 1 action with a simply connected quotient.Then M admits a metric with positive Ricci curvature.
Proof In [37] Sha and Yang put a metric of positive Ricci curvature on # c S 2 S 2 .A modification of Perelman's construction in [34] puts such a metric on # a CP 2 # b CP 2 ; see Lemma 3.10.Corollary 1.12 shows that M 5 admits a free S 1 action with quotient B 4 diffeomorphic to one of those manifolds.Gilkey, Park and Tuschmann [20] showed that if B 4 admits Ric> 0, M 5 is the total space of a principal bundle over B 4 with compact connected structure group G, and 1 .M 5 / is finite, then M admits a G-invariant metric with Ric > 0. In this case G D S 1 , 1 .M / D Z 2 and the corollary follows.
The corresponding result in the simply connected case was proved by Corro and Galaz-Garcia in [11].By Lichnerowicz's theorem, many simply connected 4-manifolds, such as a K3 surface, do not admit even positive scalar curvature.It is interesting to note that Corollary 1.15 and the results of [11] imply that total spaces with 1 D 0 or Z 2 of principal S 1 bundles over such manifolds nonetheless admit metrics of positive Ricci curvature.

The Á invariant
We use the Á invariant of the spin c Dirac operator, which we define in this section, to distinguish components of geometric moduli spaces.A manifold M is spin c if there exists a complex line bundle over M such that the frame bundle of TM ˚ , a principal SO.n/ U.1/ bundle, lifts to a principal Spin c .n/D Spin.n/ Z 2 U.1/ bundle.A manifold is spin c if and only if the second Stiefel-Whitney class w 2 .TM / is the image of an integral class c 2 H 2 .M; Z/ under the map H 2 .M; Z/ !H 2 .M; Z 2 /.In this case c, which we call the canonical class of the spin c structure, is the first Chern class of , which we call the canonical bundle.
Using complex representations of Spin c .n/ we form spin c spinor bundles and equip them with actions of the complex Clifford algebra bundle C`.TM /.When the dimension of M is even there is a unique irreducible such bundle S with a natural grading S D S C ˚S .Given a metric g on M and a unitary connection r on , we can construct a spinor connection r s on S , compatible with Clifford multiplication, and a spin c Dirac operator D c g;r acting on sections of S .See [31,Appendix D] for details.The Bochner-Lichnerowicz identity for this operator is where the complex two-form F r is the curvature of r.This form acts on the spinor bundle S by way of the vector bundle isomorphism ƒT M !ƒTM !C`.TM / given by g.The operator .rs / r s is nonnegative definite with respect to the L 2 inner product on a closed manifold or a compact manifold with boundary on which the Atiyah-Patodi-Singer boundary conditions have been applied.See [2, Theorem 3.9] for details.The remaining term 1  4 where the norm j j g is the operator norm on C`.TM / acting on S. In particular, ker.D c g;r / D 0 if (2.2) is satisfied.For a later purpose we note that for ! 2 2 .M; C/ and an orthonormal basis fe i g of TM with respect to g, we have .2.3/ j!j g Ä X i<j j!.e i ; e j /j: Suppose W is a spin c manifold with boundary @W D M , with and c defined on W as above.W induces a spin c structure on M with canonical class cj @W and canonical bundle j @W .Choose a metric h on W and a connection r on which are product-like near @W , ie h D hj @W C dr 2 and r D proj M .rj@W / on a collar neighborhood U Š M I , where I is an interval with coordinate r .Applying the Atiyah-Patodi-Singer boundary conditions, the Atiyah-Patodi-Singer index theorem [1] states that hj @W ;rj @W // C Á.D c hj @W ;rj @W / : Here c 1 .r/and p.g/ are the Chern-Weil Chern and Pontryagin forms constructed from the curvature tensors of the connection and metric, respectively.y A is the polynomial in the Pontryagin forms and D c hj @W ;rj @W is the spin c Dirac operator on M constructed using the induced metric and connection.
The Á invariant is an analytic invariant of the spectrum of an elliptic operator defined in [1].Given an elliptic differential operator D with spectrum f i g, we define a complex function sign.i /j i j s : One shows that the function is analytic when the real part of s is large, and Atiyah, Patodi and Singer showed that it can be analytically continued to a meromorphic function which is analytic at 0. Thus we define Á.D/ D Á.D; 0/.If a diffeomorphism preserves the spin c structure, then D c g; r is conjugate to D c g;r , and hence they have the same spectrum and the same values of Á.We will use (2.4) to calculate Á for an operator D g; x r on a manifold M by finding a suitable W with @W D M and extending g and x r to product-like h and r on W .
Kreck and Stolz combined the Á invariant with information about the Chern-Weil forms of the metric to get an invariant for metrics on .4nC3/-dimensionalspin manifolds.We prove that the Á invariant alone provides the desired invariant for certain .4nC1/-dimensionalspin c manifolds.
Theorem 2.5 Let M 4nC1 be a closed spin c manifold with canonical class c 2 H 2 .M; Z/ and canonical bundle .Suppose c and the Pontryagin classes p i .TM / are torsion and g t , where t 2 OE0; 1, is a smooth path of metrics on M with scal.g t / > 0. If r 0 and r 1 are flat unitary connections on , then Proof Modifying g t if necessary, we assume it is a constant path for t near 0 and 1.Given L 2 R >0 , define a smooth metric g on M OE0; 1 by Then g is product-like near M f0; 1g.One sees that scal.g/differs from scal.g t / by terms depending on the second fundamental form of each slice M ftg, but the second fundamental form tends to 0 as L ! 1, so for large L we have scal.g/> 0.
The difference of unitary connections on a complex line bundle is an imaginary one-form.Define ˛2 .M / such that i ˛D r 1 r 0 : Since both connections are flat, d˛D 0. Let W M OE0; 1 !M be the projection and let f W M OE0; 1 !OE0; 1 be the projection onto OE0; 1 followed by a smooth function which is 0 in a neighborhood of 0 and 1 in a neighborhood of 1. Define a connection on by Let e i be an orthonormal frame for g at a point .p;t/, such that e 1 D .1=L/@t .Then ˛.e i /: Since e i for i > 2 is tangent to M ftg, it does not depend on L. Using (2.3), for large L we have scal.g/> 2jF r j g : McFeely Jackson Goodman The definition of f ensures that r is product-like near @.M I /.Then by (2.1), D c g;r has trivial kernel and ind.D c g;r j S C / D 0.
Since F r i D 0 for i D 1; 2, scal.g i / > 0 D 2jF r i j g i and hence (2.1) implies ker D c g i ;r i D f0g.We now apply the Atiyah-Patodi-Singer index theorem (2.4).The boundary of M I is two copies of M with opposite orientations.The spectrum of the Dirac operator on M f0; 1g is the union of the spectra on M f0g and M f1g, and the Á invariant is the sum of the two Á invariants.When we change the orientation of an odd-dimensional manifold, the Dirac operator changes by a sign.Thus the Atiyah-Patodi-Singer theorem yields Since 1 c is torsion, c 1 .r/ is exact.Because r is flat near the boundary, c 1 .r/j@.M I / D 0. Furthermore, g is product-like near the boundary so p.g/j M fig D p.g i /.Since the real Pontryagin classes of M vanish, p j .gi / is exact for j > 0. By Stokes' theorem, and since the dimension of M is 4n C 1, the integral vanishes.
As a corollary we show how to use the Á invariant to detect path components of moduli spaces of metrics with curvature conditions no weaker than positive scalar curvature.
Corollary 2.6 Let M be as in Theorem 2.5.Let .gi ; r i / be a sequence of Riemannian metrics g i with Ric.g i / > 0, and flat connections r i on such that fÁ.D c g i ;r i /g i is infinite.Then M Ric>0 .M / and M scal>0 .M / have infinitely many path components.
Proof Let Diff c .M / be the set of diffeomorphisms of M which fix the spin c structure.For g 2 R scal>0 let OEg represent the image in M scal>0 and OEg c the image in R scal>0 =Diff c .M /.It follows from Ebin's slice theorem [17; 9] that if OEg i and OEg j are in the same connected component of R scal>0 =Diff c .M / then g i and g j are in the same path component of R scal>0 for some 2 Diff c .M /.Then there is a path between them maintaining positive scalar curvature, and by Theorem 2.5 and the spin c diffeomorphism invariance of Á we have Á.D c g i ;r i / D Á.D c g j ; r j / D Á.D c g j ;r j /.Since fÁ.D c g i ;r i /g is infinite, R scal>0 =Diff c .M / has infinitely many components.
Any diffeomorphism pulls back the spin c structure to another one with canonical class c, a torsion class in H 2 .M; Z/.There are finitely many such classes.The finite group H 1 .M; Z 2 / indexes the spin c structures associated to each class.Thus the orbit of the spin c structure under Diff.M / and the set Diff.M /=Diff c .M / are finite.The fibers of R scal>0 =Diff c .M / !M scal>0 are no larger than Diff.M /=Diff c .M /, implying that M scal>0 has infinitely many components.
The proof is identical for M Ric>0 since Ric > 0 implies scal > 0.
3 The Á invariant in dimension 4n C 1 with free S 1 actions In this section we prove Theorem A. We want to use the Atiyah-Patodi-Singer index theorem to calculate the Á invariant of a metric on M.Many authors have computed Á and related invariants on spin manifolds M by extending metrics to manifolds W with boundary diffeomorphic to M. If the extension has positive scalar curvature, the index of the Dirac operator will vanish.In the spin c case, we must also extend an auxiliary connection.A difficulty arises when the extended connection cannot be flat because the canonical class of the spin c structure on W is not torsion.Then the metric and connection must satisfy (2.2).The following theorem, which we prove in Section 4, illustrates how to use certain free S 1 actions on M to accomplish this.Furthermore there is a collar neighborhood V Š M OE0; N of @W Š M such that for t 2 OE0; N near 0, g W is a product metric where x r is any flat unitary connection on j @W .
Notice that here there are no restrictions on the dimension or Pontryagin classes of M, d need not be primitive, and no spin c structure is required.We next use Theorem 3.1 and (2.4) to calculate Á for S 1 -invariant metrics on certain spin c manifolds in dimensions 4n C 1.
Theorem 3.5 Let S 1 act freely on a 4n C 1 manifold M by isometries of a Riemannian metric g with scal.g/> 0. Assume 1 .M / is finite and let B D M=S 1 be the quotient.Suppose the first Chern class of the principal bundle S 1 !M !B is `d, where `is a positive even integer and w 2 .TB/ D d mod 2.
To apply Lemma 3.8 to (3.7), let ˛D e c 1 .r/=2and ˇD y A.p.g W //. Since g W is product-like near the boundary, p i .gW /j @W D p i .gM /.For i > 0, p i .gM / is exact by the assumption on the Pontryagin classes of M. Since c 1 .r/j@W D c 1 .x r/ and x r is flat, we can choose y ˛D 0. Multiplying and isolating terms of degree four yields (3.6).
We are now ready to prove Theorem A. We first construct metrics of Ric > 0 on # a CP 2 # b CP 2 .Perelman [34] constructed a metric with Ric > 0 on arbitrary connected sums of CP 2 with its standard orientation.More details on Perelman's proof can be found in [10; 8].With a slight adjustment to the construction one can reverse the orientation on some of the copies of CP 2 , proving the following.
Lemma 3.10 # a CP 2 # b CP 2 admits a metric with positive Ricci curvature for all a and b.
Proof In [34], Perelman puts a metric on # c CP 2 for all values of c.The construction involves c copies of CP 2 attached to a central S 4 by "necks" S 3 I .The metric on the necks is of the form where t is the coordinate on the interval I ; see [34, page 159].Furthermore, S 3 is represented as the product of S 2 and an interval with the top and bottom each identified to a point, and x is the coordinate on that interval, while d 2 is the standard metric on S 2 .
An orientation-reversing isometry of d 2 , such as the antipodal map, extends naturally to a diffeomorphism of W S 3 !S 3 , which induces an isometry of ds 2 .Let c D a C b, and take Perelman's metric on # c CP 2 .For b of the necks, we cut along a copy of S 3 and reglue with rather than the identity.Because reverses orientation, the resulting manifold is # a CP 2 # b CP 2 .Because induces an isometry on S 3 I , the same metrics on the pieces extend smoothly over the gluing, completing the proof of the lemma.
Let M 5 satisfy the hypotheses of Theorem A. By Lemma 1.3, M is Type III and by Theorem 1.11 M is the total space of infinitely many nonisomorphic principal From the proof of Theorem 1.11 we see that the first Chern class of k is 2d k , where for a certain infinite set of integers k.
Using the result of [20]

Metric and connection
In this section we prove Theorem 3.1.We first set up notation for the tangent space to W. We consider D 2 to be the unit disc in C. Let W M D 2 !W be the quotient map so .p;x/ D OEp; x.Then .OEp; x/ D .p/.The metric g M and the S and V OEp;x are smooth distributions on W. Note that V OEp;x is the tangent space to the fiber 1 ..p//D .fpgD 2 / and T OEp;x W D H OEp;x ˚VOEp;x .Away from the zero section of , V OEp;x is spanned by W r D .0;@ r / and W Â D .0;@ Â /: These are well-defined smooth vector fields since @ Â and @ r are S 1 -invariant vector fields on D 2 .
Fix 0 < L < 1 and define a diffeomorphism be the action field of the S 1 action on M, which spans x V p .Then, since .X ; @ Â / D 0, .X ; 0/ D W Â : Furthermore, j M f1g identifies M and @W, sending x H p to H OEp;1 and X to W Â .
We keep track of the maps in the following diagram: To construct g W and r we will use two smooth functions on the interval OE0; 1.Let f 1 W OE0; 1 !OE0; 1 be a smooth monotone function which is 0 in a neighborhood of 0, and 1 in a neighborhood of OEL; 1.
One easily sees that f 2 > 0 on .0; 1 for small .

Metric
We define a Riemannian metric at a point .p;.r;Â // 2 M D By converting to Cartesian coordinates on D 2 , one sees that g M D 2 is smooth as long as Ã is a smooth function of r 2 OE0; 1.This is easily seen to hold since for r near 0, f 2 .r/ D r r 3 .Since g M D 2 is invariant under the diagonal action of S 1 on M D 2 , it induces a metric g W on W such that g M D 2 and g W make into a Riemannian submersion.Similarly, let g B be the metric on B such that g M and g B make into a Riemannian submersion.Proof With respect to g M D 2 , x H p ˚f0g is orthogonal to X and TD 2 .Thus x H p ˚f0g is orthogonal to the vertical space of , which is spanned by .X ; @ Â /, and to the horizontal projection of TD 2 as well.It follows that with respect to g W , H OEp;x is orthogonal to V OEp;x and is the horizontal space of .Finally, we have Proof We use the paths g M .s/and g.s/ and the following lemma from [45] to replace g W near the boundary with a product metric restricting to g M at the boundary.To define our replacement path x g, we define two smooth functions: Since 1 .M / is finite, x ˛jM ˛is exact.By the defining property of the Thom form, for any point q 2 B, R 1 .q/ˆD 1.We use Stokes' theorem to compute x ˛D Z 1 .q/˛D 2 ˛.X /: We next construct a form 2 1 .W / extending 2 ˛=`.We first define a form x 2 .M D 2 /.
At .p;x/ 2 M D 2 , x ¤ 0, set x j x H p f0g D 2 `˛x H p ; x .X ; 0/ D f 1 .r/ `; x .0;@ r / D 0; x .0;@ Â / D f 1 .r/ `; where r is the radial coordinate on D 2 .This form extends smoothly to the origin of D 2 since f 1 is zero in a neighborhood of r D 0. Since r , x H p ˚f0g, ˛, @ r , @ Â and X are all preserved by the S 1 action, x is S 1 -invariant.The vertical space of is generated by .X ; @ Â /, and so x vanishes on the vertical space.It follows that there is a unique form 2 .W / such that D x .Proof Recall that f 1 .r/ D 1 for r in the image of and note that D ./j M OEL;1 D x j M OEL;1 . Thus: .X ; 0/ D x .X ; 0/ D f 1 .r/ `D 2 `˛.X /; .0; @ t / D x .0;@ r / D 0 D 2 `˛.proj M .0;@ t //: Let B be the complex line bundle with c 1 .B / D d .Given a differential form in the de Rahm cohomology class of 2 i times the first Chern class of a complex line bundle, there is a unitary connection on the line bundle whose curvature is that differential form.Thus, since ˇrepresents `d, let r B be a unitary connection on B with curvature F r B D 2 i `ˇ: We now define a connection on : r D r B i : Lemma 4.9 r is flat on U .
Proof We need to show that F r D 0. Using Lemma 4.8 it follows that r D r B i D proj U;M r B 2 i `˛Á ; and hence the curvature of the term in the parentheses is `d˛D 0: McFeely Jackson Goodman Proof Since W r D W Â D 0, F r .W r ; W Â / D i d .W r ; W Â / D i d ..0; @ r /; .0;@ Â // D i d ..0; @ r /; .0;@ Â // D i d x ..0; @ r /; .0;@ Â // D i @ r x .0;@ Â / D i f 0 1 .r/ `: Similarly, F r .W r ; X i / D i d x ..0; @ r /; .x X i ; 0// D i  From the definition of f 2 one sees that r=f 2 ! 1 as r !0. It follows that we can choose small enough that (3.2) holds, completing the proof of Theorem 3.1.
In [30,Lemma 4.2], Kreck and Stolz constructed positive scalar curvature metrics on associated disc bundles in order to calculate their invariant for spin manifolds with free S 1 actions.In their proof, they needed to assume that the S 1 orbits were geodesics.The metric g W constructed in Theorem 3.1 generalizes their method to a free isometric S 1 action without the geodesic condition. c

4 . 4 !C 4 .Using the isomorphism Pin C 4 Š
Since X and Y have primitive canonical classes, and their inverses in the cobordism group are given by reversing orientation, we conclude that every class in Spin c 4 can be represented by a simply connected manifold B with primitive canonical class d.Lemma 1.6 implies that by mapping the cobordism class of such a pair to ˇ.B; d/ we can define a homomorphism ˇW Spin c Pin Z 16 generated by a pin C structure on RP 4 we prove the following: Lemma 1.7 We have that ˇ.B; d/ D hd 2 ; OEBi C 4 ind.B; d/ mod 16 for an unknown sign D ˙1.

M 5 Q.M 5 /
D simply connected 4-manifolds B 4 such that: Type II B is spin and b 2 .B/ D b 2 .M / C 1. Type III B is nonspin, b 2 .B/ D b 2 .M / C 1 and sign.B/ D ˙OEP mod 4. Type I and M 6 2 S B is nonspin and b 2 D b 2 .M / C 1.
Type III, let 0 Ä c < 16 be an integer such that OEP D c mod 16.By [26, Theorem 3.6] we see that c D b 2 .M / C 1 mod 2. Choose l such that 0 Ä c 4l < 4:

Theorem 3 . 1
Let S 1 act freely on M by isometries of a Riemannian metric g M with scal.g M / > 0 and assume 1 .M / is finite.Let B D M=S 1 be the quotient and W W D M S 1 D 2 !B the associated disc bundle.Suppose the first Chern class of the principal S 1 bundle W M !B is `d for d 2 H 2 .B; Z/ and `2 Z.If is the complex line bundle over W with first Chern class d, then there exists a metric g W on W and a connection r on such that .3.2/ scal.gW / > `jF r j g W :

Lemma 4 . 1
The metrics g W and g B make into a Riemannian submersion.

Issue 3 (
pages 1005-1499) 2024 1005 Homological mirror symmetry for hypertoric varieties, I: Conic equivariant sheaves MICHAEL MCBREEN and BEN WEBSTER 1065 Moduli spaces of Ricci positive metrics in dimension five MCFEELY JACKSON GOODMAN 1099 Riemannian manifolds with entire Grauert tube are rationally elliptic XIAOYANG CHEN 1113 On certain quantifications of Gromov's nonsqueezing theorem KEVIN SACKEL, ANTOINE SONG, UMUT VAROLGUNES and JONATHAN J ZHU 1153 Zariski dense surface groups in SL.2k C 1; Z/ D DARREN LONG and MORWEN B THISTLETHWAITE 1167 Scalar and mean curvature comparison via the Dirac operator SIMONE CECCHINI and RUDOLF ZEIDLER 1213 Symplectic capacities, unperturbed curves and convex toric domains DUSA MCDUFF and KYLER SIEGEL 1287 Quadric bundles and hyperbolic equivalence ALEXANDER KUZNETSOV 1341 Categorical wall-crossing formula for Donaldson-Thomas theory on the resolved conifold YUKINOBU TODA 1409 Nonnegative Ricci curvature, metric cones and virtual abelianness JIAYIN PAN 1437 The homology of the Temperley-Lieb algebras RACHAEL BOYD and RICHARD HEPWORTH w 1 .TP / D w 1 .det.TP //; Let M 1 and M 2 be Type III 5-manifolds such that 1 .M i / Š Z 2 acts trivially on2 .M i / and H 2 .M i ; Z/ is torsion-free for i D 1; 2. Then M 1 is diffeomorphic to M 2 if and only if b 2 .M 1 / D b 2 .M 2 /and OEP 1 D ˙OEP 2 2 Pin C 4 ; where P i is a characteristic submanifold of M i .
26, Proposition 6.1].Let B n be a simply connected manifold and let M nC1 !B n be a nontrivial principal S 1 bundle with first Chern class kd, where d is a primitive element of H 2 .B; Z/ and k ¤ 0 is an integer.Then M is orientable, H 2 .M; Z/ is torsion-free and b 2 .M / D b 2 .B/ 1.The fundamental group 1 .M / Š Z k is generated by any S 1 fiber and acts trivially on 2 .M /.The universal cover of M is the total space of an S 1 bundle over B with first Chern class d.If k D 2, M is Type III if and only if and w 2 .TB/ D d mod 2.
19, Lemma 9; 26, Lemma 2.3] -w 2 .˚ / D w 1 ./ 2 ¤ 0. So ˚ with its natural orientation is a nontrivial complex line bundle.Since ˝ is trivial, c 1 .˚ / is torsion, and Let B 4 be a simply connected 4-manifold and let M 5 be the total space of a principal S 1 bundle over B with first Chern class 2d 2 H 2 .B; Z/ where d is a primitive element such that w 2 .TB/ D d mod 2. Then M satisfies the conditions of Theorem 1.2 with b 2 .M / D b 2 .B/ 1 and OEP D ˇ.B; d /.
[27,ed, there is a long exact sequence as in[26, page 154; 25, page 654].We see that is surjective by noting that is the composition of .The canonical class of X is dual to CP 1 CP 2 , which is nullcobordant, so .OEX / D 0. Since is surjective, .OEY / generates Spin 2 .Since CP 1 contains a regular value of g, the degree of gj equals the degree of g and is dual to g x D d Y .Giving the spin structure s used to define , we have .OEY / D OE ¤ 0.Spin.n/embedsnaturallyintobothPin˙.n/,so a spin structure induces a natural pin structure.Kirby and Taylor show that in dimension 2, the corresponding map is injective; see[27, Proposition 3.8].Let r be the Pin structure on induced by s .Using that structure, we have OE D 4 2 Pin2.Once we confirm that r is the correct structure, we conclude with (1.9) that .OEP / D 5, completing the proof of Lemma 1.7.Let r be the pin structure on † used to define .OEP /.Recall that is a diffeomorphism between the open set O D f 1 .V / † and .O/, which is with 9 discs removed.It remains only to check that r D r on O.We first recall the definition of r .Let be the nontrivial real line bundle over M and let E D TM ˚2 .Let be the complex line bundle over Y with c 1 ./DdYandletsbe spin structure on T Y ˚ used in the definition of .With the isomorphism (1.4), s induces a spin structure on E called s E .Then (1.1) shows Ej and using a canonical lift of the transition functions of 2 det.T †/ from O.2/ to Pin C .2/ we induce the pin structure r on †.Note that the normal bundle of † in P is orientable and thus w 1 .det.T †// D w 1 .det.TP /j † /:In this way we can combine the two steps and see that s E induces r on T † using the isomorphism.1.10/Ej†DT†˚5 det.T †/and a canonical lift of the transition functions of 5det.T †/ from O.5/ to Pin C .5/.The details of the canonical lifts involved can be found in[27, Lemma 1.7]; the salient fact is that each lifts the identity to the identity.Next, we note that det.T †/ and are trivial over O.The former follows because O is an open set in †, but is orientable since it is diffeomorphic to an open set in .As for the latter, we have seen that Š 2 , j P D det.TP /, and det.TP /j † D det.T †/.Since is a diffeomorphism on O and is trivial, is trivial on .O/.Let t ij be transition functions with values in SO(2) for T .As we saw in the definition of , for points in , P D TP ˚3 det.TP / and we induce a pin C structure on TP using a canonical lift of the transition functions of 3 det.TP / from O.3/ to Pin .3/.In turn, TP j † D T † ˚2 det.T †/; [31, Type II First, suppose M !B is a principal S 1 bundle.By Lemma 1.3, b 2 .B/ D b 2 .M / C 1 and by [26, Proposition 6.1], B is spin.Conversely, let B be a simply connected spin 4-manifold with b 2 .B/ D b 2 .M / C 1.It follows from [26, Proposition 6.1] that all of the total spaces of principal S 1 bundle over B with 1 D Z 2 are Type II and have second Betti number b 2 .B/ 1.By [26, Theorem 3.1] all such total spaces are diffeomorphic to M. If b 2 .M / 1 there are infinitely many primitive elements of H 2 .B; Z/ D Z b 2 .M /C1 and thus infinitely many nonisomorphic such bundles.form of B is diagonal, and so H .B; Z/ D H .# a CP 2 # b CP 2 ; Z/ for some integers a and b.Then using[31, Corollary II.2.12] again we see that w 2 .B/ D .1;1;:: : ; 1/ 2 H 2 .B; Z 2 / Š Z aCb aCb / 2 H 2 .B; Z/ Š Z aCb , where each d i is an odd integer.This completes the proof of one direction of (2) since OEP D ˇ.B; d/ D hd 2 ; OEBi D Conversely, Let B be a nonspin simply connected 4-manifold with b 2 .B/ D b 2 .M / C 1. Assume further that sign.B/ D OEP 2 Z 4 =˙.Again, H .B; Z/ D H .# a CP 2 # b CP 2 ; Z/, where b 2 .B/ D a C b and sign.
5-manifold with 1 .M / D Z 2 acting trivially on If M !B is a principal S 1 bundle, the long exact homotopy sequence implies that 1 .M / ! 1 .B/ is surjective.If 1 .B/ D Z 2 , then the Gysin sequence implies that H 3 .B/ !H 3 .M / is injective.Since M, and thus B, is orientable, H 3 .B/ D Z 2 and H 2 .M / would not be torsion-free.Thus any quotient of M by a free S 1 action is simply connected.M is Type III Suppose M !B is a principal S 1 bundle.By Lemma 1.3, b 1 .B/ D b 1 .M / C 1 and the first Chern class of the bundle is 2d, where d is a primitive element of H 2 .B; Z/ such that w 2 .TB/ D d mod 2. It follows that B is nonspin, and by [31, Corollary II.2.12] the intersection form of B is odd.By the classification of integral forms and Donaldson's theorem [16, page 5 and Theorem 1.3.1], the intersection 2 : Thus d D .d 1 ; : : : ; d B/ D a b.Choose c 2 f0; 1; 2; 3g such that ˙OEP D a b C 4c mod 16.If b 2 .M / > 0, choose k such that .4C 2 /k.kC 1/ D 4c mod 16; where D ˙1 is the sign from Lemma 1.7.If b 2 .M / D 0 then choose k D 0. Set d k D .1 C 2k; 1; : : : ; 1/ 2 H 2 .B; Z/ Š Z aCb : Then d is primitive and as above, we see that w 2 .TB/ D d mod 2. Using Lemma 1.7 we have ˇ.B; d k / D sign B C .4 C 2 /k.kC 1/ D ˙OEP mod 16: Hence, by Lemma 1.5 and Theorem 1.2, M is diffeomorphic to the total space of an S 1 bundle over B with first Chern class 2d k .In the case where b 2 .M / > 1, there are infinitely many choices of k yielding distinct classes d k , and M is diffeomorphic to infinitely many total spaces of nonisomorphic S 1 bundles over B.M is Type I Suppose M !B is a principal S 1 bundle.By Lemma 1.3, b 1 .B/ D b 1 .M / C 1 and by [26, Proposition 6.1] B is nonspin and the first Chern class of the bundle is 2d, where d is a primitive element of H 2 .B; Z/ such that w 2 .TB/ ¤ d mod 2. If M D X.q/ # S 1 .CP 2 S 1 / with q D 0; 4, then b 2 .B/ D 3 and by [26, Theorem 6.8] hd 2 ; OEBi D ˙q mod 8.If sign.B/ D ˙3, then up to orientation as above H .B; Z/ D H .# 3 CP 2 ; Z/ and w 2 .TB/ D .1;1; 1/.Thus d D .d 1 ; d 2 ; d 3 / 2 H 2 .B; Z/ Š Z 3 ; (1),ersely, Let B be a simply connected nonspin 4-manifold satisfying the conditions given by the table for Q.M /.Then H .B; Z/ D H .# a CP 2 # b CP 2 ; Z/ for some integers a; b such that aCb D b 1 .M /C1.Let .q;s/ 2 Z 8 ˚Z2 represent the cobordism class of P M in the pin c cobordism group Pin c 4 Š Z 8 ˚Z2 ; see[26, page 154].By[26, Theorem 3.6], q C s D b 2 .M / C 1 mod 2.If q D 0; 4 then[26, Theorem 3.7]implies that a C b 3, so we can assume that up to orientation a 2 and using Table1either a C b 5 or jsign.B/j < b 2 .B/, which implies b > 0. Define the following elements d k 2 H 2 .B; Z/ Š Z a ˚Zb for each k 2 Z: Theorem 6.8] the S 1 bundle over B with first Chern class 2d k is diffeomorphic to M. Again, infinitely many k yield distinct classes d k and thus nonisomorphic bundles.To prove(1), first assume M is a 5-manifold with 1 .M / D Z 2 admitting a free S 1 action with simply connected quotient B. By Lemma 1.3, M is orientable, 1 .M / acts trivially on 2 .M / and H 2 .M; Z/ is torsion-free.If b 2 .M / D 0, then b 2 .B/ D 1 and up to orientation H .B; Z/ Š H .CP 2 ; Z/ and w 2 .TB/ is nonzero.There are only two primitive classes ˙d 2 H 2 .B; Z/ Š Z, each restricting to w 2 .B/ mod 2. Thus B is of Type III and ˇ.OEB; d/ D ˙1.By Theorem 1.2, M is diffeomorphic to RP 5 .
4, then b 2 .B/ D 4 and hd 2 ; OEBi D ˙q mod 8.If sign.B/ D ˙4, then up to orientation by the argument in the Type III case, H .B; Z/ D H .# 4 CP 2 ; Z/ and d D .d 1 ; d 2 ; d 3 ; d 4 / 2 H 2 .B; Z/ Š Z 3 with at least one d i even and at least one d i odd.Again hd 2 ; OEBi ¤ 0; 4 mod 8, so jsign.B/j < 4. If q D 2, [26, Theorem 3.7] implies that a C b 3 and we can assume a 2 and define q D 2W d k D .1 C 8k; 1; 0; : : : ; 0/: If q is odd, by [26, Theorem 3.7] a C b 2, and we can assume a 1. Define q D 1W d k D .1 C 8k; 4; 0; : : : ; 0/; q D 3W d k D .1 C 8k; 2; 0; : : : ; 0/:In each case d k is primitive, w 2 .TB/ ¤ d k mod 2, and q D ˙hd 2 k ; OEBi mod 8.By[26, The form c 1 .r/represents the cohomology class c 1 ./ D d.Thus Since the terms of y A.T W / have degree 4k, with k 2 Z, and the dimension of W is 4n C 2, only terms of degree 4k C 2 in e d=2 will contribute.In those degrees, e d=2 D sinh.d=2/ as power series.Let ˆ2 H 2 .W; @W; Z/ be the Thom class of W W ! B. Then j .ˆ/D c 1 ./ D .`d/. Ã: Using the Gysin sequence it follows that H 4 .M; R/ D 0 and M , g k and B satisfy the hypotheses of Theorem 3.5 with g M D g k , d D d k , `D 2 and x r any flat connection on the canonical bundle of the spin c structure.By (3.6) we have using the fact that ˝1 3 p 1 .TB/; OEB ˛D hL.TB/; OEBi is equal to the signature of B. / is a nontrivial polynomial in k and takes on infinitely many values for the infinite set of integers k.Corollary 2.6 implies that M Ric>0 .M / has infinitely many components, completing the proof of Theorem A.Note that Corollary 2.6 also implies that M scal>0 .M / has infinitely many components.
(seeCorollary 1.15)we see that since B admits a metric of positive Ricci curvature by Lemma 3.10, k W M !B is a principal S 1 bundle, and 1 .M / is finite, then for each k M admits a metric g k with Ric.g k / > 0 such that the S 1 action corresponding to the principal bundle k W M !B acts by isometries of g k .
[45]a 4.6[45]Let g.t/ C dt 2 be a metric of positive scalar curvature on M OE0; N such that scal.g.t// > 0 and M f0g has positive mean curvature with respect to the inward normal vector @ t .Let x g.t/ be a smooth path of metrics on M such that scal.xg.t// > 0 for t 2 OE0; N and x g.t/ D g.t/ for t in a neighborhood of N. Then there exists a function ˇW OE0; N !R C such that ˇD 1 for t in a neighborhood of N, ˇD ˇ.0/ is constant for t in a neighborhood of 0, and x g.t/ C ˇ.t/ dt 2 has positive scalar curvature.
1 2 N !OE 2 f 2 .1/ 2 ; 1 such that 1 .t/D 1 for t near 0 and 1 .t/D 2 f 2 .1/ 2 for t near 1 2 N; 2 W 1 2 N; N !OE0; 1 such that 2 .t/D 0 for t near 1 2 N and 2 .t/D t for t near N: We then define a smooth path of metrics x g.t/ D g M ı 1 .t/if t 2 0; 1 2 N ; g ı 2 .t/if t 2 1 2 N; N : By Lemma 4.4 and the definition of g, scal.xg.t// > 0 for all t.Then Lemmas 4.3 and 4.6 imply that x g.t/ C ˇ.t/ dt 2 has positive scalar curvature for the function ˇ.t/ given by Lemma 4.6.For t near N, x g.t/ D g.t/ and ˇ.t/ D 1, so x g.t/ C ˇ.t/ dt 2 D g W . Thus replacing g W j V with this metric results in a new smooth metric, for which we reuse the notation g W . Since x g.t/ D g and ˇ.t/ is constant for t near 0, x g.t/ C ˇ.t/ dt 2 has the desired product structure (3.3).This proves Lemma 4.5.Let ˇ2 2 .B/ represent the image of `d in H 2 .B; R/.The Gysin sequence for an S 1 bundle shows that `d D 0, so we can choose ˛2 1 .M / such that ˇD d˛.Since ˇis S 1 -invariant, we can choose ˛to be S 1 -invariant by averaging.Proof Let ˆ2 2 .W / be a Thom form of the disc bundle W W ! B. Since under the long exact sequence map H 2 .W; @W / !H 2 .W /, we have ˇ ˆD d x for some x ˛2 1 .W /.Since ˆvanishes near @W,d x ˛jM D ˇjM D ˇD d˛: .W Â ; X i / D i d x ..0; @ Â /; .x X i ; 0// D i r