Fermat and the number of fixed points of periodic flows
aa r X i v : . [ m a t h . A T ] A p r FERMAT AND THE NUMBER OF FIXED POINTS OFPERIODIC FLOWS
LEONOR GODINHO, ´ALVARO PELAYO, AND SILVIA SABATINI
Abstract.
We obtain a general lower bound for the number of fixed points ofa circle action on a compact almost complex manifold M of dimension 2 n withnonempty fixed point set, provided the Chern number c c n − [ M ] vanishes.The proof combines techniques originating in equivariant K-theory with cele-brated number theory results on polygonal numbers, introduced by Pierre deFermat. This lower bound confirms in many cases a conjecture of Kosniowskifrom 1979, and is better than existing bounds for some symplectic actions.Moreover, if the fixed point set is discrete, we prove divisibility properties forthe number of fixed points, improving similar statements obtained by Hirze-bruch in 1999. Our results apply, for example, to a class of manifolds whichdo not support any Hamiltonian circle action, namely those for which the firstChern class is torsion. This includes, for instance, all symplectic Calabi Yaumanifolds. Introduction
Finding the minimal number of fixed points of a circle action on a compact man-ifold is one of the most pressing unsolved problems in equivariant geometry . Itis deeply connected with the question of whether there exists a symplectic non-Hamiltonian S -action on a compact symplectic manifold with nonempty and dis-crete fixed point set, which is probably the hardest and most interesting open prob-lem in the subject. Much of the activity concerning this question originated in aresult by T. Frankel [Fr59] for K¨ahler manifolds, in which he showed that a K¨ahler S -action on a compact K¨ahler manifold M is Hamiltonian if and only if it hasfixed points. In this case, this implies that the action has at least dim M + 1 fixedpoints, since they coincide with the critical points of the corresponding Hamiltonianfunction (a perfect Morse-Bott function). For the larger class of unitary manifolds,a conjecture in this direction was made by Kosniowski [Ko79] in 1979 and is openin general. Conjecture 1 (Kosniowski ’79) . Let M be a n -dimensional compact unitary S -manifold with isolated fixed points. If M does not bound equivariantly then thenumber of fixed points is greater than f ( n ) , where f ( n ) is some linear function. Date : July 11, 2018.LG and SS were partially supported by Funda¸c˜ao para a Ciˆencia e Tecnologia (FCT / Portugal)through projects EXCL/MAT-GEO/0222/2012 and POCTI/MAT/117762/2010.AP was partially supported by an NSF CAREER DMS-1055897.SS was partially supported by the Funda¸c˜ao para a Ciˆencia e Tecnologia (FCT / Portugal)through the postdoctoral fellowship SFRH/BPD/86851/2012. In the terminology of dynamical systems, circle actions are regarded as periodic flows and thefixed points of the action correspond to the equilibrium points of the flow.
In fact, Kosniowski suggested that, most likely, f ( n ) is n since this bound worksin low dimensions, leading to the conjecture that the number of fixed points on M is at least ⌊ n ⌋ + 1.Several other lower bounds were obtained in the literature, by retrieving infor-mation from a nonvanishing Chern number of the manifold. For example, Hattori[Ha84] showed that, under a certain technical condition, if c n [ M ] does not vanish(implying that c is not torsion) then any S -action on an almost complex manifoldpreserving the a.c. structure must have at least n +1 fixed points (see Theorem 2.4).Since then many other results followed [PT11, LL10, CKP12]; we review these inSection 2.It is therefore natural to study the situation in which the first Chern class isindeed torsion. In the symplectic case this condition automatically implies that themanifold cannot support any Hamiltonian circle action (see Proposition 4.3), and is,for instance, satisfied by the important family of symplectic Calabi-Yau manifolds,for which we have c = 0. Since the existence of a symplectic manifold admittinga non-Hamiltonian circle action with discrete fixed point set is still unknown, andthere is very little information on the required topological properties of the possiblecandidates, our results shed some light on this problem.In this note we make the weaker assumption that c c n − [ M ] is zero (cf. Section4.1). The choice of this Chern number is motivated by its expression in terms ofnumbers of fixed points obtained in [GoSa12, Theorem 1.2]. Interestingly, on acompact symplectic manifold of dimension 6, a circle action is non-Hamiltonian ifand only if c c [ M ] = 0 (cf. Proposition 4.2).Using this expression, we show in Theorem C that, whenever c c n − [ M ] vanishes,a lower bound B ( n ) for the number of fixed points of a circle action on an almostcomplex manifold can be obtained from the minimum values of certain integer-valued functions restricted to a set of integer points in a specific hyperplane. Theseminimization problems are then solved in Theorems D and E, using celebratednumber theory results on the possible representations of a positive integer numberas a sum of polygonal numbers (namely squares and triangular numbers). Thesewere originally stated by Fermat in 1640 and proved by Legendre, Lagrange, Euler,Gauss and Ewell (see Section 3). The lower bounds obtained are summarized inthe following theorem. Theorem A.
Let ( M, J ) be a n -dimensional compact connected almost complexmanifold equipped with a J -preserving S -action with nonempty fixed point set andsuch that c c n − [ M ] = 0 . Then the number of fixed points of the S -action is atleast B ( n ) , where B ( n ) is given as follows.For n = 2 m and r := gcd( m, , (i) if r = 1 then B ( n ) = 12 ; (ii) if r = 2 then • B ( n ) = 6 if n
28 (mod 32) , • B ( n ) = 12 otherwise; (iii) if r = 3 then • B ( n ) = 4 if all prime factors of n congruent to occur with even exponent, • B ( n ) = 8 otherwise; ERMAT AND THE NUMBER OF FIXED POINTS OF PERIODIC FLOWS 3 (iv) if r = 4 then • B ( n ) = 3 if n is a square, • B ( n ) = 6 if n is not a square and n = 4 k (8 t + 7) ∀ k, t ∈ Z > , • B ( n ) = 9 otherwise; (v) if r = 6 then • B ( n ) = 2 if n is a square, • B ( n ) = 4 if n is not a square and all prime factors of n congruent to occur with even exponent, • B ( n ) = 6 if n is not a square, at least one prime factor of n congruent to occurs with an odd exponentand n
28 (mod 32) , • B ( n ) = 8 otherwise; (vi) if r = 12 then • B ( n ) = 2 if n is a square, • B ( n ) = 3 if n is a square, • B ( n ) = 4 if none of the above holds and all prime factors of n congruent to occur with even exponent, • B ( n ) = 6 if none of the above holds and n = 4 k (8 t + 7) ∀ k, t ∈ Z > , • B ( n ) = 7 otherwise.For n = 2 m + 1 and r := gcd( m − , , (i) if r then B ( n ) = r ; (ii) if r = 6 then • B ( n ) = 4 if every prime factor of n congruent to occurs with even exponent, • B ( n ) = 8 otherwise; (iii) if r = 12 then • B ( n ) = 2 if n − is a triangular number, • B ( n ) = 4 if n − is not a triangular number and every primefactor of n congruent to occurs witheven exponent, • B ( n ) = 6 otherwise. In some dimensions, the lower bounds obtained confirm Kosniowski’s conjec-ture, and, in some cases, they are better than n + 1, the existing lower bound forHamiltonian and some almost complex actions. We give a complete list of thesedimensions in Propositions 7.1 and 7.2.Under the same vanishing condition on c c n − [ M ], the expression of this Chernnumber given in [GoSa12, Theorem 1.2] can also be used to prove divisibility results for the number of fixed points, provided that the fixed point set is discrete. Theorem B.
Let ( M, J ) be a n -dimensional compact connected almost complexmanifold equipped with a J -preserving S -action with nonempty, discrete fixed pointset M S and such that c c n − [ M ] = 0 . Then if n = 2 m is even, (1.1) | M S | ≡ r ) with r = gcd ( m, and, if n = 2 m + 1 is odd, (1.2) | M S | ≡ r ) with r = gcd ( m − , . L. GODINHO, ´A. PELAYO, AND S. SABATINI
Remark 1.1
In particular, if n = dim( M ) is odd then | M S | is always even. ⊘ Note that, when the fixed point set is discrete, the number of fixed points coin-cides with the Euler characteristic of the a.c. manifold. Coincidently, in a letter toV. Gritsenko, Hirzebruch also obtains divisibility results for the Euler characteris-tic of an almost complex manifold M satisfying c c n − [ M ] = 0 [Hi99]. By usingour methods we are able to improve his results whenever dim M c = 0 in integer cohomology, wecan combine Hirzebruch’s results with ours obtaining, in some cases, a better lowerbound for the number of fixed points (see Theorems G and H). For example, whendim M = 4 and c = 0 we prove that the number of fixed points is at least 24. Thiswill be true, in general, whenever dim M ≡ M , , ,
12 and 18. It would be interesting to know the answer to the followingquestion.
Question 1.2
Does there exist a compact almost complex S -manifold M of di-mension with c c [ M ] = 0 and with exactly fixed points? ⊘ The existence of such a manifold would also provide an example with a sharplower bound in dimension 14. In the following table we illustrate some of the resultsobtained in this work.dim M = 2 n Possible values Kosniowski’s Lower bound forof | M S | if lower bound Hamiltonian actions c c n − [ M ] = 0 ⌊ n ⌋ + 1 n + 14 ∗ , 24, 36, . . . 2 36 , 4, 6, . . . 2 48 , 12, 18, . . . 3 510 , 48, 72, . . . 3 612 , 8, 12, . . . 4 714 , 24, 36, . . . 4 816 , 6, 9, . . . 5 918 , 16, 24, . . . 5 1020 ∗ , 24, 36, . . . 6 1122 , 12, 18, . . . 6 1224 , 4, 6, . . . 7 1326 , 48, 72, . . . 7 1428 , 24, 36, . . . 8 1530 , 8, 12, . . . 8 16* if c = 0 then the possible values of | M S | are , 48, 72, . . . ERMAT AND THE NUMBER OF FIXED POINTS OF PERIODIC FLOWS 5
Acknowledgements . This paper started at the Bernoulli Center in Lausanne(EFPL) during the program on Semiclassical Analysis and Integrable Systems or-ganized by ´Alvaro Pelayo, Nicolai Reshetikhin, and San V˜u Ngo. c, from July 1 toDecember 31, 2013. We would like to thank D. McDuff and T. S. Ratiu for usefulcomments and discussions. 2.
Preliminares
We review some results which are relevant for this article, including some whichwe will need in the proofs.2.1.
Origins.
It has been a long standing problem to estimate the minimal num-ber of fixed points of a smooth circle action on a compact smooth manifold withnonempty fixed point set. If the manifold is symplectic, i.e. if it admits a closed,non-degenerate two-form ω ∈ Ω ( M ) ( symplectic form ), we say that an S -actionon ( M, ω ) is symplectic if it preserves ω . If X M is the vector field induced by the S action then we say that the action is Hamiltonian if the 1-form ι X M ω := ω ( X M , · )is exact, that is, if there exists a smooth map µ : M → R such that − d µ = ι X M ω. The map µ is called a momentum map . If a symplectic manifold is equipped witha Hamiltonian S -action then the following fact is well-known (cf. Section 10). Proposition 2.1.
A Hamiltonian S -action on a n -dimensional compact sym-plectic manifold has at least n + 1 fixed points. Since Frankel [Fr59] showed that a K¨ahler S -action on a compact K¨ahler mani-fold is Hamiltonian if and only if it has fixed points, a lower bound for the numberof fixed points of a circle action was thus obtained in the K¨ahler case. Theorem 2.2 (Frankel ’59) . A K¨ahler S -action on a n -dimensional compactconnected K¨ahler manifold is Hamiltonian if and only if it has fixed points, inwhich case it has at least n + 1 fixed points. Trying to extend this lower bound on the number of fixed points, Kosniowskiconsiders unitary S -manifolds and proposes that if M is a 2 n -dimensional unitary S -manifold with isolated fixed points which does not bound equivariantly, thenthe number of fixed points is greater than f ( n ), where f ( n ) is some linear function(Conjecture 1). In fact, Kosniowski suggests that, most likely, one has f ( n ) = n since it works in low dimensions, leading to the conjecture that the number of fixedpoints on M is at least ⌊ n ⌋ + 1, where 2 n is the dimension of M . Remark 2.3
Recall that a unitary S -manifold is a stable complex manifold M ,i.e. T M ⊕ R k has a complex structure for some k (where R k is the trivial realvector bundle over M ), equipped with an S -action that preserves this structure.Thus, every S -almost complex manifold, and hence every S -symplectic manifold,is unitary. Moreover, M bounds if it is cobordant with the empty set, meaning thatit can be realized as the oriented boundary of a smooth oriented 2 n + 1-manifoldwith boundary. In particular, if this is the case, all the Pontrjagin and Stiefel-Whitney numbers vanish. Note that if M is smooth and admits a semi-free circleaction with isolated fixed points then it bounds [LL10]. ⊘ Another related result was proved by Hattori [Ha84, Theorem 5.1] for almostcomplex manifolds. If (
M, J ) is a 2 n -dimensional almost-complex manifold , i.e. J is L. GODINHO, ´A. PELAYO, AND S. SABATINI a complex structure on the tangent bundle
T M , then one can consider the Chernclasses c j ∈ H j ( M, Z ) of T M as well as any Chern number. In particular, one cantake c n [ M ]. On the other hand, by taking the restriction of the first equivariantChern class of T M at each fixed point p ∈ M S , which one can naturally identifywith the sum of the weights of the S -isotropy representation T p M , one obtains amap c S ( M ) : M S → Z , p c S ( M )( p ) ∈ Z , called the Chern class map of M . Theorem 2.4 (Hattori) . If M is a n -dimensional almost-complex manifold suchthat c n [ M ] does not vanish and the Chern class map is injective, then any S -actionon M must have at least n + 1 fixed points. Remark 2.5
This result is improved in [LL10, Corollary 1.5] where the conditionon the injectivity of the Chern class map is removed. ⊘ Some recent contributions.
Following Kosniowski’s conjecture and the the-orems of Frankel and Hattori, many results have appeared in recent works.By using the Atiyah-Bott and Berline-Vergne localization formula in equivariantcohomology, Pelayo and Tolman [PT11] proved the following result which gener-alizes the known lower bound for the number of fixed points of Hamiltonian S -actions to some symplectic non-Hamiltonian actions. Note, however, that there areno known examples of these actions with discrete fixed point sets. Theorem 2.6 ([PT11]) . Let S act symplectically on a compact symplectic n -dimensional manifold M . If the Chern class map c S ( M ) : M S → Z is somewhereinjective then the S -action has at least n + 1 fixed points. In particular, they obtain the following lower bounds for the number of fixedpoints.
Theorem 2.7 ([PT11]) . Let S act symplectically on a compact symplectic mani-fold M with nonempty fixed point set. Then there are at least two fixed points. Inparticular, • if dim M ≥ , then there exist at least three fixed points; • if dim M ≥ , and c S ( M ) : M S → Z is not identically zero, then thereexist at least four fixed points. Following this result, Ping Li and Kefeng Liu generalized Hattori’s theorem[LL10].
Theorem 2.8 (Li-Liu) . Let M mn be an almost-complex manifold. If there existpositive integers λ , . . . , λ u with P ui =1 λ i = m such that the corresponding Chernnumber ( c λ · · · c λ u ) n [ M ] is nonzero, then any S -action on M must have at least n + 1 fixed points. Another related result was obtained by Cho, Kim and Park [CKP12]. Let f : X → Y be a map between sets then f is somewhere injective if there exists y ∈ Y such that f − ( { y } ) is the singleton. ERMAT AND THE NUMBER OF FIXED POINTS OF PERIODIC FLOWS 7
Theorem 2.9 (Cho-Kim-Park) . Let M be a n -dimensional unitary S -manifoldand let i , i , . . . , i n be non-negative integers satisfying i + 2 i + · · · + ni n = n . If M does not bound equivariantly and c i c i · · · c i n n = 0 then M must have at leastmax { i , · · · , i n } + 1 fixed points. Fermat’s famous statements
In 1640 Fermat stated (without proof) that every positive integer is a sum ofat most 4 squares and a sum of at most 3 triangular numbers, where square andtriangular numbers are those respectively described by k and k ( k +1)2 , with k =0 , , , , . . . .Lagrange, in 1770, proved the part of Fermat’s theorem regarding squares, ob-taining his celebrated Four Squares Theorem [D, p. 279]. Theorem 3.1 (Lagrange’s Four Squares Theorem) . Every nonegative integer isthe sum of or fewer squares. In 1798 Legendre proved a much deeper statement which described exactly whichnumbers needed all four squares [D, p. 261].
Theorem 3.2 (Legendre’s Three Squares Theorem) . The set of positive integersthat are not sums of three or fewer squares is the set (cid:8) m ∈ Z > : m = 4 k (8 t + 7) , for some k, t ∈ Z > (cid:9) . After this, it was natural to think which numbers could be written as sum oftwo squares. A complete answer to this question was given by Euler [D, p. 230].
Theorem 3.3 (Euler) . A positive integer m > can be written as a sum of twosquares if and only if every prime factor of m which is congruent to occurs with even exponent. Example 3.4
The integer m = = 5 · can be written as a sum of twosquares. In particular, 245 = 4 · + 7 = 14 + 7 . As the number m = is notdivisible by 4 and is congruent to 1 (mod 8), one concludes that it is the sum of 3or fewer squares. However, since 105 = 3 · · + 2 + 1 . Since m = = 4 ·
15 = 4 · (8 + 7),we know from Theorem 3.2 that it cannot be represented as a sum of 3 or fewersquares so we really need 4 squares. For example, 60 = 6 + 4 + 2 + 2 . ⊘ Let us now see what happens with triangular numbers. The part of Fermat’sstatement regarding these numbers was first proved by Gauss [D, p. 17].
Theorem 3.5 (Gauss) . Every nonegative number is the sum of three or fewertriangular numbers.
After this result, Ewell [E92] gave a simple description of those numbers thatare sums of two triangular numbers.
Theorem 3.6 (Ewell) . A positive integer m can be represented as a sum of twotriangular numbers if and only if every prime factor of m + 1 which is congruentto occurs with even exponent. L. GODINHO, ´A. PELAYO, AND S. SABATINI
Example 3.7
Taking m = one obtains 4 m + 1 = 425 = 5 ·
17 and so m can be written as a sum of two triangular numbers. For instance, 106 = 105 + 1 = · + · . On the other hand, if one takes m = , then 4 m + 1 = 237 = 3 · m cannot be written as a sum of 2 triangular numbers.For instance we have 59 = 28 + 21 + 10 = · + · + · . ⊘ A minimization problem
The hypothesis c c n − [ M ] = 0 . Let us now return to our initial problemof obtaining lower bounds for the number of fixed points of an almost complexcircle action. In most of the results presented in Section 2 the lower bounds forthe number of fixed points of a circle action are obtained by retrieving informationfrom a nonvanishing Chern number. In Theorem 2.6, the crucial hypothesis for theestablishment of the lower bound is the existence of a value k of the Chern map c ( M ) for which X p ∈ M S c ( M )( p ) = k p = 0 , where Λ p is the product of the weights in the isotropy representation T p M . This istrivially achieved whenever the Chern map is somewhere injective [PT11] but alsowhen c n [ M ] = 0 as pointed out in [LL10, Lemma 3.1], leading to Theorem 2.4.In this work we focus on the situation in which a particular Chern numbervanishes. The only known expressions of Chern numbers in terms of number offixed points concern c n [ M ] (which equals the Euler characteristic and the number offixed points, when the fixed point set is discrete) and c c n − [ M ] [GoSa12, Theorem1.2]. Thus the natural candidate is c c n − [ M ] = 0.Note that c c n − [ M ] = 0 is satisfied under the stronger condition that c or c n − are torsion in integer cohomology. In the case in which c is torsion, the followingLemma proves that the results in [Ha84, PT11] cannot be applied. Lemma 4.1.
Let ( M, J ) be a compact almost complex manifold such that c isa torsion element in H ( M, Z ) . If M admits a J -preserving circle action with adiscrete fixed point set then the Chern class map c S ( M ) : M S → Z is identicallyzero.Proof. Since c is a torsion element in H ( M, Z ), there exists k ∈ Z such that kc = 0. Then the restriction of the equivariant extention k c S ∈ H S ( M, Z ) tothe fixed point set is constant, implying that the Chern class map is constant. Since c S ( M )( p ) coincides with the sum of the isotropy weights at p ∈ M S , Proposition2.11 in [Ha84] implies that X p ∈ M S c S ( M )( p ) = 0and so this constant must be zero. (cid:3) Note that if M is a 6-dimensional compact connected symplectic manifold, theaction is Hamiltonian if and only if c c [ M ] = 0. Indeed, we have the followingproposition. ERMAT AND THE NUMBER OF FIXED POINTS OF PERIODIC FLOWS 9
Proposition 4.2.
Suppose that S acts symplectically on a compact connected --dimensional symplectic manifold M with nonempty discrete fixed point set. Thenthe S -action is Hamiltonian if and only if c c [ M ] = 0 .Proof. The result follows from [Fe01] and the fact that, when dim( M ) = 6, one hasTodd( M ) = Z M c c . (cid:3) In general, if the manifold is symplectic and c is torsion in integer cohomology,then, necessarily, the action is non-Hamiltonian. Proposition 4.3.
Let ( M, ω ) be a compact symplectic manifold such that c istorsion in integer cohomology. Then M does not admit any Hamiltonian circleaction with isolated fixed points.Proof. This follows immediately from Proposition 3.21 in [Ha84], by using a resultof Feldman [Fe01] which states that the Todd genus associated to M is either 1or 0, according to whether the action is Hamiltonian or not. Alternatively we canuse Lemma 4.1 since, if the action is Hamiltonian, then c S ( M )( p ) = 0 at both theminimum and the maximum points of the momentum map. (cid:3) Therefore, our results naturally apply to a class of compact symplectic manifoldsthat do not support any Hamiltonian circle action with isolated fixed points, namelysymplectic manifolds whose first Chern class is torsion. For example, symplecticCalabi Yau manifolds, i.e. symplectic manifolds with c = 0 [FP09].4.2. Tools.
Let us then see how to obtain a lower bound for the number of fixedpoints of a J -preserving circle action on an almost complex manifold ( M, J ) satis-fying c c n − [ M ] = 0.The first result that we need is the expression of c c n − in terms of numbers offixed points. Theorem 4.4 ([GoSa12]) . Let ( M, J ) be a n -dimensional compact connected al-most complex manifold equipped with an S -action which preserves the almost com-plex structure J and has a nonempty discrete fixed point set. For every i = 0 , . . . , n ,let N i be the number of fixed points with exactly i negative weights in the isotropyrepresentation T p M . Then (4.1) c c n − [ M ] := Z M c c n − = n X i =0 N i (cid:16) i ( i −
1) + 5 n − n (cid:17) , where c and c n − are respectively the first and the ( n − Chern classes of M . Remark 4.5 If M is a 2 n -dimensional symplectic and the S -action is Hamil-tonian then the number N i of fixed points with exactly i negative weights in thecorresponding isotropy representations coincides with the 2 i -th Betti number b i ( M )of M . Consequently the expression for c c n − [ M ] given in (4.1) becomes(4.2) c c n − [ M ] = n X i =0 b i ( M ) (cid:16) i ( i −
1) + 5 n − n (cid:17) . For example, if dim M = 4, equation (4.2) gives(4.3) c [ M ] = 10 b ( M ) − b ( M ) , where we used the fact that b ( M ) = b ( M ). ⊘ The minimization problem.
For each m ∈ Z > let us consider the functions F , F : Z m +1 → Z defined by F ( N , . . . , N m ) := N m + 2 m X k =1 N m − k ;(4.4) F ( N , . . . , N m ) := 2 m X k =0 N k ;(4.5) G ( N , . . . , N m ) := − mN m + 2 m X k =1 (6 k − m ) N m − k ; G ( N , . . . , N m ) := m X k =0 (cid:16) k ( k + 1) − ( m − (cid:17) N m − k . Moreover, for i ∈ { , } , let(4.6) Z i := (cid:8) ( N , . . . , N m ) ∈ ( Z > ) m +1 | F i ( N , . . . , N m ) > , G i ( N , . . . , N m ) = 0 (cid:9) . Then we have the following result.
Theorem C.
Let ( M, J ) be a n -dimensional compact connected almost complexmanifold equipped with a J -preserving S -action with nonempty, discrete fixed pointset and such that c c n − [ M ] = 0 .For m := ⌊ n ⌋ , let F , F : Z m +1 → Z be the functions defined respectively in (4.4) and (4.5) , and let Z , Z be the sets given in (4.6) . Then the S -action hasat least B ( n ) fixed points, where B ( n ) := min Z F if n is even ;min Z F if n is odd . Proof.
By Proposition 2.11 in [Ha84], we know that N i = N n − i for every i ∈ Z .Thus, as the total number of fixed points is n X k =0 N k , it follows that F ( N , . . . , N m ) and F ( N , . . . , N m ) count the total number of fixedpoints when n = 2 m and n = 2 m + 1 respectively, and, since the fixed point set isnonempty, we must have F > F > c c n − [ M ] = 0, the constraints G = 0 and G = 0are obtained from Theorem 4.4, according to whether n is odd or even. Indeed, let g : Z × Z → Z be the map g ( i, n ) = 6 i ( i −
1) + 5 n − n . ERMAT AND THE NUMBER OF FIXED POINTS OF PERIODIC FLOWS 11 If n = 2 m , since N i = N n − i , we have0 = n X i =0 N i g ( i, n ) = − mN m + m X k =1 (cid:16) g ( m − k, m ) + g ( m + k, m ) (cid:17) N m − k = − mN m + 2 m X k =1 (6 k − m ) N m − k = G ( N , . . . , N m ) . Analogously, if n = 2 m + 1, we have0 = n X i =0 N i g ( i, n ) = m X k =0 N m − k (cid:16) g ( m − k, m + 1) + g ( m + k + 1 , m + 1) (cid:17) = 2 m X k =0 (cid:16) k ( k + 1) − m + 1 (cid:17) N m − k = 2 G ( N , . . . , N m ) . (cid:3) A lower bound when n is even Here we compute the minimal value B ( n ) of the function F restricted to Z ,obtaining a lower bound for the number of fixed points of the S -action when n is even . Theorem D.
Let n = 2 m be an even positive integer and let B ( n ) be the minimumof the function F restricted to Z , where F and Z are respectively defined by (4.4) and (4.6) . Then B ( n ) can take all values in the set { , , , , , , , } . Inparticular, if r := gcd (cid:0) n , (cid:1) ( = gcd ( m, ), we have that:(i) if r = 1 then B ( n ) = 12 ;(ii) if r = 2 then • B ( n ) = 6 if n
28 (mod 32) , • B ( n ) = 12 otherwise;(iii) if r = 3 then • B ( n ) = 4 if all prime factors of n congruent to occur with even exponent, • B ( n ) = 8 otherwise;(iv) if r = 4 then • B ( n ) = 3 if n is a square, • B ( n ) = 6 if n is not a square and n = 4 k (8 t + 7) ∀ k, t ∈ Z > , • B ( n ) = 9 otherwise;(v) if r = 6 then • B ( n ) = 2 if n is a square, • B ( n ) = 4 if n is not a square and all prime factors of n congruent to occur with even exponent, • B ( n ) = 6 if n is not a square, at least one prime factor of n congruent to occurs with an odd exponentand n
28 (mod 32) , • B ( n ) = 8 otherwise; (vi) if r = 12 then • B ( n ) = 2 if n is a square, • B ( n ) = 3 if n is a square, • B ( n ) = 4 if none of the above holds and all prime factors of n congruent to occur with even exponent, • B ( n ) = 6 if none of the above holds and n = 4 k (8 t + 7) ∀ k, t ∈ Z > , • B ( n ) = 7 otherwise.Proof. In Z we have G = − mN m + 2 m X k =1 (6 k − m ) N m − k = 0 , and so, in this set,(5.1) N m = 2 m X k =1 (cid:18) k m − (cid:19) N m − k ∈ Z > . Hence, to find min Z F , we start by substituting (5.1) in (4.4), obtaining(5.2) F = 12 m m X k =1 k N m − k . Since F is integer valued on Z n +1 , we need12 m m X k =1 k N m − k ∈ Z . As N , . . . , N m − ∈ Z , this is equivalent to requiring m X k =1 k N m − k ≡ mr ) , with r := gcd ( m,
12) = gcd ( n , ∈ { , , , , , } . Note that this also impliesthat the expression on the right hand side of (5.1) is an integer and that(5.3) F ≡ r ) . We then want to find the smallest positive value of m X k =1 k N m − k which is a multiple of mr and such that(5.4) m X k =1 (cid:18) k m − (cid:19) N m − k > , so that (5.1) is satisfied. Then, by (5.2), the minimum B ( n ) of F on Z is obtainedby multiplying this value by m . Remark 5.1
Note that, when m
6, condition (5.4) is always satisfied. Hence, thesmallest multiple of mr that satisfies all the required conditions is mr itself (takingfor instance N m − = mr , N m = − m ) r and all other N i s equal to 0), leading to B ( n ) = mr · m = 12 r , whenever n = 2 m with m . ERMAT AND THE NUMBER OF FIXED POINTS OF PERIODIC FLOWS 13 ⊘ In general, we see that (5.4) is equivalent to m X k =1 k N m − k > m m X k =1 N m − k , so our goal is to find the smallest positive multiple of mr which can be written as m X k =1 k N m − k and is greater or equal to m m X k =1 N m − k . In other words, for each m , we want to find the smallest value of l ∈ Z > such that(5.5) l · mr = m X k =1 k N m − k > m m X k =1 N m − k . Note that the first sum in (5.5) is a sum of squares, possibly with repetitions(whenever one of the N m − k s is greater than 1), and that the sum on the right handside of (5.5) is precisely the number of squares used in this representation of l · mr as a sum of squares. We then want to find the smallest value of l ∈ Z > such that(5.6) m X k =1 N m − k lr , where P mk =1 N m − k is the smallest number of squares that is needed to represent thepositive integer l · mr as a sum of squares. We can then use the results in Section 3.When r = 1 condition (5.6) becomes(5.7) m X k =1 N m − k l. Since, by Theorem 3.1, we know that every positive integer can be written as asum of 4 or fewer squares, (5.7) can be achieved with l = 1, since mr = m can bewritten as a sum of 4 or fewer squares and then m X k =1 N m − k l = 6 . We conclude that, when r = 1, we always have B ( n ) = m · mr = 12.When r = 2, condition (5.6) becomes(5.8) m X k =1 N m − k l. Hence, if mr = m can be written as a sum of 3 or fewer squares, (5.8) can beachieved with l = 1. Otherwise we need l = 2, since then, by Theorem 3.1, thenumber mr = m can be written as a sum of 4 or fewer squares and then m X k =1 N m − k l = 6 . Note that, since r = 2, the number m cannot be a multiple of 4 and so the condition m = 4 k (8 t + 7) for all k, t ∈ Z > in Theorem 3.2 is, in this situation, equivalent to m = 8 t + 7 for all t ∈ Z > which, in turn, is equivalent to m
14 (mod 16) (i.e. n
28 (mod 32)). Hence,by Theorem 3.2, we conclude that, when r = 2, we have B ( n ) = m · m = 6 if n B ( n ) = m · m = 12 otherwise.When r = 3, condition (5.6) becomes(5.9) m X k =1 N m − k l. Hence, if mr is a square or a sum of 2 squares, (5.9) can be achieved with l = 1.Otherwise we need l = 2, since then, by Theorem 3.1, the number mr = m can bewritten as a sum of 4 or fewer squares. Hence, by Theorem 3.3, we conclude that,when r = 3, B ( n ) = m · m = 4 if all prime factors of m congruent to 3 (mod 4)occur with even exponent and B ( n ) = m · m = 8 otherwise.When r = 4, condition (5.6) becomes(5.10) m X k =1 N m − k l . Hence, if mr = m is a square (or, equivalently, if m is a square), (5.10) can beachieved with l = 1. Otherwise, if mr = m can be written as a sum of 3 or fewersquares, (5.10) can be achieved with l = 2. Otherwise, we need l = 3 since then, byTheorem 3.2, the number m = m can be written as a sum of 4 or fewer squares.Hence, by Theorem 3.2, we conclude that, when r = 4, we have B ( n ) = m · m = 3if m is a square, B ( n ) = m · m = 6 if m is not a square and m = 4 k (8 t + 7) for all k, t ∈ Z > (which, since n is even, is equivalent to n = 4 k (8 t + 7) for all k, t ∈ Z > ),and B ( n ) = m · m = 9 in all other cases.When r = 6, condition (5.6) becomes(5.11) m X k =1 N m − k l. Hence, if mr = m is a square, then (5.11) can be achieved with l = 1. Otherwise,if mr = m is a square or a sum of 2 squares, (5.11) can be achieved with l = 2.If this is not the case and mr = m is a sum of 3 or fewer squares, then (5.11) canbe achieved with l = 3. If this also does not hold then (5.11) can only be achievedwith l = 4 since then, by Theorem 3.1 the number mr = m can be written as asum of 4 or fewer squares.Note that, since r = 6, the number m cannot be a multiple of 4. Hence, condition m = 4 k (8 t + 7) for all k, t ∈ Z > in Theorem 3.2 is, in this situation, equivalent to m = 8 t + 7 for all t ∈ Z > which, in turn, is equivalent to n = 28 (mod 32). Hence, by Theorems 3.2 and 3.3,we conclude that, when r = 6, we have B ( n ) = m · m = 2 if m = n is a square;otherwise B ( n ) = m · m = 4 if all prime factors of m = n congruent to 3 (mod 4)occur with even exponent; if none of these holds then B ( n ) = m · m = 6 if n = 28(mod 32) and B ( n ) = m · m = 8 otherwise.When r = 12, condition (5.6) becomes(5.12) m X k =1 N m − k l . Hence, even if mr were a square, condition (5.12) could never be achieved with l = 1.If mr = m = n is a square, (5.12) can be achieved with l = 2. If this is not thecase and mr = m is a square (or, equivalently, if m is a square), then (5.12) can beachieved with l = 3. (Note that if m is a square then m is not a square.) If thisalso does not hold and mr = m is a square or a sum of two squares, then (5.12) canbe achieved with l = 4. In none of the above holds and mr = m is a square or asum of two squares then (5.12) could be achieved with l = 5. Note, however, thatif m is not a square nor a sum of two squares then, by Theorem 3.3, at least oneprime factor of m is congruent to 3 (mod 4) and occurs with odd exponent. Then,since 5 = 3 (mod 4), the number m also has this prime factor occurring with thesame odd exponent and so, in this situation, m cannot be written as a sum of 2or fewer squares, implying that this case is impossible.If none of the above conditions are true but mr = m = n is a sum of 3 or fewersquares, then (5.12) can be achieved with l = 6. If still mr = m cannot be writtenas a sum of 3 or fewer squares then mr = m can, and so condition (5.12) can beachieved with l = 7. Indeed, if m cannot be written as a sum of 3 or fewer squares,then m k (8 t + 7) for some k, t ∈ Z > , and k > m is multiple of 4); then7 m
12 = 143 · k − (8 t + 7) , and so 8 t + 7 = 0 (mod 3), implying that t = 1 (mod 3). Hence,7 m
12 = 143 · k − (24 t ′ + 15) = 14 · k − (8 t ′ + 5) = 4 k − (8 t ′′ + 70) = 4 k − (8 t ′′′ + 6)for some t ′ , t ′′ , t ′′′ ∈ Z > and so, by Theorem 3.2, the number m can be representedby a sum of 3 or fewer squares.We conclude, by Theorems 3.2 and 3.3 that, when r = 12, we have B ( n ) = m · m = 2 if m = n is a square, B ( n ) = m · m = 3 if m = n is a square,and B ( n ) = m · m = 4 if neither n nor n are squares and all prime factors of n congruent to 3 (mod 4) occur with even exponent. If none of these conditions holdthen B ( n ) = m · m = 6, if m = 4 k (8 t + 7) (or, equivalently, n = 4 k (8 t + 7)) forany k, t ∈ Z > , and B ( n ) = 7 otherwise. (cid:3) Remark 5.2
In the Appendix we provide examples that show that all the caseslisted in Theorem D are possible. ⊘ Lower bound when n is odd Here we compute the minimal value B ( n ) of the function F restricted to Z ,obtaining a lower bound for the number of fixed points of the S -action when n is odd . Theorem E.
Let n = 2 m +1 be an odd positive integer and let B ( n ) be the minimumof the function F restricted to the set Z , where F and Z are respectively definedby (4.5) and (4.6) . Then B ( n ) can take all values in the set { , , , , , } . Inparticular, if r = gcd ( ⌊ n ⌋ − , ( = gcd ( m − , ), we have:(i) if r then B ( n ) = r ;(ii) if r = 6 then • B ( n ) = 4 if every prime factor of n congruent to occurs with even exponent, • B ( n ) = 8 otherwise;(iii) if r = 12 then • B ( n ) = 2 if n − is a triangular number, • B ( n ) = 4 if n − is not a triangular number and every primefactor of n congruent to occurs witheven exponent, • B ( n ) = 6 otherwise.Proof. In Z we have G = (1 − m ) N m + m X k =1 (cid:16) k ( k + 1) − ( m − (cid:17) N m − k = 0 . If m = 1 then G = 12 N = 0 implies that N = 0 and so the minimum of F =2 N on Z is B (3) = 2 (attained with N = 0 and N = 1). Note that here n − = 0is a triangular number and we assume r = gcd ( m − ,
12) = gcd (0 ,
12) = 12.If m = 1 then on Z we have(6.1) N m = m X k =1 (cid:18) k ( k + 1) m − − (cid:19) N m − k ∈ Z > . Hence, to find min Z F , we start by substituting (6.1) in (4.5), obtaining(6.2) F = 24 m − m X k =1 k ( k + 1)2 N m − k . Since F is integer valued and N m ∈ Z , we need12 m − m X k =1 k ( k + 1)2 N m − k ∈ Z . Since N , . . . , N m − ∈ Z , this is equivalently to requiring m − X k =1 k ( k + 1)2 N m − k ≡ m − r ) , ERMAT AND THE NUMBER OF FIXED POINTS OF PERIODIC FLOWS 17 with r := gcd ( m − ,
12) = gcd ( ⌊ n ⌋ − , ∈ { , , , , , } . Note that, by(6.2) this also implies that(6.3) F ≡ r ) . We then want to find the smallest positive value of m X k =1 k ( k + 1)2 N m − k which is a multiple of m − r and such that(6.4) m X k =1 (cid:18) k ( k + 1) m − − (cid:19) N m − k > , so that (6.1) is satisfied. Then, by (6.2), the minimum B ( n ) of F on Z is obtainedby multiplying this value by m − . Remark 6.1
Note that, when m
13, condition (6.4) is always satisfied. Hence,the smallest multiple of m − r that satisfies all the required conditions is m − r itself,leading to B ( n ) = m − r · m − r , whenever n = 2 m + 1 with m . ⊘ In general, we see that (6.4) is equivalent to m X k =1 k ( k + 1)2 N m − k > m − m X k =1 N m − k , so our goal is to find the smallest positive multiple of m − r which can be written as m X k =1 k ( k + 1)2 N m − k and is greater or equal to m − m X k =1 N m − k . In other words, for each m , we want to find the smallest value of l ∈ Z > such that(6.5) l · m − r = m X k =1 k ( k + 1)2 N m − k > m − m X k =1 N m − k . Note that the first sum in (5.5) is a sum of triangular numbers, possibly withrepetitions (whenever one of the N m − k s is greater than 1), and that the sum onthe right hand side of (6.5) is precisely the number of triangular numbers used inthis representation of l · m − r as a sum of triangular numbers. We then want to findthe smallest value of l ∈ Z > such that(6.6) m X k =1 N m − k lr , where P mk =1 N m − k is the smallest number of triangular numbers that is neededto represent the positive integer l · m − r as a sum of triangular numbers. We cantherefore use the results in Section 3 concerning these numbers.Since, by Theorem 3.5, we know that every positive integer can be written as asum of 3 or fewer triangular numbers, condition (6.6) can be achieved with l = 1whenever r B ( n ) = m − · m − r = r .When r = 6, condition (6.6) becomes(6.7) m X k =1 N m − k l. Hence, if m − r = m − can be written as a sum of 2 or fewer triangular numbers, (6.7)can be achieved with l = 1. Otherwise we need l = 2, since then, by Theorem 3.5,the number mr = m can be written as a sum of 3 or fewer squares and so m X k =1 N m − k l = 4 . Hence, by Theorem 3.6, we conclude that B ( n ) = m − · m − = 4 if every primefactor of 4 (cid:18) m − (cid:19) + 1 = 2 m + 13 = n B ( n ) = m − · m − = 8otherwise.When r = 12, condition (6.6) becomes(6.8) m X k =1 N m − k l. Hence, if m − r is a triangular number, then (6.8) can be achieved with l = 1.Otherwise, if m − r = m − can be written as a sum of 2 or fewer triangularnumbers, (6.8) can be achieved with l = 2. If this is not the case, we need l = 3since then, by Theorem 3.5, the number m − = m − can be written as a sum of3 or fewer triangular numbers.Hence, by Theorem 3.6, we conclude that B ( n ) = m − · m − = 2 if m − is atriangular number, B ( n ) = m − · m − = 4 if every prime factor of m +13 congruentto 3 (mod 4) occurs with even exponent and B ( n ) = m − · m − = 6 in all othercases. (cid:3) Remark 6.2
In the Appendix we provide examples that show that all the caseslisted in Theorem E are possible. ⊘ Comparing with other bounds
Although our lower bound B ( n ) does not, in general, increase with n , there aresome values of n for which B ( n ) is better than the lower bound ⌊ n ⌋ + 1 proposedby Kosniowski [Ko79]. Indeed, the following is an easy consequence of Theorems Dand E. ERMAT AND THE NUMBER OF FIXED POINTS OF PERIODIC FLOWS 19
Proposition 7.1.
Let B ( n ) be the lower bound for the number of fixed points of a J -preserving circle action on a n -dimensional compact connected almost complexmanifold ( M, J ) with c c n − [ M ] = 0 obtained in Theorems D and E. Then B ( n ) > ⌊ n ⌋ + 1 if and only if dim M ∈ { , , , , , , , , , , , , , , , , , } . In particular, Kosniowski’s conjecture is true for these dimensions whenever c c n − [ M ] = 0 . There are also some values of n for which B ( n ) is greater than n and we recoverthe lower bound for K¨ahler (Hamiltonian) actions. Proposition 7.2.
Let B ( n ) be the lower bound for the number of fixed points of a J -preserving circle action on a n -dimensional compact connected almost complexmanifold ( M, J ) with c c n − [ M ] = 0 obtained in Theorems D and E. Then B ( n ) > n + 1 if and only if dim M ∈ { , , , , , , } . Divisibility results for the number of fixed points
In a letter to V. Gritsenko, Hirzebruch [Hi99] obtains divisibility results for theChern number c n [ M ] (the Euler characteristic of the manifold) under the assump-tion c c n − [ M ] = 0 (or under the stronger assumption that c = 0 in integercohomology). In particular he proves the following result. Theorem 8.1 (Hirzebruch) . Let M be a n -dimensional stably almost complexmanifold. If c c n − [ M ] = 0 then • if n ≡ or , the Chern number c n [ M ] is divisible by ; • if n ≡ , or , the Chern number c n [ M ] is divisible by ; • if n ≡ or , the Chern number c n [ M ] is divisible by . If an almost complex manifold is equipped with an S -action preserving thealmost complex structure with a nonempty discrete fixed point set, we know that c n [ M ] is equal to the number of fixed points of the action (see for example [GoSa12,Section 3]). Therefore, we can also obtain divisibility results for c n [ M ] from theexpressions of the functions F and F in (5.2) and (6.2) in the proofs of Theorems Dand E (since F and F count the number of fixed points), and it is straightforwardto see that Theorem B is a direct consequence of (5.3) and (6.3).In particular, we improve Hirzebruch’s divisibility factors for c n [ M ] whenever n Theorem F.
Let ( M, J ) be a n -dimensional compact connected almost complexmanifold equipped with a J -preserving S -action with nonempty, discrete fixed pointset M S . If c c n − [ M ] = 0 and n then • if n ≡ , then | M S | is divisible by ; • if n ≡ or , then | M S | is divisible by ; • if n ≡ , or , then | M S | is divisible by ; • if n ≡ or , then | M S | is divisible by . Proof. If n = 2 m is even, we can write n ≡ k (mod 8) with k ∈ { , , , } and m ≡ k (mod 4). Moreover, n m r = gcd ( m, r = 4 if k = 0 ,r = 1 if m is odd (i.e. if k = 1 or 3), r = 2 if k = 2 . The result for even values of n then follows from Theorem B since the numberof fixed points is divisible by r . Note that, if n ≡ m ≡ r = 12 if k = 0, r = 3 if m is odd, r = 6 if k = 2 , and we recover Hirzebruch’s divisibility factors in Theorem 8.1.If n = 2 m + 1 is odd, we can write n ≡ k + 1 (mod 8) with k ∈ { , , , } and m − ≡ k − n m − r = gcd ( m − , r = 4 if k = 1 ,r = 1 if m − k = 0 or 2), r = 2 if k = 3 . The result for odd values of n then follows from Theorem B since the number offixed points is divisible by r . Note that, if n ≡ m − ≡ r = 12 if k = 1, r = 3 if m − r = 6 if k = 3 , and we recover Hirzebruch’s divisibility factors in Theorem 8.1. (cid:3) ERMAT AND THE NUMBER OF FIXED POINTS OF PERIODIC FLOWS 21
In summary, we have the following table. n (mod 8) | M S | is divisible by0 1 if n ≡ n ≡ n ≡ n ≡ n ≡ n ≡ n ≡ n ≡ c = 0 in integer cohomology, Hirzebruch wasable to improve his divisibility factor for c n [ M ] when n = 2 m with m ≡ Proposition 8.2 (Hirzebruch) . If M is a n -dimensional stably almost complexmanifold with c = 0 and even n = 2 m with m ≡ , then c n [ M ] ≡ . Knowing this, we are also able to improve our divisibility factor for | M S | underthis condition. Theorem G.
Let ( M, J ) be a n -dimensional compact connected almost complexmanifold equipped with a J -preserving S -action with nonempty, discrete fixed pointset M S . If c = 0 with n ≡ and n , then | M S | is divisibleby .Proof. By Theorem F and Proposition 8.2, we have that | M S | ≡ | M S | ≡ (cid:3) Using this result we can improve the lower bound for the number of fixed pointsgiven by B ( n ), provided the assumptions of Theorem G are satisfied. Theorem H.
Let ( M, J ) be a n -dimensional compact connected almost complexmanifold equipped with a J -preserving S -action with nonempty, discrete fixed pointset M S . If c = 0 with n ≡ and n , then the number offixed points is at least . Remark 8.3 If n = 2 m with n ≡ n B ( n ) = 12 since gcd ( m,
12) = 1 ( m is odd and is not amultiple of 3). ⊘ Examples
We will now show that some of the lower bounds obtained in Theorems D and Efor the number of fixed points are sharp. For that, we will first state the followinglemma which gives a way of producing infinitely many manifolds with c c n − [ M ] =0. Lemma 9.1.
Let M m and N n be compact almost complex manifolds satisfying c c m − [ M ] = c c n − [ N ] = 0 . Then c c m + n − [ M × N ] = 0 .Proof. This follows from the fact that if, for any almost complex manifold M m with c m [ M ] = 0, we set γ ( M ) := c c m − [ M ] c m [ M ] , we have γ ( M × N ) = γ ( M ) + γ ( N ) (see [S, Section 3]). (cid:3) Example 9.2
There exists a 4 dimensional almost complex manifold manifold( N , J ) with c [ N ] = 0 that admits a J -preserving circle action with 12 fixedpoints (note that, since n = 2, we have gcd ( n ,
12) = 1 and B (2) = 12). Indeed,from (4.3) we can just take N = CP CP , the 9-point blow-up of CP since b ( N ) = 10 and b ( N ) = b ( N ) = 1 , so that, by (4.3), c [ N ] = 10 b ( N ) − b ( N ) = 0 and b ( N ) + b ( N ) + b ( N ) = 12 . Taking the standard Hamiltonian circle action on CP (with 3 isolated fixed points)and blowing up successively at index 2 fixed points, we can obtain a Hamiltoniancircle action on N with exactly 12 fixed points. ⊘ Example 9.3
For dim M = 6 we can take M = S with the almost complexstructure induced by a vector product in R and equipped with the S -actioninduced by the action on R = R ⊕ C given by λ · ( t, z , z , z ) = ( t, λ n z , λ m z , λ − ( n + m ) z ) , λ ∈ S , with t ∈ R , z , z , z ∈ C , m, n ∈ Z \ { } and m + m = 0. This action has exactly2 fixed points and N = N = 1 (note that B (3) = 2). ⊘ Example 9.4
In any dimension, since we can write every even positive integer2 n > n = 2(2 k + 3 l ) = 4 k + 6 l, for some k, l ∈ Z > , we can take M = ( N ) k × ( S ) l , where N is the S -manifold in Example 9.2 and S has the action in Example 9.3,to obtain an example of dimension 2 n . By Lemma 9.1 this almost complex manifoldsatisfies c c n − [ M ] = 0 , ERMAT AND THE NUMBER OF FIXED POINTS OF PERIODIC FLOWS 23 and the diagonal circle action preserves the almost complex structure and has 2 l × k fixed points.If k = l = 1 then dim M = 10 and the action has a minimal number of fixedpoints. Indeed, it has 24 fixed points and B (5) = 24.If k = 0 and l = 2 then dim M = 12 and the action has exactly 4 fixed points soit also has a minimal number of fixed points (since B (6) = 4).If k = 0 and l = 3 then dim M = 18 and the action has 8 fixed points which isalso a minimal number ( B (9) = 8). ⊘ Remark 9.5
It would be very interesting to find out if there exists an 8 dimensionalalmost complex manifold ( M , J ) with a J preserving circle action with exactly B (4) = 6 fixed points. If this example could be constructed then M × S wouldgive us a minimal example with B (7) = 12 fixed points. ⊘ Example 9.6
Returning to Example 9.4, we see that, although the S -manifolds( N ) k × ( S ) l do not always have a minimal number of fixed points, | M S | = 2 l × k ,is always divisible by the factors predicted in Theorems 8.1 and F.Indeed, if n is even and k = 0, then | M S | is a multiple of 12 and so it is divisibleby all the factors predicted in Theorems 8.1 and F. If n is even and k = 0 thennecessarily n ≡ n = 3 l is even, we have l >
1, and so | M S | is amultiple of 4, again divisible by all the factors predicted in Theorem 8.1.If n is odd then necessarily l >
1. If k = 0 then | M S | is a multiple of 24 andso it is divisible by all the factors predicted in Theorems 8.1 and F. If k = 0 thennecessarily n ≡ n = 3 l and 2 l fixed points. If l = 1 then n ≡ | M S | is divisible by 2 (the factor predicted in Theorem 8.1); if l = 2 then n ≡ | M S | is divisible by 4, as predicted in Theorem 8.1;if l > | M S | is a multiple of 8 and so it is divisible by all the factors givenby Theorem 8.1. ⊘ Final remarks
The lower bound for the number of fixed points in Proposition 2.1 follows fromthe fact that the momentum map µ is a Morse-Bott function, whose set of criticalpoints Crit( µ ) is a submanifold of M , and coincides with the fixed point set of theaction. Thus, if it is not zero dimensional, there are infinitely many critical pointsof µ and the result is obvious. If it is zero dimensional, then µ is a perfect Morsefunction (i.e. the Morse inequalities are equalities) because of the following classicalresult: If f is a Morse function on a compact and connected manifold whose criticalpoints have only even indices, then it is a perfect Morse function [Ni07, Corollary2.19 on page 52].Let N k ( µ ) be the number of critical points of µ of index k . The total number ofcritical points of µ is n X k =0 N k ( µ ) = n X k =0 b k ( M ) , where b k ( M ) := dim (cid:0) H k ( M, R ) (cid:1) is the k th Betti number of M . The classes [ ω k ]are nontrivial in H k ( M, R ) for k = 0 , . . . , n , so b k ( M ) >
1, and hence the numberof critical points of µ is at least n + 1.One can try to use Theorem 10.1 below to deduce a result analogous to Propo-sition 2.1 for circle valued momentum maps by replacing the Morse inequalities bythe Novikov inequalities (see [Pa06, Chapter 11, Proposition 2.4], [Fa04, Theorem2.4]), if all the critical points of µ are non-degenerate. Theorem 10.1 (McDuff, ’88) . Let the circle S act symplectically on a compactconnected symplectic manifold ( M, σ ) . Then either the action admits a standardmomentum map or, if not, there exists a S -invariant symplectic form ω on M thatadmits a circle valued momentum map µ : M → S . Moreover, µ is a Morse-Bott-Novikov function and each connected component of M S = Crit( µ ) has even index.If σ is integral, then ω = σ . The number of critical points of the circle-valued momentum map µ in Theo-rem 10.1 is then P nk =0 N k ( µ ). This number is at least n X k =0 (cid:16) ˆb k ( M ) + ˆq k ( M ) + ˆq k − ( M ) (cid:17) , where ˆb k ( M ) is the rank of the Z (( t ))-module H k ( f M , Z ) ⊗ Z [ t,t − ] Z (( t )), ˆq k ( M ) isthe torsion number of this module, and f M is the pull back by µ : M → R / Z ofthe principal Z -bundle t ∈ R [ t ] ∈ R / Z . Unfortunately, this lower bound can bezero. We refer to [PR12, Sections 3 and 4] for a detailed proof of Theorem 10.1 and[PR12, Remark 6] for further details. ERMAT AND THE NUMBER OF FIXED POINTS OF PERIODIC FLOWS 25
Appendix A. Tables n m r = gcd ( m, B ( n )
13 1 12
10 2 6 20
28 (mod 32)
14 2 12
27 3 4 n = 3 n = 3
16 4 3 n = 4
20 4 6 n is not a square and n = 4 · = 4 k (8 t + 7) , ∀ k, t ∈ Z >
56 4 9 n = 56 is not a square and n = 4 ·
54 6 2 n = 3
30 6 4 n = 5 is not a square and n = 2 ·
90 6 6 n = 15 is not a square, n = 2 · · n = 20 + 5 ·
126 6 8 n = 21 is not a square, n = 2 · · n = 28 + 7 ·
24 12 2 n = 2
36 12 3 n = 6 is not a square and n = 6
12 12 4 n , n are not squares and n = 2
72 12 6 n = 12 and n = 72 are not squares, n = 2 · n = 4 (8 + 1)
504 12 7 n = 84, n = 504 are not squares, n = 2 · · n = 4 (8 · Table A.1.
Examples that illustrate all possible lower bounds of | M S | listed in Theorem D when n = dim M is even (by increas-ing order of r ) n m r = gcd ( m − , B ( n )
19 6 4 n = 13
31 6 8 n = 3 ·
37 12 2 n − = ·
25 12 4 n − = 2 is not triangularand n = 17
49 12 6 n − = 4 is not triangularand n = 3 · Table A.2.
Examples that illustrate the possible lower boundsof | M S | when n = dim M is odd, for nontrivial cases listed intheorem E ( r = 6 or 12) References [CKP12] Cho, H. W., Kim, J. H., and Park, H. C., On the conjecture of Kosniowski,
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