by Chen He B.S., Zhejiang University, China M.S. in Mathematics, Northeastern University, US A dissertation submitted to

Localization of certain torus actions on odd-dimensional manifolds and its applications by Chen He B.S., Zhejiang University, China M.S. in Mathematics, Northeastern University, US A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 4, 2017 Dissertation directed by Jonathan Weitsman Professor of Mathematics Northeastern University

Abstract of Dissertation Let torus T act on a compact smooth manifold M, if the equivariant cohomology HT (M) is a free module of HT (pt), then according to the Chang-Skjelbred Lemma, H T (M) can be localized to the 1-skeleton M 1 consisting of fixed points and 1-dimensional orbits. Goresky, Kottwitz and MacPherson considered the case where M is an algebraic manifold and M 1 is 2-dimensional, and introduced a graphic description of its equivariant cohomology. In this thesis, firstly we will compute the equivariant cohomology of 3-dimensional closed manifolds with circle actions, then we will give graphic descriptions of equivariant cohomology of certain class of torus actions on odd-dimensional manifolds with 3-dimensional 1-skeleton. i

Acknowledgments I want to thank my advisors Victor Guillemin and Jonathan Weitsman for guidance and encouragement. Victor suggested the thesis project to me and helped me sort out my research ideas and gain mathematical maturity. Jonathan inspired me with his mathematical thoughts and helped me be on the right track of research progress. I would like to thank Shlomo Sternberg for enlightening me in his summer course of Symplectic Geometry and for introducing me to work with Victor and Jonathan. I wish to thank Alex Suciu and Maxim Braverman for teaching many advanced courses that I was lucky to attend and for joining in the defence committee of my thesis. Last but not the least, I owe my gratitude to my parents and friends for their constant encouragement. ii

Table of Contents Abstract of Dissertation Acknowledgments i ii 1 Introduction 1 1.1 Torus actions and equivariant cohomology...................... 2 1.1.1 Torus actions and isotropy weights...................... 2 1.1.2 Some basics of Equivariant cohomology................... 2 1.2 GKM theory in even dimension............................ 5 1.2.1 GKM condition in even dimension...................... 5 1.2.2 The geometry and cohomology of 2d S 1 -manifolds............. 5 1.2.3 The GKM graph and GKM theorem in even dimension........... 6 1.3 S 1 -actions on closed 3d manifolds.......................... 10 1.3.1 Equivariant tubular neighbourhoods of principal orbits........... 10 1.3.2 Equivariant tubular neighbourhoods of exceptional orbits.......... 10 1.3.3 Equivariant tubular neighbourhoods of special exceptional orbits...... 11 1.3.4 Equivariant tubular neighbourhoods of fixed points............. 11 1.3.5 Patching: from local to global........................ 12 2 Equivariant cohomology of 3d S 1 -manifolds 15 2.1 More basic facts about equivariant cohomology................... 15 2.2 A short exact sequence................................ 17 2.3 The ring and module structure............................ 22 2.4 The vector-space structure.............................. 26 2.5 Equivariant Betti numbers and Poincaré series.................... 27 2.6 Equivariant formality................................. 28 3 Localization of odd-dimensional manifold 33 3.1 Minimal 1-skeleton condition in odd dimension................... 33 3.2 The geometry and cohomology of 3d S 1 -manifolds................. 34 3.3 GKM graph and GKM theorem in odd dimension.................. 36 iii

4 Some applications 39 4.1 Some direct examples................................. 39 4.2 Odd-dimensional real and oriented Grassmannians................. 41 4.2.1 Torus actions and 1-skeleton of odd-dimensional Grassmannians...... 41 4.2.2 Equivariant cohomology of odd-dimensional real Grassmannian...... 43 4.2.3 Equivariant cohomology of odd-dimensional oriented Grassmannian.... 45 Bibliography 47 iv

Chapter 1 Introduction Let torus T act on a compact smooth manifold M. The T -equivariant cohomology of M is defined using the Borel construction H T (M) = H ((M ET )/T ), where ET = (S ) dim T and the coefficient of cohomology will always be Q throughout the paper. By this definition, if we denote t as the dual Lie algebra of T, then H T (pt) = H (ET/T ) = H ((CP ) dim T ) = St is a polynomial ring in dim T variables. The trivial map M pt induces a homomorphism H T (pt) HT (M) and gives H T (M) a H T (pt)-module structure. For every point p M, its stabilizer is defined as T p = {t T t p = p}, and its orbit is O p = T/Tp. If we set the i-th skeleton M i = {p dim O p i}, then this gives an equivariant stratification M 0 M 1 M dim T = M on M, where the 0-skeleton M 0 is exactly the fixedpoint set M T. If HT (M) is a free H T (pt)-module, also called equivariantly formal in [GKM98], Chang and Skjelbred [CS74] proved that H T (M) only depends on the fixed-point set M T and 1-skeleton M 1 : H T (M) = H T (M 1 ) = ( Im ( H T (M K ) H T (M T ) )) where the intersection is taken over all corank-1 subtorus K of T. The Chang-Skjelbred isomorphism enables one to describe the equivariant cohomology H T (M) as a sub-ring of H T (M T ), subject to certain algebraic relations determined by the 1-skeleton M 1. For example, Goresky, Kottwitz and MacPherson [GKM98] considered torus actions on algebraic varieties when the fixed-point set M T is finite and the 1-skeleton M 1 is a union of spheres S 2. They proved that the cohomology HT (M) can be described in terms of congruence relations on a regular graph determined by the 1-skeleton M 1. Since then, various GKM-type theorems were proved, for instance, by Brion [Br97] on equivariant Chow groups, by Knutson&Rosu [KR03], Vez- 1

CHAPTER 1. INTRODUCTION zosi&vistoli [VV03] on equivariant K-theory, and by Guillemin&Holm [GH04] on Hamiltonian symplectic manifold with non-isolated fixed points. Recent generalization of GKM-type theorem is due to Goertsches,Nozawa&Töben [GNT12] on Cohen-Macaulay actions on K-contact manifolds, and Goertsches&Mare [GM14] on non-abelian actions. In this thesis, we will try to develop a graphic description of equivariant cohomology for manifolds (possibly non-orientable) in odd-dimensional cases. 1.1 Torus actions and equivariant cohomology First we will recall some definitions and classical theorems regarding torus actions, equivariant cohomology (cf. [B72, Hs75, AP93]). 1.1.1 Torus actions and isotropy weights Throughout the paper, a manifold M is always assumed to be compact, smooth and boundaryless. Let torus T act on a manifold M, we will denote M T as the fixed-point set. For any point p in a connected component C of M T, there is the isotropy representation of T on the tangent space T p M, which splits into weighted spaces T p M = V 0 V [α1 ] V [αr] where the nonzero distinct weights [α 1 ],..., [α r ] t Z /±1 are determined only up to signs. Comparing with the tangent-normal splitting T p M = T p C N p C, we get that T p C = V 0 and N p C = V [α1 ] V [αr]. Since N p C = V [α1 ] V [αr] is of even dimension, the dimensions of M and components of M T will be of the same parity. If dim M is even, the smallest possible components of M T could be isolated points. If dim M is odd, the smallest possible components of M T could be isolated circles. Since T acts on the normal space N p C by rotation, this gives the normal space N p C an orientation. Moreover, if M is oriented, then any connected component C of M T has an induced orientation. For any subtorus K of T, we get two more actions automatically: the sub-action of K on M and the residual action of T/K on M K. 1.1.2 Some basics of Equivariant cohomology Given an action of torus T on M, comparing HT (M) with H T (M T ), the Borel localization theorem says: Theorem 1.1.1 (Borel Localization Theorem). The restriction map HT (M) HT (M T ) 2

CHAPTER 1. INTRODUCTION is a H T (pt)-module isomorphism modulo H T (pt)-torsion. Inspired by this localization theorem, we can hope for more connections between the manifold M and its fixed-point set M T, if H T (M) is actually H T (pt)-free. Definition 1.1.2. An action of T on M is equivariantly formal if H T (M) is a free H T (pt)-module. For equivariantly formal action, the Borel localization theorem gives an embedding of H T (M) into H T (M T ). Moreover, the image can be described as: Theorem 1.1.3 (Chang-Skjelbred Lemma, [CS74]). If M is equivariantly formal T -action, the equivariant cohomology H T (M) only depends on the fixed-point set M T and 1-skeleton M 1 : H T (M) = H T (M 1 ) = ( Im ( H T (M K ) H T (M T ) )) where the intersection is taken over all corank-1 subtorus K of T. Remark 1.1.4. More general results, named Atiyah-Bredon long exact sequence, appeared earlier in Atiyah s 1971 lecture notes [A74] for equivariant K-theory and later in Bredon [B74] for equivariant cohomology. action is: A direct consequence of the Borel localization theorem 1.1.1 for equivariantly formal group Corollary 1.1.5 (Existence of fixed points). If an action of T on M is equivariantly formal, then the fixed-point set M T is non-empty. Proof. According to the Borel localization theorem 1.1.1, the H T (M T ) will be of the same nonzero H T (pt)-rank as H T (M). Therefore, M T is non-empty. Using the techniques of spectral sequences, equivariant formality amounts to the degeneracy at E 2 level of the Leray-Serre sequence of the fibration M (M ET )/T BT. In the case of torus action, there is a much more applicable criterion for equivariant formality, (cf. [AP93] Theorem 3.10.4). Theorem 1.1.6 (Cohomology inequality and equivariant formality). If a torus T acts on M, then dim H (M T ) dim H (M), where equality holds if and only if the action is equivariantly formal. A sufficient condition for equivariant formality is that 3

CHAPTER 1. INTRODUCTION Corollary 1.1.7. If a T -manifold M has a T -invariant Morse-Bott function f such that Crit(f) = M T, then it is equivariantly formal. Proof. The cohomology H (M) can be computed from Morse-Bott-Witten cochain complex generated on the critical submanifold Crit(f). Hence dim H (M T ) = dim H (Crit(f)) dim H (M). The above theorem 1.1.6 says this inequality is actually an equality and hence the T -manifold M is equivariantly formal. Example 1.1.8. When M is equipped with a symplectic form, a Hamiltonian T -action and a moment map µ : M t, then µ ξ gives a Morse-Bott function for any generic ξ t and has Crit(µ ξ ) = M T, therefore M is T -equivariantly formal. Restricting to any subtorus K of T acting on M, we get Proposition 1.1.9 (Inheritance of equivariant formality). An action of torus T on M is equivariantly formal if and only if for any subtorus K of T, both the sub-action of K on M and the residual action of T/K on M K are equivariantly formal. Proof. Notice that after choosing a subtorus K, the three actions of T on M, K on M and T/K on M K give us the sequence of inequalities dim H (M T ) dim H (M K ) dim H (M) Thus, we see that the equality dim H (M T ) = dim H (M) holds if and only if both of the two intermediate equalities dim H (M T ) = dim H (M K ) and dim H (M K ) = dim H (M) hold, which is just a restatement of the proposition. Combining the Proposition 1.1.9 on inheritance of equivariant formality with the Corollary 1.1.5 on existence of fixed points, we get the inheritance of fixed points: Corollary 1.1.10 (Inheritance of fixed points). If an action of torus T on M is equivariantly formal, then for any subtorus K of T, every connected component of M K has T -fixed points. Proof. By the inheritance of equivariant formality, the residual action of T/K on any connected component C of M K is also equivariantly formal. Then by the existence of fixed points, C T = C T/K is non-empty. 4

CHAPTER 1. INTRODUCTION 1.2 GKM theory in even dimension Goresky, Kottwitz and MacPherson[GKM98] originally considered their theory for algebraic manifolds. Their ideas also work for general even-dimensional manifolds M 2n with torus action. When a T -action on M is equivariantly formal, a simple application of the Borel localization theorem 1.1.1 implies the non-emptiness of the fixed-point set M T. Then the Chang-Skjelbred isomorphism H T (M) = H T (M 1) = ( Im ( H T (M K ) H T (M T ) )) says that one can study the equivariant cohomology HT (M) by understanding (1) The fixed-point set M T (2) The 1-skeleton M 1 1.2.1 GKM condition in even dimension To apply the Chang-Skjelbred Lemma, Goresky, Kottwitz and MacPherson[GKM98] considered the smallest possible 0-skeleton M T and 1-skeleton M 1. Definition 1.2.1 (GKM condition in even dimension). An action of torus T on M 2n is GKM if the fixed-point set M T is non-empty and the 1-skeleton M 1 is at most of 2-dimensional. Or equivalently, (1) The fixed-point set M T consists of non-empty isolated points. (2) The 1-skeleton M 1 is of dimension at most 2. Or equivalently, at each fixed point p M T, the non-zero weights [α 1 ],..., [α n ] t Z /±1 of the isotropy T -representation T T pm are pair-wise linearly independent. From the condition (1), we get HT (M T ) = p M T St. From the condition (2), at each fixed point p, we get pair-wise independent weights [α 1 ],..., [α n ] t Z /±1 of the isotropy T -representation. If we denote T α i as the subtorus of T with Lie sub-algebra t αi = Ker α i, then the component C αi of M Tα i containing p will be of dimension 2 with the residual action of the circle T/T αi, i.e. a non-trivial S 1 -action on 2-dimensional surface with non-empty isolated fixed points. 1.2.2 The geometry and cohomology of 2d S 1 -manifolds According the classification of 2-dimensional compact S 1 -manifolds with non-empty fixed points, there are two such manifolds. 5

CHAPTER 1. INTRODUCTION Lemma 1.2.2 (see [Au04] subsection I.3.a). If S 1 acts effectively on a surface M with non-empty isolated fixed points, then M is S 2 with two fixed points RP 2 with one fixed point, and an exceptional orbit S 1 /(Z/2Z) where RP 2 as the Z/2Z quotient of S 2, has the induced S 1 -action from S 2. Using equivariant Mayer-Vietoris sequence, we see that the S 1 -actions on S 2 and RP 2 are both equivariantly formal, with equivariant cohomology H S 1 (S 2 ) = { (f N, f S ) Q[u] Q[u] f N (0) = f S (0) } H S 1 (RP 2 ) = Q[u] Transferring to the T -action on S 2 or RP 2 with subtorus T α acting trivially and the residual circle T/T α -action equivariantly formal, the equivariant cohomology is H T (S 2 [α] ) = H T/T α (S 2 [α] ) H T α (pt) H T (RP 2 [α] ) = St = { (f N, f S ) St St f N f S mod α } giving relations of elements of H T (M) expressed in terms of H T (M T ). 1.2.3 The GKM graph and GKM theorem in even dimension In the 1-skeleton M 1, each S 2 has two fixed points, and each RP 2 has one fixed point. This observation leads to a graphic representation of the relation among M T and M 1. Definition 1.2.3 (GKM graph in even dimension). The GKM graph of a GKM action of torus T on M 2n consists of Vertices There are two types of vertices for each fixed point in M T Empty dot for each RP 2 M 1 Edges & Weights A solid edge with weight [α] for each S[α] 1 joining two s representing its two fixed points, and a dotted edge with weight [β] for each RP[β] 2 joining a to an empty dot. 6

CHAPTER 1. INTRODUCTION Remark 1.2.4. The GKM graphs were originally defined for orientable even-dimensional manifolds and hence only have one type of vertices. The above GKM graph with two type of vertices for possibly non-orientable even-dimensional manifolds are due to Goertsches and Mare [GM14]. Remark 1.2.5. By the GKM condition 1.2.1, a fixed point has exactly n pair-wise linearly independent weights. Thus each, representing a fixed point, is joined by exactly n edges to s or empty dots. Note that each empty dot belongs to a unique RP 2 and will have exactly one edge joining it to the fixed point of that RP 2. α 2 α n 1 α 1 α n Figure 1.1: Each has exactly n edges Goresky, Kottwitz and MacPherson[GKM98] originally gave graphic descriptions for certain class of algebraic manifolds with torus actions. Goertsches and Mare [GM14] observed that those ideas also work for certain class of non-orientable even-dimensional manifolds with torus action. Theorem 1.2.6 (GKM theorem in even dimension, [GKM98, GM14]). If the action of torus T on a (possibly non-orientable) manifold M 2n is equivariantly formal and GKM, then we can construct its GKM graph G, with vertex set V = M T and weighted edge set E, such that the equivariant cohomology has a graphic description HT (M) = { f : V St f p f q mod α for each solid edge pq with weight α in E } Proof. Combining Chang-Skjelbred Lemma and and the equivariant cohomology of S 2 and RP 2, we get the GKM theorem. Remark 1.2.7. The RP 2 s in the 1-skeleton M 1 don t contribute to the congruence relations. We can erase all the dotted edges in the GKM graph, and call the remaining graph as the effective GKM graph. Remark 1.2.8. Note that in this paper we are working in Q coefficients. However, if we want to get a GKM-type theorem for much subtler coefficients like Z, the RP 2 s in the 1-skeleton M 1 and their 7

CHAPTER 1. INTRODUCTION corresponding dotted edges in the GKM graph are as crucial as the S 2 s and their corresponding solid edges. Remark 1.2.9. If M 2n has a T -invariant stable almost complex structure, then the isotropy weights α 1,..., α n t Z are determined with signs, and its GKM graph can be made into a directed graph. Moreover, as explained by Guillemin and Zara [GZ01], there is a set of congruence relations between the bouquets of isotropy weights for each edge, and they call it the connection of the GKM graph. Remark 1.2.10. We have assumed M to be connected. Suppose its GKM graph G has l connected components G 1,..., G l. Note the assignment of polynomials on vertices from the same G i with the same constant rational number gives all the elements in the graphic description of HT 0 (M). Thus we have dim HT 0 (M) = l = 1, i.e. the graph G is also connected. Example 1.2.11. Toric manifolds are GKM manifolds. Example 1.2.12. For the sphere S 2n, we use the coordinates (x, z 1,..., z n ) where x is a real variable, z i s are complex variables. Let T n act on S 2n by (e iθ 1,..., e iθn ) (x, z 1,..., z n ) = (x, e iθ 1 z 1,..., e iθn z n ) with fixed-point set (S 2n ) T n = {(±1, 0,..., 0)}. Since dim H ((S 2n ) T n ) = 2 = dim H (S 2n ), the T n action on S 2n is equivariantly formal by the formality criterion Theorem 1.1.6. Let α 1,..., α n be the standard integral basis of t Z = Zn, then each fixed point has the unsigned isotropy weights [α 1 ],..., [α n ]. This means the action is GKM and the GKM graph consists of two vertices with n edges weighted [α 1 ],..., [α n ] joining them. The equivariant cohomology is then H T n(s2n ) = {(f, g) St St f g mod n i=1 α i}. Example 1.2.13. RP 2n as the quotient of S 2n by the Z/2Z action e πi (x, z 1,..., z n ) = ( x, z 1,..., z n ) also inherits a T n -action from that on S 2n, discussed in previous section. The fixed-point set is (RP 2n ) T n = {(±1, 0,..., 0)}/(Z/2Z), a single point. Since dim H ((RP 2n ) T n ) = 1 = dim H (RP 2n ), the T n action on RP 2n is equivariantly formal by the formality criterion Theorem 1.1.6 with the unsigned isotropy weights [α 1 ],..., [α n ] at the only fixed point. This means the action is GKM and the GKM graph consists of a single vertex with n dotted edges weighted [α 1 ],..., [α n ], and the effective GKM graph is a single vertex without edges. The equivariant cohomology is then HT n(rp 2n ) = St. Example 1.2.14. Let s consider the real Grassmannian G 2k (R 2n ). Write the coordinates on R 2n as (x 1, y 1,..., x n, y n ). Let T n act on R 2n so that the i-th S 1 -component of T n exactly rotates the i-th pairs of real coordinates (x i, y i ) and leaves the remaining coordinates free, hence we can write 8

CHAPTER 1. INTRODUCTION R 2n = n i=1 R2 [α i ] for their decompositions into weighted subspaces, where [α i] t Z / ± 1. These actions induce T n actions on G 2k (R 2n ). In order to determine the fixed-point set and 1-skeleton of the action T n G 2k (R 2n ), we only need to observe that fixed-points of T n G 2k (R 2n ) are exactly the T n -subrepresentations of R 2n = n i=1 R2 [α i ]. It s not hard to show that the GKM graph of T n G 2k (R 2n ) consists of Vertices G 2k (R 2n ) T n = { S {1, 2,..., n} #S = k } Edges If S\{i} = S \{j}, then there are two edges with weights [α j α i ] and [α j +α i ] connecting S and S Denote S = G 2k (R 2n ) T n as the collection of k-element subset of {1, 2,..., n}. Then the GKM description then can be given as the following collection of maps: { } f : S Q[α 1,..., α n ] f S f S mod αj 2 αi 2 for S\{i} = S \{j} Example 1.2.15. Let M 2n M 2n be a T -equivariant finite covering with deck transformation group Γ. If the T -action on M is equivariantly formal and GKM, then according to the evendimensional GKM theorem 1.2.6, the equivariant cohomology HT ( M) concentrates on even degrees, so does its ordinary cohomology H ( M). Since H (M) = H ( M) Γ, the ordinary cohomology H (M) also concentrates on even degrees, which means the T -action on M is equivariantly formal. The isotropy weights at T -fixed points of M are inherited from M, hence T -action on M is also GKM. Restricting the covering to fixed points and 1-skeleta M T M T, M1 M 1 and denoting G, G as the GKM graphs of M, M, we can view the GKM graph G as G/Γ in the following sense: the Γ-orbits of vertices in G one-to-one correspond to the vertices in G; the free Γ-orbits of solid edges in G one-to-one correspond to solid edges in G; the Γ-orbits of empty vertices and dotted edges in G form part of the empty vertices and dotted edges in G; the non-free Γ-orbits of solid edges in G form the remaining empty vertices and dotted edges in G. Example 1.2.16. As an application, we can revisit Guillemin, Holm and Zara s [GHZ06] GKM description of homogeneous space G/K where rank G = rank K and K is connected. We can actually drop the assumption of K being connected. Let K 0 be the identity component of K and fix a maximal torus T of K 0. Under the left action of T, both G/K 0 and G/K are equivariantly formal and GKM with GKM graphs denoted as G G/K0, G G/K. Note the covering G/K 0 G/K is T -equivariant with deck transformation group K/K 0, we get the relation between the GKM graphs as G G/K = G G/K0 /(K/K 0 ). 9

CHAPTER 1. INTRODUCTION Remark 1.2.17. Besides the application to covering spaces, the even-dimensional GKM theorem 1.2.6 for possibly non-orientable manifolds can also be readily applied to related fibrations developed by Guillemin, Sabatini and Zara [GSZ12]. 1.3 S 1 -actions on closed 3d manifolds The idea of classifying effective S 1 -actions in dimension 3 is the same as in dimension 2 by listing all the possible equivariant tubular neighbourhoods of non-principal orbits, and then try to patch them together. But one more dimension for the isotropic representations provides a longer list of equivariant tubular neighbourhoods. 1.3.1 Equivariant tubular neighbourhoods of principal orbits For a point x of principal type, its isotropy group is the identity group {1} with a trivial isotropic representation {1} R 2. So an equivariant tubular neighbourhood of S 1 x can be written as S 1 {1} D = S 1 D, with the S 1 -action concentrating entirely on the S 1 -factor. So the orbit space of this tubular neighbourhood is (S 1 D)/S 1 = S 1 /S 1 D = D, a smooth local chart. 1.3.2 Equivariant tubular neighbourhoods of exceptional orbits The union of exceptional orbits will be denoted as E. For an exceptional orbit S 1 /Z m with stabilizer Z m = {e 2πki m, k = 1, 2,..., m}, its isotropic representation of Z m is 2-dimensional. Such a 2-dimensional effective Z m -representation could preserve the orientation by rotating: Z m rotate C : e 2πki m 2πki z = (e m ) n z where the orbit invariants (m, n), also called Seifert invariants, are coprime positive integers, and 0 < n < m. The resulting equivariant tubular neighbourhood is S 1 Zm D, whose orbit space is an orbifold disk (S 1 Zm D)/S 1 = D/Z m where the central orbifold point pt/z m corresponds to the exceptional orbit S 1 /Z m. 10

CHAPTER 1. INTRODUCTION 1.3.3 Equivariant tubular neighbourhoods of special exceptional orbits Besides rotating, a 2-dimensional effective Z m -representation could also reverse the orientation by reflection: Z 2 reflect R 2 : e πi (x, y) = ( x, y) This case requires the Z m to be Z 2. Because of the reverse of orientation, we call such an orbit S 1 /Z 2 a special exceptional orbit. The union of all such special exceptional orbits will be denoted as SE. If we use the open square I I = {(x, y) 1 < x, y < 1} as a neighbourhood in R 2, an equivariant tubular neighbourhood of the special exceptional orbit S 1 /Z 2 can be written as S 1 Z2 (I I), the orbit space by Z 2 of the solid torus S 1 (I I). Note that the reflection Z 2 reflect I I : e πi (x, y) = ( x, y) only affects the first I-factor, so we can split the second I-factor out of the orbit space S 1 Z2 (I I): S 1 Z2 (I I) = S 1 (I I)/(e iθ, x, y) ( e iθ, x, y) ( ) = S 1 I/(e iθ, x) ( e iθ, x) I = Möb I where we write Möb for short of the Möbius band S 1 Z2 I. Because the set of points with stabilizer Z 2 in the Möbius band S 1 Z2 I is Möb (Z2 ) = S 1 Z2 {0} = S 1 /Z 2 a circle, the set of points with stabilizer Z 2 in Möb I is (Möb I) (Z2 ) = S 1 /Z 2 I of dimension 2. Thus, if a 3d S 1 -manifold M has a special exceptional orbit S 1 /Z 2, then the connected component of M (Z2 ) that contains this orbit will be of dimension 2 and is acted freely by S 1 /Z 2, hence has to be S 1 /Z 2 S 1 according the list of 2d S 1 -manifolds. Now an equivariant tubular neighbourhood of this torus S 1 /Z 2 S 1 will be a bundle of Möbius band over S 1, which is actually a product bundle Möb S 1, cf. Raymond [Ra68]. Notice that the S 1 -action concentrates entirely on the factor of Möbius band, so the orbit space is (Möb I)/S 1 = Möb/S 1 I = [0, 1) I with a boundary circle {0} S 1. 1.3.4 Equivariant tubular neighbourhoods of fixed points The set of fixed points will be denoted as F. For a fixed point x with stabilizer S 1, its isotropic representation is of dimension 3. There is only one such effective 3-dimensional S 1 -representation S 1 C R by acting on the C-factor rotationally and acting on the R-factor trivially. So an equivariant tubular neighbourhood of x can be written as D I, with fixed point set {0} I, an interval. We can continue to glue along this fixed interval to form S 1, a connected 11

CHAPTER 1. INTRODUCTION component of the fixed point set. Now an enlarged equivariant tubular neighbourhood of the fixed circle S 1 is going to be a disk bundle over the S 1, which is actually a product bundle D S 1, cf. Raymond [Ra68]. Notice that the S 1 -action concentrates entirely on the D-factor, so the orbit space is (D S 1 )/S 1 = D/S 1 S 1 = [0, 1) S 1 with a boundary circle {0} S 1. 1.3.5 Patching: from local to global First, we can summarize all the local pictures into a list Principal Exceptional Special exceptional Singular Stabilizer S 1 x {1} Z m Z 2 S 1 Isotropic representation {1} C Z m rotate C Z 2 reflect R 2 S 1 rotate C R Orbit S 1 x S 1 S 1 /Z m S 1 /Z 2 pt Equivariant neighbourhood S 1 D S 1 Zm D Möb I D I Orbit neighbourhood D D/Z m [0, 1) I [0, 1) I Component of orbits of same type S 1 /Z m S 1 /Z 2 S 1 pt S 1 Enlarged equivariant neighbourhood S 1 Zm D Möb S 1 D S 1 Enlarged orbit neighbourhood D/Z m [0, 1) S 1 [0, 1) S 1 From the above list, we see that, passing to the orbit space, the local neighbourhood of an exceptional orbit S 1 /Z m contributes to an orbifold neighbourhood D/Z m. Both the local neighbourhoods of special exceptional orbits and the local neighbourhoods of fixed circles give rise to half closed, half open annuli [0, 1) S 1 with circle boundaries {0} S 1. Theorem 1.3.1 (Orbit space of closed 3d S 1 -manifold, [Ra68, OR68]). For a compact closed 3d effective S 1 -manifold M, the orbit space M = M/S 1 is a 2d orbifold surface, possibly with boundaries. The orbifold surface M has finite number of interior orbifold points with Seifert invariants {(m 1, n 1 ),..., (m l, n l )}, and boundary M = F SE/S 1 coming from the fixed circles and special exceptional orbits. To express M = M/S 1 and its orbifold points into numeric data, let s denote ɛ as the orientability of the 2d orbit space M = M/S 1, g the genus, (f, s) the numbers of circles formed from fixed circles and components of special exceptional orbits respectively. As for the total space M, after specifying the neighbourhoods of non-principal orbits, there is an obstruction integer b of finding a cross section over the principal part of the orbit space. The 12

CHAPTER 1. INTRODUCTION theorem by Orlik and Raymond says that, these invariants completely classify the 3d S 1 -manifolds, after adding some constraints within these invariants. The following version is taken from Orlik s lecture notes [Or72]. Theorem 1.3.2 (Equivariant classification of closed 3d S 1 -manifolds, [Ra68, OR68]). Let S 1 act effectively and smoothly on a closed, connected smooth 3d manifold M. Then the orbit invariants { b; (ɛ, g, f, s); (m1, n 1 ),..., (m l, n l ) } determine M up to equivariant diffeomorphisms, subject to the following conditions (1) b = 0, if f + s > 0 b Z, if f + s = 0 and ɛ = o, orientable b Z 2, if f + s = 0 and ɛ = n, non-orientable b = 0, if f + s = 0, ɛ = n and m i = 2 for some i (2) 0 < n i < m i, (m i, n i ) = 1 if ɛ = o 0 < n i m i 2, (m i, n i ) = 1 if ɛ = n Conversely, any such set of invariants can be realized as a closed 3d manifold with an effective S 1 -action. Remark 1.3.3. Raymond s idea of proving this classification theorem is as follows: given any two closed 3d S 1 -manifolds M, M with the same orbit invariants, firstly we can establish an orbifold diffeomorphism between M/S 1 and M/S 1. Secondly, we can lift this orbifold diffeomorphism to E F SE Ē F SE between the three types of non-principal orbits and extend this map to a tubular neighbourhood of the non-principal orbits. Finally, we can extend this map to all the principal orbits using local cross sections, which actually gives a global S 1 -diffeomorphism if the principal Euler numbers b, b are the same. Remark 1.3.4. When M has neither fixed point nor special exceptional orbit, i.e. f = s = 0, then this is the case of Seifert manifolds. Remark 1.3.5. The invariants in M = { b; (ɛ, g, f, s); (m 1, n 1 ),..., (m l, n l ) } mostly come from the orbit space M = M/S 1 except the invariant b. Therefore the constraint (b = 0, if f + s > 0) says that if the orbifold M has boundaries, then M = { b = 0; (ɛ, g, f, s); (m 1, n 1 ),..., (m l, n l ) } is determined by the orbifold M/S 1 and the assignment of its boundary circles being either from fixed components or special exceptional components. 13

CHAPTER 1. INTRODUCTION Remark 1.3.6. The above classification is up to equivariant diffeomorphisms. But Orlik and Raymond also discussed in certain conditions, more than one S 1 -actions can appear on the same 3d manifold. For an orientable S 1 -manifold M, the orbit space M = M/S 1 will be orientable, i.e. ɛ = o, and there will be no special exceptional orbits, i.e. s = 0. Corollary 1.3.7 (Classification of closed orientable 3d S 1 -manifolds, [Ra68, OR68]). If a closed 3d S 1 -manifold is oriented and the S 1 -action preserves the orientation. Then the orbit invariants { b; (ɛ = o, g, f, s = 0); (m1, n 1 ),..., (m l, n l ) } determine M up to equivariant diffeomorphisms, subject to the following conditions (1) b = 0, if f > 0 b Z, if f = 0 (2) 0 < n i < m i, (m i, n i ) = 1 14

Chapter 2 Equivariant cohomology of 3d S 1 -manifolds The classification of 3d S 1 -manifolds (possibly with boundaries) in terms of numeric invariants and graphs gives us an S 1 -equivariant stratification of every such manifold and enables us to calculate all kinds of topological data. For example, the fundamental groups, ordinary cohomology with Z or Z p coefficients have been computed extensively for closed 3d S 1 -manifolds in literature [JN83, BZ03], and now can be generalized to 3d S 1 -manifolds with boundaries, using the classification Theorems 1.3.2. But not much has been discussed for S 1 -equivariant cohomology, which is the goal of current section. In the following subsections, we will first prove our core Theorem 2.2.5 in full generality. When we explore more delicate computational invariants, we will try to keep the presentation of results in a manageable way but perhaps with a slight loss of generality. 2.1 More basic facts about equivariant cohomology In the following discussion, the coefficient of cohomology will always be Q unless otherwise mentioned. For a group action of G on M, the equivariant cohomology ring is defined using the Borel construction HG (M) = H (EG G M), where H ( ) is the ordinary simplicial cohomology theory, EG is the universal principal G-bundle and EG G M is the associated bundle with fibre M. The pull-back π : HG (pt) H G (M) of the trivial map π : M pt gives H G (M) a module structure of the ring HG (pt). 15

CHAPTER 2. EQUIVARIANT COHOMOLOGY OF 3D S 1 -MANIFOLDS In general, the equivariant cohomology HG (M) is not the same as the ordinary cohomology H (M/G) of the orbit space M/G. If we choose any fibre inclusion ι : M EG M and pass to the orbit spaces ῑ : M/G EG G M, then the pull-back ῑ : H G (M) = H (EG G M) H (M/G) gives a natural map between H G (M) and H (M/G). We will need some basic facts to compute equivariant cohomology, see the expository survey [Go] for details. The first set of facts is about equivariant cohomology of homogeneous space, i.e. space with one single orbit: Fact 2.1.1. Let G be a compact Lie group, and H a closed Lie subgroup. Denote BG = EG/G and BH = EH/H for the classifying space of G-bundles and H-bundles respectively. Then, H G (pt) = H (EG/G) = H (BG) H G (G/H) = H H (pt) = H (BH) The second set of facts is about equivariant cohomology of extremal types of group actions: Fact 2.1.2. Let a compact Lie group G act on a compact manifold M. If the action G M is free, then H G (M) = H (M/G). If the action G M is trivial, then H G (M) = H (M) H G (pt). In particular, when G = S 1, there are three types of orbits: S 1, S 1 /Z m, S 1 /S 1. For a principal orbit, H S 1 (S 1 ) = H (pt). For an exceptional orbit S 1 /Z m, the classifying space BZ m = S /Z m is the infinite Lens space with cohomology in Q-coefficient the same as H (pt). For a fixed point S 1 /S 1, the classifying space BS 1 = CP is the infinite projective space with cohomology Q[u] a polynomial ring, where the parameter u is the generator of H 2 (CP 1 ) in degree 2. Principal orbit Exceptional orbit Singular orbit Orbit O S 1 S 1 /Z m S 1 /S 1 HS (O, Q) 1 H (pt, Q) H (pt, Q) Q[u] The third set of facts enables us to compute equivariant cohomology by deforming, cutting and pasting, similar to the computation in ordinary cohomology: Fact 2.1.3. Let U 1, U 2 be two G-spaces, and A, B be two G-subspaces of a G-space X. Homotopy invariance If ϕ : U 1 H G (U 1) is an isomorphism. U2 is a G-homotopic equivalence, then ϕ : H G (U 2) = 16

CHAPTER 2. EQUIVARIANT COHOMOLOGY OF 3D S 1 -MANIFOLDS Mayer-Vietoris sequence If X = A B is the union of interiors of A and B, then there is a long exact sequence: HG(X) i HG(A) i HG(B) i HG(A i B) H i+1 (X) δ G Remark 2.1.4. Besides the Borel model of equivariant cohomology, there are also Cartan model and Weil model (cf. Guillemin-Sternberg [GS99]) using equivariant de Rham theory. In this paper, we prefer the Borel model because the homotopy invariance and Mayer-Vietoris sequence are more natural for Borel model, from the topological rather than the differential point of view. The fourth set of facts deals with equivariant cohomology of product spaces: Fact 2.1.5. Let G M and H N be two group actions on manifolds. Then, for the product action G H M N, we get HG H(M N) = HG(M) HH(N) Especially, for the action G M N where N is acted by G trivially, we get HG(M N) = HG(M) H (N) 2.2 A short exact sequence Let S 1 act effectively on a compact connected 3d manifold M, possibly with boundary. We will compute the equivariant cohomology group HS (M, Q) by cutting and pasting, with the help 1 of the classification theorem from previous sections. As we have seen from the previous computation of HS (O) for each S 1 -orbit O, the S 1-1 equivariant cohomology in Q coefficient does not distinguish principal orbit S 1 from exceptional orbit S 1 /Z m or special exceptional orbit S 1 /Z 2. However, there is big difference between the S 1 -equivariant cohomology of fixed point and non-fixed orbit. If a 3d S 1 -manifold M does not have fixed points, we would hope that its S 1 -equivariant cohomology is the ordinary cohomology of the orbit space M/S 1. Actually, a more general statement is true due to Satake [Sa56]. The version here is taken from Duistermaat s lecture notes [Du94]. Definition 2.2.1. An action of a compact Lie group G on a manifold M is locally free, if for every x M, the isotropy group G x is finite. 17

CHAPTER 2. EQUIVARIANT COHOMOLOGY OF 3D S 1 -MANIFOLDS Theorem 2.2.2 (Satake [Sa56]). If a compact Lie group G acts locally freely on a compact manifold M, then M/G is an orbifold, and HG (M, R) = H (M/G, R). We can certainly apply the Theorem of Satake to our special case of S 1 -actions. However, there is a subtlety in Satake s definition of H (M/S 1, R) for the orbifold M/S 1 in terms of orbifold differential forms (cf. [Sa56, Du94]). Moreover, because of the use of differential forms, the above theorem is originally stated for R-coefficients not for Q-coefficients. In our definition of H (M/S 1, Q), we will simply use the ordinary simplicial cohomology for the topological space M/S 1 by forgetting its orbifold structure. Our method of calculating equivariant cohomology is based on the equivariant Meyer-Vietoris sequence and induction on the number of non-principal components which is finite because of the compactness of M. Proposition 2.2.3. Let S 1 act effectively on a compact connected 3d manifold M, possibly with boundary. If M does not have fixed points, then HS (M, Q) = H (M/S 1, Q). 1 Proof. We will proceed by induction on the number of non-principal components. To begin with, suppose M does not have non-principal component. Since we assume there is no fixed point, then S 1 acts on M freely and hence HS (M) = H (M/S 1 ). 1 Now suppose the proposition is true for any 3d fixed-point-free S 1 -manifold with k 0 non-principal components, and suppose M has k + 1 non-principal components. Let C be a nonprincipal component together with an equivariant tubular neighbourhood N, then the complement M = M N has k non-principal components and HS (M ) = H (M /S 1 ) according to our 1 assumption. Let s also denote L = M N The equivariant Mayer-Vietoris sequence for the union M = M N and the ordinary Mayer- Vietoris sequence for the union M/S 1 = M /S 1 N/S 1 gives: H 1 (M ) H 1 (N) H 1 (L) H S 1 S 1 S 1 S (M) H 1 S (M ) H 1 S (N) H 1 S (L) 1 H 1 (M /S 1 ) H 1 (N/S 1 ) H 1 (L/S 1 ) H (M/S 1 ) H (M /S 1 ) H (N/S 1 ) H (L/S where the second and the fifth vertical maps are isomorphisms, because the intersection L = M N does not touch non-principal orbits and consists of only principal orbits. According to the Five Lemma in homological algebra, in order to prove that the middle vertical map is an isomorphism, we now need to prove the first and the fourth maps are isomorphisms. But 18

CHAPTER 2. EQUIVARIANT COHOMOLOGY OF 3D S 1 -MANIFOLDS we already have the isomorphism H S 1 (M ) = H (M /S 1 ). So we only need to prove H S 1 (N) = H (N/S 1 ). For a 3d fixed-point-free S 1 -manifold M, according to our detailed discussion in Section 1.3, there are three cases for a non-fixed, non-principal component C, its equivariant neighbourhood N and orbit space N/S 1. Note that, for each case, there is an equivariant deformation retraction N C, so we have H S 1 (N) = H S 1 (C). Also recall that we have calculated H S 1 (S 1 /Z m, Q) = H (pt, Q). C S 1 /Z m S 1 /Z 2 S 1 N S 1 Zm D 2 Möb S 1 N/S 1 D 2 /Z m I S 1 H S 1 (N) = H S 1 (C) H (pt) H (S 1 ) H (N/S 1 ) H (D 2 /Z m ) H (S 1 ) For the second and the third case, it is clear that H S 1 (N) = H (N/S 1 ). For the first case, the orbit space D 2 /Z m, viewed as an ice-cream cone, has a deformation retract to the cone s tip pt, so H S 1 (N) = H (pt) = H (D 2 /Z m ) = H (N/S 1 ). If a 3d S 1 -manifold M has fixed points, then the number of fixed components will be finite due to the compactness of M, and every fixed component is a circle S 1 according to our discussion in the previous Subsection 1.3.4. The calculation of S 1 equivariant cohomology of a general 3d S 1 -manifold M will be carried out by doing induction on the number of connected components of these fixed points. The beginning case of no fixed points is just the previous Proposition 2.2.3. Suppose now that an S 1 -manifold M has k > 0 connected components of fixed points. Let s choose any such connected component F, with its equivariant neighbourhood N. According to Subsection 1.3.4, we have N = D F. If we set the complement M = M N, then M is attached equivariantly by M and N = D F along S 1 F. The Mayer-Vietoris sequence of equivariant cohomology groups then gives HS (M, Q) H 1 S (M, Q) H 1 S (D F, Q) H 1 S (S 1 F, Q) H +1 (M, Q) 1 S 1 However, since the S 1 -action on D F and S 1 F concentrates on their first components respectively, we have: 19

CHAPTER 2. EQUIVARIANT COHOMOLOGY OF 3D S 1 -MANIFOLDS H S 1 (D F ) H S 1 (S 1 F ) H S 1 (D) H (F ) H S 1 (S 1 ) H (F ) Q[u] H (F ) H (F ) : f(u) α f(0) α where the upper 2 vertical isomorphisms are because of the cohomology of product spaces, the lower left vertical isomorphism is because of homotopy between D and pt, and the lower right vertical isomorphism is because the S 1 is a principal orbit. The bottom map is obviously surjective, so is the top map H S 1 (D F ) H S 1 (S 1 F ). This means that the long exact sequence actually stops at H S 1 (M ) H S 1 (D F ) H S 1 (S 1 F ) 0. We then conclude that the long exact sequence reduces into the following short exact sequence: ( ) 0 HS (M) H 1 S (M ) Q[u] H (F ) H (F ) 0 1 where we have replaced the H S 1 (D F ) and H S 1 (S 1 F ) by Q[u] H (F ) and H (F ) respectively. We can now consider all the k components of fixed points F 1, F 2,..., F k, together with their equivariant tubular neighbourhood N 1, N 2,..., N k. If we set the complement M = M i N i, an S 1 -manifold without fixed points, then there is a short exact sequence of cohomology groups: ) 0 HS (M) H 1 S (M 1 ) i (Q[u] H (F i ) i H (F i ) 0 ( ) Since M is fixed-point-free, H S 1 (M, Q) = H (M /S 1, Q) by Proposition 2.2.3. To understand the orbit space M /S 1, we can compare it with the orbit space M/S 1. Lemma 2.2.4. Following the above notation, the two orbit spaces M /S 1 and M/S 1 are topologically homotopic. Especially, H (M /S 1, Q) = H (M/S 1, Q). Proof. Since the majority of M /S 1 and M/S 1 is isomorphic, we only need to check what happens in an equivariant neighbourhood N near an S 1 -fixed component F of M. Let N be an equivariant neighbourhood slightly larger than N. If we choose local S 1 - equivariant coordinates properly, we can write N = D 1 F and N = D 1 2 F, where D 1 and D 1 2 are 2-dimensional disks of radii 1 and 1 2, such that S1 acts on the disks by standard rotation. 20

CHAPTER 2. EQUIVARIANT COHOMOLOGY OF 3D S 1 -MANIFOLDS Now N N = (D 1 D 1 ) F and N = D 1 F are equivariant neighbourhoods of 2 M = M N and M respectively. Their orbit spaces by the S 1 -action give neighbourhoods (N N)/S 1 and N /S 1 of M /S 1 and M/S 1 respectively. and However, ( (N N)/S 1 = (D 1 D 1 )/S 1) F = [ 1 2 2, 1) F N /S 1 = ( D 1 /S 1) F = [0, 1) F are homotopic. Thus M /S 1 and M/S 1 are homotopic. Finally, we can combine all the above discussions and get: Theorem 2.2.5. Let M be a compact connected 3d effective S 1 -manifold(possibly with boundary), and F be its fixed-point set(possibly empty), then there is a short exact sequence of cohomology groups in Q coefficients: ( ) 0 HS (M) H (M/S 1 ) Q[u] H (F ) H (F ) 0 ( ) 1 Proof. If the fixed-point set F is not empty, then we can use the short exact sequence ( ), and the replacement H S 1 (M ) = H (M /S 1 ) = H (M/S 1 ) because of the Lemma 2.2.4. If the fixed-point set F = is empty, then H (F ) = 0. We just use the Proposition 2.2.3 which says H S 1 (M) = H (M/S 1 ). Remark 2.2.6. To be more specific about the maps involved in the above short exact sequence ( ): 1. H S 1 (M) H (M/S 1 ) is the natural map between equivariant cohomology of M and ordinary cohomology of M/S 1 2. H S 1 (M) Q[u] H (F ) is the equivariant restriction map from M to its fixed-point set F 3. H (M/S 1 ) H (F ) is the restriction map from M/S 1 to its boundary formed by F 4. Q[u] H (F ) H (F ) is the evaluation map given by f(u) α f(0)α 21

CHAPTER 2. EQUIVARIANT COHOMOLOGY OF 3D S 1 -MANIFOLDS 2.3 The ring and module structure By the short exact sequence ( ) of Theorem 2.2.5, we have the inclusion of cohomology groups: H S 1 (M) H (M/S 1 ) ( Q[u] H (F ) ). Since this inclusion is the direct sum of two restriction maps of cohomology rings, it preserves ring structure. Therefore, we can describe the ring structure of H S 1 (M) explicitly in terms of elements and constraints in H (M/S 1 ) and Q[u] H (F ). For simplicity, we will focus on closed 3d S 1 -manifolds. If M does not have fixed points, then the Proposition 2.2.3 says that its equivariant cohomology ring is the cohomology ring of the orbit space. Thus we will only be interested in the case where M has non-empty set of fixed points. According to the classification theorem, we can write M = { b = 0; (ɛ, g, f, s); (m 1, n 1 ),..., (m l, n l ) } with f > 0. Topologically, M/S 1 is a 2d surface of genus g, with f + s > 0 boundary circles. Let s first give a description of the involved cohomologies H (M/S 1 ) and Q[u] H (F i ). The orbit space M/S 1 as a topological 2d surface of genus g, has f boundary circles f i=1 F i from fixed components and s boundary circles s j=1 SE j from the orbit spaces of special exceptional components. For a fixed circle F i = S 1, 1 i f, we write H (F i, Q) = Qδ i Qθ i, where δ i and θ i are generators of H 0 (F i, Z) and H 1 (F i, Z) respectively. Similarly, for SE j = S 1, 1 j s, we write H (SE j, Q) = Qδ f+j Qθ f+j. If the orbit space M/S 1 is orientable, i.e. ɛ = o, though ±θ i are both generators for H 1 (F i, Z), we only choose θ i compatible with the boundary orientation on F i. The same rule of choice also applies to θ f+j. Moreover, we can write Q[u] H (F i ) = Q[u]δ i Q[u]θ i such that every element of Q[u] H (F i ) can be expressed as p i (u)δ i + q i (u)θ i for polynomials p i (u), p i (u) Q[u]. Using the classic calculation of cohomology of 2d surfaces with boundaries, the cohomology H (M/S 1 ) has two different descriptions according to whether M/S 1 is orientable or not. If M/S 1 is an orientable surface of genus g with f + s > 0 boundary circles, then it is homotopic to a wedge of 2g + f + s 1 circles. Let s denote α k, β k, 1 k g for the generators of H 1 ( ) of the 2g circles used in the polygon presentation of the surface M/S 1. Then we can write H (M/S 1 ) as a sub-ring of Qδ 0 g ( ) ( k=1 Qαk Qβ k f i=1 Qθ ( ) i) s j=1 Qθ f+j, such that every element of H (M/S 1 ) can be expressed as Dδ 0 + k (A kα k + B k β k ) + i C iθ i + j C f+jθ f+j for D, A k, B k, C i, C f+j Q, under the constraint that k (A k + B k ) + i C i + j C f+j = 0. Moreover, we have the restriction maps to each fixed circle F i : Q[u] H (F i ) H (F i ) : p i (u)δ i + q i (u)θ i p i (0)δ i + q i (0)θ i 22

CHAPTER 2. EQUIVARIANT COHOMOLOGY OF 3D S 1 -MANIFOLDS and H (M/S 1 ) H (F i ) : Dδ 0 + g (A k α k + B k β k ) + k=1 f C i θ i + i=1 s C f+j θ f+j Dδ i + C i θ i If M/S 1 is a non-orientable surface of genus g with f + s > 0 boundary circles, then it is homotopic to a wedge of g + f + s 1 circles. We can denote α k, 1 k g for the generators of H 1 ( ) of the g circles used in the polygon presentation of the surface M/S 1. The description of the cohomology H (M/S 1 ) together with the restriction maps is similar to the orientable case, with the only difference that there is no β k, B k for the non-orientable case. Following the above notations, we get Theorem 2.3.1. For a closed 3d S 1 -manifold M = { b = 0; (ɛ = o, g, f, s); (m 1, n 1 ),..., (m l, n l ) } with f > 0 and an orientable orbit space M/S 1, an element of its equivariant cohomology H S 1 (M) can be written as ( Dδ 0 + g (A k α k + B k β k ) + k=1 f C i θ i + i=1 s C f+j θ f+j ; j=1 ) in H (M/S 1 ) i (Q[u] H (F i ), under the relations j=1 f ) (p i (u)δ i + q i (u)θ i ) i=1 ( ) 1. g k=1 (A k + B k ) + f i=1 C i + s j=1 C f+j = 0 2. p 1 (0) = p 2 (0) = = p f (0) = D 3. q i (0) = C i for each i Breaking the equivariant cohomology H S 1 (M) into different degrees, we have H 0 S 1 (M) = Q H 1 S 1 (M) is a subgroup of H 1 (M/S 1 ) i H 1 (F i ) consisting of elements ( g (A k α k + B k β k ) + k=1 f C i θ i + i=1 s C f+j θ f+j ; j=1 f ) C i θ i under the constraint g k=1 (A k + B k ) + f i=1 C i + s j=1 C f+j = 0. ) H 2 (M) = S 1 i (Q[u] + H (F i ) where Q[u] + consists of polynomials without constant terms. i=1 23

CHAPTER 2. EQUIVARIANT COHOMOLOGY OF 3D S 1 -MANIFOLDS Proof. The expression ( ) of elements of H S 1 (M) comes from the description of cohomologies H (M/S 1 ) and Q[u] H (F i ). The relations (1)(2)(3) are due to the theorem 2.2.5 that HS (M) ) 1 is the kernel of the restriction map H (M/S 1 ) i (Q[u] H (F i ) i H (F i ). Thus, the images of restrictions are the same: p 1 (0) = p 2 (0) = = p f (0) = D, and q i (0) = C i. Since the relations (1)(2)(3) only live in degree less than 2, we get the description of H S 1 (M) in different degrees. Remark 2.3.2. For a closed 3d S 1 -manifold M = { b = 0; (ɛ = n, g, f, s); (m 1, n 1 ),..., (m l, n l ) } with f > 0 and a non-orientable orbit space M/S 1, the explicit expression of elements of H S 1 (M) is almost the same as the oriented case, with the only modification that there is no β k, B k term. Theorem 2.3.3. For a closed 3d S 1 -manifold M = { b = 0; (ɛ, g, f, s); (m 1, n 1 ),..., (m l, n l ) } with f > 0, the graded ring structure of H S 1 (M) is as follows: 1. H 0 S 1 (M) H S 1 (M) multiplication. 2. H 1 S 1 (M) H 1 S 1 (M) HS (M) and H 1 S (M) H 0 1 S (M) H 1 S (M) are just scalar 1 H 2 S 1 (M) is a zero map 3. HS 1 (M) H 2 (M) H 3 (M) fits into a commutative diagram: 1 S 1 S 1 H 1 S 1 (M) H 2 S 1 (M) H 3 S 1 (M) ( ) ( i H 1 ( (F i ) i Q[u]+ H (F i ) )) ( ) i Q[u] + H (F i ) where the left map is the restriction map HS 1 (M) 1 i HS 1 (F 1 i ) = i H 1 (F i ) tensored with the identification H 2 (M) ) S = 1 i (Q[u] + H (F i ), and the bottom map is the componentwise multiplication in i ) (Q[u] H (F i ). 4. H 2 (M) H 2 (M) H 2 (M) is just the component-wise multiplication of S 1 S 1 S 1 i (Q[u] + ) H (F i ) Proof. We will explain the above breakdown one by one for the case when M/S 1 is orientable. 1. This is clear. 24