Circle Actions Research Articles

Multiplicative structure in equivariant cohomology

We introduce the notion of a strongly homotopy-comultiplicative resolution of a module coalgebra over a chain Hopf algebra, which we apply to proving a comultiplicative enrichment of a well-known theorem of Moore concerning the homology of quotient spaces of group actions. The importance of our enriched version of Moore’s theorem lies in its application to the construction of useful cochain algebra models for computing multiplicative structure in equivariant cohomology.In the special cases of homotopy orbits of circle actions on spaces and of group actions on simplicial sets, we obtain small, explicit cochain algebra models that we describe in detail.

CIRCLE ACTION AND SOME VANISHING RESULTS ON MANIFOLDS

Kawakubo and Uchida showed that, if a closed oriented 4k-dimensional manifold M admits a semi-free circle action such that the dimension of the fixed point set is less than 2k, then the signature of M vanishes. In this note, by using G-signature theorem and the rigidity of the signature operator, we generalize this result to more general circle actions. Combining the same idea with the remarkable Witten–Taubes–Bott rigidity theorem, we explore more vanishing results on spin manifolds admitting such circle actions. Our results are closely related to some earlier results of Conner–Floyd, Landweber–Stong and Hirzebruch–Slodowy.

The rigidity of Dolbeault-type operators and symplectic circle actions

Following the idea of Lusztig, Atiyah-Hirzebruch and Kosniowski, we note that the Dolbeault-type operators on compact, almost-complex manifolds are rigid. When the circle action has isolated fixed points, this rigidity result will produce many identities concerning the weights on the fixed points. In particular, it gives a criterion to determine whether or not a symplectic circle action with isolated fixed points is Hamiltonian. As applications, we simplify the proofs of some known results related to symplectic circle actions, due to Godinho, Tolman-Weitsman and Pelayo-Tolman, and generalize some of them to more general cases.

Families of monotone symplectic manifolds constructed via symplectic cut and their Lagrangian submanifolds

We describe families of monotone symplectic manifolds constructed via the symplectic cutting procedure of Lerman from the cotangent bundle of manifolds endowed with a free circle action. We also give obstructions to the monotone Lagrangian embedding of some compact manifolds in these symplectic manifolds.

The Segal conjecture for topological Hochschild homology of complex cobordism

We study the C_p-equivariant Tate construction on the topological Hochschild homology THH(B) of a symmetric ring spectrum B by relating it to a topological version R_+(B) of the Singer construction, extended by a natural circle action. This enables us to prove that the fixed and homotopy fixed point spectra of THH(B) are p-adically equivalent for B = MU and BP. This generalizes the classical C_p-equivariant Segal conjecture, which corresponds to the case B = S.

ON POSITIVELY CURVED FOUR-MANIFOLDS WITH S1-SYMMETRY

It is well known by the work of Hsiang and Kleiner that every closed oriented positively curved four-dimensional manifold with an effective isometric S1-action is homeomorphic to S4or CP2. As stated, it is a topological classification. The primary goal of this paper is to show that it is indeed a diffeomorphism classification for such four-dimensional manifolds. The proof of this diffeomorphism classification also shows an even stronger statement that every positively curved simply connected four-manifold with an isometric circle action admits another smooth circle action which extends to a two-dimensional torus action and is equivariantly diffeomorphic to a linear action on S4or CP2. The main strategy is to analyze all possible topological configurations of effective circle actions on simply connected four-manifolds by using the so-called replacement trick of Pao.

Nonnegatively curved fixed point homogeneous manifolds in low dimensions

Let G be a compact Lie group acting isometrically on a compact Riemannian manifold M with nonempty fixed point set M G . We say that M is fixed-point homogeneous if G acts transitively on a normal sphere to some component of M G . Fixed-point homogeneous manifolds with positive sectional curvature have been completely classified. We classify nonnegatively curved fixed-point homogeneous Riemannian manifolds in dimensions 3 and 4 and determine which nonnegatively curved simply-connected 4-manifolds admit a smooth fixed-point homogeneous circle action with a given orbit space structure.

Type III 1 equilibrium states of the Toeplitz algebra of the affine semigroup over the natural numbers

We complete the analysis of KMS-states of the Toeplitz algebra T ( N ⋊ N × ) of the affine semigroup over the natural numbers, recently studied by Raeburn and the first author, by showing that for every inverse temperature β in the critical interval 1 ⩽ β ⩽ 2 , the unique KMS β -state is of type III 1. We prove this by reducing the type classification from T ( N ⋊ N × ) to that of the symmetric part of the Bost–Connes system, with a shift in inverse temperature. To carry out this reduction we first obtain a parametrization of the Nica spectrum of N ⋊ N × in terms of an adelic space. Combining a characterization of traces on crossed products due to the second author with an analysis of the action of N ⋊ N × on the Nica spectrum, we can also recover all the KMS-states of T ( N ⋊ N × ) originally computed by Raeburn and the first author. Our computation sheds light on why there is a free transitive circle action on the extremal KMS β -states for β > 2 that does not ostensibly come from an action of T on the C ⁎-algebra.

On the orders of periodic diffeomorphisms of 4-manifolds

This paper initiated an investigation on the following question: Suppose a smooth 4-manifold does not admit any smooth circle actions. Does there exist a constant $C>0$ such that the manifold support no smooth $\Z_p$-actions of prime order for $p>C$? We gave affirmative results to this question for the case of holomorphic and symplectic actions, with an interesting finding that the constant $C$ in the holomorphic case is topological in nature while in the symplectic case it involves also the smooth structure of the manifold.

Twisted cyclic theory, equivariant KK-theory and KMS states

Given a C*-algebra A with a KMS weight for a circle action, we construct and compute a secondary invariant on the equivariant K-theory of the mapping cone of , both in terms of equivariant KK-theory and in terms of a semifinite spectral flow. This in particular puts the previously considered examples of Cuntz algebras [Carey, Phillips, Rennie, Twisted cyclic theory and the modular index theory of Cuntz algebras] and SUq(2) [Carey, Rennie, Tong, J. Geom. Phys. 59: 1431–1452, 2009] in a general framework. As a new example we consider the Araki–Woods IIIλ representations of the Fermion algebra.

Constructing integrable systems of semitoric type

Let (M, ω) be a connected, symplectic 4-manifold. A semitoric integrable system on (M, ω) essentially consists of a pair of independent, real-valued, smooth functions J and H on M, for which J generates a Hamiltonian circle action under which H is invariant. In this paper we give a general method to construct, starting from a collection of five ingredients, a symplectic 4-manifold equipped a semitoric integrable system. Then we show that every semitoric integrable system on a symplectic 4-manifold is obtained in this fashion. In conjunction with the uniqueness theorem proved recently by the authors, this gives a classification of semitoric integrable systems on 4-manifolds, in terms of five invariants.

The number of exceptional orbits of a pseudofree circle action on S 5

Let M be a closed Riemannian manifold of dimension 5 which admits a Riemannian metric of nonnegative sectional curvature. The aim of this short paper is to show that under certain lower bound of the orders of isotropy subgroups, every pseudofree and isometric S1-action on M cannot have more than five exceptional circle orbits. As a consequence, we conclude that a pseudofree and isometric S1-action on a 5-sphere S5 with a Riemannian metric of nonnegative sectional curvature cannot have more than five exceptional circle orbits. This gives a result related to the Montgomery–Yang problem. In addition, we also give some further related result about nonnegatively curved manifolds of dimension 5 with an isometric but not necessarily pseudofree circle action.

Algebraic Montgomery–Yang problem: the non-cyclic case

Montgomery–Yang problem predicts that every pseudofree circle action on the 5-dimensional sphere has at most 3 non-free orbits. Using a certain one-to-one correspondence, Kollár formulated the algebraic version of the Montgomery–Yang problem: every projective surface S with quotient singularities such that the second Betti number b 2(S) = 1 has at most 3 singular points if its smooth locus S 0 is simply connected. We prove the conjecture under the assumption that S has at least one non-cyclic singularity. In the course of the proof, we classify projective surfaces S with quotient singularities such that (i) b 2(S) = 1, (ii) \({H_1(S^0, \mathbb{Z}) = 0}\), and (iii) S has 4 or more singular points, not all cyclic, and prove that all such surfaces have \({\pi_1(S^0)\cong \mathfrak{A}_5}\), the icosahedral group.

CIRCLE ACTIONS, CENTRAL EXTENSIONS AND STRING STRUCTURES

The caloron correspondence can be understood as an equivalence of categories between G-bundles over circle bundles and LG ⋊ρ S1-bundles where LG is the group of smooth loops in G. We use it, and lifting bundle gerbes, to derive an explicit differential form based formula for the (real) string class of an LG ⋊ρ S1-bundle.

The Log-Concavity Conjecture for the Duistermaat-Heckman Measure Revisited

Karshon constructed the first counterexample to the log-concavity conjecture for the Duistermaat-Heckman measure: a Hamiltonian six-manifold whose fixed-points set is the disjoint union of two copies of T 4 . In this article, for any closed symplectic four-manifold N with b + > 1, we show that there is a Hamiltonian six-manifold M such that its fixed-points set is the disjoint union of two copies of N and such that its Duistermaat-Heckman function is not log-concave. On the other hand, we prove that if there is a torus action of complexity 2 such that all the symplectic reduced spaces taken at regular values satisfy the condition b + = 1, then its Duistermaat-Heckman function has to be log-concave. As a consequence, we prove the log-concavity conjecture for Hamiltonian circle actions on six manifolds such that the fixed-points sets have no 4-dimensional components, or only have 4-dimensional pieces with b + = 1.

Topological properties of Hamiltonian circle actions

This paper studies Hamiltonian circle actions, i.e. circle subgroups of the group Ham(M,ω) of Hamiltonian symplectomorphisms of a closed symplectic manifold (M,ω). Our main tool is the Seidel representation of π1(Ham(M,ω)) in the units of the quantum homology ring. We show that if the weights of the action at the points at which the moment map is a maximum are sufficiently small then the circle represents a nonzero element of π1(Ham(M,ω)). Further, if the isotropy has order at most two and the circle contracts in Ham(M,ω) then various symmetry properties hold. For example, the image of the normalized moment map is a symmetric interval [−a, a]. If the action is semifree (i.e. the isotropy weights are 0 or ±1) then we calculate the leading order term in the Seidel representation, an important technical tool in understanding the quantum cohomology of manifolds that admit semifree Hamiltonian circle actions. If the manifold is toric, we use our results about this representation to describe the basic multiplicative structure of the quantum cohomology ring of an arbitrary toric manifold. There are two important technical ingredients; one relates the equivariant cohomology of M to the Morse flow of the moment map, and the other is a version of the localization principle for calculating Gromov–Witten invariants on symplectic manifolds with S1-actions.

Fixed points of symplectic periodic flows

AbstractThe study of fixed points is a classical subject in geometry and dynamics. If the circle acts in a Hamiltonian fashion on a compact symplectic manifold M, then it is classically known that there are at least $\frac {1}{2}\,{\dim M}+1$ fixed points; this follows from Morse theory for the momentum map of the action. In this paper we use Atiyah–Bott–Berline–Vergne (ABBV) localization in equivariant cohomology to prove that this conclusion also holds for symplectic circle actions with non-empty fixed sets, as long as the Chern class map is somewhere injective—the Chern class map assigns to a fixed point the sum of the action weights at the point. We complement this result with less sharp lower bounds on the number of fixed points, under no assumptions; from a dynamical systems viewpoint, our results imply that there is no symplectic periodic flow with exactly one or two equilibrium points on a compact manifold of dimension at least eight.

Local Gromov-Witten invariants and tautological sheaves on Hilbert schemes

We study the local Gromov-Witten invariants of \( \mathcal{O} \)(k)⊕\( \mathcal{O} \)(−k−2) → ℙ1 by localization techniques and the Marino-Vafa formula, using suitable circle actions. They are identified with the equivariant Riemann-Roch indices of some power of the determinant of the tautological sheaves on the Hilbert schemes of points on the affine plane. We also compute the corresponding Gopakumar-Vafa invariants and make some conjectures about them.

Complex contact manifolds and S 1 actions

We prove rigidity and vanishing theorems for several holomorphic Euler characteristics on complex contact manifolds admitting holomorphic circle actions preserving the contact structure. Such vanishings are reminiscent of those of LeBrun and Salamon on Fano contact manifolds but under a symmetry assumption instead of a curvature condition.

Calabi-Yau cones from contact reduction

We consider a generalization of Einstein–Sasaki manifolds, which we characterize in terms both of spinors and differential forms, that in the real analytic case corresponds to contact manifolds whose symplectic cone is Calabi-Yau. We construct solvable examples in seven dimensions. Then, we consider circle actions that preserve the structure and determine conditions for the contact reduction to carry an induced structure of the same type. We apply this construction to obtain a new hypo-contact structure on S 2 × T 3.