arXiv subjects

Dusa McDuff

Publications and source records attributed to Dusa McDuff.

At least 55 records · Page 3Linked to original sources

arXiv2008

Comparing Absolute and relative Gromov--Witten invariants

This note compares the usual (absolute) Gromov-Witten invariants of a symplectic manifold with the invariants that count the curves relative to a (symplectic) divisor D. We give explicit examples where these invariants differ even though it seems at first that they should agree, for example when counting genus zero curves in a class \be such that \be\cdot D=0. The main tool is the decomposition formula in the form developed by A. Li--Ruan.

math.SG↗

arXiv2007

The symplectomorphism group of a blow up

We study the relation between the symplectomorphism group Symp M of a closed connected symplectic manifold M and the symplectomorphism and diffeomorphism groups Symp \TM and Diff \TM of its one point blow up \TM. There are three main arguments. The first shows that for any oriented M the natural map from pi_1(M) to pi_0(Diff \TM) is often injective. The second argument applies when M is simply connected and detects nontrivial elements in the homotopy group pi_1(Diff \TM) that persist into the space of self homotopy equivalences of \TM. Since it uses purely homological arguments, it applies to c-symplectic manifolds (M,a), that is, to manifolds of dimension 2n that support a class a in H^2(M;R) such that a^n\ne 0. The third argument uses the symplectic structure on M and detects nontrivial elements in the (higher) homology of BSymp \TM using characteristic classes defined by parametric Gromov--Witten invariants. Some results about many point blow ups are also obtained. For example we show that if M is the 4-torus with k-fold blow up \TM_k (where k>0) then pi_1(Diff \TM_k) is not generated by the groups pi_1\Symp (\TM_k, \Tom) as \Tom ranges over the set of all symplectic forms on \TM_k.

math.SG↗

arXiv2007

Groupoids, branched manifolds and multisections

Cieliebak, Mundet i Riera and Salamon recently formulated a definition of branched submanifold of Euclidean space in connection with their discussion of multivalued sections and the Euler class. This note proposes an intrinsic definition of a weighted branched manifold Z that is obtained from the usual definition of oriented orbifold groupoid by relaxing the properness condition and adding a weighting. We show that if Z is compact, finite dimensional and oriented, then it carries a fundamental class [Z]. Adapting the construction of Liu and Tian, we also show that the fundamental class [X] of any oriented orbifold X may be represented by a map from Z to X, where the branched manifold Z is unique up to a natural equivalence relation. This gives further insight into the structure of the virtual moduli cycle in the new polyfold theory recently constructed by Hofer, Wysocki and Zehnder.

math.SG↗

arXiv2006

Geometric variants of the Hofer norm

This note discusses some geometrically defined seminorms on the group $\Ham(M, ω)$ of Hamiltonian diffeomorphisms of a closed symplectic manifold $(M, ω)$, giving conditions under which they are nondegenerate and explaining their relation to the Hofer norm. As a consequence we show that if an element in $\Ham(M, ω)$ is sufficiently close to the identity in the $C^{2}$-topology then it may be joined to the identity by a path whose Hofer length is minimal among all paths, not just among paths in the same homotopy class relative to endpoints. Thus, true geodesics always exist for the Hofer norm. The main step in the proof is to show that a "weighted" version of the nonsqueezing theorem holds for all fibrations over $S^2$ generated by sufficiently short loops.Further, an example is given showing that the Hofer norm may differ from the sum of the one sided seminorms.

math.SG↗

arXiv2006

Enlarging the Hamiltonian group

This paper investigates ways to enlarge the Hamiltonian subgroup Ham of the symplectomorphism group Symp(M) of the symplectic manifold (M, ω) to a group that both intersects every connected component of Symp(M) and characterizes symplectic bundles with fiber M and closed connection form. As a consequence, it is shown that bundles with closed connection form are stable under appropriate small perturbations of the symplectic form. Further, the manifold (M,ω) has the property that every symplectic M-bundle has a closed connection form if and only if the flux group vanishes and the flux homomorphism extends to a crossed homomorphism defined on the whole group Symp(M). The latter condition is equivalent to saying that a connected component of the commutator subgroup [Symp,Symp] intersects the identity component of Symp only if it also intersects Ham. It is not yet clear when this condition is satisfied. We show that if the symplectic form vanishes on 2-tori the flux homomorphism extends to the subgroup of Symp acting trivially on π_1(M). We also give an explicit formula for the Kotschick--Morita extension of Flux in the monotone case. The results in this paper belong to the realm of soft symplectic topology, but raise some questions that may need hard methods to answer.

math.SG↗

arXiv2006

Symplectic Structures on Fiber Bundles

Let $π: P\to B$ be a locally trivial fiber bundle over a connected CW complex $B$ with fiber equal to the closed symplectic manifold $(M,\om)$. Then $π$ is said to be a symplectic fiber bundle if its structural group is the group of symplectomorphisms $\Symp(M,\om)$, and is called Hamiltonian if this group may be reduced to the group $\Ham(M,\om)$ of Hamiltonian symplectomorphisms. In this paper, building on prior work by Seidel and Lalonde, McDuff and Polterovich, we show that these bundles have interesting cohomological properties. In particular, for many bases $B$ (for example when $B$ is a sphere, a coadjoint orbit or a product of complex projective spaces) the rational cohomology of $P$ is the tensor product of the cohomology of $B$ with that of $M$. As a consequence the natural action of the rational homology $H_k(\Ham(M))$ on $H_*(M)$ is trivial for all $M$ and all $k > 0$. Added: The erratum makes a small change to Theorem 1.1 concerning the characterization of Hamiltonian bundles.

math.SG↗

arXiv2005

Topological properties of Hamiltonian circle actions

This paper studies Hamiltonian circle actions, i.e. circle subgroups of the group Ham(M,\om) of Hamiltonian symplectomorphisms of a closed symplectic manifold (M,\om). Our main tool is the Seidel representation of π_1(\Ham(M,\om)) 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 π_1(\Ham(M,\om)). Further, if the isotropy has order at most two and the circle contracts in \Ham(M,\om) then the homology of M is invariant under an involution. 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 S^1-actions.

math.SG↗

arXiv2005

On nearly semifree circle actions

Recall that an effective circle action is semifree if the stabilizer subgroup of each point is connected. We show that if $(M, \om)$ is a coadjoint orbit of a compact Lie group $G$ then every element of $π_1(G)$ may be represented by a semifree $S^1$-action. A theorem of McDuff--Slimowitz then implies that $π_1(G)$ injects into $π_1(\Ham(M, \om))$, which answers a question raised by Weinstein. We also show that a circle action on a manifold $M$ which is semifree near a fixed point $x$ cannot contract in a compact Lie subgroup $G$ of the diffeomorphism group unless the action is reversed by an element of $G$ that fixes the point $x$. Similarly, if a circle acts in a Hamiltonian fashion on a manifold $(M,ω)$ and the stabilizer of every point has at most two components, then the circle cannot contract in a compact Lie subgroup of the group of Hamiltonian symplectomorphism unless the circle is reversed by an element of $G$

math.SG↗

arXiv2005

Homotopy properties of Hamiltonian group actions

Consider a Hamiltonian action of a compact Lie group H on a compact symplectic manifold (M,w) and let G be a subgroup of the diffeomorphism group Diff(M). We develop techniques to decide when the maps on rational homotopy and rational homology induced by the classifying map BH --> BG are injective. For example, we extend Reznikov's result for complex projective space CP^n to show that both in this case and the case of generalized flag manifolds the natural map H_*(BSU(n+1)) --> H_*(BG) is injective, where G denotes the group of all diffeomorphisms that act trivially on cohomology. We also show that if lambda is a Hamiltonian circle action that contracts in G = Ham(M,w) then there is an associated nonzero element in pi_3(G) that deloops to a nonzero element of H_4(BG). This result (as well as many others) extends to c-symplectic manifolds (M,a), ie, 2n-manifolds with a class a in H^2(M) such that a^n is nonzero. The proofs are based on calculations of certain characteristic classes and elementary homotopy theory.

math.SG↗

arXiv2004

Symplectomorphism groups and almost complex structures

This paper studies groups of symplectomorphisms of ruled surfaces for symplectic forms with varying cohomology class. This class is characterized by the ratio R of the size of the base to that of the fiber. By considering appropriate spaces of almost complex structures, we investigate how the topological type of these groups changes as R increases. If the base is a sphere, this changes precisely when R passes an integer, and for general bases it stabilizes as R goes to infinity. Our results extend and make more precise some of the conclusions of Abreu--McDuff concerning the rational homotopy type of these groups for rational ruled surfaces.

math.SG↗

arXiv2004

Lectures on Groups of Symplectomorphisms

These notes combine material from short lecture courses given in Paris, France, in July 2001 and in Srni, the Czech Republic, in January 2003. They discuss groups of symplectomorphisms of closed symplectic manifolds (M,\om) from various points of view. Lectures 1 and 2 provide an overview of our current knowledge of their algebraic, geometric and homotopy theoretic properties. Lecture 3 sketches the arguments used by Gromov, Abreu and Abreu-McDuff to figure out the rational homotopy type of these groups in the cases M= CP^2 and M=S^2\times S^2. We outline the needed J-holomorphic curve techniques. Much of the recent progress in understanding the geometry and topology of these groups has come from studying the properties of fibrations with the manifold M as fiber and structural group equal either to the symplectic group or to its Hamiltonian subgroup Ham(M). The case when the base is S^2 has proved particularly important. Lecture 4 describes the geometry of Hamiltonian fibrations over S^2, while Lecture 5 discusses their Gromov-Witten invariants via the Seidel representation. It ends by sketching Entov's explanation of the ABW inequalities for eigenvalues of products of special unitary matrices. Finally in Lecture 6 we apply the ideas developed in the previous two lectures to demonstrate the existence of (short) paths in Ham(M,\om) that minimize the Hofer norm over all paths with the given endpoints.

math.SG↗

arXiv2004

A survey of the topological properties of symplectomorphism groups

The special structures that arise in symplectic topology (particularly Gromov--Witten invariants and quantum homology) place as yet rather poorly understood restrictions on the topological properties of symplectomorphism groups. This article surveys some recent work by Abreu, Lalonde, McDuff, Polterovich and Seidel, concentrating particularly on the homotopy properties of the action of the group of Hamiltonian symplectomorphisms on the underlying manifold M. It sketches the proof that the evaluation map π_1(Ham(M))\to π_1(M) given by {ϕ_t}\mapsto {ϕ_t(x_0)} is trivial, as well as explaining similar vanishing results for the action of the homology of Ham(M) on the homology of M. Applications to Hamiltonian stability are discussed.

math.SG↗

arXiv2001

Hofer-Zehnder capacity and length minimizing Hamiltonian paths

We use the criteria of Lalonde and McDuff to show that a path that is generated by a generic autonomous Hamiltonian is length minimizing with respect to the Hofer norm among all homotopic paths provided that it induces no non-constant closed trajectories in M. This generalizes a result of Hofer for symplectomorphisms of Euclidean space. The proof for general M uses Liu-Tian's construction of S^1-invariant virtual moduli cycles. As a corollary, we find that any semifree action of S^1 on M gives rise to a nontrivial element in the fundamental group of the symplectomorphism group of M. We also establish a version of the area-capacity inequality for quasicylinders.

math.SG↗

arXiv2000

Cohomological properties of ruled symplectic structures

This survey presents some recent results by the authors and Polterovich on the topological properties of ruled symplectic manifolds. The bundle M \to P \to B that is associated with a ruled manifold has the group of Hamiltonian symplectomorphisms of M as structure group if the base is simply connected. Thus information about Hamiltonian bundles gives stability results for ruled structures as well as obstructions to their existence.

math.SG↗

arXiv1999

Topology of symplectomorphism groups of rational ruled surfaces

Let $M$ be either $S^2\times S^2$ or the one point blow-up $\cp# \bcp$ of $\cp$. In both cases $M$ carries a family of symplectic forms $\om_\la$, where $\la > -1$ determines the cohomology class $[\om_\la]$. This paper calculates the rational (co)homology of the group $G_\la$ of symplectomorphisms of $(M,\om_\la)$ as well as the rational homotopy type of its classifying space $BG_\la$. It turns out that each group $G_\la$ contains a finite collection $K_k, k = 0,...,\ell = \ell(\la)$, of finite dimensional Lie subgroups that generate its homotopy. We show that these subgroups "asymptotically commute", i.e. all the higher Whitehead products that they generate vanish as $\la\to \infty$. However, for each fixed $\la$ there is essentially one nonvanishing product that gives rise to a "jumping generator" $w_\la$ in $H^*(G_\la)$ and to a single relation in the rational cohomology ring $H^*(BG_\la)$. An analog of this generator $w_\la$ was also seen by Kronheimer in his study of families of symplectic forms on 4-manifolds using Seiberg--Witten theory. Our methods involve a close study of the space of $\om_\la$-compatible almost complex structures on $M$.

math.SG↗

arXiv1999

Quantum Homology of fibrations over $S^2$

This paper studies the (small) quantum homology and cohomology of fibrations $p: P\to S^2$ whose structural group is the group of Hamiltonian symplectomorphisms of the fiber $(M,\om)$. It gives a proof that the rational cohomology splits additively as the vector space tensor product $H^*(M)\otimes H^*(S^2)$, and investigates conditions under which the ring structure also splits, thus generalizing work of Lalonde-McDuff-Polterovich and Seidel. The main tool is a study of certain operations in the quantum homology of the total space $P$ and of the fiber $M$, whose properties reflect the relations between the Gromov-Witten invariants of $P$ and $M$. In order to establish these properties we further develop the language introduced in [Mc3] to describe the virtual moduli cycle (defined by Liu-Tian, Fukaya-Ono, Li-Tian, Ruan and Siebert).

math.SG↗

arXiv1998

Almost Complex Structures on $S^2\times S^2$

In this note we investigate the structure of the space $\Jj$ of smooth almost complex structures on $S^2\times S^2$ that are compatible with some symplectic form. This space has a natural stratification that changes as the cohomology class of the form changes and whose properties are very closely connected to the topology of the group of symplectomorphisms of $S^2\times S^2$. By globalizing standard gluing constructions in the theory of stable maps, we show that the strata of $\Jj$ are Fréchet manifolds of finite codimension, and that the normal link of each stratum is a finite dimensional stratified space. The topology of these links turns out to be surprisingly intricate, and we work out certain cases. Our arguments apply also to other ruled surfaces, though they give complete information only for bundles over $S^2$ and $T^2$.

math.SG↗

arXiv1997

Topological rigidity of Hamiltonian loops and quantum homology

This paper studies the question of when a loop $ϕ$ in the group Symp$(M,ω)$ of symplectomorphisms of a symplectic manifold $(M,ω)$ is isotopic to a loop that is generated by a time-dependent Hamiltonian function. (Loops with this property are said to be Hamiltonian.) Our main result is that Hamiltonian loops are rigid in the following sense: if $ϕ$ is Hamiltonian with respect to $ω$, and if $ϕ'$ is a small perturbation of $ϕ$ that preserves another symplectic form $ω'$, then $ϕ'$ is Hamiltonian with respect to $ω'$. This allows us to get some new information on the structure of the flux group, i.e. the image of $π_1(Symp(M,ω))$ under the flux homomorphism. We give a complete proof of our result for some manifolds, and sketch the proof in general. The argument uses methods developed by Seidel for studying properties of Hamiltonian loops via the quantum homology of $M$.

dg-ga↗