Celebratio Mathematica

D. McDuff and S. Tolman: On nearly semifree circle actions. Preprint, March 2005. ArXiv math/0503467 techreport

Recall that an effective circle action is semifree if the stabilizer subgroup of each point is connected. We show that if \( (M,\omega) \) is a coadjoint orbit of a compact Lie group \( G \) then every element of \( \pi_1(G) \) may be represented by a semifree \( S^1 \)-action. A theorem of McDuff–Slimowitz then implies that \( \pi_1(G) \) injects into \( \pi_1(\operatorname{Ham}(M, \omega)) \), 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 \( (M,\omega) \) 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 \( G \).

D. McDuff and S. Tolman: “Topological properties of Hamiltonian circle actions,” Int. Math. Res. Pap. 2006 (2006), pp. 1–77. MR 2210662 Zbl 1123.53044 article

This paper studies Hamiltonian circle actions, that is, circle subgroups of the group \( \mathrm{Ham}(M,\omega) \) of Hamiltonian symplectomorphisms of a closed symplectic manifold \( (M,\omega) \). Our main tool is the Seidel representation of \( \pi_1(\mathrm{Ham}(M,\omega)) \) 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 \( \pi_1(\mathrm{Ham}(M,\omega)) \). Further, if the isotropy has order at most two and the circle contracts in \( \mathrm{Ham}(M,\omega) \), 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 \( \pm 1 \)), 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.

D. McDuff and S. Tolman: “Polytopes with mass linear functions, I,” Int. Math. Res. Not. 2010 : 8 (2010), pp. 1506–1574. MR 2628835 Zbl 1202.52010 ArXiv 0807.0900 article

Let \( \Delta \) be an \( n \)-dimensional polytope that is simple, that is, exactly \( n \) facets meet at each vertex. An affine function is “mass linear” on \( \Delta \) if its value on the center of mass of \( \Delta \) depends linearly on the positions of the supporting hyperplanes. On the one hand, we show that certain types of symmetries of \( \Delta \) give rise to nonconstant mass linear functions on \( \Delta \). On the other hand, we show that most polytopes do not admit any nonconstant mass linear functions. Further, if every affine function is mass linear on \( \Delta \), then \( \Delta \) is a product of simplices. Our main result is a classification of all smooth polytopes of dimension \( \leq 3 \) which admit nonconstant mass linear functions. In particular, there is only one family of smooth three-dimensional polytopes — and no polygons — that admit “essential mass linear functions,” that is, mass linear functions that do not arise from the symmetries described above. In part II, we will complete this classification in the four-dimensional case. These results have geometric implications. Fix a symplectic toric manifold \( (M,\omega,T,\Phi) \) with moment polytope \( \Delta = \Phi(M) \). Let \[ \mathrm{Symp}_0(M,\omega) \] denote the identity component of the group of symplectomorphisms of \( (M,\omega) \). Any linear function \( H \) on \( \Delta \) generates a Hamiltonian \( \mathbb{R} \) action on \( M \) whose closure is a subtorus \( T_H \) of \( T \). We show that if the map \[ \pi_1(T_H)\to\pi_1(\mathrm{Symp}_0(M,\omega)) \] has finite image, then \( H \) is mass linear. Combining this fact and the claims described above, we prove that in most cases, the induced map \[ \pi_1(T)\to\pi_1(\mathrm{Symp}_0(M,\omega)) \] is an injection. Moreover, the map does not have finite image unless \( M \) is a product of projective spaces. Note also that there is a natural maximal compact connected subgroup \[ \mathrm{Isom}_0(M)\subset\mathrm{Symp}_0(M,\omega) ;\] there is a natural compatible complex structure \( J \) on \( M \), and \( \mathrm{Isom}_0(M) \) is the identity component of the group of symplectomorphisms that also preserve this structure. We prove that if the polytope \( \Delta \) supports no essential mass linear functions, then the induced map \[ \pi(\mathrm{Isom}_0(M))\to\pi_1(\mathrm{Symp}_0(M,\omega)) \] is injective. Therefore, this map is injective for all four-dimensional symplectic toric manifolds and is injective in the six-dimensional case unless \( M \) is a \( \mathbb{C}P^2 \) bundle over \( \mathbb{C}P^1 \).

D. McDuff and S. Tolman: “Polytopes with mass linear functions, II: The four-dimensional case,” Int. Math. Res. Not. 2013 : 15 (2013), pp. 3509–3599. MR 3089735 ArXiv 1106.1623 article

This paper continues the analysis begun in part I of the structure of smooth moment polytopes \( \Delta\subset t^* \) that support a mass linear function \( H\in t \). As explained there, besides its purely combinatorial interest, this question is relevant to the study of the homomorphism \[ \pi_1(T^n)\to \pi_1(\mathrm{Symp}(M_{\Delta},\omega_{\Delta})) \] from the fundamental group of the torus \( T^n \) to that of the group of symplectomorphisms of the \( 2n \)-dimensional symplectic toric manifold \( (M_{\Delta},\omega_{\Delta})) \) associated to \( \Delta \). In Part I, we made a general investigation of this question and classified all mass linear pairs \( (\Delta,H) \) in dimensions up to 3. The main result of the current paper is a classification of all four-dimensional examples. Along the way, we investigate the properties of general constructions such as fibrations, blow ups, and expansions (or wedges), describing their effect on both moment polytopes and mass linear functions. We end by discussing the relation of mass linearity to Shelukhin’s notion of full mass linearity. The two concepts agree in dimensions up to and including 4. However, full mass linearity may be the more natural concept when considering the question of which blowups preserve mass linearity.

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