One-connectivity and finiteness of Hamiltonian S1S^1-manifolds with minimal fixed sets

Let the circle act effectively in a Hamiltonian fashion on a compact symplectic manifold $(M, \omega)$. Assume that the fixed point set $M^{S^1}$ has exactly two components, $X$ and $Y$, and that $\dim(X) + \dim(Y) +2 = \dim(M)$. We first show that $X$, $Y$ and $M$ are simply connected. Then we show that, up to $S^1$-equivariant diffeomorphism, there are finitely many such manifolds in each dimension. Moreover, we show that in low dimensions, the manifold is unique in a certain category. We use techniques from both areas of symplectic geometry and geometric topology

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 \pi_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 \pi_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.Comment: significantly revised, some new results added, others moved to SG/0503467, 56 pages, no figure

Cohomology over complete intersections via exterior algebras

A general method for establishing results over a commutative complete intersection local ring by passing to differential graded modules over a graded exterior algebra is described. It is used to deduce, in a uniform way, results on the growth of resolutions of complexes over such local rings.Comment: 18 pages; to appear in "Triangulated categories (Leeds, 2006)", LMS lecture notes series

Semifree circle actions, Bott towers, and quasitoric manifolds

A Bott tower is the total space of a tower of fibre bundles with base CP^1 and fibres CP^1. Every Bott tower of height n is a smooth projective toric variety whose moment polytope is combinatorially equivalent to an n-cube. A circle action is semifree if it is free on the complement to fixed points. We show that a (quasi)toric manifold (in the sense of Davis-Januszkiewicz) over an n-cube with a semifree circle action and isolated fixed points is a Bott tower. Then we show that every Bott tower obtained in this way is topologically trivial, that is, homeomorphic to a product of 2-spheres. This extends a recent result of Ilinskii, who showed that a smooth compact toric variety with a semifree circle action and isolated fixed points is homeomorphic to a product of 2-spheres, and makes a further step towards our understanding of a problem motivated by Hattori's work on semifree circle actions. Finally, we show that if the cohomology ring of a quasitoric manifold (or Bott tower) is isomorphic to that of a product of 2-spheres, then the manifold is homeomorphic to the product.Comment: 22 pages, LaTEX; substantially revise

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.Comment: 28 pages. Final version (cosmetic changes, slight reorganization), to appear in JPA