The purpose of this article is to characterize symplectic and Hamiltonian circle actions on symplectic manifolds in terms of symplectic embeddings of Riemann surfaces. More precisely, it is shown that (1) if (M, ω) admits a Hamiltonian S 1-action, then there exists a two-sphere S in M with positive symplectic area satisfying ‹c 1(M, ω), [S]› > 0, and (2) if the action is non-Hamiltonian, then there exists an S 1-invariant symplectic 2-torus T in (M, ω) such that ‹c 1(M, ω), [T]› = 0. As applications, the authors give a very simple proof of the following well-known theorem which was proved by Atiyah-Bott, Lupton-Oprea, and Ono: Suppose that (M, ω) is a smooth closed symplectic manifold satisfying c 1(M, ω) = λ·[ω] for some λ ∈ R and G is a compact connected Lie group acting effectively on M preserving ω. Then (1) if λ < 0, then G must be trivial, (2) if λ = 0, then the G-action is non-Hamiltonian, and (3) if λ > 0, then the G-action is Hamiltonian.
The authors compute the (rational) Betti number of real toric varieties associated to Weyl chambers of type B, and furthermore show that their integral cohomology is p-torsion free for all odd primes p.
Let X be a topological space. In this survey the authors consider several types of configuration spaces, namely, the classical (usual) configuration spaces F n(X) and D n(X), the orbit configuration spaces F G n (X) and F G n (X)/S n with respect to a free action of a group G on X, and the graph configuration spaces F Г n (X) and F Г n (X)/H, where Г is a graph and H is a suitable subgroup of the symmetric group S n. The ordered configuration spaces F n(X), F G n (X), F Г n (X) are all subsets of the n-fold Cartesian product n Π 1 X of X with itself, and satisfy F G n (X) → F n(X) → F Г n (X) → n Π 1 X. If A denotes one of these configuration spaces, the authors analyse the difference between A and n Π 1 X from a topological and homotopical point of view. The principal results known in the literature concern the usual configuration spaces. The authors are particularly interested in the homomorphism on the level of the homotopy groups of the spaces induced by the inclusion ι: A → n Π 1 X, the homotopy type of the homotopy fibre I ι of the map ι via certain constructions on various spaces that depend on X, and the long exact sequence in homotopy of the fibration involving I ι and arising from the inclusion ι. In this respect, if X is either a surface without boundary, in particular if X is the 2-sphere or the real projective plane, or a space whose universal covering is contractible, or an orbit space S k/G of the k-dimensional sphere by a free action of a Lie group G, the authors present recent results obtained by themselves for the first case, and in collaboration with Golasiński for the second and third cases. The authors also briefly indicate some older results relative to the homotopy of these spaces that are related to the problems of interest. In order to motivate various questions, for the remaining types of configuration spaces, a few of their basic properties are described and proved. A list of open questions and problems is given at the end of the paper.
The authors study torsion in the integral cohomology of a certain family of 2n-dimensional orbifolds X with actions of the n-dimensional compact torus. Compact simplicial toric varieties are in our family. For a prime number p, the authors find a necessary condition for the integral cohomology of X to have no p-torsion. Then it is proved that the necessary condition is sufficient in some cases. The authors also give an example of X which shows that the necessary condition is not sufficient in general.
A theorem of Lambrechts and Stanley is used to find the rational cohomology of the complement of an embedding S 4n−1 → S 2n ×S m as a module and demonstrate that it is not necessarily determined by the map induced on cohomology by the embedding, nor is it a trivial extension. This demonstrates that the theorem is an improvement on the classical Lefschetz duality.
The authors study the properties of virtual Temperley-Lieb algebra and show how the f-polynomial of virtual knot can be derived from a representation of the virtual braid group into the virtual Temperley-Lieb algebra, which is an approach similar to Jones’s original construction.
The author constructs a family of manifolds, one for each n ≥ 2, having a nontrivial Massey n-product in their cohomology for any given n. These manifolds turn out to be smooth closed 2-connected manifolds with a compact torus Tm-action called moment-angle manifolds Z P, whose orbit spaces are simple n-dimensional polytopes P obtained from an n-cube by a sequence of truncations of faces of codimension 2 only (2-truncated cubes). Moreover, the polytopes P are flag nestohedra but not graph-associahedra. The author also describes the numbers β −i,2(i+1)(Q) for an associahedron Q in terms of its graph structure and relates it to the structure of the loop homology (Pontryagin algebra) H *(ΩZ Q), and then studies higher Massey products in H *(Z Q) for a graph-associahedron Q.
Let X be a closed, simply-connected, smooth, spin 4-manifold whose intersection form is isomorphic to 2k(−E s) ⊕ lH, where H is the hyperbolic form. In this paper, the authors prove that if there exists a locally linear pseudofree ℤ3-action on X, then Sign(g,X) ≡ −k mod 3. They also investigate the smoothability of locally linear ℤ3-action satisfying above congruence. In particular, it is proved that there exist some nonsmoothable locally linear ℤ3-actions on certain elliptic surfaces.
In this paper, it is shown that for a 3-dimensional small cover M over a polytope P, there are only 2-torsions in H 1(M; Z). Moreover, the mod 2 Betti number growth of finite covers of M is studied.
This paper deals with two things. First, the cohomology of canonical extensions of real topological toric manifolds is computed when coefficient ring G is a commutative ring in which 2 is unit in G. Second, the author focuses on a specific canonical extensions called doublings and presents their various properties. They include existence of infinitely many real topological toric manifolds admitting complex structures, and a way to construct infinitely many real toric manifolds which have an odd torsion in their cohomology groups. Moreover, some questions about real topological toric manifolds related to Halperin’s toral rank conjecture are presented.
Davis and Januszkiewicz introduced (real and complex) universal complexes to give an equivalent definition of characteristic maps of simple polytopes, which now can be seen as “colorings”. The author derives an equivalent definition of Buchstaber invariants of a simplicial complex K, then interprets the difference of the real and complex Buchstaber invariants of K as the obstruction to liftings of nondegenerate simplicial maps from K to the real universal complex or the complex universal complex. It was proved by Ayzenberg that real universal complexes can not be nondegenerately mapped into complex universal complexes when dimension is 3. This paper presents that there is a nondegenerate map from 3-dimensional real universal complex to 4-dimensional complex universal complex.
The author gives an explicit formula on the Ehrhart polynomial of a 3-dimensional simple integral convex polytope by using toric geometry.
This paper mainly deals with the question of equivalence between equivariant cohomology Chern numbers and equivariant K-theoretic Chern numbers when the transformation group is a torus. By using the equivariant Riemann-Roch relation of Atiyah-Hirzebruch type, it is proved that the vanishing of equivariant cohomology Chern numbers is equivalent to the vanishing of equivariant K-theoretic Chern numbers.
The authors consider the difference of Reidemeister traces and difference cochain of given two self-maps, and find out a relation involving these two invariants. As an application, an inductive formula of the Reidemeister traces for self-maps on a kind of CW-complex, including spherical manifolds is obtained.
Let M be a compact connected 3-submanifold of the 3-sphere S 3 with one boundary component F such that there exists a collection of n pairwise disjoint connected orientable surfaces S = {S 1, · · ·, S n} properly embedded in M, ∂S = {∂S 1, · · ·, ∂S n} is a complete curve system on F. We call S a complete surface system for M, and ∂S a complete spanning curve system for M. In the present paper, the authors show that the equivalent classes of complete spanning curve systems for M are unique, that is, any complete spanning curve system for M is equivalent to ∂S. As an application of the result, it is shown that the image of the natural homomorphism from the mapping class group M(M) to M(F) is a subgroup of the handlebody subgroup H n.