Finite Abelian actions on surfaces

Edmonds showed that two free orientation preserving smooth actions φ₁ and φ₂ of a finite Abelian group G on a closed connected oriented smooth surface M are equivalent by an equivariant orientation preserving diffeomorphism iff they have the same bordism class [M,φ₁]=[M,φ₂] in the oriented bordism group Ω₂(G) of the group G. In this paper, we compute the bordism class [M,φ] for any such action of G on M and we determine for a given M, the bordism classes in Ω₂(G) that are representable by such actions of G on M. This will enable us to obtain a formula for the number of inequivalent such actions of G on M. We also determine the “weak” equivalence classes of such actions of G on M when all the p-Sylow subgroups of G are homocyclic (i.e. of the form n(Z/pαZ)).     

Orientation-reversing actions on lens spaces and Gaussian integers

A finite group action on a lens space L(p,q) has ‘type OR’ if it reverses orientation and has an invariant Heegaard torus whose sides are interchanged by the orientation-reversing elements. In this paper we enumerate the actions of type OR up to equivalence. This leads to a complete classification of geometric finite group actions on amphicheiral lens spaces L(p,q) with p>2. The family of actions of type OR is partially ordered by lifting actions via covering maps. We show that each connected component of this poset may be described in terms of a subset of the lattice of Gaussian integers ordered by divisibility. This results in a correspondence equating equivalence classes of actions of type OR with pairs of Gaussian integers.     

6-dimensional manifolds with effective T4-actions

Suppose the four dimensional torus T⁴ acts effectively on a 6-manifold M so that the orbit space M¹ is a closed 2-disk, and there exist no exceptional orbits, and the isotropy groups span T⁴. Then the fundamental group of M is a finite abelian group with at most two generators. In this paper, we obtain a homology classification of manifolds of this type under an additional hypothesis that one of the two generators is trivial. We then use this result to obtain a complete classification of simply connected 6-manifolds supporting effective T⁴-actions.     

The t-singular homology of orbifolds

For an orbifold M we define a new homology group, called t-singular homology group t-Hq(M) by using singular simplicies intersecting ‘transversely’ with ΣM. The rightness of this homology group is ensured by the facts that the 1-dimensional homology group t-H₁(M) is isomorphic to the abelianization of the orbifold fundamental group π₁(M,x₀). If M is a manifold, t-Hq(M) coincides with the usual singular homology group. We prove that it is a ‘b-homotopy’ invariant of orbifolds and develop many algebraic tools for the calculations. Consequently we calculate the t-singular homology groups of several orbifolds.     

How tightly can you fold a sphere?

Consider a compact, connected Lie group G acting isometrically on a sphere Sn of radius 1. The quotient of Sn by this group action, Sn/G, has a natural metric on it, and so we may ask what are its diameter and q-extents. These values have been computed for cohomogeneity one actions on spheres. In this paper, we compute the diameters, extents, and several q-extents of cohomogeneity two orbit spaces resulting from such actions, and we also obtain results about the q-extents of Euclidean disks. Additionally, via a simple geometric criterion, we can identify which of these actions give rise to a decomposition of the sphere as a union of disk bundles. In addition, as a service to the reader, we give a complete breakdown of all the isotropy subgroups resulting from cohomogeneity one and two actions.     

A disjoint disks property for 3-manifolds

We show that the map separation property (MSP), a concept due to H.W. Lambert and R.B. Sher, is an appropriate analogue of J.W. Cannon’s disjoint disks property (DDP) for the class $\text{C}$ of compact generalized 3-manifolds with zero-dimensional singular set, modulo the Poincaré conjecture. Our main result is that the Poincaré conjecture (in dimension three) is equivalent to the conjecture that every X?$\text{C}$ with the MSP is a topological 3-manifold.     

Compressing thin spheres in the complement of a link

Let L be a link in S³ that is in thin position but not in bridge position and let P be a thin level sphere with compressing disk D. We introduce the idea of alternating level spheres for D and show that all such spheres are thin and their widths are monotone decreasing. This allows us to generalize a result of Wu by giving a bound on the number of disjoint irreducible compressing disks P can have in terms of the width of P, including identifying thin spheres with unique compressing disks. We also give conditions under which P must be incompressible on some side or be weakly incompressible. In particular we show that the thin level sphere of second lowest width is weakly incompressible. If P is strongly compressible we describe how a pair of compressing disks must lie relative to the link.     

Singular fibers and characteristic classes

Let be a proper generic map between smooth manifolds with dimN−dimM=−1. We explicitly calculate the cohomology class dual to the closure of the set of points in N over which lies a specific singular fiber in terms of characteristic classes of M and N.     

Pre-Bloch invariants of 3-manifolds with boundary

Let M be an oriented hyperbolic 3-manifold with finite volume. In [W.D. Neumann, J. Yang, Bloch invariants of hyperbolic 3-manifolds, Duke Math. J. 96 (1999) 29-59. [9]], Neumann and Yang defined an element β(M) of Bloch group B(C) for M. For this β(M), volume and Chern-Simons invariant of M is represented by a transcendental function. In this paper, we define β(M,ρ,C,o)∈P(C) for an oriented 3-manifold M with boundary, a representation of its fundamental group , a pants decomposition C of ∂M and an orientation o on simple closed curves of C. Unlike in the case of finite volume, we construct an element of pre-Bloch group P(C), and we need essentially the pants decomposition on the boundary. The volume makes sense for β(M,ρ,C,o) and we can describe the variation of volume on the deformation space.     

Induced Hamiltonian maps on the symplectic quotient

Let (M,ω) be a symplectic manifold and G a compact Lie group that acts on M. Assume that the action of G on M is Hamiltonian. Then a G-equivariant Hamiltonian map on M induces a map on the symplectic quotient of M by G. Consider an autonomous Hamiltonian H with compact support on M, with no non-constant closed trajectory in time less than 1 and time-1 map fH. If the map fH descends to the symplectic quotient to a map Φ(fH) and the symplectic manifold M is exact and Ham(M,ω) has no short loops, we prove that the Hofer norm of the induced map Φ(fH) is bounded above by the Hofer norm of fH.     

Three-dimensional Lie group actions on compact (4n+3)-dimensional geometric manifolds

The (4n+3)-dimensional sphere S⁴ⁿ⁺³ can be viewed as the boundary of the quaternionic hyperbolic space and the group PSp(n+1,1) of quaternionic hyperbolic isometries extends to a real analytic transitive action on S⁴ⁿ⁺³. We call the pair (PSp(n+1,1),S⁴ⁿ⁺³) a spherical Q C-C geometry. A manifold M locally modelled on this geometry is said to be a spherical Q C-C manifold. We shall classify all pairs (G,M) where G is a three-dimensional connected Lie group which acts smoothly and almost freely on a compact spherical Q C-C manifold M, preserving the geometric structure. As an application, we shall determine all compact 3-pseudo-Sasakian manifolds admitting spherical Q C-C structures.     

Local torus actions modeled on the standard representation

We introduce the notion of a local torus action modeled on the standard representation (for simplicity, we call it a local torus action). It is a generalization of a locally standard torus action and also an underlying structure of a locally toric Lagrangian fibration. For a local torus action, we define two invariants called a characteristic pair and an Euler class of the orbit map, and prove that local torus actions are classified topologically by them. As a corollary, we obtain a topological classification of locally standard torus actions, which includes the topological classifications of quasi-toric manifolds by Davis and Januszkiewicz and of effective T²-actions on four-dimensional manifolds without nontrivial finite stabilizers by Orlik and Raymond. We discuss locally toric Lagrangian fibrations from the viewpoint of local torus actions. We also investigate the topology of a manifold equipped with a local torus action when the Euler class of the orbit map vanishes.     

Finite Groups Acting on Homology 3-Spheres: On a Result of M. Reni

 We give a list including all finite groups G which admit smooth orientation preserving actions on homology 3-spheres (arbitrary actions, i.e. possibly with fixed points; if the action is free then the group G has periodic cohomology and the classification of such groups is well known). The main work in this direction is due to M. Reni. In the present paper, we complete and extend his results for the case of nonsolvable groups G. Received 19 March 2001; in revised form 15 September 2001     

Properly discontinuous actions of subgroups in amenable algebraic groups and its application to affine motions

This note will concern properly discontinuous actions of subgroups in real algebraic groups on contractible manifolds. Let (π,X,ρ) be such an action, where ρ:π→ Diff(X) is a homomorphism. We assume that ? extends to a smooth action of a real algebraic group G containing π. If such π has a nontrivial radical (i.e., unique maximal normal solvable subgroup), then we can apply the method of Seifert construction [14],[17] to yield that the quotient π\X supports the structure of an injective Seifert fibering with typical (resp. exceptional) fiber diffeomorphic to a solv (resp. infrasolv)-manifold (when π acts freely). When G is an amenable algebraic group, we can say about a uniqueness property for such actions. Namely, let (π_i, X_i, ρ_i) be actions as above (i= 1,2). Then, given an isomorphism f of π₁ onto ?₂, there is a diffeomorphism h: X₁→X₂ such that h(ρ₁(r)x)=ρ₂(f(r)h(x).As an application, we try to decide the structure of affine motions of some euclidean space $\text{R}$ⁿ. First we verify the conjecture of [17, 4 5], i.e., a compact complete affinely flat manifold admits a maximal toral action if its fundamental group has a nontrivial center. Second, a compact complete affinity flat manifold whose fundamental group is virtually polycyclic supports the structure of an infrasolvmanifold. This structure varies depending on its solvable kernel (if it is abelian or nilpotent, it must be a euclidean space form or an infranilmanifold respectively). If a group of the affine group A(n) acts properly discontinuously and with compact quotient of $\text{R}$ⁿ, then it is called an affine crystallographic group. Finally, we can say so far as to a uniqueness property that two virtually polycyclic affine crystallographic groups are conjugate inside Diff($\text{R}$ⁿ) if they are isomorphic (cf.[8]).     

Bredon homology and ramified covering G-maps

Let G be a finite group. The objective of this paper is twofold. First we prove that the cellular Bredon homology groups with coefficients in an arbitrary coefficient system M are isomorphic to the homotopy groups of certain topological abelian group. And second, we study ramified covering G-maps of simplicial sets and of simplicial complexes. As an application, we construct a transfer for them in Bredon homology, when M is a Mackey functor. We also show that the Bredon-Illman homology with coefficients in M satisfies the equivariant weak homotopy equivalence axiom in the category of G-spaces.     

On algebraic Riccati equations associated with M-matrices

We consider the algebraic Riccati equation for which the four coefficient matrices form an M-matrix K. When K is a nonsingular M-matrix or an irreducible singular M-matrix, the Riccati equation is known to have a minimal nonnegative solution and several efficient methods are available to find this solution. In this paper we are mainly interested in the case where K is a reducible singular M-matrix. Under a regularity assumption on the M-matrix K, we show that the Riccati equation still has a minimal nonnegative solution. We also study the properties of this particular solution and explain how the solution can be found by existing methods.     

Proper smooth G-manifolds have complete G-invariant Riemannian metrics

Assume G is a Lie group, K is a compact subgroup of G and M is a proper smooth G-manifold. Using properties of the regular representations L²(G) and L²(K), we first prove results about extending certain representations and embedding homogeneous spaces smoothly into Hilbert G-spaces. We then prove that M can be embedded as a closed smooth G-invariant submanifold of some Hilbert G-space. It follows that M admits a complete G-invariant smooth Riemannian metric.     

Proofs of Conjectures about Singular Riemannian Foliations

A singular foliation on a complete Riemannian manifold M is said to be Riemannian if each geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets. We prove that if the distribution of normal spaces to the regular leaves is integrable, then each leaf of this normal distribution can be extended to be a complete immersed totally geodesic submanifold (called section), which meets every leaf orthogonally. In addition the set of regular points is open and dense in each section. This result generalizes a result of Boualem and solves a problem inspired by a remark of Palais and Terng and a work of Szenthe about polar actions. We also study the singular holonomy of a singular Riemannian foliation with sections (s.r.f.s. for short) and in particular the tranverse orbit of the closure of each leaf. Furthermore we prove that the closures of the leaves of a s.r.f.s on M form a partition of M which is a singular Riemannian foliation. This result proves partially a conjecture of Molino.     

Topological,affine and isometric actions on flat riemannian manifolds II

Explicit examples of finite subgroups of the group of homotopy classes of self-homotopy equivalences of some flat Riemannian manifolds which cannot be lifted to effective actions are given. It is also shown that no finite subgroups of the kernel of π₀(Homeo(M))→Out π₁(M) can be lifted back to Homeo(M), for a large class of flat manifolds M. Some results of an earlier paper by the authors are refined and related to recent work of R. Schoen and S.T. Yau.     

Classes of general H-matrices

Let M(A) denote the comparison matrix of a square H-matrix A, that is, M(A) is an M-matrix. H-matrices such that their comparison matrices are nonsingular are well studied in the literature. In this paper, we study characterizations of H-matrices with either singular or nonsingular comparison matrices. The spectral radius of the Jacobi matrix of M(A) and the generalized diagonal dominance property are used in the characterizations. Finally, a classification of the set of general H-matrices is obtained.