Equivariant homotopy and deformations of diffeomorphisms

We present a way of constructing and deforming diffeomorphisms of manifolds endowed with a Lie group action. This is applied to the study of exotic diffeomorphisms and involutions of spheres and to the equivariant homotopy of Lie groups.     

Affine diffeomorphisms of translation surfaces: Periodic points, Fuchsian groups, and arithmeticity

We study translation surfaces with rich groups of affine diffeomorphisms—“prelattice” surfaces. These include the lattice translation surfaces studied by W. Veech. We show that there exist prelattice but nonlattice translation surfaces. We characterize arithmetic surfaces among prelattice surfaces by the infinite cardinality of their set of points periodic under affine diffeomorphisms. We give examples of translation surfaces whose periodic points and Weierstrass points coincide.     

Periodic homotopy and conjugacy idempotents

A self-map on the CW complex is a periodic homotopy idempotent if for some and the iterates and are homotopic. Geoghegan and Nicas defined the rotation index of such a map. They proved that for , the homotopy idempotent splits if and only if , while for , the index divides . We extend this to arbitrary and , and generalize various results related to the splitting of homotopy idempotents on CW complexes and conjugacy idempotents on groups.

     

On 4-manifolds homotopy equivalent to surface bundles over surfaces

We show that a closed 4-manifold is homotopy equivalent to the total space of a surface bundle over a surface if the obviously necessary conditions on the fundamental group and Euler characteristic hold. When the base is the 2-sphere we need also conditions on the characteristic classes of the manifold. (Our results are incomplete when the base is the projective plane.) In most cases we can show the manifold is s-cobordant to the total space of the bundle.     

Cremona transformations and diffeomorphisms of surfaces

We show that the action of Cremona transformations on the real points of quadrics exhibits the full complexity of the diffeomorphisms of the sphere, the torus, and of all non-orientable surfaces. The main result says that if X is rational, then Aut(X), the group of algebraic automorphisms, is dense in Diff(X), the group of self-diffeomorphisms of X.     

Energy-minimal diffeomorphisms between doubly connected Riemann surfaces

Let \(M\) and \(N\) be doubly connected Riemann surfaces with boundaries and with nonvanishing conformal metrics \(\sigma \) and \(\rho \) respectively, and assume that \(\rho \) is a smooth metric with bounded Gauss curvature \({\mathcal {K}}\) and finite area. The paper establishes the existence of homeomorphisms between \(M\) and \(N\) that minimize the Dirichlet energy. Among all homeomorphisms \(f :M{\overset{{}_{ \tiny {\mathrm{onto}} }}{\longrightarrow }} N\) between doubly connected Riemann surfaces such that \({{\mathrm{Mod\,}}}M \leqslant {{\mathrm{Mod\,}}}N\) there exists, unique up to conformal automorphisms of M, an energy-minimal diffeomorphism which is a harmonic diffeomorphism. The results improve and extend some recent results of Iwaniec et al. (Invent Math 186(3):667–707, 2011), where the authors considered bounded doubly connected domains in the complex plane w.r. to Euclidean metric.     

Right-veering diffeomorphisms of compact surfaces with boundary

We initiate the study of the monoid of right-veering diffeomorphisms on a compact oriented surface with nonempty boundary. The monoid strictly contains the monoid of products of positive Dehn twists. We explain the relationship to tight contact structures and open book decompositions. Mathematics Subject Classification (1991) Primary 57M50, secondary 53C15     

Lagrangian tori in homotopy elliptic surfaces

Let denote the symplectic four-manifold, homotopy equivalent to the rational elliptic surface, corresponding to a fibred knot in constructed by R. Fintushel and R. J. Stern in 1998. We construct a family of nullhomologous Lagrangian tori in and prove that infinitely many of these tori have complements with mutually non-isomorphic fundamental groups if the Alexander polynomial of has some irreducible factor which does not divide for any positive integer . We also show how these tori can be non-isotopically embedded as nullhomologous Lagrangian submanifolds in other symplectic -manifolds.

     

Quandle homotopy invariants of knotted surfaces

Given a finite quandle, we introduce a quandle homotopy invariant of knotted surfaces in the 4-sphere, modifying that of classical links. This invariant is valued in the third homotopy group of the quandle space, and is universal among the (generalized) quandle cocycle invariants. We compute the second and third homotopy groups, with respect to “regular Alexander quandles”. As a corollary, any quandle cocycle invariant using the dihedral quandle of prime order is a scalar multiple of Mochizuki 3-cocycle invariant. As another result, we determine the third quandle homology group of the dihedral quandle of odd order.     

Quantum hyperbolic invariants for diffeomorphisms of small surfaces

An earlier article [Bonahon, F., Liu, X. B.: Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms. Geom. Topol., 11, 889-937 (2007)] introduced new invariants for pseudo-Anosov diffeomorphisms of surface, based on the representation theory of the quantum Teichmu¨ller space. We explicitly compute these quantum hyperbolic invariants in the case of the 1-puncture torus and the 4-puncture sphere.