曲面映射类群上的扭转余有限拓扑

Twistor Topology of the Fermat Cubic

We describe topologically the discriminant locus of a smooth cubic surface in the complex projective space ${\mathbb{CP}}^3$ that contains 5 fibres of the projection ${\mathbb{CP}}^3 \longrightarrow S^4$.

Fractional Dehn twists in knot theory and contact topology

Fractional Dehn twists give a measure of the difference between the relative isotopy class of a homeomorphism of a bordered surface and the Thurston representative of its free isotopy class. We show how to estimate and compute these invariants. We discuss the the relationship of our work to stabilization problems in classical knot theory, general open book decompositions, and contact topology. We include an elementary characterization of overtwistedness for contact structures described by open book decompositions.

Generalized Dehn twists in low-dimensional topology

The generalized Dehn twist along a closed curve in an oriented surface is an algebraic construction which involves intersections of loops in the surface. It is defined as an automorphism of the Malcev completion of the fundamental group of the surface. As the name suggests, for the case where the curve has no self-intersection, it is induced from the usual Dehn twist along the curve. In this expository article, after explaining their definition, we review several results about generalized Dehn twists such as their realizability as diffeomorphisms of the surface, their diagrammatic description …

Elements of Torelli topology: I. The rank of abelian subgroups

By Torelli topology the author understands aspects of the topology of surfaces (potentially) relevant to the study of Torelli groups. The present paper is devoted to a new approach to the results of W. Vautaw about Dehn multi-twists in Torelli groups and abelian subgroups of Torelli groups. The new proofs are more transparent and lead to stronger estimates of the rank of an abelian subgroup when some additional information is available. An unexpected application of our methods is a new proof of the algebraic characterization of the Dehn twist about separating circles and the Dehn-Johnson twist…

超共形引力与微分同胚群的拓扑

Superconformal Gravity And The Topology Of Diffeomorphism Groups

本文推广扭曲四维超对称杨-米尔斯理论，通过路径积分构造光滑四流形族的唐纳森不变量推广，并证明其可解释为耦合共形超引力背景场的拓扑扭曲路径积分，从而揭示微分同胚群的拓扑信息。

Exotic spheres and the topology of symplectomorphism groups

We show that, for certain families $ϕ_{\mathbf{s}}$ of diffeomorphisms of high-dimensional spheres, the commutator of the Dehn twist along the zero-section of $T^*S^n$ with the family of pullbacks $ϕ^*_{\mathbf{s}}$ gives a noncontractible family of compactly-supported symplectomorphisms. In particular, we find examples: where the Dehn twist along a parametrised Lagrangian sphere depends up to Hamiltonian isotopy on its parametrisation; where the symplectomorphism group is not simply-connected, and where the symplectomorphism group does not have the homotopy-type of a finite CW-complex. We sho…