Equivariant framings, lens spaces and Contact structures

Equivariant Homology of Generating Functions and Orderability of Lens Spaces

In her PhD thesis Milin developed an equivariant version of the contact homology groups constructed by Eliashberg, Kim and Polterovich and used it to prove an equivariant contact non-squeezing theorem. In this article we re-obtain the same result in the setting of generating functions, starting from the homology groups studied in arXiv:0901.3112. As Milin showed, this result implies orderability of lens spaces.

Equivariant embeddings of rational homology balls

We generalise theorems of Khodorovskiy and Park-Park-Shin, and give new topological proofs of those theorems, using embedded surfaces in the 4-ball and branched double covers. These theorems exhibit smooth codimension-zero embeddings of certain rational homology balls bounded by lens spaces.

Holomorphic curves in the 6-pseudosphere and cyclic surfaces

The space $\mathbf{H}^{4,2}$ of vectors of norm -1 in $\mathbb{R}^{4,3}$ has a natural pseudo-Riemannian metric and a compatible almost complex structure. The group of automorphisms of both of these structures is the split real form $G_2'$. In this paper we consider a class of holomorphic curves in $\mathbf{H}^{4,2}$ which we call alternating. We show that such curves admit a so called Frenet framing. Using this framing, we show that the space of alternating holomorphic curves which are equivariant with respect to a surface group are naturally parameterized by certain $G_2'$-Higgs bundles. Thi…

透镜空间与曲面奇点上的实紧接触结构

Real tight contact structures on lens spaces and surface singularities

本文分类了实心环面上的实紧接触结构，证明 $S^3$ 和 $\mathbb{R}P^3$ 上实紧结构唯一，并给出实透镜空间 $L(p,\pm 1)$ 上实紧结构的计数上下界，通过等变接触手术、实开书分解和实代数曲面奇点构造下界实例。

Equivariant Rho-Invariants and Instanton Homology of Torus Knots

The equivariant rho-invariants studied in this paper are a version of the classical rho-invariants of Atiyah, Patodi, and Singer in the presence of an isometric involution. We compute these rho-invariants for all involutions on the 3-dimensional lens spaces with 1-dimensional fixed point sets, as well as for some involutions on Brieskorn homology spheres. As an application, we compute the generators and Floer gradings in the singular instanton chain complex of (p,q)-torus knots with odd p and q.

Contact Structures on elliptic 3-manifolds

We show that an oriented elliptic 3-manifold admits a universally tight positive contact structure iff the corresponding group of deck transformations on $S^3$ preserves a standard contact structure pointwise. We also relate univerally tight contact structures on 3-manifolds covered by $S^3$ to the exceptional isomorphism $SO(4)=(SU(2)\times SU(2))/{\pm 1}$. The main tool used is equivariant framings of 3-manifolds.