Articles written in Proceedings – Mathematical Sciences

• Limits of functions and elliptic operators

We show that a subspaceS of the space of real analytical functions on a manifold that satisfies certain regularity properties is contained in the set of solutions of a linear elliptic differential equation. The regularity properties are thatS is closed inL2 (M) and that if a sequence of functions fn in ƒn converges inL2(M), then so do the partial derivatives of the functions ƒn.

• Homeomorphisms and the homology of non-orientable surfaces

We show that, for a closed non-orientable surfaceF, an automorphism ofH1(F, ℤ) is induced by a homeomorphism ofF if and only if it preserves the (mod 2) intersection pairing. We shall also prove the corresponding result for punctured surfaces.

• Splittings of Free Groups, Normal Forms and Partitions of Ends

Splittings of a free group correspond to embedded spheres in the 3-manifold $M=\sharp_k S^2\times S^1$. These can be represented in a normal form due to Hatcher. In this paper, we determine the normal form in terms of crossings of partitions of ends corresponding to normal spheres, using a graph of trees representation for normal forms. In particular, we give a constructive proof of a criterion determining when a conjugacy class in $\pi_2(M)$ can be represented by an embedded sphere.

• Relative symplectic caps, 4-genus and fibered knots

We prove relative versions of the symplectic capping theorem and sufficiency of Giroux’s criterion for Stein fillability and use these to study the 4-genus of knots. More precisely, suppose we have a symplectic 4-manifold $X$ with convex boundary and a symplectic surface $\Sigma$ in $X$ such that $\delta\Sigma$ is a transverse knot in $\delta X$. In this paper, we prove that there is a closed symplectic 4-manifold $Y$ with a closed symplectic surface $S$ such that $(X, \Sigma)$ embeds into $(Y, S)$ symplectically. As a consequence we obtain a relative version of the symplectic Thom conjecture. We also prove a relative version of the sufficiency part of Giroux’s criterion for Stein fillability, namely, we show that a fibered knot whose mondoromy is a product of positive Dehn twists bounds a symplectic surface in a Stein filling. We use this to study 4-genus of fibered knots in $\mathbb S^3$. Further, we give a criterion for quasipositive fibered knots to be strongly quasipositive.

• # Proceedings – Mathematical Sciences

Volume 130, 2020
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019