Immersed projective planes, Arf invariants and even 4-manifolds

Local surgery formulas for quantum invariants and the Arf invariant

A formula for the Arf invariant of a link is given in terms of the singularities of an immersed surface bounded by the link. This is applied to study the computational complexity of quantum invariants of 3--manifolds.

Higher Order Intersections in Low-Dimensional Topology

We show how to measure the failure of the Whitney trick in dimension 4 by constructing higher- order intersection invariants of Whitney towers built from iterated Whitney disks on immersed surfaces in 4-manifolds. For Whitney towers on immersed disks in the 4-ball, we identify some of these new invariants with previously known link invariants like Milnor, Sato-Levine and Arf invariants. We also define higher- order Sato-Levine and Arf invariants and show that these invariants detect the obstructions to framing a twisted Whitney tower. Together with Milnor invariants, these higher-order invaria…

An integral generalization of the Gusein-Zade--Natanzon theorem

A few years ago N.A'Campo invented a construction of a link from a real curve immersed into a disk. In the case of the curve originating from the real morsification method the link is isotopic to the link of the corresponding singularity. There are some curves which do not occur in the singularity theory. In this article we describe the Casson invariant of A'Campo's knots as a J^{+/-}-type invariant of the immersed curves. Thus we get an integral generalization of the Gusein-Zade--Natanzon theorem which says that the Arf invariant of a singularity is equal to J^{-}/2(mod 2) of the correspondin…

Milnor Invariants and Twisted Whitney Towers

This paper describes the relationship between the first non-vanishing Milnor invariants of a classical link and the intersection invariant of a twisted Whitney tower. This is a certain 2-complex in the 4-ball, built from immersed disks bounded by the given link in the 3-sphere together with finitely many `layers' of Whitney disks. The intersection invariant is a higher-order generalization of the intersection number between two immersed disks in the 4-ball, well known to give the linking number of the link on the boundary, which measures intersections among the Whitney disks and the disks boun…

A 3-manifold complexity via immersed surfaces

We define an invariant, which we call surface-complexity, of closed 3-manifolds by means of Dehn surfaces. The surface-complexity of a manifold is a natural number measuring how much the manifold is complicated. We prove that it fulfils interesting properties: it is subadditive under connected sum and finite-to-one on P2-irreducible manifolds. Moreover, for P2-irreducible manifolds, it equals the minimal number of cubes in a cubulation of the manifold, except for the sphere S3, the projective space RP3 and the lens space L41, which have surface-complexity zero. We will also give estimations of…

通过浸入曲面定义的紧致3-流形复杂度

A complexity of compact 3-manifold via immersed surfaces

本文通过Dehn曲面定义紧致3-流形的曲面复杂度不变量，证明其满足连通和次可加性及有限对一性质，并建立其与立方剖分、理想三角剖分和Matveev复杂度之间的关系。