Topological Index Theory for Surfaces in 3-manifolds
نویسنده
چکیده
The disk complex of a surface in a 3-manifold is used to define its topological index. Surfaces with well-defined topological index are shown to generalize well-known classes, such as incompressible, strongly irreducible, and critical surfaces. The main result is that one may always isotope a surface H with topological index n to meet an incompressible surface F so that the sum of the indices of the components of H \ N(F ) is at most n. This theorem and its corollaries generalize many known results about surfaces in 3-manifolds, and often provides more efficient proofs. The paper concludes with a list of questions and conjectures, including a natural generalization of Hempel’s distance to surfaces with topological index ≥ 2.
منابع مشابه
The Nielsen coincidence theory on topological manifolds
We generalize the coincidence semi-index introduced in [D-J] to pairs of maps between topological manifolds. This permits extending the Nielsen theory to this class of maps. Introduction. In this paper we generalize the coincidence semi-index theory, introduced in [D-J] in the smooth case, to pairs of maps between topological manifolds. It will be based on the topological transversality lemma (...
متن کاملTopological Quantum Field Theories for Surfaces with Spin Structure
Reened quantum invariants for closed three-manifolds with links and spin structures are extended to a Topological Quantum Field Theory. By a `universal con-struction', one associates, to surfaces with structure, modules which are shown to be free of nite rank. These modules satisfy the multiplicativity axiom of TQFT in an extended Z=2-graded sense, and their ranks are given by a spin reened ver...
متن کاملLength equivalent hyperbolic manifolds
In the case when M is an orientable Riemannian manifold of negative curvature L (M) and L(M) are related to the eigenvalue spectrum of M; that is, the set E (M) of all eigenvalues of the Laplace–Beltrami operator acting on L2(M) counted with multiplicities. For example, in this setting it is known that E (M) determines L(M) (see [3]). In the case of closed hyperbolic surfaces, the stronger stat...
متن کاملCombinatorics of the Surface Category and Tqfts
The (1+1)-dimensional cobordism category of closed 1-dimensional manifolds and oriented surfaces is a most basic mathematical structure. It has played a fundamental role in string theory and conformal field theory. We give a brief account of its structure and indicate what structure it gives vector spaces and other algebraic objects on which it (or an embellished version of it) acts. We explain...
متن کاملOn the Existence of High Index Topologically Minimal Surfaces
The topological index of a surface was previously introduced by the first author as the topological analogue of the index of an unstable minimal surface. Here we show that surfaces of arbitrarily high topological index exist. Consider a compact, connected, two sided surface S properly embedded in a compact, orientable 3-manifold M . The disk complex Γ(S) is the simplicial complex defined as fol...
متن کامل