Orientable 3-dimensional Manifold Research Articles

Atomic Embeddability, Clustered Planarity, and Thickenability

We study the atomic embeddability testing problem, which is a common generalization of clustered planarity ( c-planarity , for short) and thickenability testing, and present a polynomial-time algorithm for this problem, thereby giving the first polynomial-time algorithm for c-planarity. C-planarity was introduced in 1995 by Feng, Cohen, and Eades as a variant of graph planarity, in which the vertex set of the input graph is endowed with a hierarchical clustering and we seek an embedding (crossing free drawing) of the graph in the plane that respects the clustering in a certain natural sense. Until now, it has been an open problem whether c-planarity can be tested efficiently. The thickenability problem for simplicial complexes emerged in the topology of manifolds in the 1960s. A 2-dimensional simplicial complex is thickenable if it embeds in some orientable 3-dimensional manifold. Recently, Carmesin announced that thickenability can be tested in polynomial time. Our algorithm for atomic embeddability combines ideas from Carmesin’s work with algorithmic tools previously developed for weak embeddability testing. We express our results purely in terms of graphs on surfaces, and rely on the machinery of topological graph theory. Finally, we give a polynomial-time reduction from atomic embeddability to thickenability thereby showing that both problems are polynomially equivalent, and show that a slight generalization of atomic embeddability to the setting in which clusters are toroidal graphs is NP-complete.

Cohomology jump loci of 3-manifolds

The cohomology jump loci of a space $X$ are of two basic types: the characteristic varieties, defined in terms of homology with coefficients in rank one local systems, and the resonance varieties, constructed from information encoded in either the cohomology ring, or an algebraic model for $X$. We explore here the geometry of these varieties and the delicate interplay between them in the context of closed, orientable 3-dimensional manifolds and link complements. The classical multivariable Alexander polynomial plays an important role in this analysis. As an application, we derive some consequences regarding the formality and the existence of finite-dimensional models for such 3-manifolds.

Ricci flow on open 3–manifolds and positive scalar curvature

We show that an orientable 3-dimensional manifold M admits a complete riemannian metric of bounded geometry and uniformly pos- itive scalar curvature if and only if there exists a finite collection F of spherical space-forms such that M is a (possibly infinite) connected sum where each summand is diffeomorphic to S2xS1 or to some mem- ber of F. This result generalises G. Perelman's classification theorem for compact 3-manifolds of positive scalar curvature. The main tool is a variant of Perelman's surgery construction for Ricci flow.

Eilenberg-Moore spectral sequence calculation of function space cohomology

The purpose of this article is to introduce an Eilenberg-Moore spectral sequence converging to the cohomology algebra of a function space with an adjunction space as its source. Computability of the spectral sequence is shown by determining explicitly the mod p cohomology algebra of the function space Open image in new window of maps from a non-orientable surface S to the classifying space of a simply-connected Lie group G whose homology is p-torsion free. Let M be a closed orientable 3-dimensional manifold. Applying the spectral sequence obtained from a Heegaard splitting of the manifold M, we also prove that Open image in new window is a direct summand of Open image in new window.

Nielsen Zeta Function, 3-Manifolds, and Asymptotic Expansions in the Nielsen Theory

We prove that the Nielsen zeta function is either a rational function or a radical of a rational function for orientation-preserving homeomorphisms of closed orientable 3-dimensional manifolds which are special Haken or Seifert manifolds. In the case of a pseudo-Anosov homeomorphism of a surface, we find an asymptotics for the number of twisted conjugacy classes or for the number of Nielsen fixed-point classes with norm at most x. Bibliography: 20 titles.

Zeta functions, 3-manifolds and asymptotics in Nielsen theory

We prove that the Nielsen zeta function is a rational function or a radical of a rational function for orientation preserving homeomorphisms on closed orientable 3-dimensional manifolds which are special Haken or Seifert manifolds. In the case of pseudo-Anosov homeomorphism of surface we compute an asymptotic for the number of twisted conjugacy classes or for the number of Nielsen fixed point classes whose norm is at most x.

Dehn surgery on knots

The Dehn surgery construction on knots in orientable 3-dimensional manifolds is introduced. We describe the parameters on which this construction depends, various situations in which it arises, and some of the main results and open research problems relating to it.

The Topological Classification of the Lens Spaces

The n-dimensional lens spaces can be classified under semi-linear equivalence by the torsion invariant of Reidemeister [1]. It seems desirable to prove the topological invariance of this classification without reference to the Hauptverinutung. In this paper we carry out such a proof for n = 3, using a method which, intuitively speaking, distinguishes manifolds by the differences in their knot theories. This method was proposed by Fox [16, p. 247] at the Princeton Bicentennial Conference of 1946, and later applied [17, p. 455] to re-prove the semi-linear classification of the lens spaces. Associated with every simple closed polygon k in a complex K there is a principal ideal (A) of the 1-dimensional Betti group ring of K k; A is known as the Alexander polynomial of k. [2, 3, 4, 5]. Let M be an orientable triangulable 3-dimensional manifold, y an element of H1(M), k a simple closed curve which represents y and which is polygonal in some triangulation of M. Let A be the Alexander polynomial of k, T1(M k) and B1(M k) the torsion subgroup and Betti group of H1(M -k), i: H1(M k) H1(M) the injectionhomomorphism, and *: B1(M k) H1(M)/iT1(M k) the homomorphism induced by i. Then we prove: (i) H1(M k) and iT1(M k) depend only upon e; (ii) A* depends only upon e. Unlike the Reidemeister torsion, the invariant A* is applicable to all orientable 3-dimensional manifolds. However, if one is interested only in the lens spaces, one may restrict oneself to the case H1(M k) infinite cyclic, which allows considerable simplification of the proof (cf., ?4.). As an additional example we give the classification of the topological sums of two 3-dimensional lens spaces. In what follows, manifold will mean compact orientable 3-dimensional manifold in the sense of [6]; similarly for bounded manifold. The term polygonal curve will mean polygonal in some triangulation of the given space; it should be emphasized that the assumption that two or more curves are polygonal does not necessarily postulate their polygonality in the same triangulation unless specifically stated. The reader is also cautioned that all groups, including homology groups, will be written multiplicatively to avoid confusion with group ring notation. Also, homomorphisms will frequently be written as superscripts, that is, f (x) as xf and f(x1) as x-. 163