The edge-minimal polyhedral maps of Euler characteristic .
The end curve theorem for normal complex surface singularities
We prove the “End Curve Theorem,” which states that a normal surface singularity with rational homology sphere link is a splice quotient singularity if and only if it has an end curve function for each leaf of a good resolution tree. An “end curve function” is an analytic function whose zero set intersects in the knot given by a meridian curve of the exceptional curve corresponding to the given leaf. A “splice quotient singularity” is described by giving an explicit set of equations describing...
The equation [B,(A-1)(A,B)] = 0 and virtual knots and links
Let A, B be invertible, non-commuting elements of a ring R. Suppose that A-1 is also invertible and that the equation [B,(A-1)(A,B)] = 0 called the fundamental equation is satisfied. Then this defines a representation of the algebra ℱ = A, B | [B,(A-1)(A,B)] = 0. An invariant R-module can then be defined for any diagram of a (virtual) knot or link. This halves the number of previously known relations and allows us to give a complete solution in the case when R is the quaternions.
The equivalence of -groupoids and crossed complexes
The equivalence of -groupoids and cubical -complexes
The equivariant Dehn's lemma and loop theorem.
The existence of negatively Ricci curved metrics on three manifolds.
The extended complexity of three-manifolds.
The extension of Johnson's homomorphism form the Torelli group to the mapping class group.
The Finiteness Obstructions for Nilpotent Spaces lie in D (...).
The forcing relation for horseshoe braid types.
The Fukumoto-Furuta and the Ozsváth-Szabó invariants for spherical 3-manifolds
We show that the Fukumoto-Furuta invariant for a rational homology 3-sphere M, which coincides with the Neumann-Siebenmann invariant for a Seifert rational homology 3-sphere, is the same as the Ozsváth-Szabó's correction term derived from the Heegaard Floer homology theory if M is a spherical 3-manifold.
The fundamental group and Galois coverings of hexagonal systems in 3-space.
The fundamental group and the spectrum of the Laplacian.
The fundamental group of compact manifolds without conjugate points.
The Fundamental Group of Fibered Knot Cobordisms.
The genera, reflexibility and simplicity of regular maps
This paper uses combinatorial group theory to help answer some long-standing questions about the genera of orientable surfaces that carry particular kinds of regular maps. By classifying all orientably-regular maps whose automorphism group has order coprime to , where is the genus, all orientably-regular maps of genus for prime are determined. As a consequence, it is shown that orientable surfaces of infinitely many genera carry no regular map that is chiral (irreflexible), and that orientable...
The genus problem for one-relator products of locally indicable groups.
The geometric genus of hypersurface singularities
Using the path lattice cohomology we provide a conceptual topological characterization of the geometric genus for certain complex normal surface singularities with rational homology sphere links, which is uniformly valid for all superisolated and Newton non-degenerate hypersurface singularities.
The geometry of abstract groups and their splittings.
A survey of splitting theorems for abstract groups and their applications. Topics covered include preliminaries, early results, Bass-Serre theory, the structure of G-trees, Serre's applications to SL2 and length functions. Stallings' theorem, results about accessibility and bounds for splittability. Duality groups and pairs; results of Eckmann and collaborators on PD2 groups. Relative ends, the JSJ theorems and the splitting results of Kropholler and Roller on PDn groups. Notions of quasi-isometry,...