Cellular Homology

This chapter introduces more basic concepts of algebraic topology and describes datatypes and algorithms for implementing them on a computer. The basic concepts include: chain complex, chain mapping, chain homotopy, homology of a (simplicial or cubical or permutahedral or CW-) space, persistent homology of a filtered space, cohomology ring of a space, van Kampen diagrams, excision. These are illustrated using computer examples involving digital images, protein backbones, high-dimensional point cloud data, knot complements, discrete groups, and random simplicial complexes.

A uniform model for almost convexity and rewriting systems

We introduce a topological property for finitely generated groups, called stackable, that implies the existence of an inductive procedure for constructing van Kampen diagrams with respect to a particular finite presentation. We also define algorithmically stackable groups, for which this procedure is an algorithm. This property gives a common model for algorithms arising from both rewriting systems and almost convexity for groups.

A GENERALIZATION OF ADJAN'S THEOREM ON EMBEDDINGS OF SEMIGROUPS

In this paper we investigate the question of possibility to injectively map a semigroup into a group. Adjan's theorem provides a sufficient condition for such a map to exist for semigroups with relations li = ri, where both li and ri are not empty. Presence of defining relations of the form l = 1 makes many combinatorial properties of semigroups significantly more complex. However, we generalize Adjan's theorem to the class of semigroups with defining relations of both kinds. We use Remmers's approach to exploit Van Kampen diagrams as major tool to abstract from the algebraic combinatorics behind the relations and, instead, work with more tangible objects, such as graphs on the plane.

THE SUBWORD REVERSING METHOD

We summarize the main known results involving subword reversing, a method of semigroup theory for constructing van Kampen diagrams by referring to a preferred direction. In good cases, the method provides a powerful tool for investigating presented (semi)groups. In particular, it leads to cancellativity and embeddability criteria for monoids and to efficient solutions for the word problem of monoids and groups of fractions. The text includes some new results about mixed reversing (combination of left- and right-reversings) and about the combinatorial distance of braids.

SOME DUALITY CONJECTURES FOR FINITE GRAPHS AND THEIR GROUP THEORETIC CONSEQUENCES

We pose some graph theoretic conjectures about duality and the diameter of maximal trees in planar graphs, and we give innovations in the following two topics in geometric group theory, where the conjectures have applications.Central extensions. We describe an electrostatic model concerning how van Kampen diagrams change when one takes a central extension of a group. Modulo the conjectures, this leads to a new proof that finitely generated class $c$ nilpotent groups admit degree $c+1$ polynomial isoperimetric functions.Filling functions. We collate and extend results about interrelationships between filling functions for finite presentations of groups. We use the electrostatic model in proving that the gallery length filling function, which measures the diameter of the duals of diagrams, is qualitatively the same as a filling function DlogA, concerning the sum of the diameter with the logarithm of the area of a diagram. We show that the conjectures imply that the space-complexity filling function filling length essentially equates to gallery length. We give linear upper bounds on these functions for a number of classes of groups including fundamental groups of compact geometrizable 3-manifolds, certain graphs of groups, and almost convex groups. Also we define restricted filling functions which concern diagrams with uniformly bounded vertex valence, and we show that, assuming the conjectures, they reduce to just two filling functions—the analogues of non-deterministic space and time.

EMBEDDING THEOREMS FOR GROUPS PRESENTED VIA PARTIAL AUTOMORPHISMS: A GENERALIZATION OF SEMIDIRECT PRODUCTS AND HNN EXTENSIONS

We examine a certain embedding problem for groups that have a presentation described by partial automorphisms. Semidirect products and HNN extensions have such a presentation. The embedding problem is closely related to systems of partial automorphisms, which are formalized by the concept of inverse semigroups. A necessary and sufficient condition for a group to be embedded in a certain sense is obtained by geometric methods using van Kampen diagrams.