Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory. Vassily Olegovich Manturov

^k−1, n) is simply connected for k > 3 but if we take some restricted configuration space C′(

^k−1, n), it will not be simply connected any more and leads to a meaningful notion of higher dimensional braid (Chapter 11).

      What sort of the restriction do we impose? On the plane, we consider just braids, so we say that no two points coincide. In

³ we forbid collinear triples, in

⁴ we forbid coplanar quadruple of points.

      When I showed the spaces I study to my coauthor, Jie Wu, he said: look, these are k-regular embeddings, they go back to Carol Borsuk. Indeed, after looking at some papers by Borsuk, I saw similar ideas were due to P.L. Chebysheff ([Borsuk, 1957; Kolmogorov, 1948]).

      By the way, once Wu looked at the group

, he immediately asked about the existence of simplicial group structure on such groups, the joint project we are working on now with S. Kim, J. Wu, F. Li.

      An interested reader may ask whether such braids exist not only for

^k (or

Pk), but also for other spaces. This question we shall touch on later.

      From the algebraic point of view, why are these groups good, how are they related to other groups, how to solve the word and the conjugacy problems, etc.?

      It is impossible to describe all directions of the

group theory in the preface, the reader will find many directions in the unsolved problem list; I just mention some of them.

      For properties of

, we can think of them as n-strand braids with k-fold strand intersection.

      Like

, there are nice “strand forgetting” and “strand deletion” maps to

and

, see Fig. 0.4.

      The groups

have lots of epimorphisms onto free products of cyclic groups; hence, invariants constructed from them are powerful enough and easy to compare.

      For example, the groups

are commensurable with some Coxeter groups of special type, see Fig. 0.5, which immediately solves the word problem for them.

      As Diamond lemma works for Coxeter groups, it works for

, and in many other places throughout the book.

      After a couple of years of study of

ⁿ and

Pn, but what if we consider just braids on a 2-surface? What can we study then? The property “three points belong to the same line” is not quite good even in the metrical case because even if we have a Riemannian metric on a 2-surface of genus g, there may be infinitely many geodesics passing through two points. Irrational cables may destroy the whole construction.

      Then I decided to transform the “

-point of view” to make it more local and more topological. Assume we have a collection of points in a 2-surface and seek

-property: four points belong to the same circle.

      Consider a 2-surface of genus g with N points on it. We choose N to be sufficiently large and put points in a position to form the centers of Voronoï cells. It is always possible for the sphere g = 0, and for the plane we may think that all our points live inside a triangle forming a Voronoï tiling of the latter.

      Fig. 0.4Maps from

to

and

      We are interested in those moments when the combinatorics of the Voronoï tiling changes, see Fig. 0.7.

      This corresponds to a flip, the situation when four nearest points belong to the same circle. This means that no other point lies inside the circle passing through these four, see [Gelfand, Kapranov and Zelevinsky, 1994].

      The most interesting situation of codimension 2 corresponds to five points belonging to the same circle.

      Fig. 0.5The Cayley graph of the group

and the Coxeter group C(3, 2)

      Fig. 0.6Flips on a pentagon

      This leads to the relation:

      Note that unlike the case of

, here we have five terms, not ten. What is the crucial difference? The point is that if we have five points in the neighbourhood of a circle, then every quadruple of them appears to be on the same circle twice, but one time the fifth point is outside the circle, and one time it is inside the circle. We denote the set {1, . . . , n} by n and introduce the following

      Definition 0.2. The group is the group given by group presentation generated by subject to the following relations:

      Fig. 0.7Voronoï tiling change

      (1),

      (2),

      (3)