##
**Invitation to topological robotics.**
*(English)*
Zbl 1148.55011

Zurich Lectures in Advanced Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-054-8/pbk). x, 133 p. (2008).

This book is mostly about a number of interesting subspaces of \((E^m)^n\), thought of as the space of collections (configurations when the points are disjoint) of \(n\) points in \(m\)-space. Of particular interest are the cases \(m=2\) and \(m=3\). The subspaces are generally defined by some set of metric restrictions on the points.

The first chapter focuses on the case where \(m=2\) and the \(n\) points are the vertices of a (possibly singular or degenerate) polygon whose sides are fixed in length. Theorem 1.7 from this chapter describes explicitly how the vector \(l\) of lengths of the sides of the polygon determines the Betti numbers (the Poincaré polynomial) of this space, \(M_l\), whose diffeomorphism class incidentally is independent of permutations of the side lengths. The natural involution of reflection in the \(x\)-axis plays a role in the equivariant Morse theory which is used in the proof of 1.7. The conditions under which the graded isomorphism class of the cohomology algebra of \(M_l\) determines \(l\) are also studied in this chapter along with “random linkages” in the cases \(m=2\) and \(m=3\).

Chapter 2 is concerned with the configuration spaces of \(n\) distinct and indistinguishable points in a finite simplicial polyhedron, denoted \(X\). These spaces arise in the study of collision avoidance in the control of robots. If one forms a power series (the Euler-Gal series) in \(t\) with the coefficient of \(t^n\) being the Euler characteristic of the space of \(n\) indistinguishable points in \(X\), then one has Theorem 2.1 which states that this power series is in fact a rational function \(P(t)/Q(t)\). In addition the constant terms of \(P\) and \(Q\) are 1, and the degree of \(P-\) the degree of \(Q\) is the Euler characteristic of \(X\). Various implications of this theorem are drawn and the behavior of the Euler-Gal series with respect to the Grothendieck ring determined.

Chapter 3 is entirely devoted to a proof (mainly due to R. Connelly, E. Demaine and G. Rote in [Discrete Comput. Geom. 30, 205–239 (2003; Zbl 1046.52016)]) of a solution to the Carpenter’s Rule problem. This problem asks whether a polygonal arc in the plane can be straightened without changing side lengths or intersecting sides, i.e. through a homotopy which maintains the polygonal and non-singular character of the arc and maintains the side lengths as well. The answer, as is shown here, is yes it can.

The last chapter is concerned with the actual algorithms and programs for moving in the spaces studied, that is for moving from one configuration to another without leaving the space. Such motions are studied in some detail as sections of the (fiber) map of the path space to the endpoints of the motion. Notions of domains of continuity and complexity related to the Schwarz genus and the Lusternik-Schnirelmann category play a central role in the discussion and results. Certain cohomology operations are utilized to calculate topological complexity, especially for the case of Lens spaces.

The first chapter focuses on the case where \(m=2\) and the \(n\) points are the vertices of a (possibly singular or degenerate) polygon whose sides are fixed in length. Theorem 1.7 from this chapter describes explicitly how the vector \(l\) of lengths of the sides of the polygon determines the Betti numbers (the Poincaré polynomial) of this space, \(M_l\), whose diffeomorphism class incidentally is independent of permutations of the side lengths. The natural involution of reflection in the \(x\)-axis plays a role in the equivariant Morse theory which is used in the proof of 1.7. The conditions under which the graded isomorphism class of the cohomology algebra of \(M_l\) determines \(l\) are also studied in this chapter along with “random linkages” in the cases \(m=2\) and \(m=3\).

Chapter 2 is concerned with the configuration spaces of \(n\) distinct and indistinguishable points in a finite simplicial polyhedron, denoted \(X\). These spaces arise in the study of collision avoidance in the control of robots. If one forms a power series (the Euler-Gal series) in \(t\) with the coefficient of \(t^n\) being the Euler characteristic of the space of \(n\) indistinguishable points in \(X\), then one has Theorem 2.1 which states that this power series is in fact a rational function \(P(t)/Q(t)\). In addition the constant terms of \(P\) and \(Q\) are 1, and the degree of \(P-\) the degree of \(Q\) is the Euler characteristic of \(X\). Various implications of this theorem are drawn and the behavior of the Euler-Gal series with respect to the Grothendieck ring determined.

Chapter 3 is entirely devoted to a proof (mainly due to R. Connelly, E. Demaine and G. Rote in [Discrete Comput. Geom. 30, 205–239 (2003; Zbl 1046.52016)]) of a solution to the Carpenter’s Rule problem. This problem asks whether a polygonal arc in the plane can be straightened without changing side lengths or intersecting sides, i.e. through a homotopy which maintains the polygonal and non-singular character of the arc and maintains the side lengths as well. The answer, as is shown here, is yes it can.

The last chapter is concerned with the actual algorithms and programs for moving in the spaces studied, that is for moving from one configuration to another without leaving the space. Such motions are studied in some detail as sections of the (fiber) map of the path space to the endpoints of the motion. Notions of domains of continuity and complexity related to the Schwarz genus and the Lusternik-Schnirelmann category play a central role in the discussion and results. Certain cohomology operations are utilized to calculate topological complexity, especially for the case of Lens spaces.

Reviewer: Lee P. Neuwirth (Princeton)

### MSC:

55R80 | Discriminantal varieties and configuration spaces in algebraic topology |

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |

57R70 | Critical points and critical submanifolds in differential topology |

58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |

68T40 | Artificial intelligence for robotics |

70B15 | Kinematics of mechanisms and robots |

70Q05 | Control of mechanical systems |