Se p 20 01 Braids , trees , and operads

The space of unordered configurations of distinct points in the plane is aspherical, with Artin’s braid group as its fundamental group. Remarkably enough, the space of ordered configurations of distinct points on the real projective line, modulo projective equivalence, has a natural compactification (as a space of equivalence classes of trees) which is also (by a theorem of Davis, Januszkiewicz, and Scott) aspherical. The classical braid groups are ubiquitous in modern mathematics, with applications from the theory of operads to the study of the Galois group of the rationals. The fundamental groups of these new configuration spaces are not braid groups, but they have many similar formal properties. This talk [at the Gdansk conference on algebraic topology 05-06-01] is an introduction to their study. 1. The bubbletree operad and quantum cohomology 1.0 Work on conformal field theories leads physicists to an interest in configuration spaces ConfigCP1 ∼ Config n C . of points on the complex projective line. They are most interested in the quotients of these spaces by the action of PGl2(C). The points are noncoincident, so both the spaces and the group are noncompact, and taking the quotient is tricky: it leads naturally to a compactification M0,n(C) ∼ Config (CP1)/PGl2(C) . The physicists discovered a repulsive potential among these points: pushing two together creates a bubble onto which they escape [13]. 1.1 Thus M0,n(C) is the moduli space of marked genus zero stable algebraic curves (which have (at worst) double points, and at least three marked points on each irreducible component). Example: M0,4(C) ∼= CP1 via the classical cross-ratio. Note, M0,3(C) is a point: a configuration of three points on CP1 is rigid. These spaces are very nice in some ways: they are compact manifolds, with cohomology concentrated in even dimension, and no torsion [8]. 1991 Mathematics Subject Classification. Primary 55R80, Secondary 14N35, 20F36. The author was supported in part by the NSF. 1