Braids, trees, and operads Jack Morava Abstract.The space of unordered configurations of distinct points in the plane is aspherical, with Artin's braid group as its fundamental group. * *Re- markably enough, the space of ordered configurations of distinct points * *on the real projective line, modulo projective equivalence, has a natural compa* *ctifi- cation (as a space of equivalence classes of trees) which is also (by a * *theorem of Davis, Januszkiewicz, and Scott) aspherical. The classical braid grou* *ps are ubiquitous in modern mathematics, with applications from the theory of o* *per- ads to the study of the Galois group of the rationals. The fundamental g* *roups of these new configuration spaces are not braid groups, but they have ma* *ny similar formal properties. This talk [at the Gdansk conference on algebr* *aic 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 configu- ration spaces Confign+1CP1 ~ ConfignC . of points on the complex projective line. They are most interested in the quoti* *ents 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 __ n M 0,n(C) ~ Config(CP1)=PGl 2(C) . The physicists discovered a repulsive potential among these points: pushing two together creates a bubble onto which they escape [13]. __ 1.1 Thus M 0,n(C) is the moduli space of marked genus zero stable algebraic cur* *ves (which have (at worst) double points, and at least three marked points on each irreducible component). __ __ Example: M 0,4(C) ~=CP1 via the classical cross-ratio. Note, M 0,3(C) is a poin* *t: a configuration of three points on CP1 is rigid. These spaces are very nice in some ways: they are compact manifolds, with coho- mology concentrated in even dimension, and no torsion [8]. ____________ 1991 Mathematics Subject Classification. Primary 55R80, Secondary 14N35, 20* *F36. The author was supported in part by the NSF. 1 2 JACK MORAVA Operads, by example: 1.2 An operad O* = {Ok, k 1} is a collection of spaces together with some composition maps P On x Oi1x . .x.Oin! Oi (where i = ik) satisfying some axioms . . . __ ex. i) M 0,*+1(C) ex. ii) Endn(X) = Maps(Xn, X) is the endomorphism operad of an object X in a monoidal category. Com- position sends a map from Xn to X and a tuple of maps from Xi* to X to a composition Xi = Xi1x . .X.in! Xn ! X . A morphism O* ! End*(X) of operads makes X into an O*-algebra. ex. iii) Brn = Artin's braid group on n strings defines the braid operad, with cabling Brnx Bri1x . .B.rin! Bri as composition. Monoidal functors preserve operads; hence the homology of an operad (in spaces) is an operad in graded modules [9]. 1.3 Theorem (WDVV, Kontsevich [5]): The_(rational)_homology of a smooth projective algebraic variety V is an H*(M 0,*+1(C))-operad algebra. [This led to the solution of the 19th-century enumerative geometry problem of classification of lines with specified incidence in CP2.] The construction of this algebra structure uses the Gromov-Witten invariants: There are (compact) moduli spaces GWk(V ) of holomorphic maps from genus zero stable curves with k marked points, to V . These spaces have many components, indexed by degree [h : C ! V ] 2 H2(V, Z). There is also an evaluation map __ k GWk(V ) ! M 0,k(C) x V which defines a cycle __ k GWk 2 H*(M 0,k(C)) H*(V ) . [Actually the coefficients lie in the Novikov ring = Q[H2(V, Z)], but this wi* *ll be supressed.] Using Poincar'e duality, we can rewrite GWk+1 as an element of __ k Hom (H*(M 0,k+1(C)), Hom (H*(V ) , H*(V ))) which then defines a morphism __ H*(M 0,k+1(C)) ! Endk(H*(V )) of operads, QED. __ In particular, the point M 0,3(C) defines a quantum multiplication H*(V, ) H*(V, ) ! H*(V, ) which is usually not standard . . . BRAIDS, TREES, AND OPERADS 3 2. Devadoss's mosaic operad and Fukaya's Lagrangian cohomology 2.0 The moduli space [2] __ Confign(RP1)=PGl 2(R) ~ M 0,n(R) of configurations of points on the circle can be pictured as a space of trees or mosaics_of hyperbolic polygons. The points have an intrinsic cyclic order, and {M 0,*(R)} is naturally a cyclic operad [6]. I am indebted to Sasha Voronov, for pointing out that the analogous complex operad is also cyclic! 2.1 Fukaya considers a compact symplectic manifold (M, !) together with an oriented Lagrangian submanifold L (i.e. of half the dimension of M, such that !* *|L = 0; some subtle issues involving the Stiefel-Whitney class w2(L) will be ignored* *.) I conjecture that the following is a theorem; something slightly weaker (cf. belo* *w) has been proved by Fukaya and his school [4]: For a generic almost-complex structure compatible with !, there are compact ori- ented moduli spaces F Ok of pseudo-holomorphic hyperbolic polygons (P, @P ) ! (M, L) together with evaluation maps __ k F Ok ! M 0,k(R) x L __ which define an action of H*(M 0,*+1(R)) on H*(L, ) (where now = Q[H2(M, Z)]* *). 2.2 The (co)homology of these spaces is not known (but cf. [14]). However, we can draw some beautiful pictures [2,3]. __ Grothendieck,_in_his_Esquisse,_says M 0,5(C) is `un petit joyaux'. Its real po* *ints M 0,5(R) map to M 0,4(R) x M 0,4(R) = T 2by selecting two distinct subsets of f* *our points. To get the full space, we need to blow up (ie, add crosscaps) at the th* *ree configurations defined by triple coincidences, resulting in T 2#3RP 2. There is a more symmetric picture, defined by blowing up four points on RP 2. Both pictures are tesselated by pentagons, though this is easier to see in the second picture. This is a regular polytope with twelve pentagonal faces: it is * *the dodecahedron's `evil twin'. 2.3 In general, there is a surjective map __ k xDk Kk-3 ! M 0,k(R) (where Dk is the dihedral group of order 2k) which is 2n to 1 on codimension n faces: in general, these moduli spaces are tesselated by Stasheff associahedra. There is a commutative diagram Config*(R) ----! Config*(C) ?? ? y ?y __ __ M 0,*+1(R)----! M 0,*+1(C) 4 JACK MORAVA The space in the upper right corner is homotopy-equivalent to the little disks * *op- erad, and the space in the upper left corner is the classical A1 operad { *xK*-* *1} (made permutative, ie endowed with an action of the symmetric group. The left vertical map defines the tesselation; thus the mosaic operad is a kind of (quas* *i- commutative) quotient of the A1 operad. Fukaya shows that the A1 operad acts on Floer cohomology, but I believe that action passes through this quotient. The diagram above is a fiber product of spaces, but it is not quite a fiber product* * of operads. 2.4 Theorem of Davis, Januszkiewicz, and_Scott [1]: this tesselation defines a piecewise negatively curved metric on M 0,*(R); these spaces are therefore K(ß,* * 1)'s! Remark: Devadoss [3] has recently shown (using similar methods) that the Fulton- MacPherson compactification of Config*(RP1) is also aspherical! 3. Operads in groups (and groupoids) 3.0 Observation: The fundamental group of an operad is an operad in groups . . . provided you're careful about basepoints. Example {1, . .,.n 2 C} (with that order) defines a basepoint * 2 Confign(C); b* *ut the natural action of n moves it around (by changing the order). Recall that a space has a fundamental groupoid, with respect to a system of basepoints: it is a category, with the points as objects, and homotopy classes * *of paths between them as morphisms. Note also that a surjective homomorphism OE : G ! H of groups defines a groupoid [H=G] with H as set of objects, and mor (h0, h1) = {g 2 G | OE(g)h0 = h1} as morphisms. With this notation, ß(Confign(C) rel n(*)) ~=[ n=Brn] where Brn ! n is the standard homomorphism. Thus the fundamental groupoid of the little disks operad is the braid operad. I owe thanks to Dan Christensen* * and J.P. Meyer, for explaining this to me. 3.1 Definition: The braid category B has integers n as objects, with Brn as its endomorphisms. [There are no morphisms between distinct integers.] This is a (universal) braided monoidal category, with tensor product B x B ! B defined by juxtaposition (n, m) 7! n + m. A standard construction [7] defines a functor [ n=Brn] ! Func(Bn, B) which makes the category B an algebra over the braid operad. More generally, any braided monoidal category is an algebra over the braid operad. BRAIDS, TREES, AND OPERADS 5 __ The operad M 0,*+1(R) defines a similar category: there is an exact sequence __ __ ß1(M 0,*+1(R)) æ ß1(M 0,*+1(R)h *) i * in which the fundamental group of the homotopy quotient plays the role of the braid group. There is a similar tensor category D, which is a kind of universal example of an algebra over the associated operad in groupoids. 3.2 Here are some questions and speculations: i) do these fundamental groups act in some natural way on Fukaya's cohomology? ii) does D have an interpretation in terms of cyclic operads with a trace in the sense of Markl [11]? iii) are these fundamental groups in some sense Galois groups for solutions of Calogero-Moser systems [of points moving on the line [12]] analogous to the role played by the braid groups in the Knizhnik-Zamolodchikov equations? iv) The Grothendieck-Teichmüller group [10] is in some sense the automorphisms of the braid operad._Do automorphisms of the operad in groupoids defined by the moduli spaces M 0,*(R) have any similar properties? __ n iv) Does the rank of H1(M 0,k+1(R)) equal 3 ? References 1. M. Davis, T. Januszkiewicz, R. Scott, Nonpositive curvature of blow-ups. Sel* *ecta Math. 4 (1998) 491-547 2. S. Devadoss, Tessellations of moduli spaces and the mosaic operad, in Homoto* *py invariant algebraic structures, Contemp. Math. 239 (1998) 91-114 3. __-, A space of cyclohedra, available at math.ohio-state.edu/ devadoss; Disc* *rete and Combinatorial Geometry, to appear 4. K. Fukaya, Y-G Oh, H. Ohta, K. Ono, Lagrangian intersection Floer theory: an* *omaly and obstruction, Kyoto, Dept. of Math. 00-17 5. E. Getzler, Operads and moduli spaces of genus 0 Riemann surfaces, in The mo* *duli space of curves, 199-230, Progr. Math. 129 (1995) 199-230, Birkhäuser 6. __-, M. Kapranov, Cyclic operads and cyclic homology, in Geometry, Topology,* * and Physics for Raoul Bott (1994) 7. C. Kassel, Quantum groups, Springer Graduate Texts 155 (1995) 8. S. Keel, Intersection theory of moduli space of stable N-pointed curves of g* *enus zero. Trans. AMS 330 (1992) 545-574 9. T. Kimura, J. Stasheff, A. Voronov, Homology of moduli of curves and commuta* *tive homotopy algebras, in The Gelfand Mathematical Seminars (1993-95), Birkhäuser, 1996. 10.P. Lochak, L. Schneps, The Grothendieck-Teichmüller group and automorphisms * *of braid groups, in The Grothendieck theory of dessins d'enfants, London Math. Soc. L* *ecture Notes 200 (1994) 323-358 11.M. Markl, Simplex, associahedron, and cyclohedron, in Higher homotopy struct* *ures in topology and mathematical physics 235-265, Contemp. Math. 227, AMS (1999) 12.J. Moser, Various aspects of integrable Hamiltonian systems, in Dynamical sy* *stems (Bres- sanone, 1978) 233-289, Progr. Math. 8, Birkhäuser (1980) 13.T.H. Parker, J.G. Wolfson, Pseudo-holomorphic maps and bubble trees. J. Geom* *. Anal. 3 (1993) 63-98 14.M. Yoshida, The democratic compactification of configuration spaces of point* * sets on the real projective line, Kyushu J. of Math. 50 (1996) 493-512 Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 212* *18 E-mail address: jack@math.jhu.edu