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