TESSELLATIONS OF MODULI SPACES AND THE MOSAIC
OPERAD
SATYAN L. DEVADOSS
Abstract.We construct a new (cyclic) operad of mosaics defined by poly-
gons_with marked diagonals. Its underlying (aspherical) spaces are the se*
*ts
M n0(R) which are naturally tiled by Stasheff associahedra. We describe t*
*hem
as iterated blow-ups and show that their fundamental groups form an operad
with similarities to the operad of braid groups.
Acknowledgments.This paper is a version of my doctorate thesis under Jack Morav*
*a,
to whom I am indebted for providing much guidance and encouragement. Thanks
also go out to F. Hirzebruch, T. Januszkiewicz, and R. Scott for sharing useful
insights. I am especially grateful to Jim Stasheff for bringing up questions ab*
*out
the blow-up arrangements and for his overall enthusiasm about this work.
1.The Operads
1.1. The notion of an operad was created for the study of iterated loop spaces
[13]. Since then, operads have been used as universal objects representing a wi*
*de
range of algebraic concepts. We give a brief definition and provide classic exa*
*mples
to highlight the issues to be discussed.
Definition 1.1.1.An operad {O(n) | n 2 N} is a collection of objects O(n) in a
monoidal category endowed with certain extra structures:
i) O(n) carries an action of the symmetric group Sn.
ii)There are composition maps
(1.1) O(n) x O(k1) x . .x.O(kn) ! O(k1+ . .+.kn)
which satisfy certain well-known axioms, cf. [14].
This paper will be concerned mostly with operads in the context of topological
spaces, where the objects O(n) will be equivalence classes of geometric objects.
Example 1.1.2.These objects can be pictured as trees (Figure 1a). A tree is
composed of corollas1 with one external edge marked as a root and the remaining
external edges as leaves. Given trees s and t, basic compositions are defined *
*as
s Oit, obtained by grafting the root of s to ithleaf of t. This grafted piece o*
*f the
tree is called a branch.
Example 1.1.3.There is a dual picture in which bubbles replace corollas, marked
points replace leaves, and the root is denoted as a point labeled 1 (Figure 1b).
Using the above notation, the composition s Oit is defined by fusing the 1 of t*
*he
bubble s with the ithmarked point of t. The branches of the tree are now identi*
*fied
with double points, the places where bubbles intersect.
___________
1A corolla is a collection of edges meeting at a common vertex.
1
2 SATYAN L. DEVADOSS
Figure 1. Trees, Bubbles, and Polygons
1.2. Taking yet another dual, we can define an operad structure on a collection
of polygons (modulo an appropriate equivalence relation) as shown in Figure 1c.
Each bubble corresponds to a polygon, where the number of marked and double
points become the number of sides; the fusing of points is associated with the *
*gluing
of faces. The nicest feature of polygons is that, unlike corollas and bubbles, *
*the
iterated composition of polygons yields a polygon with marked diagonals (Figure*
* 2).
Figure 2
Unlike the rooted trees, this mosaic operad is cyclic in the sense of Getzler*
* and
Kapranov [6, x2]. The most basic case (Figure 3) shows how two polygons, with
sides labeled a and b respectively, compose to form a new polygon. The details *
*of
this operad are made precise in x3.1.
Figure 3. Polygon composition
1.3. In the work of Boardman and Vogt [2, x2.6], an operad is presented using
m dimensional cubes Im Rm . An element C(n) of this little cubes operad is
the space of an ordered collection of n cubes linearly embedded by fi: Im ,! Im*
* ,
with disjoint interiors and axes parallel to Im . The fi's are uniquely determi*
*ned
by the 2n-tuple of points (a1; b1; : :;:an; bn) in Im , corresponding to the im*
*ages of
the lower and upper vertices of Im . An element oe 2 Sn acts on C(n) by permuti*
*ng
the labeling of each cube:
(a1; b1; : :;:an; bn) 7! (aoe(1); boe(1); : :;:aoe(n); boe(n)):
TESSELLATIONS OF MODULI SPACES AND THE MOSAIC OPERAD 3
Figure 4. Little cubes composition
The composition operation (1.1)is defined by taking n spaces C(ki) (each having*
* ki
embedded cubes) and embedding them as an ordered collection into C(n). Figure 4
shows an example for the two dimensional case when n = 4.
Boardman showed that the space of n distinct cubes in Rm is homotopically
equivalent to Confign(Rm ), the configuration space on n distinct points in Rm *
*.2
When m = 2, Confign(R2) is homeomorphic to Cn - , where is the thick
diagonal {(x1; : :;:xn) 2 Cn | 9 i; j; i 6= j ; xi = xj}. Since the action of S*
*n on
Cn - is free, taking the quotient yields another space (Cn - )=Sn. It is well-
known that both these spaces are aspherical, having all higher homotopy groups
vanish [4]. The following short exact sequence of fundamental groups results:
ss1(Cn - ) ae ss1((Cn - )=Sn) i Sn:
___________
2The equivariant version of this theorem is proved by May in [13, x4].
4 SATYAN L. DEVADOSS
But ss1 of Cn - is simply Pn, the pure braid group. Similarly, ss1 of Cn -
quotiented by all permutations of labelings is the braid group Bn. Therefore, t*
*he
short exact sequence above takes on the more familiar form:
Pn ae Bn i Sn:
We will return to these ideas in x6.
2.The Moduli Space
2.1. The moduli space of Riemann spheres with n punctures,
Mn0(C) = (Confign(CP1))=PGl2(C);
has_beennstudied extensively [10]. It has a Deligne-Mumford compactification
M 0(C), a smooth variety of complex dimension n - 3. In fact, this variety is
defined over the integers; we will look at the real points of this space.
__n
Definition 2.1.1.The moduli space M 0(R) of configurations of n smooth points
on punctured stable real algebraic curves of genus zero is a compactification o*
*f the
quotient ((RP1)n - )=PGl2(R); where is the thick diagonal.
Remark.This is an action of a non-compact group on a non-compact space. Geo-
metric invariant theory gives a natural compactification for this quotient, def*
*ined
combinatorially in terms of bubble trees or algebraically as a moduli space of *
*real
algebraic curves of genus zero with n points, which are stable in the sense tha*
*t they
have only finitely many automorphisms.
A point of Mn0(R)_cannbe visualized as a bubble (that is, RP1) with n distinct
marked points. In M 0(R), however, these marked points are allowed to `collide'*
* in
the following sense: As two adjacent points p1 and p2 of the bubble come closer
together and try to collide, the result is a new bubble fused to the old at the*
* point
of collision (a double point), where the marked points p1 and p2 are now on the
new bubble (Figure 5). Note that each bubble must have at least three marked or
double points in order to be stable.
Figure 5
The mosaic operad encapsulates all the information of the bubbles, enabling o*
*ne
to look at the situation above from the vantage point of polygons. Having n mar*
*ked
points on a circle now corresponds to an n-gon; when two adjacent sides p1 and *
*p2
of the polygon try to collide, a diagonal of the polygon is formed such that p1*
* and
p2 lie on one side of the diagonal (Figure 6). This can be generalized: A place*
* where
k + 1 marked points on RP1 collide corresponds to having a diagonal d partition*
* an
n-gon in such a way that k + 1 points lie_onnone side of d and n - k - 1 points*
* lie
on the other. Therefore, the cells of M 0(R) correspond to polygons with diagon*
*als.
The exact formulation of this appears in x4.
TESSELLATIONS OF MODULI SPACES AND THE MOSAIC OPERAD 5
Figure 6
2.2. There_arentwo elegant approaches in better understanding this space. One *
*is
to view M 0(R) in terms of hyperplane arrangements as formulated by Kapranov
[8, x4.3] and described by Davis, Januszkiewicz, and Scott [5, x0.1].
Definition 2.2.1.Let V n Rn-1 be the hyperplane defined by xi = 0. For
1 i < j n - 1, let Hnij V nbe the hyperplane defined by xi = xj. The
braid arrangement is the collection of subspaces of V ngenerated by all possible
intersections of the Hnij.
Notation.If Hn denotes the collection of subspaces Hnij, then Hn cuts V ninto
simplicial cones. Let P(V n) be the projective sphere in V n, that is, RPn-3. L*
*et
Bn be the intersection of Hn with P(V n) and let bk be a k-dimensional irreduci*
*ble
component of Bn. The arrangement Bn cuts P(V n) into open (n - 3)-simplicies;
the irreducible components {bk} comprise the boundaries of these simplicies.
Definition 2.2.2.Replace bk with sbk, the sphere bundle associated to the normal
bundle of bk P(V n). This process yields a manifold with boundary. Then
projectify sbk into pbk, the projective sphere bundle. This defines a manifold
without boundary, called the blow-up of P(V n) along bk.
Remark.Replacing bk with sbkfor any dimension k creates a new manifold with
boundary. However, blowing up along bk defines a new manifold for all dimensions
except codim one. That is, for codim one, projectifying sbkinto pbkannuls the
process of replacing bk with sbk.
Proposition 2.2.3.[8, x4.3] The iterated_blow-up_ofnP(V n) along the cells {bk}
in increasing order of dimension yields M 0(R).3
Therefore, the compactification of Mn0(R) is obtained by replacing the set {b*
*k}
with {pbk}. The closure of Mn0(R) in P(V n), denoted by bMn0(R), is obtained by
replacing the set {bk} with {sbk}; this procedure truncates each simplex of P(V*
* n)
into an n - 3 dimensional polytope.4 The resulting polytope after truncation is
called the associahedron Kn-1. Each Kn-1 is naturally given a Riemann metric
coming from Mn0(R). The properties of the associahedra are examined in x3.
Historical Note.Stasheff originally defined the associahedra in a purely combin*
*a-
torial manner for use in homotopy theory [17, x6]. Since then, they have contin*
*ued
to appear in a vast number of mathematical fields, gradually acquiring more and
more structure, cf. [19].
___________
3It is inessential to specify the order in which cells {bk} of the same dimen*
*sion are blown up.
4For a detailed construction of this truncation, see Appendix B of [16].
6 SATYAN L. DEVADOSS
__5
Example 2.2.4.The diagram of M 0(R) is shown in Figure 7, first found in a
different context by Brahana and Coble in 1926 [1, x1]. The arrangement B5 on
P(V 5) ' RP2yields six lines forming twelve 2-simplicies; the irreducible compo*
*nents
of dim 0 turn out to be the points {b01; : :;:b04} of triple intersection. Blow*
*ing up
along these components, we get S1 as a hexagon for sb0iand RP1 as a triangle_
for pb0i. The associahedron K4 is seen to be a pentagon, and the space M 50(R)
becomes tessellated by twelve such cells (shaded), a somewhat "evil twin" of the
dodecahedron.
5
__5 z_____"______-22
Figure 7. M 0(R) = RP # . .#.RP
2.3. Another way of looking_atnthe moduli space comes from observing the inclu-
sion S3 PGl2(R). Since M 0(R) is defined as n distinct points on RP1 quotiented
by PGl2(R), one can_fix three of these points to be 0; 1; and 1. From this pers*
*pec-
tive we_see that M30(R) is a point. When n = 4, the cross-ratio is a homeomorph*
*ism
from M 40(R) to_RP1,_thenresult of identifying three of the four points with 0;*
* 1; and
1. In general, M 0(R) becomes a manifold blown up from an n - 3 dimensional
torus, coming from the n - 3 -fold products of RP1. Therefore, the moduli space
before compactification can be defined as
((RP1)n - *)=PGl2(R);
where * = {(x1; : :;:xn) 2 (RP1)n | at least 3 points collide}.
Proposition 2.3.1.[18, x5.3] Mn0(R) is compactified by blowing up along *.
Remark.For 0 k n - 5, * coincides exactly with {bk} Bn. Therefore,
places where k + 1 points collide correspond to codim k cells bn-3-ki.
__5
Example 2.3.2.An illustration of M 0(R) from this perspective appears in Fig-
ure 8. From the five marked points on RP1, three are fixed leaving two dimensio*
*ns
TESSELLATIONS OF MODULI SPACES AND THE MOSAIC OPERAD 7
3
__5 z_____"______-22
Figure 8. M 0(R) = T2# RP # . .#.RP
__6
Figure 9. M 0(R)
to vary, say x1 and x2. The set is made up of seven lines {x1; x2 = 0; 1; 1} a*
*nd
{x1 = x2}, giving a space tessellated by six squares and six triangles. Further*
*more,
* becomes the set of three_points {x1 = x2 = 0; 1; 1}; blowing up along these
points yields the space M 50(R) tessellated by twelve pentagons.
__6
Example 2.3.3.In Figure 9, a rough sketch_of M0(R) is shown as the blow-up of a
three torus. The set * associated to M 60(R) has ten lines {xi= xj= 0; 1; 1} and
{x1 = x2 = x3}, and three points {x1 = x2 = x3 = 0; 1; 1}. The lines correspond
to the hexagonal prisms, nine cutting through the faces, and the tenth (hidden)
running through the torus from the bottom left to the top right corner. The thr*
*ee
points correspond to places where four of the prisms intersect.
8 SATYAN L. DEVADOSS
The shaded region has three squares and six pentagons_as its codim one faces.*
* In
fact, all the top dimensional cells that form M 60(R) turn out to have this pro*
*perty;
these cells are the associahedra K5.
3.The Associahedron
3.1. We now turn to defining the mosaic operad and relating its properties with
the cellular structure of the associahedron. Let S1 be the unit circle bounding
D, the conformal disk endowed with the Poincare metric. The geodesics in D
correspond to open diameters of S1 together with open circular arcs orthogonal *
*to
S1. The group of isometries on D is PGl2(R) [15, x4].
Associate S1 with RP1. Between two adjacent points of an element in Confign(R*
*P1),
draw a geodesic in D. Let eG(n; 0) be the space of such geodesics coming from
Confign(RP1). In general, let eG(n; k) be the space eG(n; 0) with k non-interse*
*cting
geodesics between non-adjacent points. Define the space G(n; k) to be eG(n; k)=*
*PGl2(R).
Henceforth, we will visualize an element of G(n; k) as an n-gon with k non-inte*
*rsecting
diagonals (Figure 10). For an arbitrary number of diagonals, we write G(n).
Figure 10
Definition 3.1.1.Given G 2 G(m; l) and Gi2 G(ni; ki) (where 1 i m), there
are composition maps
G a1Ob1G1 a2Ob2 . .a.mObm Gm 7! Gt;
P P
where Gt2 G(-m + ni; m + l + ki). The object Gtis obtained by gluing side
aiof G along side biof Gi. The symmetric group Sn acts on Gn by permuting the
labeling of the sides. These operations define the mosaic operad {G(n; k)}.
Remark.The one dimensional case of the little cubes operad is {I(n)}, the little
intervals operad. An element I(n) is defined to be an ordered collection of n
embeddings of the interval I ,! I, with disjoint interiors. The notion of trees*
* and
bubbles, shown in Figure 1, is encapsulated in this intervals operad. Furthermo*
*re,
after embedding I in R and identifying R [ 1 with RP1, we note that the mosaic
operad {G(n; k)} is a compactification of {I(n)}.
3.2. We now define the associahedron Kn-1 as a concrete geometric object and
present its connections with the mosaic operad.
Definition 3.2.1.Let A be the space of n - 3distinct points {t1; : :;:tn-3} on
the interval [0; 1] such that 0 < t1 < . .<.tn-3 < 1. Identifying R [ 1 with RP1
carries the set {0; t1; : :;:tn-3; 1; 1} of_nnpoints onto RP1 (Figure 11). Ther*
*efore,
there exists a natural inclusion of A in M_0(R).nThe associahedron Kn-1 can be
defined as the closure of the space A in M 0(R).
TESSELLATIONS OF MODULI SPACES AND THE MOSAIC OPERAD 9
Figure 11
Define GL(n; k) to be the space G(n; k) with the geodesic sides labeled with
all permutations of {1; :::; n}. The following is an immediate consequence of t*
*he
discussion in x2.1 and the construction of G(n; k).
Proposition 3.2.2.There exists a bijection between the points of bMn0(R) and the
elements of GL(n; k).
Since the associahedron Kn-1 can be identified with the cell tiling the space
bMn0(R), it can be realized as a purely combinatorial object from its CW-complex
structure. This results in
Proposition 3.2.3.An interior point of Kn-1corresponds to an element of G(n; 0),
and an interior point of a codim k face corresponds to an element of G(n; k).
The relation between the n-gon and Kn-1 is further highlighted by a work of
Lee [12], where he constructs a polytope Qn that is dual to Kn-1, with one vert*
*ex
for each diagonal and one facet for each triangulation of an n-gon. He then pro*
*ves
the symmetry group of Qn to be the dihedral group Dn. Restated, it becomes
Proposition 3.2.4.[12, x5] The symmetry group of Kn-1 is Dn.
Figure 12. Associahedra K2, K3, and K4
10 SATYAN L. DEVADOSS
Remark.Stasheff classically defined Kn-1 as a CW-ball with codim k faces corre-
sponding to using k sets of parentheses meaningfully on n - 1letters.5 Later the
associahedron was given a realization as an n - 3 dimensional convex polytope by
Milnor (unpublished). The two descriptions of the associahedron, using polygons
and parentheses, are compatible: Figure 12 illustrates K2 (point), K3 (line), a*
*nd
K4 (pentagon).
3.3. The polygon relation to the associahedron enables the use of the mosaic
operad structure on Kn-1. The following brings in the operad's role in terms of
decomposition.
Proposition 3.3.1.[17, x2] Each face of Kn-1 is a product of lower dimensional
associahedra.
In general, the codim k - 1 face of the associahedron Km-1 will decompose as
Kn1-1x . .x.Knk-1,! Km-1;
P
where ni= m + 2(k - 1) and ni 3. This parallels the mosaic operad structure
G(n1) O . .O.G(nk) 7! G(m);
where G(ni) 2 G(ni; 0); G(m) 2 G(m; k - 1), and the gluing of sides is arbitrar*
*y.
Therefore, the product in Proposition 3.3.1 is indexed by the internal vertices*
* of
the tree corresponding to the face of the associahedron.
Example 3.3.2.We look at the codim one faces of K5. The three dimensional
K5 corresponds to a 6-gon, which has two distinct ways of adding a diagonal. One
way, in Figure 13a, will allow the 6-gon to decompose into a product of two 4-g*
*ons
(K3's). Since K3 is a line, this codim one face yields a square. The other way,*
* in
Figure 13b, decomposes the 6-gon into a 3-gon (K2) and a 5-gon (K4). Taking the
product of a point and a pentagon results in a pentagon. Note how this descript*
*ion
of K5 matches the shaded region in Figure 9.
Figure 13. Codim one cells of K5
Example 3.3.3.We look at the codim one faces of K6. Similar to the ideas above,
Figure 14 shows the decomposition of the two types of codim one faces of K6, a
pentagonal prism and K5.
___________
5From the definition above, the n - 1letters can be viewed as the points {0; *
*t1; : :;:tn-3; 1}.
TESSELLATIONS OF MODULI SPACES AND THE MOSAIC OPERAD 11
Figure 14. Codim one cells of K6
4.The Tessellation
__n
4.1. We extend the combinatorial structure of the associahedra to M0(R). Propo-
sitions 3.2.2 and 3.2.3 show the correspondence between the associahedra_innbMn*
*0(R)
and GL(n; k). We investigate how these copies of Kn-1 glue to form M 0(R).
Definition 4.1.1.Let G 2 GL(n; k) and d be a diagonal of G. A twist along d,
denoted by rd(G), is the element of GL(n; k) obtained by `breaking' G along d
into two parts, reflecting (or `twisting') one of the pieces, and `gluing' them*
* back
together (Figure 15).
Figure 15. Twist along d
The twisting operation is well-defined since the diagonals of an element in G*
*L(n; k)
do not intersect. Furthermore, it does not matter which piece of the polygon is
twisted since the two results are identified by an action of Dn. It follows imm*
*edi-
ately that rd. rd = e; the identity element.
12 SATYAN L. DEVADOSS
Figure 16
Proposition 4.1.2.Two elements, G1;_G2_2nGL(n; k), representing codim k faces
of associahedra, are identified in M 0(R) if there exist diagonals d1; : :;:dr *
*of G1
such that
(rd1. .r.dr)(G1) = G2:
Proof.As two adjacent points p1 and p2 on RP1 collide, the result is a new bubb*
*le
fused to the old at a point of collision p3, where p1 and p2 are on the new bub*
*ble
(see x2.1). The location of the three points pion the new bubble is irrelevant *
*since
S3 PGl2(R). In the language of polygons, this means rd does not affect the cel*
*l,
where d is the diagonal representing the double point p3. In general, it follow*
*s that
the labels of triangles can be permuted without affecting the cell. Let G be an*
* n-gon
and let d be a diagonal partitioning G into a square and an (n - 2)-gon. Figure*
* 16
shows that since the square decomposes into triangles, the cell corresponding t*
*o G
is invariant under the action of rd. Since any partition of G by a diagonal d c*
*an
be decomposed into triangles, it follows by induction that rd does not_affect t*
*he
cell corresponding to G. |__|
Theorem 4.1.3.There exists a surjection
__n
Kn-1xDn Sn ! M0(R);
which is a bijection_on the interior of the cells. In particular, 1_2(n - 1)! c*
*opies of
Kn-1 tessellate M n0(R).
__n
Proof.Each Kn-1 in M 0(R) is associated to a particular labeling on an n-gon
(see_x3.2).nSince there exists a copy of the dihedral group Dn in PGl2(R), and *
*since
M 0(R) is defined as a quotient by PGl2(R), then two labeled n-gons (correspond*
*ing
to associahedra) are identified by an action of Dn. Therefore, there turn out t*
*o be
1_ __n
2(n - 1)! copies of Kn-1 that make up M 0(R), with n! possible labelings coming
from Sn and 2n identifications coming from Dn. This gives the bijection above
with the interior of the cells; the map is not an injection since the boundarie*
*s_of
the associahedra are glued according to Proposition 4.1.2. |__|
__5
4.2. In Figure 17, a piece of M 0(R) represented by labeled polygons with diag-
onals is shown. Note how two codim one pieces (lines) glue together and four
TESSELLATIONS OF MODULI SPACES AND THE MOSAIC OPERAD 13
__5
Figure 17. A piece of M 0(R)
codim two pieces (points) glue together. Understanding this gluing now becomes a
combinatorial problem related to GL(n; k).
Notation.Let (x; X) be_the_numbernof codim x cells in a CW-complex X. For a
fixed codim y2cell_innM0(R), and for y1 < y2, let n(y1; y2) be the number of co*
*dim
y1 cells in M 0(R) whose boundary contains the codim y2 cell. Note the number
n(y1; y2) is well-defined by Theorem 4.1.3.
Lemma 4.2.1.
(k; Kn-1)= __1__kn+-13k n -k1 + k:
Proof.This is obtained by just counting the number of n-gons with k non-interse*
*cting_
diagonals, done by A. Cayley in 1891 [3]. |__|
Lemma 4.2.2.
n(k - t; k)= 2t kt:
Proof.The boundary components of a cell corresponding to an element in GL(n; k)
are obtained by adding non-intersecting diagonals. To look at the coboundary ce*
*lls,
diagonals need to be removed. For each diagonal removed, two cells result (comi*
*ng
from the twist operation); removing t diagonals gives 2tcells. We then look_at *
*all
possible ways of removing t out of k diagonals. |__|
14 SATYAN L. DEVADOSS
Theorem 4.2.3. (
__n 0 n even
(4.1) O(M 0(R)) = n-3_
(-1) 2 (n - 2)((n - 4)!!)2n odd.
Proof.It is easy to show the following:
__n n __n
(k; M0(R)). (0; k)= (0; M0(R)). (k; Kn-1):
__n
Using Theorem 4.1.3 and Lemmas 4.2.1 and 4.2.2,_wensolve for (k; M0(R)); but
this is simply the number of codim k cells in M 0(R). Therefore,
__n n-3X n-3-k (n - 1)! 1 n - 3 n - 1 + k
O(M 0(R)) = (-1) ______k+1___ :
k=0 2 k + 1 k k
This equation can be reduced to the desired form. |___|
Remark.Prof. F. Hirzebruch has kindly informed us that he has shown,_usingn
techniques of Kontsevich and Manin [11], that the signature of M 0(C) is given *
*by
(4.1). He remarks that the equivalence of this signature with the Euler number
of the space of real points is an elementary consequence of the Atiyah-Singer G-
signature theorem.
__n
4.3. We now introduce a construction of Kapranov to give added clarity to M0(R*
*).
__n
Definition 4.3.1.[9, x4] A double cover of M0(R), denoted by eMn0(R), is obtain*
*ed
by fixing the nthmarked point on RP1 to be 1 and assigning an orientation to th*
*is
point.6
__4
Example 4.3.2.Figure 18 shows the polygon labelings of eM40(R) and M 0(R), be-
ing tiled by six and three copies of K3 respectively._In this figure, the label*
* 4 has
been set to 1. Note that the map eM40(R) ! M40(R) is the antipodal quotient.
__4
Figure 18. Me40(R) ! M0(R)
The double cover can_benconstructed using blow-ups similar to the method de-
scribed in x2.2 for M 0(R). However, instead of blowing up the projective sphere
P(V n), we blow-up S(V n), the sphere in V n. Therefore, except for the anomalo*
*us
___________
6Kapranov uses the notation "Sn-3to represent this double cover.
TESSELLATIONS OF MODULI SPACES AND THE MOSAIC OPERAD 15
case of eM40(R), the double_cover is a non-orientable manifold. Note also that *
*the
covering map eMn0(R) ! M n0(R) is the antipodal quotient, coming from the map
S(V n) ! P(V n). Being a double cover, eMn0(R) will be tiled by (n - 1)! copies*
* of
Kn-1.7 It is natural to ask how these copies of Kn-1 will glue to form eMn0(R).
Definition 4.3.3.A marked twist of an n-gon G along its diagonal d, denoted by
erd(G), is the polygon obtained by breaking G along d into two parts, reflectin*
*g the
piece that does not contain the side labeled 1, and gluing them back together.
The two polygons at the right of Figure_15 turn out to be different elements *
*in
eMn0(R), whereas they are identified in M n0(R) by an action of Dn. The followi*
*ng is
an immediate consequence of the above definitions and Theorem 4.1.3.
Corollary 4.3.4 (of Theorem 4.1.3).There exists a surjection
Kn-1xZnSn ! eMn0(R)
which is a bijection on the interior of the cells.
Remark.The spaces on the left define the classical A1 operad [6, x2.9].
Theorem 4.3.5.There exists the following commutative diagram:
(Kn-1x Sn)=er----! eMn0(R)
?? ?
y ?y
__n
(Kn-1x Sn)=r ----! M 0(R)
where the vertical maps are antipodal identifications and the horizontal maps a*
*re a
quotient by Zn.
Proof.Look at Kn-1x Sn by associating to each Kn-1 a particular labeling of an
n-gon. We obtain (Kn-1xSn)=erby gluing the associahedra along codim one faces
using er(keeping the side labeled 1 fixed). It follows that two associahedra wi*
*ll
never glue if their corresponding n-gons have 1 labeled on different sides of t*
*he
polygon. This partitions Sn into Sn-1. Zn, with each element of Zn corresponding
to 1 labeled on a particular side of the n-gon. Furthermore, Corollary 4.3.4 te*
*lls
us that each set of the (n - 1)! copies of Kn-1 glue to form eMn0(R). Therefore*
*, we
get
(Kn-1x Sn)=er= (Kn-1x Sn-1)=erx Zn = Men0(R) x Zn:
|___|
5. The Blow-Ups
__n
5.1. The only difference between M 0(R) and RPn-3 is the blow-ups, making the
study of their structures crucial. Looking at the arrangement Bn on P(V n), the*
*re
turn out to be n - 1 irreducible {b0} cells in general position. In other words*
*, these
points can be thought of as the vertices of an (n - 3)-simplex with an addition*
*al
point at the center. Between every two b0 points of Bn, there exists a b1 cell,
resulting in n-12such irreducible lines. In general, k irreducible points of B*
*n span
a k - 1 dimensional irreducible cell; restating this, we get
___________
7These copies of Kn-1are in bijection with the vertices of the permutohedron *
*Pn-1[9].
16 SATYAN L. DEVADOSS
Proposition 5.1.1.The number of the codim k irreducible components of Bn
equals
(5.1) n - 1:
k + 1
The construction above shows that around a point b0 P(V n), the structure
of Bn resembles the barycentric subdivision of an n - 3 simplex. We look at some
concrete examples to demonstrate this.
__5
Example 5.1.2.In the case of M 0(R), Figure 7 shows the b0 cells in general
position; there are four blown up points, three belonging to vertices of a 2-si*
*mplex,
and one in the center of this simplex. Between every two of these points, there
exists a b1; notice Figure 7 showing 6 such lines. Since these lines are of cod*
*im one,
they do not need to be blown up.
Looking at the structure of a blown up point b0 in B5, notice that sb0is a
hexagon and pb0is a triangle. It is no coincidence that these correspond exactl*
*y to
eM40(R) and __M40(R) (see Figure 18).
__6
Example 5.1.3.For the three dimensional M 0(R), the b0 cells and the b1 cells
need to be blown up, in that order. Choose a codim 3 cell b0; recall that a nei*
*ghbor-
hood around b0 will resemble the barycentric subdivision of a 3-simplex. Figure*
* 19
shows four tetrahedra, each being made up of six tetrahedra (some shaded), pull*
*ed
apart in space such that when glued together the result will constitute the afo*
*re-
mentioned subdivision. The barycenter is the point b0.
Figure 19. Barycentric subdivision of a 3-simplex
The left-most piece of Figure 20 shows one of the tetrahedra from Figure 19. *
*The
map f1 takes the point b0 to sb0whereas the map f2 takes each b1 to sb1. When
looking down at the resulting `blown up' tetrahedron piece, there are 6 pentago*
*ns
(shaded) with a hexagon hollowed out in the center. Taking sb1to pb1turns these
hexagons into triangles. Putting the four `blown up' tetrahedra pieces togethe*
*r,
the faces of sb0make up a two dimensional sphere tiled by 24 pentagons, with 8
hexagons (with antipodal_maps) cut out. This turns out to be eM50(R); projectif*
*ying
sb0to pb0yields M 50(R) (Figure 21).
TESSELLATIONS OF MODULI SPACES AND THE MOSAIC OPERAD 17
Figure 20
__5
Figure 21. Me50(R) ! M0(R)
__n
This pattern_seems to indicate that for M 0(R), blowing up along the b0 cells
will yield M n-10(R). But what happens in general, when a codim k cell bn-3-k is
blown up? A glimpse of the answer_was seen above with regard to the hexagons
and triangles showing up in M 60(R).
__n
5.2. To better understand M 0(R), we look at the faces of the associahedra that
surround each blown up component of Bn; this is done through the eyes of mosaic*
*s.
From the remark in x2.3 it follows that each blow-up along a codim k cell bn-3-k
corresponds to a place where k +1 marked points on RP1 collide. The discussion *
*in
x2.1 relates this to having a diagonal d partition an n-gon such that k + 1 lab*
*eled
sides {p1; : :;:pk+1} lie on one side and {pk+2; : :;:pn} lie on the other. Usi*
*ng the
mosaic operad structure, d decomposes the n-gon into G1O G2, where G1 2 GL(k+2)
and G2 2 GL(n - k), with the new sides diof Gicoming from d. Note that G1O G2
corresponds to the product of associahedra Kk+1x Kn-k-1.
Fix a particular labeling of G2 using the elements {pk+2; : :;:pn}. There are
(k+1)! different ways in which {p1; : :;:pk+1} can be arranged to label G1. How*
*ever,
since twisting is allowed along d1, we get 1_2(k + 1)! different labelings of G*
*1, each
__k+2
corresponding to a Kk+1. But observe that this is exactly how one gets M 0 (R),
where the associahedra_glue as defined in x4.1. Therefore, for a fixed labeling*
* of
G2, we get M k+20(R) x Kn-k-1; looking at all possible labelings gives
18 SATYAN L. DEVADOSS
__n
Theorem 5.2.1.Blowing up along a codim k cell bn-3-k of M 0(R) results in
__k+2 __n-k
(5.2) M 0 (R) x M0 (R):
Notice how this was obtained for a particular partition of {p1; : :;:pn}. The*
*re-
fore, the number of places that (5.2)occurs becomes
(5.3) n
k + 1
corresponding to partitioning {p1; : :;:pn} into two sets of order k+1 and n - *
*k -.1
However, (5.3)alone does not tell the whole story since it fails to match up wi*
*th
the result of (5.1). This is due to a sense of `duality'_that is present:_The c*
*onstruc-
tion_above is_unable to distinguish between the spaces M k+20(R) x Mn-k0(R) and
M n-k0(R) x Mk+20(R). Since
n = n - 1 + n - 1
k + 1 k + 1 n - k - 1
we see (5.3)actually gives the sum of the number of codim k and codim n - k - 2
components. By fixing one of the n points_tonbe 1, we see through this duality:
Replacing the codim k cell bn-3-k of M 0(R) with sbn-3-know results in
(5.4) Mek+20(R) x __Mn-k0(R):
__n-1
Example 5.2.2.The blown up b0 cells yields eMn-10(R) for sb0and M 0 (R) for
pb0, matching the earlier observations of x5.1. Furthermore, (5.1)shows there to
be n - 1 such structures.
Example 5.2.3.Although blowing up along codim one components_does not affect
the resulting manifold, we observe their presence_in M 50(R). From (5.1),_we ge*
*t six
such bn-4 cells;_these turn out to be eM30(R) x M 40(R) for sbn-4and M 40(R) for
pbn-4. The M 40(R)'s are seen in Figure 7 as the six lines cutting through RP2.
Note that every line is broken into six parts, each part being a K3.
__6
Example 5.2.4.The space M 0(R), illustrated in Figure 9,_moves a dimension
higher.8 There are ten b1 cells, each having eM40(R) x M40(R) for sb1. These ar*
*e the
hexagonal prisms that cut through the three torus as described in Example 2.3.3.
__n-k __n
5.3. The question arises as to why M 0 (R) appears_in M 0(R). The answer lies
in the braid arrangement of hyperplanes. Taking M60(R) as an example, blowing up
along each b0 in B6 uses the following procedure: A small spherical neighborhoo*
*d is
drawn around b0 and the inside of the sphere is removed, resulting in sb0. Obse*
*rve
that the sphere (which we denote as S) is engraved with great arcs coming from *
*B6.
Projectifying, sb0becomes pb0, and S becomes the projective sphere PS. Amazingl*
*y,
the engraved arcs on PS are B5, and PS can be thought of as P(V 5). Furthermore,
blowing up along the cells b1 of B6 corresponds to blowing up along the cells b0
of B5 in PS ! As before, this new etching on PS translates into an even lower
dimensional braid arrangement, B4.
___________
8Although this figure is not constructed from the braid arrangement, it is ho*
*meomorphic to
the structure described by the braid arrangement.
TESSELLATIONS OF MODULI SPACES AND THE MOSAIC OPERAD 19
__n
It is not hard to see how this generalizes in the natural way: For M0(R), the*
* iter-
ated blow-ups along the cells {b0} up to_{bn-5} in turn create_braid arrangemen*
*ts
within braid arrangements. Therefore, M n-k0(R) is seen in M n0(R).
6.The Fundamental Group
6.1. Coming full circle, we look at the connections between the results from t*
*he
little cubes and the mosaic operads.
Definition 6.1.1.[4, x3] Let Jn-1 to be the right-angled Coxeter group of Kn-1.
That is, Jn-1 is generated by reflections {si} which correspond bijectively to *
*the
codim one faces {i} of Kn-1, where sisj= sjsiif and only if i\ j6= ;.
Let G 2 GL(n; 0) be an n-gon having sides with the cyclic labeling {1; 2; : :*
*;:n}.
Define G 2 GL(n; 1) to be the set {gi} of polygons G with one diagonal; Lemma *
*4.2.1
shows the order of G to be 1_2(n2-3n). From x3.2, we have the following biject*
*ions:
(6.1) {si} ! {i} ! {gi}:
For g1; g2 2 G , create a new polygon (with two diagonals) by superimposing t*
*he
images of g1 and g2 on each other such that the labelings match (Figure 22). Two
elements of G commute (as images in Jn-1) if and only if the diagonals of their
superimposed polygons do not intersect. Furthermore, if the two elements do com-
mute, the codim two cell of intersection in Kn-1 corresponds to the superimposed
polygons. This follows immediately from the properties of the associahedron.
Figure 22. Superimpose
__ThenCoxeter group Jn-1 is used to better understand the homotopy groups of
M 0(R). However, Davis, Januszkiewicz, and_Scottnhave shown in the following
theorem that the homotopy properties of M0(R) are completely encapsulated in its
fundamental group.
__n
Theorem 6.1.2.[5, x5.1] M 0(R) is aspherical.
6.2. We now introduce a conjecture of Januszkiewicz,_whichntogether with the
machinery above is used to analyze ss1(M 0(R)).
Conjecture 6.2.1.[7] When n exceeds 3,
__n
(6.2) ss1(M 0(R)) ae Dn n Jn-1 i Sn:
Remark.By Proposition 3.2.4, the dihedral group Dn acts on Kn-1. From (6.1),
it follows that Dn acts on Jn-1, preserving the relations.
20 SATYAN L. DEVADOSS
Figure 23. Pair-of-pants
Consider the subset G* G whose diagonals partition the sides of the n-
gon into a set of order two (say with labels a and b) and of order n - 2. Let
OE : G ! Snbe defined as follows: Elements of G* are mapped to the transpositi*
*on
(a; b); the rest of G is mapped to the identity. To check that this map is wel*
*l-
defined, we must verify that the commutativity property of G (coming from Jn-1)
must hold in OE(G ). Since OE maps G - G* to the identity, we need only restri*
*ct
to elements of G*. As noted above, elements of G* commute only when their
superimposed polygon has non-intersecting diagonals. But this is only possible
when the corresponding partitioned labels {a1; b1} and {a2; b2} are all distinc*
*t. It
follows that the transposition elements (a1; b1) and (a2; b2) commute in Sn. The
bijection of (6.1)extends to define a homomorphism OE : Jn-1 ! Sn.
Conjecture 6.2.2.The map taking Jn-1 to Sn in (6.2)is OE.
Let G be the n-gon with cyclic labeling {1; 2; : :;:n} defined above. As Dn a*
*cts
on G, it permutes the n labels; therefore, there is a natural inclusion i : Dn *
*,! Sn.
For elements d 2 Dn and c 2 Jn-1, let be the map DnnJn-1 i Sn of (6.2)defined
as follows: (d; c) = i(d) . OE(c). This map is surjective since the transposi*
*tions
{(1; 2); : :;:(n - 1; n); (n; 1)} generate Sn. It immediately follows that
ker = { ( OE(c-1); c ) | c 2 OE-1(Dn) } = OE-1(Dn):
__n
Therefore, ss1(M 0(R)) is isomorphic to OE-1(Dn):
6.3. There exists a natural pullback of Sn+1 to Sn by fixing of one the n+1 po*
*ints
to be 1, thereby breaking symmetry. Using (6.2), the pullback of Dn+1n Jn fits
into the following commutative diagram:
"Jn ----! Sn
?? ?
y ?y
Dn+1n Jn ----! Sn+1
TESSELLATIONS OF MODULI SPACES AND THE MOSAIC OPERAD 21
The pair-of-pants product (Figure 23) takes m + 1 and 1 + n marked_points on
RP1 to m + 1 + n marked points. The operad structure on the spaces M n+10(R), i*
*ts
simplest case corresponding to the pair-of-pants product, defines composition m*
*aps
J"mx "Jn! "Jm+n
analogous to the juxtaposition map of braids. We can thus construct a monoidal
category which has finite ordered sets as its objects and the group "Jnas the a*
*u-
tomorphisms of a set of cardinality n, all other morphism sets being empty. Note
the following similarity between the braid group Bn obtained from the little cu*
*bes
operad and the `quasibraids' "Jnobtained from the mosaics operad:
ss1(Cn - ) ae Bn i Sn
____________________
ss1(((RP1)n+1- )=PGl2(R))ae "Jn i Sn
There are deeper analogies between these structures which have yet to be studie*
*d.
References
1.H. R. Brahana, A. M. Coble, Maps of twelve countries with five sides with a *
*group of order
120 containing an Ikosahedral subgroup, Amer. J. Math. 48 (1926) 1-20.
2.J. M. Boardman, R. M. Vogt, Homotopy invariant algebraic structures on topol*
*ogical
spaces, Lecture Notes in Math. 347 (1973).
3.A. Cayley, On the partitions of a polygon, Proc. Lond. Math. Soc. 22 (1890-9*
*1) 237-262.
4.R. Charney, M. Davis, Finite K(ss; 1)'s for Artin Groups, Prospects in Topol*
*ogy (ed. F.
Quinn), Annals of Math. Studies 138 (1995) 110-124.
5.M. Davis, T. Januszkiewicz, R. Scott, Nonpositive curvature of blowups, Ohio*
* State preprint
97-3.
6.E. Getzler, M. M. Kapranov, Cyclic operads and cyclic homology, Geometry, To*
*pology,
and Physics for Raoul Bott, ed. S. T. Yau, International Press (1994).
7.T. Januszkiewicz, Private communication with Jack Morava.
8.M. M. Kapranov, Chow quotients of Grassmannians. I, Adv. in Sov. Math. 16 (1*
*993) 29-110.
9.M. M. Kapranov, The permutoassociahedron, Mac Lane's coherence theorem, and *
*asymptotic
zones for the KZ equation, J. Pure Appl. Alg. 85 (1993) 119-142.
10.S. Keel, Intersection theory of moduli spaces of stable n-pointed curves, Tr*
*ans. Amer. Math.
Soc. 330 (1992) 545-574.
11.M. Kontsevich, Y. Manin, Gromov-Witten classes, quantum cohomology, and enum*
*erative
geometry, Comm. Math. Phys. 164 (1994) 525-562.
12.C. Lee, The associahedron and triangulations of the n-gon, European J. Combi*
*n. 10 (1989)
551-560.
13.J. P. May, The geometry of iterated loop spaces, Lecture Notes in Math. 271 *
*(1972).
14.J. P. May, Definitions: operads, algebras and modules, Contemp. Math. 202 (1*
*997) 1-7.
15.J. G. Ratcliffe, Foundations of hyperbolic manifolds, Grad. Text. in Math. 1*
*49, Springer-
Verlag (1994).
16.J. D. Stasheff (Appendix B coauthored with S. Shnider), From operads to `phy*
*sically' inspired
theories, Contemp. Math. 202 (1997) 53-81.
17.J. D. Stasheff, Homotopy associativity of H-spaces I, Trans. Amer. Math. Soc*
*. 108 (1963)
275-292.
18.M. Yoshida, Hypergeometric functions, my love, Vieweg (1997).
19.G. M. Ziegler, Lectures on polytopes, Grad. Text. in Math. 152, Springer-Ver*
*lag (1995).
Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
E-mail address: devadoss@math.jhu.edu