A Space of Cyclohedra

The real points of the Deligne-Knudsen-Mumford moduli space M0 of marked points on the sphere have a natural tiling by associahedra. We extend this idea to construct an aspherical space tiled by cyclohedra. We explore the structure of this space, coming from blow-ups of hyperplane arrangements, as well as discuss possibilities of its role in knot theory and mathematical physics. To the memory of Rodica Simion. 1. Associahedra and Cyclohedra 1.1. We present an overview of two important polytopes. Definition 1.1.1. Let A(n) be the poset of all dissections of a convex (n + 1)-gon using non-intersecting diagonals, ordered such that a ≺ a′ if a is obtained from a′ by adding new diagonals. The associahedron Kn is a convex polytope of dim n − 2 whose face poset is isomorphic to A(n). Figure 1. Associahedron K4 using brackets and polygons. The construction of the polytope Kn is given by Lee [14] and Haiman (unpublished). Stasheff originally defined the associahedron for use in homotopy theory in connection with associativity properties of H-spaces [18, §2]. There is a well-known bijection between dissections of a convex polygon and partial bracketings of letters. Figure 1 shows an example of this relationship. 1991 Mathematics Subject Classification. Primary 14P25, Secondary 05B45, 52B11. 1Mention of diagonals will henceforth mean non-intersecting ones. 1 2 SATYAN L. DEVADOSS 1.2. The cyclohedron Wn originally manifested itself in the work of Bott and Taubes and later given its name by Stasheff. A construction of the polytope Wn is given by Markl [15, §1] based on the following. Definition 1.2.1. Let B(n) be the poset of all partial bracketings of n letters arranged on a circle, ordered such that b ≺ b′ if b is obtained from b′ by adding new pairs of brackets. The cyclohedron Wn is a convex polytope of dim n − 1 whose face poset is isomorphic to B(n). It was the clever idea of Simion to come up with an alternate poset isomorphic to B(n); in fact, she provides a construction of Wn using this poset [17, §2]. It is formulated in terms of centrally symmetric 2n-gons, where a (non-intersecting) diagonal on such a polygon will either mean a pair of centrally symmetric diagonals or a diameter of the polygon. Proposition 1.2.2. The poset of non-intersecting diagonals on a centrally symmetric 2ngon, ordered such that b ≺ b′ if b is obtained from b′ by adding (non-intersecting) diagonals, is isomorphic to B(n). Figure 2 shows an example of the two descriptions of W3. One difference between Kn and Wn is that for the codim one faces of the associahedron, we do not place brackets around all n variables. In contrast, the cyclohedron allows this since one can distinguish the cyclic manner in which the n variables are combined. Figure 2. Cyclohedron W3 using brackets and polygons. 1.3. It is worthwhile to explore some properties of Wn and its relationship with associahedra. It was noted by Stasheff that the boundary faces of Kn can be identified with products of lower dimensional associahedra [18, §2]. He makes a similar observation for Wn. Proposition 1.3.1. [19, §4] Faces of the cyclohedron are products of lower dimensional cyclohedra and associahedra. In particular, all codim k faces of Wn have the form Wn0 ×Kn1 × · · · ×Knk , over varying values of 1 ≤ n0 < n and 1 < ni ≤ n such that ∑ ni = n+ k. A SPACE OF CYCLOHEDRA 3 Example 1.3.2. Since W5 is a four-dimensional polytope, we look at its possible codim one faces. They are given by the different ways of placing a diagonal in a 10-gon. Figure 3 illustrates the four possible types: On the far left is the product W4×K2; since K2 is simply a point, the result is W4 itself. The middle figures show W3 ×K3 and W2 ×K4, where on one hand the line segment is K3 and on the other W2. The last possible type is W1 ×K5, which is simply K5. Figure 3. Codim one faces of W5. Remark. A few observations follow: 1. For the associahedron, it is well known that the inclusions of lower dimensional faces form the structure maps of an operad [18]. Stasheff and Markl show the inclusion maps above giving the cyclohedron a right module operad structure [15, §2]. 2. The inclusionKn ↪→ Wn coming from the proposition above shows the associahedron as a face of the cyclohedron. A. Tonks has constructed an explicit map Wn → Kn+1 between polytopes of the same dimension [22], in the spirit of his map from the permutohedron Pn to Kn+1 [21]. 3. It is also noteworthy to point out a similar insight discovered by A. Ulyanov [23]. He shows n copies of Kn+1 gluing together to form Wn. For example, the line segment W2 is made up of two K3 line segments with a pair of end points identified. Figure 4 shows the example of W3 constructed using three copies of K4, and similarly, W4 from four copies of K5. Note that this gluing involves some non-trivial smoothing of adjacent faces after the gluing. 2. Coxeter Groups and Blow-Ups 2.1. The relationship between Coxeter groups and the polytopes Kn and Wn is introduced here. We begin by looking at certain hyperplane arrangements and refer the reader to K.S. Brown [3] for any underlying terminology. 4 SATYAN L. DEVADOSS