The Cyclic Cycle Complex of a Surface

In a recent preprint, Bestvina, Bux, and Margalit construct a cell complex that gives a combinatorial model for the collection of isotopy classes of oriented curve systems within a given homology class in a closed orientable surface [BBM]. A key property of the complex is that it is contractible. Here we will construct a different complex, this one a simplicial complex rather than a cell complex, with similar properties. Here are the definitions. By a cycle in a closed oriented surface S we mean a nonempty collection of finitely many disjoint oriented smooth simple closed curves. A cycle c is reduced if no subcycle of c is the oriented boundary of one of the complementary regions of c in S (using either orientation of the region). In particular, a reduced cycle contains no curves that bound disks in S , and no pairs of circles that are parallel but oppositely oriented. Define the cycle complex C(S) to be the simplicial complex having as its vertices the isotopy classes of reduced cycles in S , where a set of k + 1 distinct vertices spans a k simplex if these vertices are represented by disjoint cycles c0, · · · , ck that cut S into k+1 cobordisms C0, · · · , Ck such that the oriented boundary of Ci is ci+1−ci , subscripts being taken modulo k + 1, where the orientation of Ci is induced from the given orientation of S and −ci denotes ci with the opposite orientation. The cobordisms Ci need not be connected. The faces of a k -simplex are obtained by deleting a cycle and combining the two adjacent cobordisms into a single cobordism. One can think of a k -simplex of C(S) as a “cycle of cycles”. The ordering of the cycles c0, · · · , ck in a k -simplex is determined up to cyclic permutation Cycles that span a simplex represent the same element of H1(S) since they are cobordant. Thus we have a well-defined map π0C(S) → H1(S) . This has image the nonzero elements of H1(S) since on the one hand, every cycle representing a nonzero homology class contains a reduced subcycle representing the same class (subcycles of the type excluded by the definition of reduced can be discarded one by one until a reduced subcycle remains), and on the other hand, it is an elementary fact, left as an exercise, that a cycle that represents zero in H1(S) is not reduced. For a nonzero class x ∈ H1(S) let Cx(S) be the subcomplex of C(S) spanned by