Mixed cobinary trees

We develop basic cluster theory from an elementary point of view using a variation of binary trees which we call mixed cobinary trees. We show that the number of isomorphism classes of such trees is given by the Catalan number Cn where n is the number of internal nodes. We also consider the corresponding quiver Q_{\epsilon} of type An-1. As a special case of more general known results about the relation between c-vectors, representations of quivers and their semi-invariants, we explain the bijection between mixed cobinary trees and the vertices of the generalized associahedron corresponding to the quiver Q_{\epsilon}.Comment: 17 pages, 8 figures, v2: definition of cluster is fixed, v3: references added, Corollary 1 and referenced to it are reworde

Tangential Structures on Toric Manifolds, and Connected Sums of Polytopes

We extend work of Davis and Januszkiewicz by considering {\it omnioriented} toric manifolds, whose canonical codimension-2 submanifolds are independently oriented. We show that each omniorientation induces a canonical stably complex structure, which is respected by the torus action and so defines an element of an equivariant cobordism ring. As an application, we compute the complex bordism groups and cobordism ring of an arbitrary omnioriented toric manifold. We consider a family of examples $B_{i,j}$, which are toric manifolds over products of simplices, and verify that their natural stably complex structure is induced by an omniorientation. Studying connected sums of products of the $B_{i,j}$ allows us to deduce that every complex cobordism class of dimension >2 contains a toric manifold, necessarily connected, and so provides a positive answer to the toric analogue of Hirzebruch's famous question for algebraic varieties. In previous work, we dealt only with disjoint unions, and ignored the relationship between the stably complex structure and the action of the torus. In passing, we introduce a notion of connected sum # for simple $n$-dimensional polytopes; when $P^n$ is a product of simplices, we describe P^n# Q^n by applying an appropriate sequence of {\it pruning operators}, or hyperplane cuts, to $Q^n$.Comment: 22 pages, LaTeX2e, to appear in Internat. Math. Research Notices (2001

Relations in Grassmann Algebra Corresponding to Three- and Four-Dimensional Pachner Moves

New algebraic relations are presented, involving anticommuting Grassmann variables and Berezin integral, and corresponding naturally to Pachner moves in three and four dimensions. These relations have been found experimentally - using symbolic computer calculations; their essential new feature is that, although they can be treated as deformations of relations corresponding to torsions of acyclic complexes, they can no longer be explained in such terms. In the simpler case of three dimensions, we define an invariant, based on our relations, of a piecewise-linear manifold with triangulated boundary, and present example calculations confirming its nontriviality

On the black-box complexity of Sperner's Lemma

We present several results on the complexity of various forms of Sperner's Lemma in the black-box model of computing. We give a deterministic algorithm for Sperner problems over pseudo-manifolds of arbitrary dimension. The query complexity of our algorithm is linear in the separation number of the skeleton graph of the manifold and the size of its boundary. As a corollary we get an $O(\sqrt{n})$ deterministic query algorithm for the black-box version of the problem {\bf 2D-SPERNER}, a well studied member of Papadimitriou's complexity class PPAD. This upper bound matches the $\Omega(\sqrt{n})$ deterministic lower bound of Crescenzi and Silvestri. The tightness of this bound was not known before. In another result we prove for the same problem an $\Omega(\sqrt[4]{n})$ lower bound for its probabilistic, and an $\Omega(\sqrt[8]{n})$ lower bound for its quantum query complexity, showing that all these measures are polynomially related.Comment: 16 pages with 1 figur