Let us try to embed a graph 1, not necessarily simple, in a surface so that every odd polygon (the graph of a simple closed path of odd length), regarded as a path in the surface, reverses orientation while every even polygon preserves it. What is the smallest surface in which this is possible? That is, what is the minimum demigenus d(S)=2&/(S) over all embedding surfaces S? We call this kind o...

In this paper we give the first example of a surface bundle over a surface with at least three fiberings. In fact, for each n ≥ 3 we construct 4-manifolds E admitting at least n distinct fiberings pi : E → Σgi as a surface bundle over a surface with base and fiber both closed surfaces of negative Euler characteristic. We give examples of surface bundles admitting multiple fiberings for which th...

1. The principal object of this note is to give an extremely simple axiomatic characterization of the Euler characteristic for finite polyhedra. The idea of the construction used was suggested by the definition of the Grothendieck group of sheaves over a space [l]. Unless otherwise noted, "space" always means "triangulable space with base point," and a "pair" is a "triangulable pair of spaces w...

In this paper we consider the topological side of a problem which is the analogue of Sen’s S-duality testing conjecture for Hitchin’s moduli space M of rank 2 stable Higgs bundles of fixed determinant of odd degree over a Riemann surface Σ. We prove that all intersection numbers in the compactly supported cohomology ofM vanish, i.e. “there are no topological L harmonic forms on M”. This result ...

Walkup’s class K(d) consists of the d-dimensional simplicial complexes all whose vertex links are stacked (d − 1)-spheres. According to a result of Walkup, the face vector of any triangulated 4-manifold X with Euler characteristic χ satisfies f1 ≥ 5f0 − 15 2 χ, with equality only for X ∈ K(4). Kühnel observed that this implies f0(f0− 11) ≥ −15χ, with equality only for 2-neighborly members of K(...

The purpose of this paper is to study curve counting on Calabi-Yau 3-folds via wall-crossing phenomena in the derived category. We will study the generating series of Donaldson-Thomas-type invariants without virtual fundamental cycles, i.e. the Euler characteristics of the relevant moduli spaces. The main result is to show the Euler characteristic version of the Pandharipande-Thomas conjecture ...

We define a new version of the exterior derivative on the basic forms of a Riemannian foliation to obtain a new form of basic cohomology that satisfies Poincaré duality in the transversally orientable case. We use this twisted basic cohomology to show relationships between curvature, tautness, and vanishing of the basic Euler characteristic and basic signature.

The absolute order is a natural partial order on a Coxeter group W . It can be viewed as an analogue of the weak order on W in which the role of the generating set of simple reflections in W is played by the set of all reflections in W . By use of a notion of constructibility for partially ordered sets, it is proved that the absolute order on the symmetric group is homotopy Cohen-Macaulay. This...

Abstract. The purpose of this note is to complete the study, begun in the first author’s PhD thesis, of the topology of the poset of generalized noncrossing partitions associated to real reflection groups. In particular, we calculate the Euler characteristic of this poset with the maximal and minimal elements deleted. As we show, the result on the Euler characteristic extends to generalized non...

is number ostensibly depends upon the chosen cell decomposition, but one of the first stories we learn about homology is that, in fact, it’s an invariant of the space itself ! To remind ourselves why, let’s consider an algebraic Euler characteristic. .. Definition. Suppose E = (E∗, ∂) a bounded complex of finite dimensional rational vector spaces. e Euler characteristic of C is the alternat...