The purpose of this paper is to illustrate the relationship between the topological property of the Euler characteristic and a combinatorial object, the Möbius function, in the context of finite T0-spaces. To do this I first explain the fundamental connection between such spaces and finite partially ordered sets by proving some facts fundamental to the study of finite spaces. Then I define the ...

1. Statement of results. This note announces the solution of the following problem, a more precise version of which appears in §2: (1) Determine the linear combinations of rational Chern numbers (of almost-complex manifolds) that are invariants of oriented homotopy type (of almost-complex manifolds). Milnor, who posed the problem, conjectured that every such homotopy invariant could be expresse...

Theorem 1 of Euler’s paper of 1737 “Variae Observationes Circa Series Infinitas,” states the astonishing result that the series of all unit fractions whose denominators are perfect powers of integers minus unity has sum one. Euler attributes the Theorem to Goldbach. The proof is one of those examples of misuse of divergent series to obtain correct results so frequent during the seventeenth and ...

One of the traditional applications of Euler diagrams is as a representation or counterpart of the usual set-theoretical models of given sentences. However, Euler diagrams have recently been investigated as the counterparts of logical formulas, which constitute formal proofs. Euler diagrams are rigorously defined as syntactic objects, and their inference systems, which are equivalent to some sy...

We study localization of gravity in flat space in superstring theory. We find that an induced Einstein-Hilbert term can be generated only in four dimensions, when the bulk is a non-compact Calabi-Yau threefold with non-vanishing Euler number. The origin of this term is traced to R couplings in ten dimensions. Moreover, its size can be made much larger than the ten-dimensional gravitational Plan...

Let M be the interior of a compact 3–manifold with boundary, and let T be an ideal triangulation of M. This paper describes necessary and sufficient conditions for the existence of angle structures, semi–angle structures and generalised angle structures on (M ;T ) respectively in terms of a generalised Euler characteristic function on the solution space of the normal surface theory of (M ; T )....

The paper investigates vanishing conditions on the first cohomology module of a normalized rank 2 vector bundle F on P which force F to split. The present vanishing conditions improve other conditions known in the literature and are obtained with simple computations on the Euler characteristic function, avoiding the speciality lemma and other heavy tools.

One of the simplest and, at the same time, most important invariants of a topological space is the Euler characteristic. A generalization of the notion of the Euler characteristic to the equivariant setting, that is, to spaces with an action of a group (say, finite) is far from unique. An equivariant analogue of the Euler characteristic can be defined as an element of the ring of representation...

We develop the homotopy theory of Euler characteristic (magnitude) of a category enriched in a monoidal model category. If a monoidal model category V is equipped with an Euler characteristic that is compatible with weak equivalences and fibrations in V, then our Euler characteristic of V-enriched categories is also compatible with weak equivalences and fibrations in the canonical model structu...