1,191 research outputs found
Dimension on Discrete Spaces
In this paper we develop some combinatorial models for continuous spaces. In
this spirit we study the approximations of continuous spaces by graphs,
molecular spaces and coordinate matrices. We define the dimension on a discrete
space by means of axioms, and the axioms are based on an obvious geometrical
background. This work presents some discrete models of n-dimensional Euclidean
spaces, n-dimensional spheres, a torus and a projective plane. It explains how
to construct new discrete spaces and describes in this connection several
three-dimensional closed surfaces with some topological singularities
It also analyzes the topology of (3+1)-spacetime. We are also discussing the
question by R. Sorkin [19] about how to derive the system of simplicial
complexes from a system of open covering of a topological space S.Comment: 16 pages, 8 figures, Latex. Figures are not included, available from
the author upon request. Preprint SU-GP-93/1-1. To appear in "International
Journal of Theoretical Physics
On string topology of classifying spaces
Let G be a compact Lie group. By work of Chataur and Menichi, the homology of
the space of free loops in the classifying space of G is known to be the value
on the circle in a homological conformal field theory. This means in particular
that it admits operations parameterized by homology classes of classifying
spaces of diffeomorphism groups of surfaces. Here we present a radical
extension of this result, giving a new construction in which diffeomorphisms
are replaced with homotopy equivalences, and surfaces with boundary are
replaced with arbitrary spaces homotopy equivalent to finite graphs. The result
is a novel kind of field theory which is related to both the diffeomorphism
groups of surfaces and the automorphism groups of free groups with boundaries.
Our work shows that the algebraic structures in string topology of classifying
spaces can be brought into line with, and in fact far exceed, those available
in string topology of manifolds. For simplicity, we restrict to the
characteristic 2 case. The generalization to arbitrary characteristic will be
addressed in a subsequent paper.Comment: 93 pages; v4: minor changes; to appear in Advances in Mathematic
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