Topology (Classic Version)
Communications in Mathematical Physics. A Lorentz cobordism between two in general nondiffeomorphic 3-manifolds M 0 , M 1 is a pair M , v , where M is a differentiable 4-manifold and v is a differentiable vector field on M , such that 1 the boundary of M is the disjoint union of M 0 and M 1 , 2 v is everywhere nonzero, 3 v is interior normal on M 0 and exterior normal on M 1. We discuss the form that these changes can take, and give two methods for constructing a Lorentz cobordism between two nondiffeomorphic 3-manifolds. We comment on the possible relevance of Lorentz cobordism to the problem of gravitational collapse. Unable to display preview. Download preview PDF.
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Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects. Tearing, however, is not allowed. A circle is topologically equivalent to an ellipse into which it can be deformed by stretching and a sphere is equivalent to an ellipsoid. Similarly, the set of all possible positions of the hour hand of a clock is topologically equivalent to a circle i. There is more to topology, though. Topology began with the study of curves, surfaces, and other objects in the plane and three-space. One of the central ideas in topology is that spatial objects like circles and spheres can be treated as objects in their own right, and knowledge of objects is independent of how they are "represented" or "embedded" in space.
A topological space is a set endowed with a structure, called a topology , which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces , and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension , which allows distinguishing between a line and a surface ; compactness , which allows distinguishing between a line and a circle ; connectedness , which allows distinguishing a circle from two non-intersecting circles. The ideas underlying topology go back to Gottfried Leibniz , who in the 17th century envisioned the geometria situs and analysis situs. The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed.