Advances in Algebraic Topology Since 1950
Overview
The last half of the twentieth century saw numerous advances in the field of topology, especially in the study of manifolds, or surfaces, as well as in the theory of knots. These advances have had a significant impact on mathematics as a whole. In addition, the mathematical tools developed in these areas have proven useful in several branches of physics and may help make exploration of the Solar System more efficient, although at the expense of increased travel time.
Background
Topology is a branch of mathematics that resembles geometry in that it is concerned with the properties of shapes and surfaces. However, unlike geometry, topology is interested in examining only those properties that do not change with folding, stretching, and other manipulations that do not penetrate the surface. From a topological standpoint, a sausage, a sphere, and a cube are identical because they are all closed surfaces with no holes or penetrations. Similarly, a coffee cup and a donut are topologically identical because they each have exactly one hole. In both of these examples, it is not difficult to see how, by stretching, folding, and so forth, one shape can be turned into another.
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