Sorting geometric figures such as circles, squares, and triangles may appear straightforward, but when taken seriously by mathematicians, it becomes the complex field of topology. Topology is the study of various versions of shapes known as manifolds and their classification according to their properties.
A manifold can be a geometry of any dimension, ranging from zero-dimensional points to one-dimensional lines, two-dimensional surfaces such as the surface of a ball, and even mathematically real but difficult-to-imagine higher-dimensional spaces. Because manifolds resemble the spaces in which we exist, such as the Earth or the universe, they are studied.
Flatness is a characteristic shared by all manifolds. If standing on the surface of a manifold, it would appear level in all directions. This flatness does not, however, determine other characteristics such as curvature or the presence of cavities. From a local perspective, the surface of a doughnut and a sphere may both appear level, but they differ globally due to the doughnut's hole. This concludes our discussion of the three fundamental categories of manifolds: topological, smooth, and piecewise linear.
The simplest form of manifolds, topological manifolds exhibit continuous properties. You can trace your finger without raising it across a topological manifold. In contrast, smooth manifolds not only possess flatness and continuity, but also fluidity throughout. On account of their smoothness, calculus can be conducted on these manifolds. Piecewise linear manifolds are intermediate in complexity between topological and smooth manifolds and have polygon-like tiles that permit angles.
Different notions of equivalence are used by mathematicians to determine when two manifolds are equivalent. The homotopy equivalence holds that two manifolds are equivalent if one can be continuously deformed into the shape of the other without tearing. This lenient criterion for equivalence leads to unexpected outcomes in which ostensibly dissimilar shapes are deemed equivalent. A baseball and a solitary point are homotopy equivalent because the ball can be continuously deformed into a point. Due to its opening, a doughnut cannot be deformed into a point.
Comparing topological manifolds with homeomorphisms that preserve a sense of distance between points. Smooth manifolds require diffeomorphisms, a more complex notion of equivalence. Diffeomorphisms maintain both distance and smoothness. These concepts of equivalence are essential to the classification of manifolds.
While progress has been made by mathematicians in classifying manifolds with dimensions other than four, the classification of four-dimensional manifolds remains essentially unknown. The 1981 proof by Michael Freedman of the four-dimensional Poincaré conjecture was a significant result, but it did not resolve the smooth four-dimensional Poincaré conjecture. Any smooth four-dimensional manifold homotopy equivalent to the four-dimensional sphere is also diffeomorphic to the four-dimensional sphere, according to this hypothesis. The inability of mathematicians to determine when a smooth four-dimensional manifold is equivalent to a sphere is a consequence of their inability to solve this problem.
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