Space-time can have transformations. For four-dimensional flat space-time, ten transformations leave proper time (and proper length) between two events (with a trajectory between them) unchanged {isometry}: translation through time dimension, translation through space dimension 1, translation through space dimension 2, translation through space dimension 3, fixed-angle rotation around space dimension 1, fixed-angle rotation around space dimension 2, fixed-angle rotation around space dimension 3, no-rotation velocity change (Lorentz transformation) {boost} along space dimension 1, no-rotation velocity change along space dimension 2, and no-rotation velocity change along space dimension 3. Series of these transformations also leave proper time (and proper length) unchanged. Therefore, this transformation set forms a group {Poincaré group}, which shows rotational symmetries of empty space and special-relativity time coordinates.
The Poincaré group is about Minkowski space-time isometries and is a ten-dimensional noncompact Lie (abelian) group. The Poincaré group defines Minkowski-space-time geometry and is the relativistic-field-theory group. In quantum mechanics, particle mass (four-momentum), spin, parity, and charge are positive-energy unitary irreducible Poincaré-group representations.
In topology, complexes have Poincaré groups {first homotopy group}. Group operation traverses curve and then traverses another curve in same direction. Curves that can deform into each other have one class and are homotopic. Therefore, chain and cycle theory and group theory have equivalences.
For three-dimensional manifolds, two simplex subdivisions {Haupvermutung} are isomorphic. At least one singular point exists on even-dimensional spheres.
Mathematical Sciences>Topology>Polyhedron>Complex
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Date Modified: 2022.0224