Geometric-figure topologies {combinatorial topology} {algebraic topology} can be mathematical groups. Face, edge, and vertex numbers can relate.
If surfaces have edges, closed curves can go from one edge to the other {cross cut}.
For closed convex polyhedrons, vertex number v minus edge number e plus face number f equals two: v - e + f = 2 {Euler's law} {Euler law}.
Topologies {network, topology}| can have nodes {vertex, network} connected by lines {edge, network}. Two non-parallel surfaces intersect at an edge. Three non-parallel surfaces intersect at a vertex. Non-zero areas intersect at edges {path arc} {arc, boundary} {boundary, arc} and vertices and are finite in number.
connected
Networks can have one or more vertices that have only one edge. Networks {connected network} can have at least two path arcs at all vertexes.
map
Networks {map, network} can have path arcs that bound countries or regions.
path
Traveling along edges leads to vertex sequences {path}. Traveling around whole network can involve returning to vertex or never returning.
random
Keeping node number constant, nodes can have random numbers of random links to other nodes {random network} {exponential network}. Nodes have same average link number. Probability that node connects to N other nodes decreases exponentially with N. Variance can be small {Poisson distribution, network}.
hubs
Not keeping node number constant, random networks {scale-free network} can have nodes {hub} with more links than others. Probability that node connects to N other nodes is approximately proportional to (1/N)^2. Early nodes tend to have more links. Several nodes are critical. Affecting single and many random hubs is unlikely to break network. Internet, social networks, alliances, and cell biochemical reactions can be scale-free networks.
Geometric-figure or space topologies {point set topology} can be points related by distance functions or limits.
Figures {topological graph} can have nodes and connection paths between nodes. Nodes with even number of paths are even. Nodes with odd number of paths are odd.
Two-dimensional polygons can change into triangles {triangulation, polygon}.
Finite linear simplex combinations {complex, topology} can meet or not meet in common faces. Complexes have values {characteristic, complex} found by formulas {Euler-Poincaré formula}. Complexes {chain, simplex} {simplex chain} can be linear oriented-simplex combinations. Boundary of chain boundary equals zero.
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.
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0225