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A generator for the fundamental group is the closed curve obtained by projecting any curve connecting antipodal points in down to . The projective n {\displaystyle n} -space is compact, connected, and has a fundamental group isomorphic to the cyclic group of order 2: its universal covering space is given by the antipody ...
The fundamental objects in the projective plane are points and straight lines, and as in Euclidean geometry, every pair of points determines a unique line passing through both, but unlike in the Euclidean case in projective geometry every pair of lines also determines a unique point at their intersection (in Euclidean geometry, parallel lines ...
Generalizing the statement above, for a family of path connected spaces , the fundamental group () is the free product of the fundamental groups of the . [10] This fact is a special case of the Seifert–van Kampen theorem, which allows to compute, more generally, fundamental groups of spaces that are glued together from other spaces.
A more complicated example is the calculation of the fundamental group of a genus-n orientable surface S, otherwise known as the genus-n surface group. One can construct S using its standard fundamental polygon. For the first open set A, pick a disk within the center of the polygon. Pick B to be the complement in S of the center point of A.
The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial.
The following square is a fundamental polygon of the Klein bottle. The idea is to 'glue' together the corresponding red and blue edges with the arrows matching, as in the diagrams below. Note that this is an "abstract" gluing in the sense that trying to realize this in three dimensions results in a self-intersecting Klein bottle.
NEW YORK (Reuters) -Do Kwon, the South Korean cryptocurrency entrepreneur behind two digital currencies that lost an estimated $40 billion in 2022, pleaded not guilty on Thursday to U.S. criminal ...
The fundamental groupoid Π(X), or Π 1 (X), assigns to each ordered pair of points (p, q) in X the collection of equivalence classes of continuous paths from p to q. More generally, the fundamental groupoid of X on a set S restricts the fundamental groupoid to the points which lie in both X and S.