Concept
Fundamental group
Related concepts (42)
Knot group
In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. The knot group of a knot K is defined as the fundamental group of the knot complement of K in R3, Other conventions consider knots to be embedded in the 3-sphere, in which case the knot group is the fundamental group of its complement in . Two equivalent knots have isomorphic knot groups, so the knot group is a knot invariant and can be used to distinguish between certain pairs of inequivalent knots.
Monodromy
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of monodromy comes from "running round singly". It is closely associated with covering maps and their degeneration into ramification; the aspect giving rise to monodromy phenomena is that certain functions we may wish to define fail to be single-valued as we "run round" a path encircling a singularity.
Étale morphism
In algebraic geometry, an étale morphism (etal) is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not necessarily local isomorphisms. Despite this, étale maps retain many of the properties of local analytic isomorphisms, and are useful in defining the algebraic fundamental group and the étale topology.
Rose (topology)
In mathematics, a rose (also known as a bouquet of n circles) is a topological space obtained by gluing together a collection of circles along a single point. The circles of the rose are called petals. Roses are important in algebraic topology, where they are closely related to free groups. A rose is a wedge sum of circles. That is, the rose is the quotient space C/S, where C is a disjoint union of circles and S a set consisting of one point from each circle. As a cell complex, a rose has a single vertex, and one edge for each circle.
Hurewicz theorem
In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results of Henri Poincaré. The Hurewicz theorems are a key link between homotopy groups and homology groups. For any path-connected space X and positive integer n there exists a group homomorphism called the Hurewicz homomorphism, from the n-th homotopy group to the n-th homology group (with integer coefficients).
Maximal torus
In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie group G is a compact, connected, abelian Lie subgroup of G (and therefore isomorphic to the standard torus Tn). A maximal torus is one which is maximal among such subgroups. That is, T is a maximal torus if for any torus T′ containing T we have T = T′. Every torus is contained in a maximal torus simply by dimensional considerations.
Algebraic geometry and analytic geometry
In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The deep relation between these subjects has numerous applications in which algebraic techniques are applied to analytic spaces and analytic techniques to algebraic varieties.
Locally simply connected space
In mathematics, a locally simply connected space is a topological space that admits a basis of simply connected sets. Every locally simply connected space is also locally path-connected and locally connected. The circle is an example of a locally simply connected space which is not simply connected. The Hawaiian earring is a space which is neither locally simply connected nor simply connected. The cone on the Hawaiian earring is contractible and therefore simply connected, but still not locally simply connected.
Winding number
In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of turns. For certain open plane curves, the number of turns may be non-integer. The winding number depends on the orientation of the curve, and it is negative if the curve travels around the point clockwise.
Metaplectic group
In mathematics, the metaplectic group Mp2n is a double cover of the symplectic group Sp2n. It can be defined over either real or p-adic numbers. The construction covers more generally the case of an arbitrary local or finite field, and even the ring of adeles. The metaplectic group has a particularly significant infinite-dimensional linear representation, the Weil representation. It was used by André Weil to give a representation-theoretic interpretation of theta functions, and is important in the theory of modular forms of half-integral weight and the theta correspondence.