Semidirect Products
... For example, if G is a group of order pa q b then the p-Sylow subgroup P and q-Sylow subgroup Q are complementary. The complement of H is not unique, even if both H, K are normal. [E.g., take G = R2 as an additive group and let H, K be any two distinct lines through the origin.] Proposition 10.2. If ...
... For example, if G is a group of order pa q b then the p-Sylow subgroup P and q-Sylow subgroup Q are complementary. The complement of H is not unique, even if both H, K are normal. [E.g., take G = R2 as an additive group and let H, K be any two distinct lines through the origin.] Proposition 10.2. If ...
Topology of Open Surfaces around a landmark result of C. P.
... the n-dimensional sphere Sn . If P denotes the point at infinity we know that it has a fundamental system of neighborhoods {Uj } with each Uj homeomorphic to Rn . For n = 1, Uj \ {P } is disconnected. So, R is not connected at infinity. For n ≥ 2, Uj \ {P } is connected. Since each Uj \ {P } is of t ...
... the n-dimensional sphere Sn . If P denotes the point at infinity we know that it has a fundamental system of neighborhoods {Uj } with each Uj homeomorphic to Rn . For n = 1, Uj \ {P } is disconnected. So, R is not connected at infinity. For n ≥ 2, Uj \ {P } is connected. Since each Uj \ {P } is of t ...
5.1 The Lie algebra of a Lie group Recall that a Lie group is a group
... If a group G acts on a manifold M , one can define a homomorphism of L(G) to the algebra of vectorfields X(M ). We assume without loss of generality that the action is from the left. Let v ∈ L(G) be the generator of a one-parameter subgroup c(t). For each x ∈ M we define Kv (x) ∈ Tx M to be the vect ...
... If a group G acts on a manifold M , one can define a homomorphism of L(G) to the algebra of vectorfields X(M ). We assume without loss of generality that the action is from the left. Let v ∈ L(G) be the generator of a one-parameter subgroup c(t). For each x ∈ M we define Kv (x) ∈ Tx M to be the vect ...
Covering space
In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below. In this case, C is called a covering space and X the base space of the covering projection. The definition implies that every covering map is a local homeomorphism.Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. In Riemannian geometry for example, ramification is a generalization of the notion of covering maps. Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the fundamental group. An important application comes from the result that, if X is a ""sufficiently good"" topological space, there is a bijection between the collection of all isomorphism classes of connected coverings of X and the conjugacy classes of subgroups of the fundamental group of X.