Exercises for Math535. 1 . Write down a map of rings that gives the
... 9 . Prove that if H is a subgroup of an alegbraic group G, and H̄ is its Zariski closure, then H̄ is also an algebraic subgroup. 10∗ . Recall that for algebraic groups, if G is connected, then the commutator subgroup [G, G] is a closed algebraic subgroup. For Lie groups over C a similar statement do ...
... 9 . Prove that if H is a subgroup of an alegbraic group G, and H̄ is its Zariski closure, then H̄ is also an algebraic subgroup. 10∗ . Recall that for algebraic groups, if G is connected, then the commutator subgroup [G, G] is a closed algebraic subgroup. For Lie groups over C a similar statement do ...
Math 535 - General Topology Fall 2012 Homework 7 Solutions
... Problem 4. For parts (a) and (b), let X and Y be topological spaces, where Y is Hausdorff. a. Let f, g : X → Y be two continuous maps. Show that the subset E := {x ∈ X | f (x) = g(x)} where the two maps agree is closed in X. Solution. Since Y is Hausdorff, the diagonal ∆Y ⊆ Y × Y is closed. Conside ...
... Problem 4. For parts (a) and (b), let X and Y be topological spaces, where Y is Hausdorff. a. Let f, g : X → Y be two continuous maps. Show that the subset E := {x ∈ X | f (x) = g(x)} where the two maps agree is closed in X. Solution. Since Y is Hausdorff, the diagonal ∆Y ⊆ Y × Y is closed. Conside ...
Algebraic topology exam
... chain complexes with the above maps being f: C D, g : D E. Show that there is a long exact sequence of homology groups that arises from this situation. 2. A) State and prove the Mayer-Vietoris theorem for singular or simplicial (nonreduced) homology. B) Show that if S(X) is the suspension of a t ...
... chain complexes with the above maps being f: C D, g : D E. Show that there is a long exact sequence of homology groups that arises from this situation. 2. A) State and prove the Mayer-Vietoris theorem for singular or simplicial (nonreduced) homology. B) Show that if S(X) is the suspension of a t ...
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.