Complete three of the following five problems. In the next... assumed to be a topological space. All “maps” given in...
... assumed to be a topological space. All “maps” given in both sections are assumed to be continuous although in a particular problem you may need to establish continuity of a particular map. 1. A subset A ⊂ X is said to be locally closed if for any x ∈ X there is a neighborhood U of x such that A ∩ U ...
... assumed to be a topological space. All “maps” given in both sections are assumed to be continuous although in a particular problem you may need to establish continuity of a particular map. 1. A subset A ⊂ X is said to be locally closed if for any x ∈ X there is a neighborhood U of x such that A ∩ U ...
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... the target or range map τ : G → G : a 7→ aa−1 . The image of these maps is called the unit space and denoted G0 . If the unit space is a singleton than we regain the notion of a group. We also define Ga := σ −1 ({a}), Gb := τ −1 ({b}) and Gba := Ga ∩ Gb . It is not hard to see that Gaa is a group, w ...
... the target or range map τ : G → G : a 7→ aa−1 . The image of these maps is called the unit space and denoted G0 . If the unit space is a singleton than we regain the notion of a group. We also define Ga := σ −1 ({a}), Gb := τ −1 ({b}) and Gba := Ga ∩ Gb . It is not hard to see that Gaa is a group, w ...
Solution 3 - D-MATH
... Solution T 2 #T 2 is the surface of genus 2, Σ2 . (b) If n ≥ 3, and X, Y are connected, determine π1 (X#Y, x0 ) knowing π1 (X, x) and π1 (Y, y). Solution It is an easy application of Van Kampen Theorem that if x0 belongs to the boundary sphere than π1 (X#Y, x0 ) = π1 (X, x0 )∗π1 (Y, x0 ). Indeed we ...
... Solution T 2 #T 2 is the surface of genus 2, Σ2 . (b) If n ≥ 3, and X, Y are connected, determine π1 (X#Y, x0 ) knowing π1 (X, x) and π1 (Y, y). Solution It is an easy application of Van Kampen Theorem that if x0 belongs to the boundary sphere than π1 (X#Y, x0 ) = π1 (X, x0 )∗π1 (Y, x0 ). Indeed we ...
Problems for the exam
... 8. Is it possible to realize CP2 as a finite CW-complex with an even number of cells in every dimension? 9. Viewing S 1 ⊂ C2 as the unit complex numbers, define a continuous map φ : S1 × S1 → S1 × S1 by φ(ξ1 , ξ2 ) = (ξ1 , ξ1 ξ2 ). Is φ homotopic to the identity map? 10. Let ι : S 1 ,→ S 2 be the eq ...
... 8. Is it possible to realize CP2 as a finite CW-complex with an even number of cells in every dimension? 9. Viewing S 1 ⊂ C2 as the unit complex numbers, define a continuous map φ : S1 × S1 → S1 × S1 by φ(ξ1 , ξ2 ) = (ξ1 , ξ1 ξ2 ). Is φ homotopic to the identity map? 10. Let ι : S 1 ,→ S 2 be the eq ...
Topology III Exercise set 6 1. Show that the following are equivalent
... Topology III Exercise set 6 ...
... Topology III Exercise set 6 ...
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.