Math 8301, Manifolds and Topology Homework 8 1. Show that S

... 1. Show that S 2 is isomorphic to the universal covering space of RP2 . 2. Give a description of the universal cover of the space S 2 ∨ S 1 , obtained by gluing together S 2 and S 1 at a single point. 3. Suppose X and Y are path-connected spaces, p : Y → X is a covering map, and y ∈ Y . Let the imag ...

... 1. Show that S 2 is isomorphic to the universal covering space of RP2 . 2. Give a description of the universal cover of the space S 2 ∨ S 1 , obtained by gluing together S 2 and S 1 at a single point. 3. Suppose X and Y are path-connected spaces, p : Y → X is a covering map, and y ∈ Y . Let the imag ...

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... The universal covering space has the following universal property: If π : (X̃, x0 ) → (X, x) is a based universal cover, then for any connected based cover π 0 : (X 0 , x0 ) → (X, x), there is a unique covering map π 00 : (X̃, x0 ) → (X 0 , x0 ) such that π = π 0 ◦ π 00 . Clearly, if a universal cov ...

... The universal covering space has the following universal property: If π : (X̃, x0 ) → (X, x) is a based universal cover, then for any connected based cover π 0 : (X 0 , x0 ) → (X, x), there is a unique covering map π 00 : (X̃, x0 ) → (X 0 , x0 ) such that π = π 0 ◦ π 00 . Clearly, if a universal cov ...

Jerzy DYDAK Covering maps for locally path

... Abstract. We define Peano covering maps and prove basic properties analogous to classical covers. Their domain is always locally pathconnected but the range may be an arbitrary topological space. One of characterizations of Peano covering maps is via the uniqueness of homotopy lifting property for a ...

... Abstract. We define Peano covering maps and prove basic properties analogous to classical covers. Their domain is always locally pathconnected but the range may be an arbitrary topological space. One of characterizations of Peano covering maps is via the uniqueness of homotopy lifting property for a ...

Qualifying Exam in Topology

... 3. (a) Define the two notions: “homotopy between two maps” and “homotopy equivalence between two topological spaces.” (b) Give an example of topological spaces X and Y that have the same homotopy type but are not homeomorphic. (c) Give an example of (path-connected) topological spaces X and Y that h ...

... 3. (a) Define the two notions: “homotopy between two maps” and “homotopy equivalence between two topological spaces.” (b) Give an example of topological spaces X and Y that have the same homotopy type but are not homeomorphic. (c) Give an example of (path-connected) topological spaces X and Y that h ...

June 2010

... 1. Let X be a topological space and let f, g : X → R be continuous functions. (a): Show that the set L = {p ∈ X : f (p) ≤ g(p)} is a closed subset of X. (b): Show that the function h : X → R given by h(p) = min{f (p), g(p)} is continuous. 2. Let X be a Hausdorff space, p ∈ X, and A ⊆ X a compact sub ...

... 1. Let X be a topological space and let f, g : X → R be continuous functions. (a): Show that the set L = {p ∈ X : f (p) ≤ g(p)} is a closed subset of X. (b): Show that the function h : X → R given by h(p) = min{f (p), g(p)} is continuous. 2. Let X be a Hausdorff space, p ∈ X, and A ⊆ X a compact sub ...

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... 3. DISprove the following assertion: Let τ and τ 0 be two topologies on the same space X. Then if (X, τ ) ≈ (X, τ 0 ), then τ = τ 0 . 4. A discrete map is a continuous function d : X → D, where D is a finite set given the discrete topology. Prove that X is connected if and only if every discrete map ...

... 3. DISprove the following assertion: Let τ and τ 0 be two topologies on the same space X. Then if (X, τ ) ≈ (X, τ 0 ), then τ = τ 0 . 4. A discrete map is a continuous function d : X → D, where D is a finite set given the discrete topology. Prove that X is connected if and only if every discrete map ...

QUALIFYING EXAM IN TOPOLOGY WINTER 1996

... b) Is it true that f∗ : H1 (X) → H1 (Y ) is a monomorphism? Give either a proof or a counterexample. 3. Let X denote the set of all real numbers with the finite-complement topology, and define f : E 1 → X by f (x) = x. Show that f is continuous, but not a homeomorphism. 4. Given a continuous map f : ...

... b) Is it true that f∗ : H1 (X) → H1 (Y ) is a monomorphism? Give either a proof or a counterexample. 3. Let X denote the set of all real numbers with the finite-complement topology, and define f : E 1 → X by f (x) = x. Show that f is continuous, but not a homeomorphism. 4. Given a continuous map f : ...

HOMEWORK 7 Problem 1: Let X be an arbitrary nonempty set

... Problem 2: Let U and V be two path-connected open subsets of Rn such that U ∪ V = Rn . Show that U ∩ V is path-connected. Solution: This is a straight-up Mayer-Vietoris question. We have a long exact sequence · · · → H1 (Rn ) → H0 (U ∩ V ) → H0 (U ) ⊕ H0 (V ) → H0 (Rn ) → 0. Now H1 (Rn ) = 0 since R ...

... Problem 2: Let U and V be two path-connected open subsets of Rn such that U ∪ V = Rn . Show that U ∩ V is path-connected. Solution: This is a straight-up Mayer-Vietoris question. We have a long exact sequence · · · → H1 (Rn ) → H0 (U ∩ V ) → H0 (U ) ⊕ H0 (V ) → H0 (Rn ) → 0. Now H1 (Rn ) = 0 since R ...

. TOPOLOGY QUALIFYING EXAMINATION Time: Three hours.

... (a) Compute the fundamental group of the space obtained by identifying the points x and y pictured above on the solid three hole torus. (b) Compute the fundamental group of the space obtained by identifying the points x and y on the surface of the three hole torus. 2. (a) Define the join (or product ...

... (a) Compute the fundamental group of the space obtained by identifying the points x and y pictured above on the solid three hole torus. (b) Compute the fundamental group of the space obtained by identifying the points x and y on the surface of the three hole torus. 2. (a) Define the join (or product ...

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.