. 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 ...
Topology Ph.D. Qualifying Exam Gerard Thompson Mao-Pei Tsui April 14, 2007
... denotes the constant path based at p and the “ · ” denotes the product of (compatible) paths. (ii) Show that X is simply connected if and only if any two paths in X with the same initial and terminal points are path homotopic. 3. (i) The polygonal symbol of a certain surface without boundary is xyzy ...
... denotes the constant path based at p and the “ · ” denotes the product of (compatible) paths. (ii) Show that X is simply connected if and only if any two paths in X with the same initial and terminal points are path homotopic. 3. (i) The polygonal symbol of a certain surface without boundary is xyzy ...
WORKSHEET ON EULER CHARACTERISTIC FOR SURFACES
... In topology all polyhedra are just spheres, and we can think of edges and vertices as a graph on the sphere. Hope: maybe the above number does not depend on the graph you put on a sphere! Problem 2 Show that the above hope is too much to ask for. Find (simple) graphs on the sphere for which the abov ...
... In topology all polyhedra are just spheres, and we can think of edges and vertices as a graph on the sphere. Hope: maybe the above number does not depend on the graph you put on a sphere! Problem 2 Show that the above hope is too much to ask for. Find (simple) graphs on the sphere for which the abov ...
THE GEOMETRY OF SURFACES AND 3
... surface, not on how the surface is embedded in R3 ! If you bend a surface without stretching it, the geodesics don’t change! For example, a circular cylinder can be squashed into a cylinder over an ellipse, or a cylinder with a square cross section, and when you do that, the geodesics bend along wit ...
... surface, not on how the surface is embedded in R3 ! If you bend a surface without stretching it, the geodesics don’t change! For example, a circular cylinder can be squashed into a cylinder over an ellipse, or a cylinder with a square cross section, and when you do that, the geodesics bend along wit ...
First Page
... denotes the ratio of the surface energies of the grain boundary and of the exterior surface. We show numerically that for ≈ 1.81 < m ≤ 2, the corresponding solutions to the original geometric problem are not singlevalued as functions of x, where x varies along the unperturbed exterior surface of the ...
... denotes the ratio of the surface energies of the grain boundary and of the exterior surface. We show numerically that for ≈ 1.81 < m ≤ 2, the corresponding solutions to the original geometric problem are not singlevalued as functions of x, where x varies along the unperturbed exterior surface of the ...
Pizzas, Bagels, Pretzels, and Euler`s Magical χ
... What is topology? Given a set X , a topology on X is a collection T of subsets of X, satisfying the following axioms: 1. The empty set and X are in T. 2. T is closed under arbitrary union. 3. T is closed under finite intersection. ...
... What is topology? Given a set X , a topology on X is a collection T of subsets of X, satisfying the following axioms: 1. The empty set and X are in T. 2. T is closed under arbitrary union. 3. T is closed under finite intersection. ...
1.5 Introduction to Surface Electronics
... These terms are not specific to surfaces: they are also used for atoms and molecules generally, as the energy level that a) the next electron goes into, and b) the last electron comes from. c) Surface States and related ideas A surface state is a state localized at the surface, which decays exponent ...
... These terms are not specific to surfaces: they are also used for atoms and molecules generally, as the energy level that a) the next electron goes into, and b) the last electron comes from. c) Surface States and related ideas A surface state is a state localized at the surface, which decays exponent ...
