Basic Exam: Topology - Department of Mathematics and Statistics
... Passing standard: For Master’s level, 60% with two questions essentially complete. For Ph.D. level, 75% with three questions essentially complete. ...
... Passing standard: For Master’s level, 60% with two questions essentially complete. For Ph.D. level, 75% with three questions essentially complete. ...
The final exam
... (Write the answer as a free product of two well-known groups.) (f) The wireframe ...
... (Write the answer as a free product of two well-known groups.) (f) The wireframe ...
Topology Ph.D. Qualifying Exam Gerard Thompson Mao-Pei Tsui April 2009
... (ii) Let X be a topological space and let x0 ∈ X. Define the product of homotopy classes of loops [α] x0 based at x0 and verify in detail that this product is associative. 2. Give the definitions of deformation retract and strong deformation retract for topological spaces. Compute the fundamental gr ...
... (ii) Let X be a topological space and let x0 ∈ X. Define the product of homotopy classes of loops [α] x0 based at x0 and verify in detail that this product is associative. 2. Give the definitions of deformation retract and strong deformation retract for topological spaces. Compute the fundamental gr ...
ALGEBRAIC TOPOLOGY Contents 1. Preliminaries 1 2. The
... elements s1 s2 . . . sn , where si is an element of either G or H. Such a word can be reduced only by removing an instance of the identity element (in either G or H), or by replacing a consecutive pair of elements in the same group by their product in that group. Theorem 3.3. Let U1 , U2 be open sub ...
... elements s1 s2 . . . sn , where si is an element of either G or H. Such a word can be reduced only by removing an instance of the identity element (in either G or H), or by replacing a consecutive pair of elements in the same group by their product in that group. Theorem 3.3. Let U1 , U2 be open sub ...
Geometry and Topology, Lecture 4 The fundamental group and
... A path in a topological space X is a continuous map α : I = [0, 1] → X . Starts at α(0) ∈ X and ends at α(1) ∈ X . Proposition The relation on X defined by x0 ∼ x1 if there exists a path α : I → X with α(0) = x0 , α(1) = x1 is an equivalence relation. Proof (i) Every point x ∈ X is related to itself ...
... A path in a topological space X is a continuous map α : I = [0, 1] → X . Starts at α(0) ∈ X and ends at α(1) ∈ X . Proposition The relation on X defined by x0 ∼ x1 if there exists a path α : I → X with α(0) = x0 , α(1) = x1 is an equivalence relation. Proof (i) Every point x ∈ X is related to itself ...
Topology/Geometry Jan 2014
... 1. Consider the topological spaces Q1 = S 1 × B 2 ⊆ R4 and Q2 = B 2 × S 1 ⊆ R4 , where B 2 is the unit disc in R2 and S 1 is its boundary, the unit circle. Endow Qj with the topology induced from the standard topology on R4 , j = 1, 2. Note in particular that ∂Qj = S 1 × S 1 , j = 1, 2. Consider the ...
... 1. Consider the topological spaces Q1 = S 1 × B 2 ⊆ R4 and Q2 = B 2 × S 1 ⊆ R4 , where B 2 is the unit disc in R2 and S 1 is its boundary, the unit circle. Endow Qj with the topology induced from the standard topology on R4 , j = 1, 2. Note in particular that ∂Qj = S 1 × S 1 , j = 1, 2. Consider the ...
