Topology Ph.D. Qualifying Exam Gerard Thompson Mao-Pei Tsui April 2009
... 3. (i) The polygonal symbol of a certain surface without boundary is zy−1 xyzx−1 . Identify the surface. What is its Euler characteristic? (ii) Explain how polygons with an even number of sides may be used to classify surfaces without boundary. You do not need to give detailed proofs. 4. It is known ...
... 3. (i) The polygonal symbol of a certain surface without boundary is zy−1 xyzx−1 . Identify the surface. What is its Euler characteristic? (ii) Explain how polygons with an even number of sides may be used to classify surfaces without boundary. You do not need to give detailed proofs. 4. It is known ...
Notes 4.5
... congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent. ...
... congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent. ...
1. Basic Point Set Topology Consider Rn with its usual topology and
... Lin(Rp , Rq ) stand for the set of linear maps from Rp to Rq and observe that it is isomorphic to Rpq with respect to its natural structure as vector space over R . ...
... Lin(Rp , Rq ) stand for the set of linear maps from Rp to Rq and observe that it is isomorphic to Rpq with respect to its natural structure as vector space over R . ...
HOMEWORK ASSIGNMENT #6 SOLUTIONS
... and C are collinear but F is not the midpoint of CK. Since AC ≅ BC and I and J are the midpoints, it follows that IC ≅ JC . F is the foot of C on ED, so by Hypotenuse-Leg, we have that RICF ≅ RJCF . Let X ...
... and C are collinear but F is not the midpoint of CK. Since AC ≅ BC and I and J are the midpoints, it follows that IC ≅ JC . F is the foot of C on ED, so by Hypotenuse-Leg, we have that RICF ≅ RJCF . Let X ...
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. Intuitively, a 3-manifold can be thought of as a possible shape of the universe. Just like a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.