Topology/Geometry Jan 2014
... 3. Put your number, not your name, in the upper right hand corner of each page. If you have not received a number, please choose one (1234 for instance) and notify the graduate secretary as to which number you have chosen. ...
... 3. Put your number, not your name, in the upper right hand corner of each page. If you have not received a number, please choose one (1234 for instance) and notify the graduate secretary as to which number you have chosen. ...
PDF
... Theorem 1. Suppose X is a topological space. If K is a compact subset of X, C is a closed set in X, and C ⊆ K, then C is a compact set in X. The below proof follows e.g. [?]. A proof based on the finite intersection property is given in [?]. Proof. Let I be an indexing set and F = {Vα | α ∈ I} be an ...
... Theorem 1. Suppose X is a topological space. If K is a compact subset of X, C is a closed set in X, and C ⊆ K, then C is a compact set in X. The below proof follows e.g. [?]. A proof based on the finite intersection property is given in [?]. Proof. Let I be an indexing set and F = {Vα | α ∈ I} be an ...
Math 8301, Manifolds and Topology Homework 7
... 2. If F and F 0 are free groups, show that F ∗ F 0 is also free. 3. Suppose f : H → G is a group homomorphism. Show by universal property that the amalgamated product G ∗H {e} is always isomorphic to the quotient G/N , where N is the normal subgroup generated by the image of f . 4. Suppose X is a to ...
... 2. If F and F 0 are free groups, show that F ∗ F 0 is also free. 3. Suppose f : H → G is a group homomorphism. Show by universal property that the amalgamated product G ∗H {e} is always isomorphic to the quotient G/N , where N is the normal subgroup generated by the image of f . 4. Suppose X is a to ...
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.