Math 8301, Manifolds and Topology
Homework 3
Due in-class on Monday, Oct 1
1. Using the classication of closed, connected surfaces according to orientability and Euler characteristic, describe the two
Math 8301, Manifolds and Topology
Homework 11
Due in-class on Wednesday, December 5
1. For a space X , use the Mayer-Vietoris sequence to compute the homology groups of X S 1 .
2. For a space X with s
Math 8301, Manifolds and Topology
Homework 10
Due in-class on Monday, November 26
1. For maps f : A B and g : B C of abelian groups, show that there
is an exact sequence
0 ker(f ) ker(gf ) ker(g ) cok
Math 8301, Manifolds and Topology
Homework 7
Due in-class on Monday, Oct 29
1. (CORRECTED) Let F = x, y be a free group on two generators.
Show explicitly that the subgroup H F generated by the elemen
Math 8301, Manifolds and Topology
Homework 4
Due in-class on Friday, Oct 5
1. Suppose you are given a simplicial complex with set V of vertices and
set F of faces. Let X be the space you get by realiz
Math 8301, Manifolds and Topology
Homework 6
Due in-class on Friday, Oct 19
1. For an integer n and a real number R > 0, nd the eect of the map
w (Rw)n : S 1 C \ 0 on 1 .
2. Show that if a polynomial
Math 8301, Manifolds and Topology
Homework 1
Due in-class on Friday, Sep 14
In the following questions, # denotes the connected sum of surfaces.
1. Show that any open subset of a manifold is a manifol
Math 8301, Manifolds and Topology
Homework 8
Due in-class on Friday, November 9
1. Show that S 2 is isomorphic to the universal covering space of RP2 .
2. Give a description of the universal cover of
Math 8301, Manifolds and Topology
Homework 9
Due in-class on Friday, November 16
Part of the power of algebraic topology is in being able to actually compute
things.
Here is a list of topological spac
Math 8301, Manifolds and Topology
Homework 2
Due in-class on Friday, Sep 21
1. Show graphically that the simplicial complex with 7 vertices, generated
by the triangles below, gives rise to a space hom
Math 8301, Manifolds and Topology
Homework 5
Due in-class on Friday, Oct 12
1. (Moment of honesty) For a space X , wed like to dene a groupoid
1 (X ) as follows. The objects of 1 (X ) are the points o