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 surfaces obtained
by using the following strings to id
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 subspaces A B X , show that there is a short
exact seque
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 ) coker(f ) coker(gf ) coker(g ) 0.
2. If M is the Mbius str
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 elements
zn = (y n xy n ), as n ranges over the integers, is
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 realizing this simplicial
complex. For deniteness, well let V
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 f (z ) with complex coecients has no zeros,
then the in
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 manifold.
2. For each value of t R, decide whether the space
c
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 the space S 2 S 1 , obtained
by gluing together S 2 and
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 spaces, each of which can be triangulated. For
each of thes
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 homeomorphic to the torus.
123
245
127
246
134
257
145
347
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 of X . The
morphisms Hom1 (X ) (x, y ) are homotopy clas