## Algebraic Topology Homework

**Due Monday, April 1**

*Exercises using the Mayer-Vietoris sequence.*

1. A *simple cover* of a topological space is
a collection of open sets which cover, and which have the
property that all intersections of 1,2,3,... sets of the
covering are contractible. Show that if *X* admits
a simple cover by *N* open sets, then the singular
homology of *X* is 0 in dimension *N-1* and higher.

2. In this connection, use the homeomorphism of *S^n* with
the boundary of the *n+1*-simplex to show that *S^n*
admits a simple cover by *n+2* open sets.

3. Use the fact that *S^3* is the union of 2 solid tori
along their boundary *T^2*'s to calculate the homology
of *T^2*.

4. Generalize this argument, using Alexander's Theorem, to
calculate the homology of *S^n*X*S^m*.

5. Greenberg, Problem 17.16 (homology of surface of genus *k*).

6. Greenberg, Problem 17.19 (homology of Klein bottle).