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^nXS^m.
5. Greenberg, Problem 17.16 (homology of surface of genus k).
6. Greenberg, Problem 17.19 (homology of Klein bottle).