·  Identify the "orientation" double cover for a general nonorientable surface. 
· 
Suppose p:E → M is a covering space, with E and M connected nmanifolds. If M is orientable, show that E is orientable, and that every covering transformation preserves orientation. 
·  Problems #2, 5  9, Hatcher page 257. 
·  Problems #16, 17, 24, 25, Hatcher page 259. 
·  Prove that a homotopy equivalence CP^{2n} → CP^{2n} preserves orientation. 
·  Distinguish CP^{2} and S^{2}∨ S^{4}. 
·  Distinguish CP^{3} and S^{2}x S^{4}. 
· 
Let X be a compact, connected, orientable nmanifold, and f:X → X a continuous map.
If f_{*}:H_{n}(X;Z) → H_{n}(X;Z) is an isomorphism, show that f_{*}:H_{q}(X;Z) → H_{q}(X;Z) and f*:H^{q}(X;Z) → H^{q}(X;Z) are isomorphisms for all q. 
·  Let A be a tautly imbedded subspace of R^{3}. If the integral homology of A, H_{*}(A;Z), is finitely generated, show that H_{*}(A;Z) and H*(A;Z) are torsionfree. 
· 
Let M be a connected, closed (n1)dimensional submanifold of the nsphere S^{n}. By Alexander duality, M is orientable, and S^{n} \ M has two (open) components, with closures A and B such that A U B = S^{n}. Let i:M → A and j:M → B denote the inclusions.

·  Let W be a compact manifold with boundary. Show that the boundary of W is not a retract of W. 
·  Distinguish CP^{2} # CP^{2} and S^{2} x S^{2} by showing that the two spaces have nonisomorphic cohomology rings. 
·  Problems #1, 12, 14, 17, Hatcher page 358. 
Dan Cohen Fall 2009