|·||Identify the "orientation" double cover for a general non-orientable surface.|
||Suppose p:E → M is a covering space, with E and M connected n-manifolds. 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 CP2n → CP2n preserves orientation.|
|·||Distinguish CP2 and S2∨ S4.|
|·||Distinguish CP3 and S2x S4.|
||Let X be a compact, connected, orientable n-manifold, and f:X → X a continuous map.
If f*:Hn(X;Z) → Hn(X;Z) is an isomorphism, show that f*:Hq(X;Z) → Hq(X;Z) and f*:Hq(X;Z) → Hq(X;Z) are isomorphisms for all q.
|·||Let A be a tautly imbedded subspace of R3. If the integral homology of A, H*(A;Z), is finitely generated, show that H*(A;Z) and H*(A;Z) are torsion-free.|
||Let M be a connected, closed (n-1)-dimensional submanifold of the n-sphere Sn. By Alexander duality, M is orientable, and Sn \ M has two (open) components,|
with closures A and B such that A U B = Sn. 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 CP2 # CP2 and S2 x S2 by showing that the two spaces have non-isomorphic cohomology rings.|
|·||Problems #1, 12, 14, 17, Hatcher page 358.|
Dan Cohen Fall 2009