MATH 7520: Algebraic Topology Homework

## MATH 7520   Algebraic Topology

### Homework Problems

 · 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.  Show that the intersection of A and B is equal to M.  Show that Hq(M) is isomorphic to the direct sum of i*(Hq(A)) and j*(Hq(B)) for 0 < q < n-1.  Show that Hq(A) = Hq(B) = 0 for q greater than or equal to n-1.  If F is a field, show that i*(Hq(A;F)) is isomorphic to Hom(j*(Hn-q-1(B),F) for 0 < q < n-1.  Conclude that RPn does not imbed in Sn+1, CPn does not imbed in S2n+1, and HPn does not imbed in S4n+1 for n at least 2. · 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
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