Middle East Technical University Fall 2002 Math 744 Fundamental Techniques in Dif. Topology

PROBLEMS IN HOMOLOGY THEORY

Problem 1. Construct an example of a map which is not null-homotopic, but induces a trivial map in homology.

Problem 2. Prove that for any closed manifold M of dimension n > 0 there exists i ⩽ n such that π_{i}(M) ≠ 0.

Problem 3. Prove that ℂP^{3} and S^{2} × S^{4} are homotopy non-equivalent.

Problem 4. Prove that the multiplication in the ring H^{∗}(ΣX) is trivial for suspension ΣX over any X.

Problem 5. Prove that a group G cannot act on a contractible cell complex X if (a) X is a finite complex; (b) X is finitely-dimensional (possibly infinite) and G is a finite group.

Problem 6. Prove that [K, S^{n}] = H^{n}(K, ℤ), if dim K = n.

Problem 7. (a) Prove that ℂP^{n} ∖ S^{1} ≅ ℂP^{n−1} ∨ S^{2n−2}. (b) Determine S^{n}/S^{k} and S^{n} ∖ S^{k} up to homotopy equivalence. (c) Prove that X ∧ S^{1} = ΣX. Here X ∧ S^{1} = (X × S^{1})/(X ∨ S^{1}).

Problem 8. Assume that the universal covering of M is ℝ^{n}. Prove that [X, M] = Hom(π_{1}(X) → π_{1}(M)) for any path-connected X.

Problem 9. (a) Determine the ring structure in H^{∗}(ℂP^{2} ∖ A_{3}) and H^{∗}(ℂP^{2}, A_{3}) for a non-singular cubic A_{3} ⊂ ℂP^{2}. (b) Determine the structure of a module over H^{∗}(ℂP^{2}) in these rings. (c) Determine the ring structure in the cohomology of ℝP^{3}/ℝP^{1} and ℂP^{3}/ℂP^{1}.

Problem 10. Describe the transfer in the Gysin sequence for the natural fibrations (1) S^{2n+1} → ℂP^{n}, (2) L(n, 1) → S^{2}, (3) det: U(n) → U(1), (4) ℂP^{3} → ℍP^{1} = S^{4}.

Problem 11. Determine the transfer map f_{!} in homology for the following inclusion maps (a) ℝP^{3} → ℂP^{3}, (b) ℂP^{2} → ℂP^{3}, (c) A_{n} → ℂP^{2} (A_{n} is a non-singular curve of degree n), (d) Q → ℂP^{3} (Q is a non-singular quadric).

Problem 12. What does Lefschetz fixed point theory says about possible number of fixed points of maps ℂP^{2} → ℂP^{2}, S^{2} × S^{4} → S^{2} × S^{4} ?

Problem 13. Consider a map f: T^{2} → T^{2} with f_{∗}: π_{1}(T^{2}) → π_{1}(T^{2}), f_{∗} =

( \begin{matrix}a & b \\ c & d \end{matrix} ). Assuming that f has finitely many fixed points, determine their (alge-

braic) number.

Problem 14. Determine the obstruction for a section in the following bundles:

(a) S^{2n+1} → ℂP^{n}, (b) S^{n} → ℝP^{n}, (c) LF_{g} → F_{g}, where LF_{g} is the space of linear elements on a genus g surface F_{g}.

Problem 15. What can you say about the homotopy classes of maps, [S^{2} × S^{2} → S^{2}], [S^{2} × S^{2} → S^{3}], [ℂP^{2} → S^{2}], [ℂP^{3} → S^{2} × S^{4}], using the obstruction theory ?

Typeset by AMS-TEX