Middle East Technical University Differential Topology Math 541 Fall 2004

Final

(take-home)

Degrees of maps, transversal intersections and applications

Problem 1. Construct a homotopically non-trivial map f: X → X of degree 0 if X is

(1) a closed oriented surface F_{g} of genus g > 0;

(2) S^{n} × S^{n}.

(3) Explain why such an example does not exist for X = S^{n}.

(4) (advanced) Are there such examples for X = ℂP^{2} ?

Problem 2. Consider a smooth map, f: X → Y_{1} × Y_{2}, f(x) = (f_{1}(x), f_{2}(x)). Let y_{i} ∈ Y_{i} be regular values for f_{i} and M_{i} = f_{i}^{−1}(y_{i}), i = 1, 2. Assuming that dim X = dim(Y_{1} × Y_{2}) prove that deg(f) = M_{1} ∘ M_{2}.

Problem 3. (1) Given a smooth map f: X^{n+m} → S^{n} and a regular value s ∈ S^{n}, prove that M = f^{−1}(s) has a trivial normal bundle.

(2) Prove the converse: if M ⊂ X is a codimension n submanifold having a trivial normal bundle, then there exists a smooth map f: X → S^{n} such that M = f^{−1}(s) for some regular value s.

(3) Prove that for any map f: ℂP^{2} → ℂP^{1} the pull-back f^{−1}(x) of a regular value x ∈ ℂP^{1} is a null-homologous submanifold.

Problem 4. Consider a closed connected surface F ⊂ S^{3} and assume that a vector field V in S^{3} has no singular points and is transverse to F. Show that F is homeomorphic to a torus.

Problem 5. Consider polynomials f: ℂ^{p} → ℂ and g: ℂ^{q} → ℂ having isolated singularities at 0. Define h: ℂ^{p+q} → ℂ as h(x, y) = f(x) + g(y). Prove that h has also an isolated singularity at 0 and the Milnor number is additive, that is μ_{h}(0) = μ_{f}(0) + μ_{g}(0).

Problem 6. Prove that the following maps f have degree 0.

(1) f: S^{3} → S^{1} × S^{2} and more generally f: S^{p+q} → S^{p} × S^{q}, p, q > 0;

(2) f: S^{n} × S^{m} → S^{p} × S^{q}, if p, q, n, m > 0, p + q = n + m, {p, q} ≠ {n, m};

(3) f: X → Y which can be factorized through a sphere, X → S^{n} → Y, of dimension n ≠ dim X = dim Y;

(4) f: X → T^{n}, where X is a simply connected n-manifold and T^{n} is n-torus.

Problem 7. Assume that a map between oriented surfaces, f: F_{1} → F_{2} has nonzero degree. Prove that g(F_{1}) ⩾ g(F_{2}).

Problem 8. Prove that any map f: ℂP^{1} × ℂP^{1} → ℂP^{2} has even degree.

(Problems 6(3–4),7 and 8 require some knowledge of homology)