Differential Topology — Spring 2014

Gerald Hoehn

Problem sheet 4 February 23, 2014

Problem 1: Let M^{n} be an n-dimensional compact manifold, f : M ⟶ R^{n+1} be differentiable, and 0 ∉ f(M). Show that there exists a line though the origin of R^{n+1} which only meets finitely many points of f(M).

Problem 2: Let f : M ⟶ R^{p} be a differentiable map and N ⊂ R^{p} be a differentiable submanifold. Show that for each ϵ > 0 there exists a v ∈ R^{p} with |v| < ϵ, so that the map M ⟶ R^{p}, x ↦ f(x) + v, is transverse to N.

Hint: Consider the map M × N ⟶ R^{p}, (x, y) ↦ x − f(y).

Problem 3: Let M^{k} ⊂ R^{n+1} be a compact submanifold and n ≥ 2k. Show that, for the projection π : R^{n+1} ⟶ H^{n} onto a suitable hyperplane of R^{n+1}, the restriction π|_{H} : M ⟶ H is an immersion.

Hint: Consider the (2k − 1)-dimensional manifold P T M, whose elements are the 1-dimensional subspaces of of the tangent spaces of M, and study the canonical map α : P T M ⟶ RP^{n} (the n-dimensional real projective space).

Problem 4: Show that the ring E_{0}(R^{n}) of germs of differentiable functions on R^{n} at 0 has zero divisors.