Differential Topology — Spring 2014

Gerald Hoehn

Problem sheet 2 January 23, 2014

Problem 1: Prove the following universal properties for the differentiable sum and product:

(a) There are two differentiable maps ι_{i} : M_{i} ⟶ M_{1} ∪ M_{2} (the inclusions) such that a map f : M_{1} ∪ M_{2} ⟶ N into a manifold N is differentiable if and only if the restrictions f ∘ ι_{i} are differentiable.

(b) There are two differentiable maps π_{i} : M_{1} × M_{2} ⟶ M_{i} (the projections) such that a map f : N ⟶ M_{1} × M_{2} starting from a manifold N is differentiable if and only if the projections π_{i} ∘ f are differentiable.

(c) Explain why the properties (a) and (b) characterize M_{1} ∪ M_{2} respectively M_{1} × M_{2} up to diffeomorphism among manifolds.

Problem 2: Let M(m × n, R) be the vector space of real m × n-matrices with real entries, and M_{r}(m × n, R) the subset of matrices of rank r. Prove that M_{r}(m × n, R) is a submanifold of M(m × n, R) of dimension m · n − (m − r) · (n − r) for r ≤ min{m, n}.

Hint: a typical chart domain around a point of M_{r}(m × n, R) is given by the set U ⊂ M(m × n, R) of matrices of the form

( \begin{matrix}A & AB \\ D & DB + C \end{matrix} ) , A ∈ M(r × r, R), det(A) = 0.

Such a matrix lies in M_{r}(m × n, R) if and only if C = 0.

Problem 3: Show that if the map f : S^{n} ⟶ R is differentiable, then there exist two different points p, q ∈ S^{n}, so that T_{p}(f) and T_{q}(f) are both 0.

Problem 4: Let E^{(n)} := E_{0}(R^{n}) be the algebra of germs of differentiable functions on R^{n} at the point 0 ∈ R^{n}.

(a) Show that m_{n} := {ϕ_{0} ∈ E^{(n)} | ϕ_{0}(o) = 0} is the only maximal ideal of E^{(n)}.

(b) Show that m_{n} is generated by the germs of the coordinate functions x_{1}, …, x_{n}.

(c) Show that the ideal m_{n}^{k} (i.e., the ideal generated by the products a_{1} ⋯ a_{k} with a_{i} ∈ m_{n} for i = 1, …, k) is the ideal of germs of functions for which all partial derivatives of order < k vanish at the origin.

(d) Show that for a germ f_{0} ∈ E(R^{n}, R^{m}) with f_{0}(0) = 0 the induced map f^{∗} : E^{(m)} ⟶ E^{(n)} satisfies f^{∗}(m_{m}) ⊂ m_{n} and so one obtains a linear map f̄^{∗} : R^{m} ≅ \frac{m_{m}}{m_{m}^{2}} ⟶ \frac{m_{n}}{m_{n}^{2}} ≅ R^{n} which is given by the transpose of the Jacobi matrix ^{t}Df_{0}(0).

(e) Show that the dual space (\frac{m_{n}}{m_{n}^{2}} )^{∗} can be identified with the tangent space T_{0}(R^{n}).