Differential Topology — Spring 2014

Gerald Höhn

Problem sheet 1 January 22, 2014

Problem 1: Provide the surface of a cube {x ∈ R^{n+1} | max_{i=0, ..., n} |x_{i}| = 1} with the structure of a differentiable manifold.

Problem 2: (a) Let M be a differentiable manifold and τ : M ⟶ M a fixed point free involution (i.e., τ is a diffeomorphism with τ ∘ τ = id_{M} and τ(p) ≠ p for all p ∈ M). Show that the quotient space M/τ obtained from M by identifying points mapped to each other under τ possesses exactly one differentiable structure such that the projection π : M ⟶ M/τ is locally diffeomorphic.

(b) Prove that the real projective space RP^{n} = (R^{n+1} \ {0})/ ∼ with x ∼ y exactly if there is a λ ∈ R \ {0} such that λx = y has the structure of a differentiable manifold.

Problem 3: (a) Show that S^{n} is a submanifold of R^{n+1}.

(b) Show that if the n-dimensional manifold M is a product of spheres then there exists an embedding M ⟶ R^{n+1}.

Problem 4: Let M be a differentiable manifold and E(M) be the set of differentiable functions. Show:

(a) E(M) is an algebra (i.e., a vector space with a bilinear associative product) under the natural addition and multiplication of functions.

(b) A differentiable map f : M ⟶ N induces an algebra homomorphism f^{∗} : E(N) ⟶ E(M), h ↦ h ∘ f with the properties Id_{M}^{∗} = Id and (g ∘ f)^{∗} = f^{∗} ∘ g^{∗}.

(c) For a point p ∈ M the set M_{p} = {h ∈ E(M) | h(p) = 0} is a maximal ideal of E(M).

(d) If M is compact and M is a maximal ideal of E(M), then there exists a p ∈ M such that M = M_{p}.