Differential Topology — Spring 2014

Gerald Hoehn

Problem sheet 3 February 13, 2014

Problem 1: Let X be a set and A = {ϕ : U ⊂ X ⟶ V ⊂ R^{n}} be a system of maps with the following properties:

• The ϕ ∈ A are bijective.

• The images V = ϕ(U) are open subsets of R^{n} for all ϕ ∈ A

• For each x ∈ X there exists a ϕ ∈ A, ϕ : U ⟶ V, with x ∈ U.

• Given ϕ_{i}, ϕ_{j} ∈ A the maps ϕ_{j} ∘ ϕ_{i}^{−1} : ϕ_{i}(U_{i} ∩ U_{j}) ⟶ ϕ_{j}(U_{i} ∩ U_{j}) are homeomorphisms.

Show that there is a unique topology on X such that the ϕ ∈ A are homeomorphisms. (Thus A makes X into a topological manifold, besides the Hausdorff and 2nd-countability property.)

Problem 2: Prove that the Brieskorn manifold

X = {(z_{1}, …, z_{5}) ∈ C^{5} | |z_{1}|^{2} + ⋯ + |z_{5}|^{2} = 1, z_{1}^{2} + z_{2}^{2} + z_{3}^{2} + z_{4}^{3} + z_{5}^{5} = 0}

is a 7-dimensional differentiable manifold.

Problem 3: Let M, N and X be differentiable manifolds and f : M ⟶ X, g : N ⟶ X be differentiable maps. Assume that for all points p ∈ M and q ∈ N with f(p) = g(q) = a ∈ X one has

T_{p}f(T_{p}M) + T_{q}g(T_{q}N) = T_{a}X

Show that the fibre product of f and g:

{(p, q) ∈ M × N | f(p) = g(q)}

is a differentiable manifold.

Problem 4: (from this years qualifying exam):

In this problem all manifolds are assumed to be smooth.

(a) Define what it means for two submanifolds Y and Z of a manifold X to be transversal.

(b) Recall that an affine subspace of R^{n} is a translate of a linear subspace, and that affine subspaces are thus trivially seen to be submanifolds. Characterize, with proof, which affine subspaces of R^{3} are transversal to the unit sphere {(x, y, z) | x^{2} + y^{2} + z^{2} = 1}.

(c) Recall that if Y and Z are transversal submanifolds of X, then Y ∩ Z is a submanifold of Y with codimension in Y equal to the codimension of Z in X. Show using an example in which X is R^{3}, Y is the unit sphere, and Z is an affine subspace of R^{3} that the conclusion need not hold in the absence of transversality.