Differential Topology: Exercise Sheet 2

Prof. Dr. M. Wolf WS 2018/19

M. Heinze Sheet 2

Differential Topology: Exercise Sheet 2

Exercises (for Nov. 7th and 8th) 2.1 Real projective space

Consider the equivalence relation

x∼y :⇔ ∃λ∈R\{0}:x=λy (1) onRⁿ⁺¹\{0}. ThenRPⁿ:= (Rⁿ⁺¹\{0})/∼is called the (real) projective space.

(a) Show that RPⁿ is homeomorphic to Sⁿ/ ∼ where the relative topology and the equivalence relation

x∼y:⇔x=±y (2)

on the unit sphere Sⁿ ⊂ Rⁿ⁺¹ are used. This provides an alternative definition of RPⁿ.

In the following, we show that RPⁿ is a smooth n-manifold:

(b) Show that RPⁿ is Hausdorff.

(c) Show that RPⁿ is second countable.

(d) Define U_j :={x |x= (x₁, . . . , x_n+1)∈Rⁿ⁺¹\ {0}, x_j 6= 0} and let q:R\{0} →RPⁿ be the quotient map associated to the equivalence relation (1). Show that{q(U_j)}ⁿ_j=1 is an open cover ofRPⁿ, and that eachq(Uj) is homeomorphic to a subset of Rⁿ. (e) Show thatRPⁿis a smooth manifold. Show that the quotient mapq :Rⁿ\{0} →RPⁿ

used to define RPⁿ is smooth.

2.2 Boundary and interior of a topological manifold

Let M be an n-dimensional topological manifold with non-trivial boundary ∂M. Show that

(a) int(M) is ann-dimensional manifold, (b) ∂M is an (n−1)-dimensional manifold.

2.3 Examples of differentiable manifolds

(a) Suppose M_j are smooth m_j-manifolds, for j = 1,2. Show that there is a smooth structure onM₁×M₂ such that M₁×M₂ is a smooth (m₁+m₂)-manifold, and the canonical projectionsπ_j :M₁×M₂ →M_j are smooth, for j = 1,2.

(b) Show that any open subset of a smooth manifold is again a smooth manifold of the same dimension.

(c) A Lie group is aC^∞-manifoldGhaving a group structure such that the multiplication map

µ:G×G→G and the inverse map

ι:G→G, ι(c) = x⁻¹

are bothC^∞. Show that GL(n,R) is a Lie group and compute its dimension.