1. Let f : S 1 → S 1 be a continuous map. Prove that there exists a unique up to an
additive constant continuous function F : R → R such that
f (e2πit ) = e2πiF (t) for t ∈ R.
Then deg(f ) = F (1) − F (0). Show also that deg(f ) is a complete homotopy invariant of f .
In fact, the degree is a complete homotopy invariant of continuous maps S n → S n for
any n ≥ 1.
2. Let M m and N m be compact connected oriented manifolds without boundary, f : M →
N a continuous map. Suppose X ⊂ M is a closed set such that f is smooth on M \X, and
q ∈ N \f (X) is a regular value of f |M \X . Prove that
deg(f ) =
X
I(f, p).
p∈f −1 (q)
3. Identify the sphere S n (n ≥ 1) with I n /∂I n , where I n is the n-dimensional cube. Divide
the cube I n into m parallelepipeds. For each parallelepiped fix an orientation-preserving
diffeomorphism onto the cube. Show that together these diffeomorphisms define a continuous
map S n → S n of degree m. Conclude that there exist maps S n → S n of arbitrary degrees.
4. Let M n be a compact connected oriented manifold without boundary. Show that for
any m ∈ Z there exists a map M → S n of degree m. (Hint: choose U ⊂ M diffeomorphic
to the n-dimensional ball, identify U/∂U with S n , and use two previous exercises.)