HOMEWORK 7
(1) Compute the Euler characteristic χ(X) of all nonorientable 2-dim connected
compact varieties
(2) Let f : RPn
→ RPn
where n is even. Prove that f has a fixed point (using
L(f)).
(3) Suppose that S1
is covered by two open sets U1, U2. Prove that for some Ui
there exists a point x such that x, −x ∈ Ui. (Hint: Use Borsuk-Ulam theorem
and the distance map dist(U)) Show that this generalizes to any covering of Sn
by n + 1 open sets.
(4) Prove that no subset of Rn
is homeomorphic to Sn
.
Date: April 5, 2013.
1