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1. Prove that for n ≥ 2 k n H (R \{0}) = R, if k = 0, n − 1, 0, otherwise, as follows (see p.110 for a slightly different proof). Consider open subsets U = Rm+1 \{x | x1 = . . . = xm = 0, xm+1 ≥ 0} and V = Rm+1 \{x | x1 = . . . = xm = 0, xm+1 ≤ 0} of Rm+1 and show that they are contractible, while their intersection is homotopic to Rm \{0}. Using the Mayer-Vietoris sequence conclude that H k+1 (Rm+1 \{0}) ∼ = H k (Rm \{0}) for k ≥ 1. This reduces the proof to the computation of H 1 (Rn \{0}). If n = 2 we already know the answer since R2 \{0} is homotopic to T. For n > 2 prove that any loop in Rn \{0} is homotopic to a constant loop. Since Rn+1 \{0} is homotopic to S n we also get R, if k = 0, n, k n H (S ) = 0, otherwise. Show that the map x 7→ −x induces the identity map on H n (S n ) when n is odd, and the multiplication by −1 when n is even (hint: look at the action on the volume form). 2. For a manifold M consider the set M̃ of pairs (p, op ), where p ∈ M and op is either of the two orientations of τp (M ). Let π: M̃ → M be the map (p, op ) 7→ p. Explain how to introduce a natural smooth structure and an orientation on M̃ . The oriented manifold M̃ together with the map π: M̃ → M is called the oriented double covering of M . Show that if M is connected then M̃ has at most two connected components, and M is orientable if and only if M̃ has exactly two components. A connected manifold is called simply connected, or 1-connected, if every loop on it is homotopic to a constant loop. Show that a simply connected manifold is orientable. 3. Let G be a finite group acting freely by diffeomorphisms on a manifold M . Set N = M/G and let π: M → N be the factorization map. Show that π ∗ : Ω∗ (N ) → Ω∗ (M ) gives an isomorphism of Ω∗ (N ) onto the subcomplex Ω∗ (M )G of Ω∗ (M ) consisting of G-invariant forms. Prove that H k (N ) is isomorphic to H k (M )G , where H k (M )G is the subspace of H k (M ) consisting of G-invariant elements. 4. Identifying RPn with S n /x ∼ −x prove that R, if k = 0 or k = n is odd, k n H (RP ) = 0, otherwise. 5. Consider the two-dimensional torus T2 = R2 /Z2 . Prove that the map (x, y) 7→ (x + 21 , −y) defines a diffeomorphism f of T2 . Show that K = T2 /p ∼ f (p) is the Klein bottle. Prove that H 1 (K) = R. We shall see soon that H 2 (K) = 0 (or show this by first proving that H 2 (T2 ) = R).