A map f:Sn→Sn satisfying f(x)=f(−x) for all x is called an even map. Show that an even map Sn→Sn must have even degree, and that the degree must in fact be zero when n is even. When n is odd, show there exist even maps of any given even degree. [Hints: If f is even, it factors as a composition Sn→RPn→Sn. Using the calculation of Hn(RPn) in the text, show that the induced map Hn(Sn)→Hn(RPn) sends a generator to twice a generator when n is odd. It may be helpful to show that the quotient map RPn→RPn/RPn−1 induces an isomorphism on Hn when n is odd.]