at.algebraic topology - Do "surjective" degree zero maps exist?

It is a theorem of H. Hopf that a map between connected, closed, orientable n-manifolds of degree 0 is homotopic to a map that misses a point, when n > 2. See D. B. A. Epstein, The degree of a map. Proc. London Math. Soc. (3) 16 1966 369--383, for a "modern" discussion including the analogous situation in the non-orientable case. The same result holds for n = 2, but is more difficult and is due to Kneser. See Richard Skora, The degree of a map between surfaces.
Math. Ann. 276 (1987), no. 3, 415--423, for a thorough discussion of the non-orientable case in dimension 2.