A drawing of a graph on a surface is independently even if every pair of
nonadjacent edges in the drawing crosses an even number of times. The
Z2-genus of a graph G is the minimum g such that G has an
independently even drawing on the orientable surface of genus g. An
unpublished result by Robertson and Seymour implies that for every t, every
graph of sufficiently large genus contains as a minor a projective t×t
grid or one of the following so-called t-Kuratowski graphs: K3,t, or t
copies of K5 or K3,3 sharing at most 2 common vertices. We show that
the Z2-genus of graphs in these families is unbounded in t; in
fact, equal to their genus. Together, this implies that the genus of a graph is
bounded from above by a function of its Z2-genus, solving a problem
posed by Schaefer and \v{S}tefankovi\v{c}, and giving an approximate version of
the Hanani-Tutte theorem on orientable surfaces. We also obtain an analogous
result for Euler genus and Euler Z2-genus of graphs.Comment: 23 pages, 7 figures; a few references added and correcte