Removing even crossings on surfaces

In this paper we investigate how certain results related to the Hanani–Tutte theorem can be extended from the plane to surfaces. We give a simple topological proof that the weak Hanani–Tutte theorem is true on arbitrary surfaces, both orientable and non-orientable. We apply these results and the proof techniques to obtain new and old results about generalized thrackles, including that every bipartite generalized thrackle on a surface SS can be embedded on SS. We also extend to arbitrary surfaces a result of Pach and Tóth that allows the redrawing of a graph so as to remove all crossings with even edges. From this we can conclude that crS(G), the crossing number   of a graph GG on surface SS, is bounded by 2ocrS(G)2, where ocrS(G) is the odd crossing number   of GG on surface SS. Finally, we prove that ocrS(G)=crS(G) whenever ocrS(G)≤2, for any surface SS.