We find a graph of genus 5 and its drawing on the orientable surface of
genus 4 with every pair of independent edges crossing an even number of
times. This shows that the strong Hanani-Tutte theorem cannot be extended to
the orientable surface of genus 4. As a base step in the construction we use
a counterexample to an extension of the unified Hanani-Tutte theorem on the
torus.Comment: 12 pages, 4 figures; minor revision, new section on open problem