Consider the cubical 3-torus T^3 := R^3/Z^3 -- 3-space factored out by the group of integer translations.
Endow it with the quotient metric. (The distance between [p] := p + Z^3 and [q]:= q + Z^3 for p,q in R^3 is the least distance of the form |x-y| where x is in [p] and y is in [q].
PUZZLE: Consider two points P, Q of T^3 that are the maximum distance apart, namely sqrt(3/4). What is the topology of the locus of points of T^3 equidistant from P and Q ?
Prosaic solution follows after spoiler space. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Take P to be (1/2,1/2,1/2); the set of points to which that's nearer than any of its equivalents is [0,1]^3. Take the vertices of that cube (which are all equivalent) as Q. If v is any of them, the set of points for which P,v are closest is a half-size cube like [0,1/2]^3. These cover [0,1]^3 so we need only look at the bisectors in these subcubes and figure out how they fit together. And at this point it's easy: we have a truncated octahedron, with "truncations" identified in opposte pairs. So: three handles. -- g