The $Z_2$ invariant for filled bands in the ground states of systems withtime reversal invariance characterizes the number of stable pairs of edgestates. Here we study the $Z_2 $ invariant using band touching methodsdiscussed in a recent previous work \cite{roy2006zcq} and extend the study tothree dimensions. Band collisions preserve the $Z_2 $ invariant both in two andthree dimensions, but there are crucial differences in the two cases. In threedimensions,we find a novel fourth $Z_2 $ invariant which is characterized by a"trapped monopole" in momentum space. If the monopole charge in half theBrillouin zone is odd, then atleast one of the monopoles cannot recombine withanother monopole and vanish unlike the case when the monopole charge is even.We also point out the possibility of a three dimensional quantum spin Halleffect and discuss the connection of various topological invariants to such aneffect.
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