  We present a generalization of the double semion topological quantum field
theory to higher dimensions, as a theory of $d-1$ dimensional surfaces in a $d$
dimensional ambient space. We construct a local Hamiltonian which is a sum of
commuting projectors and analyze the excitations and the ground state
degeneracy. Defining a consistent set of local rules requires the sign
structure of the ground state wavefunction to depend not just on the number of
disconnected surfaces, but also upon their higher Betti numbers through the
semicharacteristic. For odd $d$ the theory is related to the toric code by a
local unitary transformation, but for even $d$ the dimension of the space of
zero energy ground states is in general different from the toric code and for
even $d>2$ it is also in general different from that of the twisted $Z_2$
Dijkgraaf-Witten model.
