  We geometrically construct a homology theory that generalizes the Euler
characteristic mod 2 to objects in the unoriented cobordism ring N_*(X) of a
topological space X. This homology theory Eh_* has coefficients Z/2 in every
nonnegative dimension. There exists a natural transformation N_*(X)->Eh_*(X)
that for X=pt assigns to each smooth manifold its Euler characteristic mod 2.
The homology theory is constructed using cobordism of stratifolds.
  For discrete groups G, we also define an equivariant version of the homology
theory Eh_*, generalizing the equivariant Euler characteristic.
