  The problem of extending fields that are defined on lattices to fields
defined on the continua that they become in the continuum limit is basically
one of continuous extension from the 0-skeleton of a simplicial complex to its
higher-dimensional skeletons. If the lattice in question has defects, as well
as the order parameter space of the field, then this process might be
obstructed by characteristic cohomology classes on the lattice with values in
the homotopy groups of the order parameter space. The examples from solid-state
physics that are discussed are quantum spin fields on planar lattices with
point defects or orientable space lattices, vorticial flows or director fields
on lattices with dislocations or disclinations, and monopole fields on lattices
with point defects.
