  Let L:= Z^D be the D-dimensional lattice and let A^L be the Cantor space of
L-indexed configurations in some finite alphabet A, with the natural L-action
by shifts. A `cellular automaton' is a continuous, shift-commuting self-map F
of A^L, and an `F-invariant subshift' is a closed, F-invariant and
shift-invariant subset X of A^L. Suppose x is a configuration in A^L that is
X-admissible everywhere except for some small region we call a `defect'. It has
been empirically observed that such defects persist under iteration of F, and
often propagate like `particles' which coalesce or annihilate on contact. We
construct algebraic invariants for these defects, which explain their
persistence under F, and partly explain the outcomes of their collisions. Some
invariants are based on the cocycles of multidimensional subshifts; others
arise from the higher-dimensional (co)homology/homotopy groups for subshifts,
obtained by generalizing the Conway-Lagarias tiling groups and the Geller-Propp
fundamental group.
