  We introduce the notion of a contractible subshift. This is a strengthening
of the notion of strong irreducibility, where we require that the gluings are
given by a block map. We show that a subshift is a retract of a full shift if
and only if it is a contractible SFT with a fixed point. For virtually
polycyclic groups, contractibility implies dense periodic points. We introduce
a ``homotopy theory'' framework for working with this notion, and
``contractibility'' is in fact simply an analog of the usual contractibility in
algebraic topology. We also explore the symbolic dynamical analogs of homotopy
equivalence and equiconnectedness of subshifts. Contractibility is implied by
the map extension property of Meyerovitch, and among SFTs, it implies the
finite extension property of Brice\~no, McGoff and Pavlov. We include thorough
comparisons with these classes. We also encounter some new group-geometric
notions, in particular a periodic variant of Gromov's asymptotic dimension of a
group.
