Duality and Its Consequences for Ordered Cohomology of Finite Type Subshifts*

We construct a fundamental group for subshifts of nite type and prove that its abelianization gives a homology dual to ordered cohomology. We characterize cohomology classes representable as characteristic functions of clopen sets and give examples to illustrate what can happen for inclusions among clopen sets. We show that existence of unital ordered cohomology equivalence arising from isomorphisms of cochain groups is equivalent to homeomorphism preserving closures of orbits. We also show we can identify from the cohomology of a product of nite type subshifts, and the action of automorphisms on the rst factor, the conjugacy class of the second factor.