Higher-dimensional shift equivalence and strong shift equivalence are the same over the integers

Abstract:

Let $RS(\Lambda )$ and $S(\Lambda )$ denote, respectively, the spaces of strong shift equivalences and shift equivalences over a subset $\Lambda$ of a ring which is closed under addition and multiplication. For example, let $\Lambda$ be the integers $Z$ or the nonnegative integers ${Z^ + }$. For any principal ideal domain $\Lambda$, we prove that the continuous map $RS\left ( \Lambda \right ) \to S\left ( \Lambda \right )$ is a homotopy equivalence. The methods also show that any inert automorphism, i.e., an element in the kernel of ${\pi _1}\left ( {RS\left ( {{Z^ + }} \right ),A} \right ) \to {\pi _1}\left ( {S\left ( {{Z^ + }} \right ),A} \right )$ can be represented by a closed loop in $RS\left ( {{Z^ + }} \right )$ which in $SR\left ( Z \right )$ is spanned by a triangulated $2$-disc supporting a positive $1$-cocycle. These cocycles are used in work of Kim-Roush that leads to a counterexample to Williams’ lifting problem for automorphisms of finite subsystems of subshifts of finite type.