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

Author:
J. B. Wagoner

Journal:
Proc. Amer. Math. Soc. **109** (1990), 527-536

MSC:
Primary 54H20; Secondary 28D20, 58F11

MathSciNet review:
1012941

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Abstract: Let and denote, respectively, the spaces of strong shift equivalences and shift equivalences over a subset of a ring which is closed under addition and multiplication. For example, let be the integers or the nonnegative integers . For any principal ideal domain , we prove that the continuous map is a homotopy equivalence. The methods also show that any inert automorphism, i.e., an element in the kernel of can be represented by a closed loop in which in is spanned by a triangulated -disc supporting a positive -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.

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1990-1012941-9

Keywords:
Higher-dimensional shift equivalence and strong shift equivalence,
positive -cocycle,
inert automorphism

Article copyright:
© Copyright 1990
American Mathematical Society