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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
DOI: https://doi.org/10.1090/S0002-9939-1990-1012941-9
MathSciNet review: 1012941
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-1012941-9
Keywords: Higher-dimensional shift equivalence and strong shift equivalence, positive $ 1$-cocycle, inert automorphism
Article copyright: © Copyright 1990 American Mathematical Society

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