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Topological Quantum Field Theory and Strong Shift Equivalence
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- Journal:
- Canadian Mathematical Bulletin / Volume 42 / Issue 2 / 01 June 1999
- Published online by Cambridge University Press:
- 20 November 2018, pp. 190-197
- Print publication:
- 01 June 1999
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- Article
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Given a
$\text{TQFT}$ in dimension 
$d\,+\,1$, and an infinite cyclic covering of a closed ($d\,+\,1$)-dimensional manifold 
$M$, we define an invariant taking values in a strong shift equivalence class of matrices. The notion of strong shift equivalence originated in R. Williams’ work in symbolic dynamics. The Turaev-Viro module associated to a $\text{TQFT}$ and an infinite cyclic covering is then given by the Jordan form of this matrix away from zero. This invariant is also defined if the boundary of $M$ has an
${{S}^{1}}$ factor and the infinite cyclic cover of the boundary is standard. We define a variant of a $\text{TQFT}$ associated to a finite group 
$G$ which has been studied by Quinn. In this way, we recover a link invariant due to D. Silver and S. Williams. We also obtain a variation on the Silver-Williams invariant, by using the $\text{TQFT}$ associated to $G$ in its unmodified form.
