Topological Quantum Field Theory and Strong Shift Equivalence

Gilmer, Patrick M.

Canadian mathematical bulletin, Tome 42 (1999) no. 2, pp. 190-197

Résumé

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.

Détail

DOI : 10.4153/CMB-1999-023-4

Mots-clés : 57R99, 57M99, 54H20, knot, link, TQFT, symbolic dynamics, shift equivalence