dorsal/arxiv
View SchemaInvariants for 1-dimensional cohomology classes arising from TQFT
| Authors | Patrick Gilmer |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9501004 |
| URL | https://arxiv.org/abs/q-alg/9501004 |
| DOI | 10.1016/S0166-8641(96)00090-9 |
| Journal | Topology and its Applications ,75 (1997) 217-259 |
Abstract
Let $(V,Z)$ be a Topological Quantum Field Theory over a field $f$ defined on a cobordism category whose morphisms are oriented $n+1$-manifolds perhaps with extra structure. Let $(M,\chi)$ be a closed oriented $n+1$-manifold $M$ with this extra structure together with $\chi \in H^1(M).$ Let $M_{\infty}$ denote the infinite cyclic cover of $M$ given by $\chi.$ Consider a fundamental domain $E$ for the action of the integers on $M_{\infty}$ bounded by lifts of a surface $\Sigma$ dual to $\chi,$ and in general position. $E$ can be viewed as a cobordism from $\Sigma$ to itself. We give Turaev and Viro's proof of their theorem that the similarity class of the non-nilpotent part of $Z(E)$ is an invariant. We give a method to calculate this invariant for the $(V_p,Z_p)$ theories of Blanchet,Habegger, Masbaum and Vogel when $M$ is zero framed surgery to $S^3$ along a knot K. We give a formula for this invariant when $K$ is a twisted double of another knot. We obtain formulas for the quantum invariants of branched covers of knots, and unbranched covers of 0-surgery to $S^3$ along knots.
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"abstract": "Let $(V,Z)$ be a Topological Quantum Field Theory over a field $f$ defined on\na cobordism category whose morphisms are oriented $n+1$-manifolds perhaps with\nextra structure. Let $(M,\\chi)$ be a closed oriented $n+1$-manifold $M$ with\nthis extra structure together with $\\chi \\in H^1(M).$ Let $M_{\\infty}$ denote\nthe infinite cyclic cover of $M$ given by $\\chi.$ Consider a fundamental domain\n$E$ for the action of the integers on $M_{\\infty}$ bounded by lifts of a\nsurface $\\Sigma$ dual to $\\chi,$ and in general position. $E$ can be viewed as\na cobordism from $\\Sigma$ to itself. We give Turaev and Viro\u0027s proof of their\ntheorem that the similarity class of the non-nilpotent part of $Z(E)$ is an\ninvariant. We give a method to calculate this invariant for the $(V_p,Z_p)$\ntheories of Blanchet,Habegger, Masbaum and Vogel when $M$ is zero framed\nsurgery to $S^3$ along a knot K. We give a formula for this invariant when $K$\nis a twisted double of another knot. We obtain formulas for the quantum\ninvariants of branched covers of knots, and unbranched covers of 0-surgery to\n$S^3$ along knots.",
"arxiv_id": "q-alg/9501004",
"authors": [
"Patrick Gilmer"
],
"categories": [
"q-alg",
"math.QA"
],
"doi": "10.1016/S0166-8641(96)00090-9",
"journal_ref": "Topology and its Applications ,75 (1997) 217-259",
"title": "Invariants for 1-dimensional cohomology classes arising from TQFT",
"url": "https://arxiv.org/abs/q-alg/9501004"
},
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