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European Mathematical Society Publishing House
2024-03-29 08:19:19
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=QT&vol=9&iss=2&update_since=2024-03-29
Quantum Topology
Quantum Topol.
QT
1663-487X
1664-073X
General
10.4171/QT
http://www.ems-ph.org/doi/10.4171/QT
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
9
2018
2
Absolute gradings on ECH and Heegaard Floer homology
Vinicius Gripp Barros
Ramos
Instituto Nacional de Matematica Pura e Aplicada, Rio de Janeiro, Brazil
Absolute gradings, embedded contact homology, Heegaard Floer homology
In joint work with Yang Huang, we de fined a canonical absolute grading on Heegaard Floer homology by homotopy classes of oriented 2-plane fields. A similar grading was de fined on embedded contact homology by Michael Hutchings. In this paper we show that the isomorphism between these homology theories de fined by Colin, Ghiggini, and Honda preserves this grading.
Differential geometry
Manifolds and cell complexes
207
228
10.4171/QT/107
http://www.ems-ph.org/doi/10.4171/QT/107
2
9
2018
Defining and classifying TQFTs via surgery
András
Juhász
University of Oxford, UK
Cobordism, TQFT, surgery
We give a presentation of the $n$-dimensional oriented cobordism category $\mathbf {Cob}_n$ with generators corresponding to di ffeomorphisms and surgeries along framed spheres, and a complete set of relations. Hence, given a functor $F$ from the category of smooth oriented manifolds and di ffeomorphisms to an arbitrary category $C$, and morphisms induced by surgeries along framed spheres, we obtain a necessary and su cient set of relations these have to satisfy to extend to a functor from $\mathbf {Cob}_n$ to $C$. If $C$ is symmetric and monoidal, then we also characterize when the extension is a TQFT. This framework is well-suited to defining natural cobordism maps in Heegaard Floer homology. It also allows us to give a short proof of the classical correspondence between (1+1)-dimensional TQFTs and commutative Frobenius algebras. Finally, we use it to classify (2+1)-dimensional TQFTs in terms of J-algebras, a new algebraic structure that consists of a split graded involutive nearly Frobenius algebra endowed with a certain mapping class group representation. This solves a long-standing open problem. As a corollary, we obtain a structure theorem for (2+1)-dimensional TQFTs that assign a vector space of the same dimension to every connected surface. We also note that there are $ 2^{2^{\omega}}$ nonequivalent lax monoidal TQFTs over $\mathbb C$ that do not extend to (1+1+1)-dimensional ones.
Manifolds and cell complexes
229
321
10.4171/QT/108
http://www.ems-ph.org/doi/10.4171/QT/108
2
9
2018
HOMFLY-PT and Alexander polynomials from a doubled Schur algebra
Hoel
Queffelec
Université de Montpellier, France
Antonio
Sartori
Albert-Ludwigs-Universität Freiburg, Germany
Schur algebras, knot invariants, Alexander polynomial, HOMFLY-PT polynomial, Reshetikhin–Turaev invariants
We define a generalization of the Schur algebra which gives a unified setting for a quantum group presentation of the HOMFLY-PT polynomial, together with its specializations to the Alexander polynomial and to the $\mathfrak {sl}_m$ Reshetikhin–Turaev invariant.
Nonassociative rings and algebras
Quantum theory
323
347
10.4171/QT/109
http://www.ems-ph.org/doi/10.4171/QT/109
2
9
2018
Holonomy perturbations and regularity for traceless SU(2) character varieties of tangles
Christopher
Herald
University of Nevada, Reno, USA
Paul
Kirk
Indiana University, Bloomington, USA
Holonomy perturbation, character variety, Hamiltonian twist flow, Floer homology
The traceless SU(2) character variety $R(S^2,\{a_i,b_i\}_{i=1}^n)$ of a $2n$-punctured 2-sphere is the symplectic reduction of a Hamiltonian $n$-torus action on the SU(2) character variety of a closed surface of genus $n$. It is stratified with a finite singular stratum and a top smooth symplectic stratum of dimension $4n-6$. For generic holonomy perturbations $\pi$, the traceless SU(2) character variety $R_\pi(Y,L)$ of an $n$-stranded tangle $L$ in a homology 3-ball $Y$ is stratified with a finite singular stratum and top stratum a smooth manifold. The restriction to $R(S^2,\{a_i,b_i\}_{i=1}^n)$ is a Lagrangian immersion which preserves the cone neighborhood structure near the singular stratum. For generic holonomy perturbations $\pi$, the variant $R_\pi^\natural(Y,L)$, obtained by taking the connected sum of $L$ with a Hopf link and considering SO(3) representations with $w_2$ supported near the extra component, is a smooth compact manifold without boundary of dimension $2n-3$, which Lagrangian immerses into the smooth stratum of $R(S^2,\{a_i,b_i\}_{i=1}^n)$. The proofs of these assertions consist of stratified transversality arguments to eliminate non-generic strata in the character variety and to insure that the restriction map to the boundary character variety is also generic. The main tool introduced to establish abundance of holonomy perturbations is the use of holonomy perturbations along curves $C$ in a cylinder $F\times I$, where $F$ is a closed surface. When $C$ is obtained by pushing an embedded curve on $F$ into the cylinder, we prove that the corresponding holonomy perturbation induces one of Goldman's generalized Hamiltonian twist flows on the SU(2) character variety $\mathcal{M}(F)$ associated to the curve $C$.
Manifolds and cell complexes
Differential geometry
Quantum theory
349
418
10.4171/QT/110
http://www.ems-ph.org/doi/10.4171/QT/110
6
3
2018