Journal of Topology最新文献

We establish the geometry behind the quantum $��6�j�$-symbols under only the admissibility conditions as in the definition of the Turaev–Viro invariants of 3-manifolds. As a classification, we show that the 6-tuples in the quantum $��6�j�$-symbols give in a precise way to the dihedral angles of (1) a spherical tetrahedron, (2) a generalized Euclidean tetrahedron, (3) a generalized hyperbolic tetrahedron or (4) in the degenerate case the angles between four oriented straight lines in the Euclidean plane. We also show that for a large proportion of the cases, the 6-tuples always give the dihedral angles of a generalized hyperbolic tetrahedron and the exponential growth rate of the corresponding quantum $��6�j�$-symbols equals the suitably defined volume of this generalized hyperbolic tetrahedron. It is worth mentioning that the volume of a generalized hyperbolic tetrahedron can be negative, hence the corresponding sequence of the quantum $��6�j�$-symbols could decay exponentially. This is a phenomenon that has never been seen before.

For any Legendrian knot or link in $�R^3$, we construct an $�L_{\infty }$ algebra that can be viewed as an extension of the Chekanov–Eliashberg differential graded algebra. The $�L_{\infty }$ structure incorporates information from rational symplectic field theory and can be formulated combinatorially. One consequence is the construction of a Poisson bracket on commutative Legendrian contact homology, and we show that the resulting Poisson algebra is an invariant of Legendrian links under isotopy.

We study the growth of double cosets in the class of groups with contracting elements, including relatively hyperbolic groups, CAT(0) groups and mapping class groups among others. Generalizing a recent work of Gitik and Rips about hyperbolic groups, we prove that the double coset growth of two Morse subgroups of infinite index is comparable with the orbital growth function. The same result is further obtained for a more general class of subgroups whose limit sets are proper subsets in the entire limit set of the ambient group. The limit sets under consideration are defined in a general convergence compactification, including Gromov boundary, Bowditch boundary, Thurston boundary and horofunction boundary. As an application, we confirm a conjecture of Maher that hyperbolic 3-manifolds are exponentially generic in the set of 3-manifolds built from Heegaard splitting using complexity in Teichmüller metric.

We calculate the Dehn twist action on the spaces of conformal blocks of a not necessarily semisimple modular category. In particular, we give the order of the Dehn twists under the mapping class group representations of closed surfaces. For Dehn twists about non-separating simple closed curves, we prove that this order is the order of the ribbon twist, thereby generalizing a result that De Renzi–Gainutdinov–Geer–Patureau–Mirand–Runkel obtained for the small quantum group. In the separating case, we express the order using the order of the ribbon twist on monoidal powers of the canonical end. As an application, we prove that the Johnson kernels of the mapping class groups act trivially if and only if for the canonical end the ribbon twist and double braiding with itself are trivial. We give a similar result for the visibility of the Torelli groups.

Clasper surgery induces the $�Y$-filtration $�{�\left\{�Y_n�\mathrm{IC}�\right\}�}_n$ over the monoid of homology cylinders, which serves as a 3-dimensional analogue of the lower central series of the Torelli group of a surface. In this paper, we investigate the torsion submodules of the associated graded modules of these filtrations. To detect torsion elements, we introduce a homomorphism on $��Y_n�\mathrm{IC}�/�Y_{�n�+�1�}�$ induced by the degree $��n�+�2�$ part of the LMO functor. Additionally, we provide a formula that computes this homomorphism under clasper surgery, and use it to demonstrate that every nontrivial torsion element in $��Y_6�\mathrm{IC}�/�Y_7�$ has order 3.

We solve a motivic version of the Adams conjecture with the exponential characteristic of the base field inverted. In the way of the proof,, we obtain a motivic version of mod $�k$ Dold theorem and give a motivic version of Brown's trick studying the homogeneous variety $���\left(�N_{{\mathrm{GL}}_r}�T�\right)���\setminus ��{\mathrm{GL}}_r�$ which turns out to be not stably $�A^1$-connected. We also show that the higher motivic stable stems are of bounded torsion.

We develop a general theory of pushforward operations for principal $�G$-bundles equipped with a certain type of orientation. In the case $��G�=��B�U�\left(�1�\right)��$ and orientations in twisted K-theory, we construct two pushforward operations, the projective Euler operation, whose existence was conjectured by Joyce, and the projective rank operation. We classify all stable pushforward operations in this context and show that they are all generated by the projective Euler and rank operation. As an application, we construct a graded Lie algebra structure on the homology of a commutative H-space with a compatible $��B�U�\left(�1�\right)�$-action and orientation. These play an important role in the context of wall-crossing formulas in enumerative geometry.

We consider the subspace of the homology of a covering space spanned by lifts of simple closed curves. Our main result is the existence of unbranched covers of surfaces where this is a proper subspace. More generally, for a fixed finite solvable quotient of the fundamental group we exhibit a cover whose homology is not generated by the lifts of curves in the complement of its kernel. We explain how the existing approach of Malestein and Putman (for branched covers) relates to the Magnus embedding, and by doing so we simplify their construction. We then generalise it to unbranched covers by producing embeddings of surface groups into units of certain graded associative algebras, which may be of independent interest.

The proof of [2, Lemma 5.1] is incomplete as it relies on some results in [4], the proof of which contains a gap. The goal of this note is to give a complete and self-contained proof of [2, Lemma 5.1].