Khovanov and Sazdanovic recently introduced symmetric monoidal categories parameterized by rational functions and given by quotients of categories of two-dimensional cobordisms. These ...categoriesgeneralizeDeligne'sinterpolation categories of representations of symmetric groups. In this paper, we classify indecomposable objects and identify the associated graded Grothendieck rings of Khovanov–Sazdanovic's categories through sums of representation categories over crossed products of polynomial rings over a general field. To obtain these results, we introduce associated graded categories for Krull–Schmidt categories with filtrations as a categorification of the associated graded Grothendieck ring, and study field extensions and Galois descent for Krull–Schmidt categories.
In this note we show that there exist cusped hyperbolic 3-manifolds that embed geodesically but cannot bound geometrically. Thus, being a geometric boundary is a non-trivial property for such ...manifolds. Our result complements the work by Long and Reid on geometric boundaries of compact hyperbolic 4-manifolds and by Kolpakov, Reid, and Slavich on embedding arithmetic hyperbolic manifolds.
In a previous paper, we developed general techniques for constructing a variety of pseudo-collars, as defined by Guilbault and Tinsley, with roots in earlier work by Chapman and Siebenmann. As an ...application of our techniques, we exhibited an uncountable collection of pseudo-collars, all with the same boundary and similar fundamental group systems at infinity. Construction of that family was very specific; it relied on properties of Thompson's group V. In this paper, we provide a more general approach to constructing similar collections of examples. Instead of using Thompson's group V, we base our new examples on a broader and more common collection of groups, in particular, fundamental groups of certain hyperbolic manifolds.
We compute the involutive Heegaard Floer homology of the family of three-manifolds obtained by plumbings along almost-rational graphs. (This includes all Seifert fibered homology spheres.) We also ...study the involutive Heegaard Floer homology of connected sums of such three-manifolds, and explicitly determine the involutive correction terms in the case that all of the summands have the same orientation. Using these calculations, we give a new proof of the existence of an infinite-rank subgroup in the three-dimensional homology cobordism group.
The present work introduces new perspectives in order to extend orientation-preserving finite group actions from oriented surfaces to 3-manifolds. We consider the Schur multiplier associated to a ...finite group G in terms of two-dimensional principal G-bordisms, called G-cobordisms. We are interested in the question of when a free action of a finite group on a closed oriented surface extends to a non-necessarily free action on a 3-manifold. We show the answer to this question is always affirmative for abelian, dihedral, symmetric and alternating groups. As an application of our methods, we show some particular cases for non-necessarily free actions of abelian groups and dihedral groups on surfaces.
TQFTs and quantum computing Azam, Mahmud; Rayan, Steven
Bulletin des sciences mathématiques,
September 2024, 2024-09-00, Letnik:
194
Journal Article
Recenzirano
Odprti dostop
Quantum computing is captured in the formalism of the monoidal subcategory of VectC generated by C2 — in particular, quantum circuits are diagrams in VectC — while topological quantum field theories, ...in the sense of Atiyah, are diagrams in VectC indexed by cobordisms. We initiate a program to formalize this connection. In doing so, we equip cobordisms with machinery for producing linear maps by parallel transport along curves under a connection and then assemble these structures into a higher category. Finite-dimensional complex vector spaces and linear maps between them are given a suitable higher categorical structure which we call FVectC. We realize quantum circuits as images of cobordisms under monoidal functors from these modified cobordisms to FVectC, which are computed by taking parallel transports of vectors and then combining the results in a pattern encoded in the domain category.
We prove that the stable moduli space of (n−1)-connected, n-parallelizable, (2n+1)-dimensional manifolds is homology equivalent to an infinite loopspace for n≥4,n≠7. The main novel ingredient is a ...version of the cobordism category incorporating surgery data in the form of Lagrangian subspaces.
PDF
