A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Proves Farrell-Jones Conjecture for specific groups and spectra.
Bökstedt and Madsen defined an infinite loop map from the embedded -dimensional cobordism category of Galatius, Madsen, Tillmann and Weiss to the algebraic -theory of in the sense of Waldhausen. The purpose of this paper is to establish two results in relation to this map. The first result is that it exte…
This note is an exposition of Waldhausen's proof of Waldhausen's Theorem: the three-sphere has a single Heegaard splitting, up to isotopy, in every genus. As a necessary step we also give a sketch of the Reidemeister-Singer Theorem.
We give a short proof of Waldhausen's homeomorphism theorem for orientable Haken 3-manifolds.
We give a proof of the so-called generalized Waldhausen conjecture, which says that an orientable irreducible atoroidal 3-manifold has only finitely many Heegaard splittings in each genus, up to isotopy. Jaco and Rubinstein have announced a proof of this conjecture using different methods.
We define the class of Haken --manifolds as a generalisation of Haken 3--manifolds. We prove that the interior of the universal covering of a Haken --manifold is $\RR^n$, which generalises a result of Waldhausen. The techniques used allow us to provide a new proof of Waldhausen's universal cover theorem for Haken…
In [Topology 35 (1996) 1005--1023] J H Rubinstein and M Scharlemann, using Cerf Theory, developed tools for comparing Heegaard splittings of irreducible, non-Haken manifolds. As a corollary of their work they obtained a new proof of Waldhausen's uniqueness of Heegaard splittings of S^3. In this note we use Cerf Theory …
We prove that the Waldhausen Nil-group associated to a virtually cyclic groups that surjects onto the infinite dihedral group vanishes if and only if the corresponding Farrell Nil-group associated to the canonical index two subgroup is trivial. The proof uses the transfer map to establish one direction, and uses contro…
New proof of Genauer fibration sequence in cobordism categories.
We study the boundary L_t of the Milnor fiber for the non-isolated singularities in C^3 with equation z^m - g(x,y) = 0 where g(x,y) is a non-reduced plane curve germ. We give a complete proof that L_t is a Waldhausen graph manifold and we provide the tools to construct its plumbing graph. As an example, we give the plu…
This paper examines the category C^k_{d,n} whose morphisms are d-dimensional smooth manifolds that are properly embedded in the product of a k-dimensional cube with an (d+n-k)-dimensional Euclidean space. There are k directions to compose k-dimensional cubes, so C^k_{d,n} is a (strict) k-tuple category. The geometric r…
The Waldhausen construction of Mayer-Vietoris splittings of chain complexes over an injective generalized free product of group rings is extended to a combinatorial construction of Seifert-van Kampen splittings of CW complexes with fundamental group an injective generalized free product.
Using basic properties of one-sided Heegaard splittings, a direct proof that geometrically compressible one-sided splittings of RP^3 are stabilised is given. The argument is modelled on that used by Waldhausen to show that two-sided splittings of S^3 are standard.
A graph multilink is a link with multiplicities in a homology 3-sphere whose exterior is a graph manifold. In this Note, we compute the Novikov homology of graph multilinks. As a corollary, we give a majoration for the number of Novikov modules on a given graph link.
Trisections are obtained by regluing surface-knots in 4-manifolds.
We describe a new approach to the canonical decompositions of 3-manifolds along tori and annuli due to Jaco-Shalen and Johannson (with ideas from Waldhausen) - the so-called JSJ-decomposition theorem. This approach gives an accessible proof of the decomposition theorem; in particular it does not use the annulus-torus t…
New proof of index theorem for topological manifold bundles.
We prove that cubulated hyperbolic groups are virtually special. The proof relies on results of Haglund and Wise which also imply that they are linear groups, and quasi-convex subgroups are separable. A consequence is that closed hyperbolic 3-manifolds have finite-sheeted Haken covers, which resolves the virtual Haken …
The two-category with three-manifolds as objects, h-cobordisms as morphisms, and diffeomorphisms of these as two-morphisms, is extremely rich; from the point of view of classical physics it defines a nontrivial topological model for general relativity. A rather striking amount of work on pseudoisotopy theory [Hatcher, …
A famous example of Casson and Gordon shows that a Haken 3-manifold can have an infinite family of irreducible Heegaard splittings with different genera. In this paper, we prove that a closed non-Haken 3-manifold has only finitely many irreducible Heegaard splittings, up to isotopy. This is much stronger than the Waldh…
We find a geometric invariant of isotopy classes of strongly irreducible Heegaard splittings of toroidal 3-manifolds. Combining this invariant with a theorem of R Weidmann, proved here in the appendix, we show that a closed, totally orientable Seifert fibered space M has infinitely many isotopy classes of Heegaard spli…
A Heegaard splitting of a closed, orientable three-manifold satisfies the disjoint curve property if the splitting surface contains an essential simple closed curve and each handlebody contains an essential disk disjoint from this curve [Thompson, 1999]. A splitting is full if it does not have the disjoint curve proper…
We investigate the orderability properties of fundamental groups of 3-dimensional manifolds. Many 3-manifold groups support left-invariant orderings, including all compact P^2-irreducible manifolds with positive first Betti number. For seven of the eight geometries (excluding hyperbolic) we are able to characterize whi…
For a relatively hyperbolic group, we construct a model for the universal space among -spaces with isotropy on the family VC of virtually cyclic subgroups of . We provide a recipe for identifying the maximal infinite virtually cyclic subgroups of Coxeter groups which are lattices in $O^+(n,1)= \iso(\mathbb H^…
We use controlled topology applied to the action of the infinite dihedral group on a partially compactified plane and deduce two consequences for algebraic K-theory. The first is that the family in the K-theoretic Farrell-Jones conjecture can be reduced to only those virtually cyclic groups which admit a surjection wit…
Complete invariant for surfaces in 3-sphere derived from diagrams of fundamental groups.
Developed a theory of local convexity for second order differential equations on Lie algebroids.
The study provides a structure theorem for a new class of noncompact 3-manifolds.
We introduce a theory of stochastic integration with respect to a family of semimartingales depending on a continuous parameter, as a mathematical background to the theory of bond markets. We apply our results to the problem of super-replication and utility maximization from terminal wealth in a bond market. Finally, w…
The Farrell-Jones Conjecture holds for groups acting acylindrically on trees.
This paper defines a theory of cobordism for virtual knots and studies this theory for standard and rotational virtual knots and links. Non-trivial examples of virtual slice knots are given. Determinations of the four-ball genus of positive virtual knots are given using the results of a companion paper by the author an…
The paper develops a theory of -superrings and their superschemes.
New approach connects pseudoisotopy theory to algebraic K-theory.
The weak regular coherence is a coarse property of a finitely generated group . It was introduced by G. Carlsson and this author to play the role of a weakening of Waldhausen's regular coherence as part of computation of the integral K-theoretic assembly map. A new class of metric spaces (sFDC) was introduced recent…
We study Farrell Nil-groups associated to a finite order automorphism of a ring . We show that any such Farrell Nil-group is either trivial, or infinitely generated (as an abelian group). Building on this first result, we then show that any finite group that occurs in such a Farrell Nil-group occurs with infinite mu…
Using screen distributions and lightlike transversal vector bundles we develop a theory of degenerate foliations of semi-Riemannian manifolds.
A theory of feature geometry using spectral analysis of weight matrices.
New combinatorial type helps distinguish plane curve topologies.
Survey of homology cobordism group with open problems.
Paper develops a theory for Patterson-Sullivan measures in higher rank symmetric spaces.
The paper develops a theory for clustering graphs sampled from a graphon model.
Study parametrized cobordism categories for smooth bundles, proving index theorem.
We develop a theory of `non-uniformly local' tent spaces on metric measure spaces. As our main result, we give a remarkably simple proof of the atomic decomposition.
New invariant identifies complex line arrangements with same combinatorics but different embeddings.
We develop a theory of sets with distributive products (called shelves and multi-shelves) and of their homology. We relate the shelf homology to the rack and quandle homology.
SSL theory improves representation learning from raw data.
We develop several methods that allow us to compute all-loop partition functions in perturbative Chern-Simons theory with complex gauge group G_C, sometimes in multiple ways. In the background of a non-abelian irreducible flat connection, perturbative G_C invariants turn out to be interesting topological invariants, wh…
A theory of deep learning is emerging, focusing on training dynamics and statistics.