arXiv research

Model for assembly map of bordism-invariant functors.

A modular functor is constructed from non-semisimple 3d TFTs.

We consider a category whose morphisms are bordisms of $n$-dimensional pseudomanifolds equipped with a certain additional structure (coloring). On the other hand, we consider the product $G$ of $(n+1)$ copies of infinite symmetric group. We show that unitary representations of $G$ produce functors from the category of …

3D TQFTs linked to modular categories with square root of global dimension.

The book develops a new bordism-theoretic approach to understanding orientations of moduli spaces.

Floer theory uses categories to construct 3-manifold invariants.

The paper studies the Pontrjagin dual of 4D Spin bordism.

Paper establishes equivalence between algebraic and functorial QFTs.

The paper contains a survey of train constructions for infinite symmetric groups and related groups. For certain pairs (a group $G$, a subgroup $K$), we construct categories, whose morphisms are two-dimensional surfaces tiled by polygons and colored in a certain way. A product of morphisms is a gluing of combinatorial …

We give a simple sufficient condition for Quinn's "bordism-type spectra" to be weakly equivalent to strictly associative ring spectra. We also show that Poincare bordism and symmetric L-theory are naturally weakly equivalent to monoidal functors. Part of the proof of these statements involves showing that Quinn's funct…

Quantum field theory uses Lorentzian bordisms to describe time evolution.

We investigate the relationship between the algebra of tensor categories and the topology of framed 3-manifolds. On the one hand, tensor categories with certain algebraic properties determine topological invariants. We prove that fusion categories of nonzero global dimension are 3-dualizable, and therefore provide 3-di…

Logarithmic representations of the bordism category are considered as a framework for capturing a class of additive invariants characterising Reidemeister torsions.

Study homotopy sheaves on categories and their presheaves, proving descent properties.

Open 2D TFTs extend to closed theories with circle value as Hochschild homology.

In terms of category theory, the Gromov homotopy principle for a set valued functor $F$ asserts that the functor $F$ can be induced from a homotopy functor. Similarly, we say that the bordism principle for an abelian group valued functor $F$ holds if the functor $F$ can be induced from a (co)homology functor. We examin…

Study of embedding calculus using infinite operads.

Extends RT TQFT to include surface defects and line defects.

For a fixed closed manifold $P$, we construct a cobordism category of embedded manifolds with a single Baas-Sullivan singularity of type $P$. Our main theorem identifies the homotopy type of the classifying space of this cobordism category with that of the infinite loop-space of a certain spectrum related to the spectr…

Ricci-positive manifolds span the kernel of the $\hat A$-genus in rational Spin bordism.

Smooth FFT from B-fields and D-branes.

For simple and simply-connected complex algebraic group G, we conjecture the existence of a functor eta_G from the category of 2-bordisms to the category of holomorphic symplectic varieties with Hamiltonian action, such that gluing of boundaries corresponds to the holomorphic symplectic quotient with respect to the dia…

Solves linearity problem for acyclic groups, bounds Cheeger-Gromov ρ-invariants.

Paper proves orientability of gauge theory moduli spaces using bordism theory.

We prove that smooth 1-dimensional topological field theories over a manifold are equivalent to vector bundles with connection. The main novelty is our definition of the smooth 1-dimensional bordism category, which encodes cutting laws rather than gluing laws. We make this idea precise through a smooth version of Rezk'…

New algebraic framework for studying surfaces in 3-manifolds.

This paper studies PL cobordism categories and their homotopy types.

Extends string-net theory to 3D TQFT via surface graphs and surgery.

Study of Morse functions with constraints and their bordism groups.

We study bordism groups and bordism homology theories based on pseudomanifolds and stratified pseudomanifolds. The main seam of the paper demonstrates that when we uses classes of spaces determined by local link properties, the stratified and unstratified bordism theories are identical; this includes the known examples…

Goosen extends TQFTs using generators and relations for a simple example.

New perspective on APS indices preserves orientations and gradings through bordisms.

This note provides a computation of the bordism groups of K-Witt spaces for fields K with characteristic 2. We provide a complete computation for the unoriented bordism groups. For the oriented bordism groups, a nearly complete computation is provided as well a discussion of the difficulty of resolving a remaining ambi…

The group of bordism classes of unoriented surfaces in 4-space is determined. The bordism classes are characterized by normal Euler numbers,double linking numbers, and triple linking numbers.

We introduce a framework, twisted parametrized stable homotopy theory, for describing semi-infinite homotopy types. A twisted parametrized spectrum is a section of a bundle whose fibre is the category of spectra. We define these bundles in terms of modules over a stack of parametrized spectra and in terms of diagrams o…

Study PL bordism theories with quantitative bounds on filling simplices.

Generics extended to new cohomologies.

New geometric model for equivariant cohomotopy using bordism.

We call a closed, connected, orientable manifold in one of the categories TOP, PL or DIFF chiral if it does not admit an orientation-reversing automorphism and amphicheiral otherwise. Moreover, we call a manifold strongly chiral if it does not admit a self-map of degree -1. We prove that there are strongly chiral, smoo…

Global group laws connect equivariant bordism rings to formal group laws.

We introduce characteristics into chromatic homotopy theory. This parallels the prime characteristics in number theory as well as in our earlier work on structured ring spectra and unoriented bordism theory. Here, the K(n)-local Hopkins-Miller classes $ζ_n$ take the places of the prime numbers, and this allows us to di…

This paper shows that, away from 6, the kernel of the Witten genus is precisely the ideal consisting of (bordism classes of) Cayley plane bundles with connected structure group, but only after restricting the Witten genus to string bordism. It does so by showing that the divisibility properties of Cayley plane bundle c…

We define new bordism and spin bordism invariants of certain subgroups of the mapping class group of a surface. In particular, they are invariants of the Johnson filtration of the mapping class group. The second and third terms of this filtration are the well-known Torelli group and Johnson subgroup, respectively. We i…

Minimal hypersurfaces link psc metrics and bordism.

Proves nearby Lagrangian cocores are homotopically rigid in certain dimensions.

The groups of link bordism can be identified with homotopy groups via the Pontryagin-Thom construction. B.J. Sanderson computed the bordism group of 3 component surface-links using the Hilton-Milnor Theorem, and later gave a geometric interpretation of the groups in terms of intersections of Seifert hypersurfaces and t…

Paper corrects mistakes in moduli space orientability for Spin(7)-instantons and coherent sheaves.

Chern-Simons and Reshetikhin-Turaev theories are shown equivalent.