Related papers: Cobordism obstructions to independent vector field…

Using spectral invariants of Dirac operators we construct a secondary version of the Witten genus, a bordism invariant of string manifolds in dimensions $4m-1$. We prove a secondary index theorem which relates this global-analytic…

In the first part of the paper we define a perturbative (pre-formal) geometry and formulate a theorem on the relation between the construction of a perturbative neighborhood of affine varieties and the higher tangent bundles. In the second…

Based on Morse theory for the energy functional on path spaces we develop a deformation theory for mapping spaces of spheres into orthogonal groups. This is used to show that these mapping spaces are weakly homotopy equivalent, in a stable…

Classical field theory is insensitive to the split of the field into a background configuration and a dynamical perturbation. In gauge theories, the situation is complicated by the fact that a covariant (w.r.t. the background field) gauge…

We define and develop a homotopy invariant notion for the sequential topological complexity of a map $f:X\to Y,$ denoted $TC_{r}(f)$, that interacts with $TC_{r}(X)$ and $TC_{r}(Y)$ in the same way Jamie Scott's topological complexity map…

When two smooth manifold bundles over the same base are fiberwise tangentially homeomorphic, the difference is measured by a homology class in the total space of the bundle. We call this the relative smooth structure class. Rationally and…

A generalized-homology bordism-theory is constructed, such that for certain manifold homotopy stratified sets (MHSS; Quinn-spaces) homeomorphism-invariant geometric fundamental-classes exist. The construction combines three ideas: Firstly,…

For a finite group $G$, we define an equivariant cobordism category $\mathcal{C}_d^G$. Objects of the category are $(d-1)$-dimensional closed smooth $G$-manifolds and morphisms are smooth $d$-dimensional equivariant cobordisms. We identify…

We compute the intersection cohomology of the moduli spaces $M_{r,d}$ of semistable vector bundles having rank $r$ and degree $d$ over a curve. We do this by relating the Hodge-Deligne polynomial of the intersection cohomology of $M_{r,d}$…

We obtain new invariants of topological link concordance and homology cobordism of 3-manifolds from Hirzebruch-type intersection form defects of towers of iterated p-covers. Our invariants can extract geometric information from an arbitrary…

The study of separating invariants is a recent trend in invariant theory. For a finite group acting linearly on a vector space, a separating set is a set of invariants whose elements separate the orbits of G. In some ways, separating sets…

We introduce a new family of invariants of real algebraic sets defined in terms of the topology of their complexifications and compute some of these invariants for spheres. This allows us to completely classify topological isomorphism…

In arXiv:1905.07734 we presented a construction that is an analogue of Pontryagin's for proper maps in stable dimensions. This gives a bijection between the cobordism set of framed embedded compact submanifolds in $W\times\mathbb{R}^n$ for…

A bounded curvature path is a continuously differentiable piecewise $C^2$ path with a bounded absolute curvature that connects two points in the tangent bundle of a surface. In this work, we analyze the homotopy classes of bounded curvature…

For each integer q>0 there is a cohomology theory such that the zero cohomology group of a manifold N of dimension n is a certain group of cobordism classes of proper fold maps of manifolds of dimension n+q into N. We prove a splitting…

Several intrinsic topological ways to encode connections on vector bundles on smooth complex algebraic curves will be described. In particular the notion of {\em Stokes decompositions} will be formalised, as a convenient intermediate…

Recent work by Abramsky and Brandenburger used sheaf theory to give a mathematical formulation of non-locality and contextuality. By adopting this viewpoint, it has been possible to define cohomological obstructions to the existence of…

Let $X$ be a closed oriented connected topological manifold of dimension $n\geq 5$. The structure group of $X$ is the abelian group of equivalence classes of all pairs $(f, M)$ such that $M$ is a closed oriented manifold and $f\colon M \to…

The existence of topological invariants analogous to Chern/Pontryagin classes for a standard $SO(D)$ or $SU(N)$ connection, but constructed out of the torsion tensor, is discussed. These invariants exhibit many of the features of the…

The main goal of this article is to construct some geometric invariants for the topology of the set $\mathcal{F}$ of flat connections on a principal $G$-bundle $P\,\longrightarrow\, M$. Although the characteristic classes of principal…