Related papers: On the Bauer--Furuta construction

We study vector bundles over Lie groupoids, known as VB-groupoids, and their induced geometric objects over differentiable stacks. We establish a fundamental theorem that characterizes VB-Morita maps in terms of fiber and basic data, and…

We define the equivariant family index of a family of elliptic operators invariant with respect to the free action of a bundle $\GR$ of Lie groups. If the fibers of $\GR \to B$ are simply-connected solvable, we then compute the Chern…

A diffeomorphism between the twistor spaces of two Riemannian metrics on a smooth manifold preserving the fibres is defined based on a well-known construction. It is shown that this bundle isomorphim is a holomorphic map with respect to the…

We provide a physical definition of new homological invariants $\mathcal{H}_a (M_3)$ of 3-manifolds (possibly, with knots) labeled by abelian flat connections. The physical system in question involves a 6d fivebrane theory on $M_3$ times a…

We construct a generalization of the Seiberg-Witten Floer spectrum for suitable three-manifolds $Y$ with $b_1(Y)>0$. For a cobordism between three-manifolds we define Bauer-Furuta maps on these new spectra, and additionally compute some…

We describe a deformation of the $\infty$-category of Borel $G$-spectra for a finite group $G$. This provides a new presentation of the $a$-complete real Artin--Tate motivic stable homotopy category when $G=C_2$ and gives a new…

In this work we construct certain general bundles $<\mathfrak{M},\rho,X>$ and $<\mathfrak{B},\eta,X>$ of Hausdorff locally convex spaces associated with a given Banach bundle $<\mathfrak{E},\pi,X>$. Then we present conditions ensuring the…

We investigate horizontal conformality of a differential of a map between Riemannian manifolds where the tangent bundles are equipped with Cheeger--Gromoll type metrics. As a corollary, we characterize the differential of a map as a…

We provide a framework for the study of structured manifolds with singularities and their locally determined invariants. This generalizes factorization homology, or topological chiral homology, to the setting of singular manifolds equipped…

Consider a holomorphic torus action on vector bundles over a complex manifold which lifts to a holomorphic vector bundle. When the connected components of the fixed-point set are partially ordered, we construct, using sheaf-theoretical…

We define the gauge-equivariant index of a family of elliptic operators invariant with respect to the free action of a family $\GR \to B$ of Lie groups (these families are called ``gauge-invariant families'' in what follows). If the fibers…

We construct an open set of structurally unstable three parameter families whose weak and so called moderate topological classification defined below has a numerical invariant that may take an arbitrary positive value. Here and below…

In a previous paper we have introduced the gauge-equivariant K-theory group of a bundle endowed with a continuous action of a bundle of compact Lie groups. These groups are the natural range for the analytic index of a family of…

We give a comprehensive account of an analytic approach to spectral flow along paths of self-adjoint Breuer-Fredholm operators in a type $I_{\infty}$ or $II_\infty$ von Neumann algebra ${\mathcal N}$. The framework is that of {\it odd…

Statistical manifolds, the parameter spaces of smooth families of probability density functions, are the central objects of study in information geometry. While the elementary differential geometry of Riemannian statistical manifolds is…

We construct invariants of birational maps with values in the Kontsevich--Tschinkel group and in the truncated Grothendieck groups of varieties. These invariants are morphisms of groupoids and are well-suited to investigating the structure…

To a smooth and proper morphism $\mathcal{X}\to U$ with quasicompact semiseparated target we associate a sheaf in the \'etale topology, which takes an affine $U$-scheme $V$ to the set of $V$-linear semiorthogonal decompositions (of fixed…

Let X be a smooth projective connected curve over an algebraically closed field k of positive characteristic. Let G be a reductive group over k, \gamma be a dominant coweight for G, and E be an \ell-adic \check{G}-local system on X, where…

In this manuscript we consider the extent to which an irreducible representation for a reductive Lie group can be realized as the sheaf cohomolgy of an equivariant holomorphic line bundle defined on an open invariant submanifold of a…

We study equivariant Seiberg-Witten Floer theory of rational homology $3$-spheres in the special case where the group action is given by an involution. The case of involutions deserves special attention because we can couple the involution…