Gluing Theorem Research Articles

Unfolded Seiberg–Witten Floer spectra, II: Relative invariants and the gluing theorem

Unfolded Seiberg–Witten Floer spectra, II: Relative invariants and the gluing theorem

Cut locus realizations on convex polyhedra

We prove that every positively weighted tree T can be realized as the cut locus C(x) of a point x on a convex polyhedron P, with T edge weights matching C(x) edge lengths. If T has n leaves, P has (in general) n+1 vertices. We show there is in fact a continuum of polyhedra P each realizing T for some x∈P. Three main tools in the proof are properties of the star unfolding of P, Alexandrov's gluing theorem, and a new cut-locus partition lemma. The construction of P from T is surprisingly simple.

Gluing of multiple Alexandrov spaces

Perel'man's Doubling Theorem [11] says that the doubling of an Alexandrov space is also an Alexandrov space with the same lower curvature bound. Petrunin [12] generalizes this theorem to two Alexandrov spaces being glued along their isometric boundaries. These theorems have a lot of applications to the study of spaces with lower curvature bounds. As a generalization to these results and a tool to construct and verify new Alexandrov spaces, we study the gluing of multiple Alexandrov spaces. We consider a Gluing Conjecture, which says that the gluing of finite number of Alexandrov spaces is an Alexandrov space, if and only if the gluing is by path isometry along the boundaries and the tangent cones are glued to be Alexandrov spaces.We prove the Gluing Conjecture when the complement of (n−1,ϵ)-regular points is discrete in the glued part. In particular, this implies the Gluing Conjecture in dimension 2. This is a new result in the case of gluing 2-dimensional convex sets. In our proof, we also establish some structural theory for the general gluing. Based on the tangent cone condition, the 2-point gluing over (n−1,ϵ)-regular points is classified to be locally separable, as being assumed in Petrunin's Gluing Theorem. The gluing near an un-glued (n−1,ϵ)-regular point is classified to be involutional.

A NOTE ON GUNNINGHAM’S FORMULA

Gunningham [‘Spin Hurwitz numbers and topological quantum field theory’, Geom. Topol.20(4) (2016), 1859–1907] constructed an extended topological quantum field theory (TQFT) to obtain a closed formula for all spin Hurwitz numbers. In this note, we use a gluing theorem for spin Hurwitz numbers to re-prove Gunningham’s formula. We also describe a TQFT formalism naturally induced by the gluing theorem.

Lipschitz–Volume rigidity in Alexandrov geometry

We prove a Lipschitz–Volume rigidity theorem in Alexandrov geometry, that is, if a 1-Lipschitz map f:X=⨿Xℓ→Y between Alexandrov spaces preserves volume, then it is a path isometry and an isometry when restricted to the interior of X. We furthermore characterize the metric structure on Y with respect to X when f is also onto. This implies the converse of Petrunin's Gluing Theorem: if a gluing of two Alexandrov spaces via a bijection between their boundaries produces an Alexandrov space, then the bijection must be an isometry.

Map gluing theorems for theta-continuous and s \(\theta\)-continuous maps on topological spaces

We obtain map gluing theorems on \(\theta\)-continuity and s\(\theta\)-continuity of maps on topological spaces, in terms of corresponding conditions on the restriction maps to certain subsets of the domain space. Sets containing all the points of s \(\theta\) discontinuities and \(\theta\)-discontinuities are obtained. Examples are given to illustrate the necessities of the conditions imposed on the restriction maps.

Some scales of equivalent conditions to characterize the Stieltjes inequality: the case

We prove that the weighted Stieltjes inequality urn:x-wiley:dummy:mana201200118:equation:mana201200118-math-0003can be characterized by four different scales of conditions also for the case , . In particular, a new proof of a result of G. Sinnamon is given, which also covers the case . Moreover, for the situation at hand a new gluing theorem and also an equivalence theorem of independent interest are proved and discussed. By applying our new equivalence theorem to weighted inequalities for the Stieltjes transform we obtain the four new scales of conditions for characterization of Stieltjes inequality.

Notes on Beilinson's "How to glue perverse sheaves"

The titular, foundational work of Beilinson not only gives a tech- nique for gluing perverse but also implicitly contains constructions of the nearby and vanishing cycles functors of perverse sheaves. These con- structions are completely elementary and show that these functors preserve perversity and respect Verdier duality on perverse sheaves. The work also de- nes a new, \maximal extension functor, which is left mysterious aside from its role in the gluing theorem. In these notes, we present the complete details of all of these constructions and theorems. In this paper we discuss Alexander Beilinson's \How to glue perverse sheaves (1) with three goals. The rst arose from a suggestion of Dennis Gaitsgory that the author study the construction of the unipotent nearby cycles functor R un which, as Beilinson observes in his concluding remarks, is implicit in the proof of his Key Lemma 2.1. Here, we make this construction explicit, since it is invaluable in many contexts not necessarily involving gluing. The second goal is to restructure the pre- sentation around this new perspective; in particular, we have chosen to eliminate the two-sided limit formalism in favor of the straightforward setup indicated briey in (3,x4.2) for D-modules. We also emphasize this construction as a simple demon- stration that R un ( 1) and Verdier duality D commute, and de-emphasize its role in the gluing theorem. Finally, we provide complete proofs; with the exception of the Key Lemma, (1) provides a complete program of proof which is not carried out in detail, making a technical understanding of its contents more dicult given the density of ideas. This paper originated as a learning exercise for the author, so we hope that in its

Equivariant Seiberg-Witten Floer homology

In this paper we construct, for all compact oriented three- manifolds, a U(1)-equivariant version of Seiberg-Witten Floer homology, which is invariant under the choice of metric and perturbation. We give a detailed analysis of the boundary structure of the monopole moduli spaces, compactified to smooth manifolds with corners. The proof of the independence of metric and perturbation is then obtained via an analysis of all the relevant obstruction bundles and sections, and the corresponding gluing theorems. The paper also contains a discussion of the chamber structure for the Seiberg-Witten invariant for rational homology 3-spheres, and proofs of the wall crossing formula, obtained by studying the exact sequences relating the equivariant and the non-equivariant Floer homologies and by a local model at the reducible monopole.

METRICS, CONNECTIONS AND GLUING THEOREMS (CBMS Regional Conference Series in Mathematics 89)

METRICS, CONNECTIONS AND GLUING THEOREMS (CBMS Regional Conference Series in Mathematics 89)

Inlaying vertex function and scattering amplitude

Scattering processes among strings are analyzed by using fundamental equations of three types, which divide the whole complex z-plane into various types of N punctured ring domains plus various unpunctured ring domains, where internal strings freely propagate. In order to calculate scattering amplitudes (among physical particles) in Witten’s quantum string field theory, we derive and apply the “Gluing theorem,’’ mathematical proof of which is given (in operator forms) by constructing various (inlint) conformal mapping operators.