Gropes, Towers, and Skyscrapers

2021 ◽  
pp. 171-184
Author(s):  
Mark Powell ◽  
Arunima Ray
Keyword(s):  
Manifolds With Boundary ◽  
Good Group ◽  
Positive Genus ◽  
Symplectic Basis ◽  
Orientable Surfaces

Gropes, towers, and skyscrapers are carefully defined. These are the objects that the rest of Part II studies and seeks to construct. All three are 4-manifolds with boundary, obtained from stacking thickened surfaces on top of one another. Gropes are constructed from thickened orientable surfaces with positive genus, each stage attached to a symplectic basis of curves for the homology of the previous stage. Towers have an additional type of stage obtained from plumbed thickened discs. A skyscraper is the endpoint compactification of an infinite tower. An introduction to endpoint compactifications is included. The notion of a good group is also defined.

scholarly journals ON LIE BIALGEBRAS OF LOOPS ON ORIENTABLE SURFACES

2008 ◽  
Vol 17 (03) ◽  
pp. 351-359 ◽  
Author(s):  
ATTILIO LE DONNE
Keyword(s):  
Negative Answer ◽  
Orientable Surface ◽  
Lie Bialgebra ◽  
Lie Bialgebras ◽  
Genus Zero ◽  
Homotopy Classes ◽  
Positive Genus ◽  
Simple Class ◽  
Orientable Surfaces

Goldman [2] and Turaev [4] found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of loops on an orientable surface. Chas [1] by the aid of the computer, found a negative answer to Turaev's question about the characterization of the classes with cobracket zero as multiples of simple classes, in every surface of negative Euler characteristic and positive genus. However, she left open Turaev's conjecture, namely if, for genus zero, every class with cobracket zero is a multiple of a simple class. The aim of this paper is to give a positive answer to this conjecture.

scholarly journals Persistence modules, symplectic Banach–Mazur distance and Riemannian metrics

2021 ◽  
pp. 2150040
Author(s):  
Vukašin Stojisavljević ◽  
Jun Zhang
Keyword(s):  
Cotangent Bundle ◽  
Isometric Embedding ◽  
Large Scale ◽  
Riemannian Metrics ◽  
Closed Geodesics ◽  
Persistence Modules ◽  
Large Scale Geometry ◽  
Nonlinear Version ◽  
Positive Genus ◽  
Orientable Surfaces

We use persistence modules and their corresponding barcodes to quantitatively distinguish between different fiberwise star-shaped domains in the cotangent bundle of a fixed manifold. The distance between two fiberwise star-shaped domains is measured by a nonlinear version of the classical Banach–Mazur distance, called symplectic Banach–Mazur distance and denoted by [Formula: see text] The relevant persistence modules come from filtered symplectic homology and are stable with respect to [Formula: see text] Our main focus is on the space of unit codisc bundles of orientable surfaces of positive genus, equipped with Riemannian metrics. We consider some questions about large-scale geometry of this space and in particular we give a construction of a quasi-isometric embedding of [Formula: see text] into this space for all [Formula: see text] On the other hand, in the case of domains in [Formula: see text], we can show that the corresponding metric space has infinite diameter. Finally, we discuss the existence of closed geodesics whose energies can be controlled.

Yb(OTf)3-mediated regioselective hydroamination of ynamides with anilines or p-toluenesulfonamide

Synthesis ◽  
2021 ◽  
Author(s):  
Xiaobao Zeng ◽  
Qingyun Gu ◽  
Wenjing Dai ◽  
Yushan Xie ◽  
Xinyi Liu ◽  
...  
Keyword(s):  
Rare Earth ◽  
Functional Group ◽  
High Yield ◽  
Diverse Range ◽  
Good Group ◽  
Rare Earth Salts

A rare-earth salts Yb(OTf)3-catalyzed regioselective hydroamination of ynamides with anilines or p-toluenesulfonamide has been developed. This protocol provided facile access to a diverse range of amidines with good group functional group tolerance in moderate to high yield.

scholarly journals A New “Good and Bad Groups-Based Optimizer” for Solving Various Optimization Problems

2021 ◽  
Vol 11 (10) ◽  
pp. 4382
Author(s):  
Ali Sadeghi ◽  
Sajjad Amiri Doumari ◽  
Mohammad Dehghani ◽  
Zeinab Montazeri ◽  
Pavel Trojovský ◽  
...  
Keyword(s):  
Optimization Algorithm ◽  
Optimization Problems ◽  
Search Algorithm ◽  
Fitness Function ◽  
Optimization Algorithms ◽  
Gravitational Search Algorithm ◽  
Gray Wolf ◽  
Good Group ◽  
Good Ability ◽  
Whale Optimization

Optimization is the science that presents a solution among the available solutions considering an optimization problem’s limitations. Optimization algorithms have been introduced as efficient tools for solving optimization problems. These algorithms are designed based on various natural phenomena, behavior, the lifestyle of living beings, physical laws, rules of games, etc. In this paper, a new optimization algorithm called the good and bad groups-based optimizer (GBGBO) is introduced to solve various optimization problems. In GBGBO, population members update under the influence of two groups named the good group and the bad group. The good group consists of a certain number of the population members with better fitness function than other members and the bad group consists of a number of the population members with worse fitness function than other members of the population. GBGBO is mathematically modeled and its performance in solving optimization problems was tested on a set of twenty-three different objective functions. In addition, for further analysis, the results obtained from the proposed algorithm were compared with eight optimization algorithms: genetic algorithm (GA), particle swarm optimization (PSO), gravitational search algorithm (GSA), teaching–learning-based optimization (TLBO), gray wolf optimizer (GWO), and the whale optimization algorithm (WOA), tunicate swarm algorithm (TSA), and marine predators algorithm (MPA). The results show that the proposed GBGBO algorithm has a good ability to solve various optimization problems and is more competitive than other similar algorithms.

scholarly journals Invariants of Stable Maps between Closed Orientable Surfaces

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 215
Author(s):  
Catarina Mendes de Jesus S. ◽  
Pantaleón D. Romero
Keyword(s):  
Point Of View ◽  
Stable Maps ◽  
Smooth Maps ◽  
Orientable Surfaces

In this paper, we will consider the problem of constructing stable maps between two closed orientable surfaces M and N with a given branch set of curves immersed on N. We will study, from a global point of view, the behavior of its families in different isotopies classes on the space of smooth maps. The main goal is to obtain different relationships between invariants. We will provide a new proof of Quine’s Theorem.

scholarly journals An Obata-type theorem on compact Einstein manifolds with boundary

2021 ◽  
Author(s):  
Kazuo Akutagawa
Keyword(s):  
Boundary Condition ◽  
Smooth Boundary ◽  
Type Theorem ◽  
Type Inequality ◽  
Einstein Metric ◽  
Manifolds With Boundary ◽  
Einstein Manifolds ◽  
Conformal Class ◽  
Totally Geodesic ◽  
Yamabe Constant

AbstractWe show a kind of Obata-type theorem on a compact Einstein n-manifold $$(W, \bar{g})$$ ( W , g ¯ ) with smooth boundary $$\partial W$$ ∂ W . Assume that the boundary $$\partial W$$ ∂ W is minimal in $$(W, \bar{g})$$ ( W , g ¯ ) . If $$(\partial W, \bar{g}|_{\partial W})$$ ( ∂ W , g ¯ | ∂ W ) is not conformally diffeomorphic to $$(S^{n-1}, g_S)$$ ( S n - 1 , g S ) , then for any Einstein metric $$\check{g} \in [\bar{g}]$$ g ˇ ∈ [ g ¯ ] with the minimal boundary condition, we have that, up to rescaling, $$\check{g} = \bar{g}$$ g ˇ = g ¯ . Here, $$g_S$$ g S and $$[\bar{g}]$$ [ g ¯ ] denote respectively the standard round metric on the $$(n-1)$$ ( n - 1 ) -sphere $$S^{n-1}$$ S n - 1 and the conformal class of $$\bar{g}$$ g ¯ . Moreover, if we assume that $$\partial W \subset (W, \bar{g})$$ ∂ W ⊂ ( W , g ¯ ) is totally geodesic, we also show a Gursky-Han type inequality for the relative Yamabe constant of $$(W, \partial W, [\bar{g}])$$ ( W , ∂ W , [ g ¯ ] ) .

Sign in / Sign up

Export Citation Format

Share Document