三角几何的基本群(英)

利用Building理论获得一种计算某些三角几何的基本群的新的方法．这种方法能够容易地计算出无限多有限三角几何的基拓扑基本群．     

The Weil-��tale fundamental group of a number field II

We define the fundamental group underlying the Weil-étale cohomology of number rings. To this aim, we define the Weil-étale topos as a refinement of the Weil-étale sites introduced by Lichtenbaum (Ann Math 170(2):657–683, 2009). We show that the (small) Weil-étale topos of a smooth projective curve defined in this paper is equivalent to the natural definition. Then we compute the Weil-étale fundamental group of an open subscheme of the spectrum of a number ring. Our fundamental group is a projective system of locally compact topological groups, which represents first degree cohomology with coefficients in locally compact abelian groups. We apply this result to compute the Weil-étale cohomology in low degrees and to prove that the Weil-étale topos of a number ring satisfies the expected properties of the conjectural Lichtenbaum topos.     

Algebraic topology of special Lagrangian manifolds

In this paper, we prove various results on the topology of the Grassmannian of oriented 3-planes in Euclidean 6-space and compute its cohomology ring. We give self-contained proofs. These spaces come up when studying submanifolds of manifolds with calibrated geometries. We collect these results here for the sake of completeness. As applications of our algebraic topological study we present some results on special Lagrangian-free embeddings of surfaces and 3-manifolds into the Euclidean 4 and 6-space.     

On fundamental groups related to the Hirzebruch surface F_1

Given a projective surface and a generic projection to the plane,the braid monodromy factorization(and thus,the braid monodromy type)of the complement of its branch curve is one of the most important topological invariants,stable on deformations.From this factorization,one can compute the fundamental group of the complement of the branch curve,either in C~2 or in CP~2.In this article,we show that these groups,for the Hirzebruch surface F_1,(a,b),are almost-solvable.That is, they are an extension of a solvable group,which strengthen the conjecture on degeneratable surfaces.     

On the Fundamental Groups of Galois Covering Spaces of the Projective Plane

We compute fundamental groups of the complements of a class of real curves in the complex projective plane. As a result, we obtain a new Zariski pair for arrangements of conics. As an application, we give a method for the computations of the fundamental groups of resolutions of Galois covering spaces of the projective plane ramifying along a special type of curves.     

Fundamental solutions for hermite and subelliptic operators

In this article, we introduce a geometric method based on multipliers to compute heat kernels for operators with potentials. Using the heat kernel, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point on Euclidean space and on Heisenberg groups. As a consequence, we obtain the fundamental solutions for the sub-laplacian □ _J in a family of quadratic submanifolds. The research is partially supported by a William Fulbright Reserch Grant and a Competitive Research Grant at Georgetown University.     

On the Möbius Function of a Pointed Graded Lattice

In this paper, we compute the Möbius function of pointed integer partition and pointed ordered set partition using topological and analytic methods. We show that the associated order complex is a wedge of spheres and compute the associated reduced homology group for each subposet. In addition, we compute the Möbius function of pointed graded lattice and use our method to compute the Möbius function of pointed direct sum decomposition of vector spaces.     

Computing the topological entropy of continuous maps with at most three different kneading sequences with applications to Parrondo’s paradox

We introduce an algorithm to compute the topological entropy of piecewise monotone maps with at most three different kneading sequences, with prescribed accuracy. As an application, we compute the topological entropy of 3-periodic sequences of logistic maps, disproving a commutativity formula for topological entropy with three maps, and analyzing the dynamics Parrondo’s paradox in this setting.     

Rewriting Systems and Geometric Three-Manifolds

The fundamental groups of most (conjecturally, all) closed three-manifolds with uniform geometries have finite complete rewriting systems. The fundamental groups of a large class of amalgams of circle bundles also have finite complete rewriting systems. The general case remains open.     

A rapid Generalized Method of Bisection for solving Systems of Non-linear Equations

Summary A rapid Generalized Method of Bisection for solving Systems of Non-linear Equations is presented in this paper, based on the non-zero value of the topological degree. Further, while the method does not compute the topological degree, it takes care of keeping its non-zero value during the bisections and thus results in a fast bisection algorithm.     

Fibrations of topological stacks

In this note we define fibrations of topological stacks and establish their main properties. When restricted to topological spaces, our notion of fibration coincides with the classical one. We prove various standard results about fibrations (long exact sequence for homotopy groups, Leray–Serre and Eilenberg–Moore spectral sequences, etc.). We prove various criteria for a morphism of topological stacks to be a fibration, and use these to produce examples of fibrations. We prove that every morphism of topological stacks factors through a fibration and construct the homotopy fiber of a morphism of topological stacks. As an immediate consequence of the machinery we develop, we also prove van Kampen?s theorem for fundamental groups of topological stacks.     

Projektive Darstellung der Moulton-Ebenen

The Moulton planes can be characterized as 2-dimensional topological projective planes having a 4-dimensional collineation group, which fixes exactly one nonincident point-line-pair aw. We give a representation of these geometries on the real protective plane such that a and W coincide with the origin and the line of infinity. This representation shows that the collineation groups of nonisomorphic Moulton planes act differently, although they are isomorphic as topological groups.     

Singular homology groups of fuzzy topological spaces

Let L be a completely distributive lattice with order reversing involution, and (X, τ) an L-fuzzy topological space. The purpose of this paper is to introduce the fundamental concept of fuzzy algebraic topology-the singular homology groups of the L-fuzzy topological space, in such a way that they take the (usual) cubical singular homology groups of a topological space as a special case. Also, we shall prove that they are L-fuzzy homeomorphic invariants.     

Quotients of uniform spaces

This paper develops the basic theory of quotients of uniform spaces via sufficiently nice group actions. We generalize and unify two fundamental constructions: quotients of topological groups via closed normal subgroups and quotients of metric spaces via actions by isometries. Basic results about inverse limits of topological groups are extended to inverse limits of group actions on uniform spaces, and notions of prodiscrete action and generalized covering map are introduced.     

Computing topological entropy for periodic sequences of unimodal maps

In this paper we introduce an algorithm which allows us to compute the topological entropy of a class of piecewise monotone continuous interval maps. The algorithm can be applied to a class of economic models called duopolies, and it can be useful to compute the topological entropy of periodic sequences of continuous maps which have been used in some population growth models.     

The Fibered Isomorphism Conjecture for Complex Manifolds

In this paper we show that the Fibered Isomorphism Conjecture of Farrell and Jones, corresponding to the stable topological pseudoisotopy functor, is true for the fundamental groups of a class of complex manifolds. A consequence of this result is that the Whitehead group, reduced projective class groups and the negative K-groups of the fundamental groups of these manifolds vanish whenever the fundamental group is torsion free. We also prove the same results for a class of real manifolds including a large class of 3-manifolds which has a finite sheeted cover fibering over the circle.     

A retraction theorem for topological fundamental groups with application to the Hawaiian earring

A characterization of regular topological fundamental groups yields a ‘no retraction theorem’ for spaces constructed in similar fashion to the Hawaiian earring.     

The Fundamental Group as the Structure of a Dually Affine Space

This paper dualizes the setting of affine spaces as originally introduced by Diers for application to algebraic geometry and expanded upon by various authors, to show that the fundamental groups of pointed topological spaces appear as the structures of dually affine spaces. The dual of the Zariski closure operator is introduced, and the 1-sphere and its copowers together with their fundamental groups are shown to be examples of complete objects with respect to the Zariski dual closure operator.