Coincidences of projections and linear n-valued maps of tori

We prove that the Nielsen fixed point number N(φ) of an n-valued map φ:X?X of a compact connected triangulated orientable q-manifold without boundary is equal to the Nielsen coincidence number of the projections of the graph of φ, a subset of X×X, to the two factors. For certain q×q integer matrices A, there exist “linear” n-valued maps Φn,A,σ:Tq?Tq of q-tori that generalize the single-valued maps fA:Tq→Tq induced by the linear transformations TA:Rq→Rq defined by TA(v)=Av. By calculating the Nielsen coincidence number of the projections of its graph, we calculate N(Φn,A,σ) for a large class of linear n-valued maps.     

The Reidemeister zeta function with applications to Nielsen theory and a connection with Reidemeister torsion

In this paper we study, the Reidemeister zeta function. We prove rationality and functional equations of the Reidemeister zeta function of an endomorphism of finite group. We also obtain these results for eventually commutative endomorphisms. These results are applied to the theory of Reidemeister and Nielsen numbers of self-maps of topological spaces. Our method is to identify the Reidemeister number of a group endomorphism with the number of fixed points in the unitary dual. As a consequence, we show that the Reidemeister torsion of the mapping torus of the unitary dual is a special value of the Reidemeister zeta function. We also prove certain congruences for Reidemeister numbers which are equivalent to a Euler product formula for the Reidemeister zeta function. The congruences are the same as those found by Dold for Lefschetz numbers.     

Wecken property for roots

We show that maps from a compact space into a topological manifold which have geometric Nielsen root number zero satisfy the Wecken property, i.e., such that .

     

Splitting Obstruction Groups in Codimension 2

The splitting obstruction groups depend functorially on the square of fundamental groups. In the paper the problem of splitting along a submanifold of codimension two under some restrictions on the square of fundamental groups is considered. New exact sequences and commutative diagrams containing Wall groups, splitting obstruction groups, and surgery obstruction groups for manifold pairs are obtained. Examples of computation of splitting obstruction groups and natural maps are considered.     

向量场的Nielsen数

对于紧致流形M上的任意一个向量场X，定义了一个由向量场X确定的自映射fX：M→M，使得向量场X的奇异点均为fX的不动点．证明了向量场的Nielsen数是不依赖于向量场选取的量．     

Coincidence Wecken Property for Nilmanifolds

Let f, g: X → Y be maps from a compact infra-nilmanifold X to a compact nilmanifold Y with dim X ≥ dim Y. In this note, we show that a certain Wecken type property holds, i.e., if the Nielsen number N(f, g) vanishes then f and g are deformable to be coincidence free. We also show that if X is a connected finite complex X and the Reidemeister coincidence number R(f, g) = ∞ then f ～ f' so that C(f', g) = {x ∈ X | f'(x) = g(x)} is empty.     

拓扑熵的一个下界估计

设Ｘ为局部闭路可缩的紧致空间，ｆ为Ｘ的自映射，ｈ（ｆ）为ｆ的拓扑熵，Ｒ∞（ｆ）为ｆ的渐近Ｒｅｉｄｅｍｅｉｓｔｅｒ数，则有ｈ（ｆ）≥ｌｏｇＲ∞（ｆ）．     

Obstruction Theory for Extensions of Categorical Groups

For any categorical group H, we introduce the categorical group O ut(H) and then the well-known group exact sequence 1Z(H)HAut(H)Out(H)1 is raised to a categorical group level by using a suitable notion of exactness. Breen's Schreier theory for extensions of categorical groups is codified in terms of homomorphism to O ut(H) and then we develop a sort of Eilenberg–Mac Lane obstruction theory that solves the general problem of the classification of all categorical group extensions of a group G by a categorical group H, in terms of ordinary group cohomology.     

Obstruction Theory for Objects in Abelian and Derived Categories

ABSTRACT

In this article, we develop the obstruction theory for lifting complexes, up to quasi-isomorphism, to derived categories of flat nilpotent deformations of abelian categories. As a particular case we also obtain the corresponding obstruction theory for lifting of objects in terms of Yoneda Ext-groups. In an appendix we prove the existence of miniversal derived deformations of complexes.     

Computation of Nielsen numbers for maps of closed surfaces

Let be a closed surface, and let be a map. We would like to determine Nielsen fixed point theory provides a lower bound for , called the Nielsen number, which is easy to define geometrically and is difficult to compute.

We improve upon an algebraic method of calculating developed by Fadell and Husseini, so that the method becomes algorithmic for orientable closed surfaces up to the distinguishing of Reidemeister orbits. Our improvement makes tractable calculations of Nielsen numbers for many maps on surfaces of negative Euler characteristic. We apply the improved method to self-maps on the connected sum of two tori including classes of examples for which no other method is known. We also include the application of this algebraic method to maps on the Klein bottle . Nielsen numbers for maps on were first calculated (geometrically) by Halpern. We include a sketch of Halpern's never published proof that for all maps on .

     

An Interpretation of the Zermelo-Fraenkel Set Theory and the Kelley-Morse Set Theory in a Positive Theory

An interesting positive theory is the GPK theory. The models of this theory include all hyperuniverses (see [5] for a definition of these ones). Here we add a form of the axiom of infinity and a new scheme to obtain GPK_∞⁺. We show that in these conditions, we can interprete the Kelley-Morse theory (KM) in GPK_∞⁺ (Theorem 3.7). This needs a preliminary property which give an interpretation of the Zermelo-Fraenkel set theory (ZF) in GPK_∞⁺. We also see what happens in the original GPK theory. Before doing this, we first need to study the basic properties of the theory. This is done in the first two sections.     

On the Consistency of a Positive Theory

In positive theories, we have an axiom scheme of comprehension for positive formulas. We study here the “generalized positive” theory GPK_∞⁺. Natural models of this theory are hyperuniverses. The author has shown in [2] that GPK_∞⁺ interprets the Kelley Morse class theory. Here we prove that GPK_∞⁺ + AC_WF (AC_WF being a form of the axiom of choice allowing to choose elements in well-founded sets) and the Kelley-Morse class theory with the axiom of global choice and the axiom “On is ramifiable” are mutually interpretable. This shows that GPK_∞⁺ + AC_WF is a “strong” theory since “On is ramifiable” implies the existence of a proper class of inaccessible cardinals.     

不动点大类与 Nielsen数

通过引进 π１ （ Ｘ,ｘ０ ） 的同态 ｆπ 的不动子群 Ｆｉｘｆπ,在 Ｈ ＝ Ｆｉｘｆπ ·ｋｅｒｆ π为π１ （ Ｘ,ｘ０ ） 的正规子群时定义了不动点大类,得到不动点大类数是有限的．在曲面 Ｘ的正则复迭空间 Ｘ Ｈ 为有限叶时只要姜子 群相对于 Ｈ 极大,ｆ 的 Ｎｉｅｌｓｅｎ 数就可以用一维同调群计算     

一种点熵的估计与计算

对紧度量空间X上的连续自映射f：X→X，Hurley利用一点逆像的(n，ε)，分离子集，引入了熵hp(f)和hm(f)。按照拓扑熵观点，它们也度量了系统(X，f)的复杂性，本文将以Nielsen根类理论为工具，首先给出hp(f)的一个恰当的下界估计；然后作为该结果的应用，我们具体计算了环面上一类自映射的熵hp(f)，同时得到了该空间上自映射同伦类熵的一个下界估计。     

Some developments in Nielsen fixed point theory

We give a brief survey of some developments in Nielsen fixed point theory. After a look at early history and a digress to various generalizations, we confine ourselves to several topics on fixed points of self-maps on manifolds and polyhedra. Special attention is paid to connections with geometric group theory and dynamics, as well as some formal approaches.     

具分布时滞的一类Logistic方程的周期正解

利用重合度定理,建立了下面具分布时滞的Logistic方程x'(t)=x(t)∑ni=1ri(t)1-(1)/(K(t))∫t-∞x(s)dsRi(t,s)正周期存在的充分条件.其中ri(t),K(t)和Ri(t,s)是以ω＞0为周期的正周期函数.     

On Some Results in Probabilistic Number Theory

The paper presents an overview of main methods and results of the value distribution problem of additive arithmetical functions.