Estimating Nielsen numbers on infrasolvmanifolds
Estimating Nielsen numbers on infrasolvmanifolds
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DOI：
10.2140/pjm.1992.154.345
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发表时间：
1992-06
影响因子：
0.6
通讯作者：
C. McCord
中科院分区：
文献类型：
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作者：
C. McCord
关于selfmap /: X -»X的不动点数目的一个众所周知的下界是Nielsen数N(f)。不幸的是，尼尔森的数据很难计算。另一方面，Lefschetz数L(f)是易于计算的，但它没有给出不动点数目的下界。本文研究了空间X上保证N(f) = \L(f)\或N(f) > \L(f)\的条件。通过考虑Nielsen和Lefschetz符合数，我们证明了紧次流形（其基本群具有有限指标的正规可解子群的非球面流形）上所有自映射的N(f) > \L(f)\。此外，对于基础流形，有一个计算N(f)的Lefschetz数公式。1. 估计尼尔森的数据。考虑一个连续自映射/:X -> X，设Fix（/）表示不动点集{X e X\f(X) = X}。不动点理论的一个基本问题是估计（最好是从下面）这个集合的基数。Nielsen数N(f)提供了这样一个估计：它是一个整数同伦不变量，对于所有映射g同伦到/，它提供了g不动点数目的下界。这种估计对于除负欧拉特征曲面外的所有紧流形都是尖锐的。它的一个缺点是很难从它的定义中计算出N(f)，因此必须寻找其他方法。至少，由于Nielsen数提供了原始拓扑对象|Fix(/)|的下界，因此找到N(f)的下界将是有用的。我们将把寻找N(f)的下界称为估计N(f)的问题，而寻找N(f)的精确值的其他代数拓扑方法将称为计算N(f)的问题。Lefschetz数L(f)是一个（合理地）可计算的不变量，但一般来说，L(f)与N(f)或|Fix(/)|之间没有关系。计算尼尔森数的一种方法是在空间X或地图上找到允许N(f)和L(f)相关的条件。例如，Jiang条件是映射上的一个条件，当满足时，从L(f)计算N(f)。
A well-known lower bound for the number of fixed points of a selfmap /: X —» X is the Nielsen number N(f). Unfortunately, the Nielsen number is difficult to calculate. The Lefschetz number L(f), on the other hand, is readily computable, but does not give a lower bound for the number of fixed points. In this paper, we investigate conditions on the space X which guarantee either N(f) = \L(f)\ or N(f) > \L(f)\ . By considering the Nielsen and Lefschetz coincidence numbers, we show that N(f) > \L(f)\ for all self-maps on compact infrasolvmanifolds (aspherical manifolds whose fundamental group has a normal solvable subgroup of finite index). Moreover, for infranilmanifolds, there is a Lefschetz number formula which computes N(f). 1. Estimating Nielsen numbers. Consider a continuous self-map /: X -> X. Let Fix(/) denote the fixed point set {x e X\f(x) = x} . One of the fundamental problems of fixed point theory is to estimate (preferably from below) the cardinality of this set. The Nielsen number N(f) provides such an estimate: it is an integer homotopy invariant which provides a lower bound on the number of fixed points of g, for all maps g homotopic to /. This estimate is sharp for all compact manifolds save surfaces of negative Euler characteristic. Its one drawback is that it is very difficult to compute N(f) from its definition, so that other means must be sought. At least, since the Nielsen number provides a lower bound for the original topological object |Fix(/)|, it would be useful to find lower bounds for N(f). We will refer to the search for lower bounds to N(f) as the problem of estimating N(f) while the search for other algebraic-topological means of finding the exact value of N(f) will be referred to as the problem of computing N(f). The Lefschetz number L(f) is a (reasonably) computable invariant, but in general, there is no relation between L(f) and either N(f) or |Fix(/)|. One approach to computing the Nielsen number is to find conditions on either the space X or the map / which allow N(f) and L(f) to be related. The Jiang condition, for example, is a condition on the map / which, when satisfied, computes N(f) from L(f)