Topological Fixed Point Theory
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1. Introduction
Fixed point theory is very simple, but is based on fundamentals in Mathematics. For any continuous function
In Banach theorem, a metric on
In this article, we intend to introduce briefly topological fixed point theory. After dinner tonight, let us drink a cup of coffee. Stirring the coffee in the cup with a tea spoon, we have a look at the movement of the top of the coffee. Probably we can feel that there are points (or parts) of the coffee not moving at all. According to Brouwer theorem, such points always exist. More precisely speaking, we can regard this movement as a continuous function
Even though we can observe points of the coffee not moving, it is hard to grasp the position of such non-moving points. Brouwer theorem has nothing to do with the position of fixed points. Changing the way of stirring the coffee, fixed points may occur on the edge of the coffee and in this case, every part of the coffee you look at is moving because the fixed points occurred in the place where the coffee touches the cup ([4]).
2. Brief History
In 1880s, H Poincar´e introduced topological methods in studying nonlinear problems of analysis. One of main ideas was to use fixed point results. Together with the study under the topological setting derived from Poincar´e’s analysis motivation, L E J Brouwer’s fixed point theorem came into the world. In 1920s, the fixed point theory became a major branch of topology and afterwards it consistently became a major theme of the research.
S Lefschetz used homology theory in the study of fixed point theory and H Hopf developed it further and generalised successfully Brouwer’s results. Lefschetz introduced the Lefschetz number
If this number is not equal to 0, then it is proven that f has a fixed point. Because the Lefschetz number is defined using the homology homomorphism, it is a homotopy invariant. Consequently, if the Lefschetz number
In 1920s, J Nielsen concentrated on not only the existence problem of fixed points but also the problem of finding the (minimal) number of fixed points in the homotopy classes in a series of papers. The Nielsen theory is to study this homotopy invariant nowadays known as the Nielsen number
In 1930–40s, even though the fixed point theory was not a main stream of topology, there were important progress inside the fixed point theory itself. K Reidemeister and his student F Wecken introduced the Reidemeister trace, which combine the notion of the Lefschetz number and the Nielsen number. When the space
In this case,
That is,
On the other hand, the Reidemeister trace of
The number of nonzero terms in this sum is the Nielsen number. In particular, Wecken proved that in manifolds of dimension greater than or equal to 3, the Nielsen number is the minimal number of fixed points in the homotopy class. H Schirmer extended Wecken’s result from fixed points to coincidences: in manifolds of dimension greater than or equal to 3, the Nielsen coincidence number is the minimal number of coincidences in the homotopy class.
Nielsen theory and Wecken theory raised many questions concerning the minimal number of fixed points of continuous functions in their homotopy classes, and finally in 1980’s B Jiang solved many such questions (using Thurston theory on surfaces [12]). Upon Jiang’s successful results, the fixed point theory again became the target of attention and until now, it is a research area in which many researchers are working actively and which is related to various topics such as Floer homology. ([3]).
3. Some Research Problems
The most useful tools among mathematical tools used in the fixed point theory is the (co)homology theories. The topological fixed point theory is not simply about the equation
Here the vertical homomorphisms are the Poincar´e duality isomorphisms. Letting
If this homotopy invariant
Fix
Per
HPer
In this case, we define the minimal numbers of coincidence points up to homotopy as follows:
NP
NF
An important problem is to find these homotopy invariants NP
In order to deal with the size of fixed points in the topological fixed point theory, we need to assume that the spaces involved are compact. The spaces which are close to compact spaces are finitely dominated (FD) spaces. According to J H C Whitehead and J Milnor’s conjecture, the FD spaces having torsion-free fundamental group are homotopic to compact spaces. When the space is
Many interesting and challenging problems are scattered in the Nielsen theory. But we will briefly introduce only 3 research topics based on the above discussion.
3.1. Fixed point theory and coincidence theory
One of important and difficult problems in the Nielsen theory is the computation of the Nielsen fixed point number and the Nielsen coincidence number. The Lefschetz number and the Reidemeister number are defined algebraically, however the Nielsen number is defined geometrically and so its computation is very difficult. But for any continuous function
By the results of W Killing and H Hopf, the study of Riemannian flat manifolds is the same as that of the Euclid space forms problem. That is, an
The computation of the Nielsen number on torus has raised questions, Problem of Anosov relation and Problem of computation method, of computing the Nielsen number on some classes of manifolds containing torus. Some known results until now are given below and also a related research problem is given.
• D V Anosov [1] (nilmanifolds):
• J Jezierski-C K McCord [16] (nilmanifolds):
• [14, 10] (Special solvmanifolds of type
It is known that the Anosov relation
• Averaging Formula for fixed points and coincidences [11]:
Problem. Derive the averaging formula for the Nielsen number on (infra-)solvmanifolds.
3.2. Application to dynamical systems
One of the important examples in higher dimensional dynamics is an Anosov dynamical system. A linear transformation with integer coefficients induces a map on the standard torus and when this induced map is an Anosov automorphism, this map has a very rich dynamical structure. Namely, this map is a chaotic map. In particular, the set of periodic points of this map is a dense subset of the torus ([6]). The study of Anosov systems on (infra-)nilmanifolds has big progressed by A Manning ([15]). On the other hand, when the automorphism is replaced by an endomorphism, among other dynamical structures on infra-nilmanifolds, we have studied the density of the periodic points ([9]). That is, the eventually periodic points form a dense subset. Now we may raise the following question.
Problem. Every nil-endomorphism has dense periodic points.
On the other hand, S Smale defined the Lefschetz zeta function, which reflects periodic point structure of corresponding dynamical system, i.e., what kind of periods of periodic point classes for a discrete dynamical system
In a similar way, we can define the Nielsen zeta function and the Reidemeister zeta function.
It can be shown easily that the Lefschetz zeta function is a rational function. A Felshtyn showed that the Nielsen zeta function and the Reidemeister zeta function are rational functions on nilmanifolds and then he raised the question that the Nielsen zeta function and the Reidemeister zeta function are rational functions on infranilmanifolds. Using the averaging formula (3.1), it is claimed in [5] that the Nielsen zeta function and the Reidemeister zeta function on infranilmanifolds are still rational. More generally,
Problem. ([7]). Determine on which spaces and for which functions the rationality and the algebraicity of the Nielsen zeta function and the Reidemeister zeta function.
In the topological discrete dynamical systems, the change of the minimal number of periodic points and the minimal period of periodic points are observed in the homotopy category. Hence computing the Nielsen type numbers NP
3.3. Bass conjecture
First of all we shall introduce briefly on Bass conjecture. For any group
Problem. (Bass Conjecture). If
R Geoghegan developed the Nielsen theory on FD (finitely dominated) spaces. An FD space is a space from which there is a retraction onto a compact space under homotopy. Main idea was to lift the Nielsen theory from compact spaces to FD spaces. The Nielsen number of the identity on compact spaces is either 0 or 1.
Problem. (Geoghegan Conjecture). The Nielsen number of the identity on FD spaces is either 0 or 1.
When G is a finitely presented group, Geoghegan showed that his conjecture and Bass conjecture are equivalent. To understand this, we need to understand C T C Wall’s finiteness obstruction and so any detailed explanation is not appropriate here and the readers refer to the reference [8]. We close this article by emphasising again that this conjecture is the most important problem in the Nielsen theory.
References
[1] D. V. Anosov, Nielsen numbers of mappings of nil-manifolds, Uspekhi Mat. Nauk 40(4) (244) (1985) 133–134.
[2] R. B. S. Brooks, R. F. Brown, J. Pak and D. H. Taylor, Nielsen numbers of maps of tori, Proc. Amer. Math. Soc. 52 (1975) 398–400.
[3] R. Brown, Fixed point theory, History of Topology (North-Holland, Amsterdam, 1999), pp. 271–299.
[4] R. Brown and R. Greene, An interior fixed point property of the disc, Amer. Math. Monthly 101 (1994) 39–47.
[5] K. Dekimpe and G.-J. Dugardein, Nielson zeta functions for maps on infra-nilmanifolds are rational, arXiv:1302.5512.
[6] R. L. Devaney, An Introduction to Chaotic Dynamical Systems (Addison-Wesley Publishing Company, 1989).
[7] A. Fel´shtyn, Dynamical zeta functions, Nielsen theory and Reidemeister torsion,Mem. Amer.Math. Soc. 147(699) Amer. Math. Soc., Providence, R.I., 2000).
[8] R. Geoghegan, Fixed points in finitely dominated compacta: the geometric meaning of a conjecture of H. Bass, in Shape Theory and Geometric Topology, Lecture Notes in Math., Vol. 870 (Springer, New York, 1981), pp. 6–22.
[9] K. Y. Ha, H. J. Kim and J. B. Lee, Eventually periodic points of infra-nil endomorphisms, Fixed Point Theory and Appl. (2010).
[10] K. Y. Ha, J. B. Lee and P. Penninckx, Anosov theorem for coincidences on special solvmanifolds of type (R), Proc. Amer. Math. Soc. 139 (2011) 2239–2248.
[11] K. Y. Ha, J. B. Lee and P. Penninckx, Formulas for the Reidemeister, Lefschetz and Nielsen coincidence number of maps between infranilmanifolds, Fixed Point Theory and Appl. (2012).
[12] B. Jiang, Fixed points of surface homeomorphisms, Bull. Amer. Math. Soc. 5 (1981) 176–178.
[13] B. Jiang, Lectures on Nielsen fixed point theory, Contemporary Math. 14 (Amer. Math. Soc., Providence, R.I., 1983).
[14] S. W. Kim and J. B. Lee, Anosov theorem for coincidences on nilmanifolds, Fund. Math. 185 (2005) 247–259.
[15] A. Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math. 96 (1974) 422–429.
[16] C. K. McCord, Lefschetz and Nielsen coincidence numbers on nilmanifolds and solvmanifolds, II,
Topology Appl. 75 (1997) 81–92.
[17] H. Schirmer, Mindestzahlen von Koinzidenzpunkten, J. Reine Angew. Math. 194 (1955) 21–39.
[18] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967) 747–817.
(Translated from the Newsletter of Korean Mathematical Society 141 (2012) 6-11)
Sogang University, Korea
Professor Jong Bum Lee received his BS and MS degrees from the Sogang University respectively in 1980 and 1982. He obtained his PhD degree from the University of Michigan in 1990. He is currently a professor at the Sogang University. His research interests include transformation group theory and Nielsen fixed point theory.
[This article originally appeared in the Asia Pacific Mathematics Newsletter, Volume 3 No. 3 (July 2013), published by World Scientific. It has been republished here with a special permission from World Scientific.]