Nielsen Theory

Robert F. Brown

Department of Mathematics, University of California at Los Angeles
Email: rfb@math.ucla.edu
WWW: http://www.math.ucla.edu/~rfb/

The customary procedure of Nielsen theory consists of defining an equivalence relation on the set of solutions and then identifying the "essential" equivalence classes in such a way that the Nielsen number, defined as the number of essential classes, is a homotopy invariant lower bound for the minimum number. In Nielsen fixed point theory, fixed points x and x' of a given map f: X --> X are equivalent if there is a path w connecting them such that the paths w and f o w are homotopic relative to the endpoints. The fixed point equivalence classes are classified as essential or not by means of the fixed point index of algebraic topology. The resulting Nielsen number N(f) has the required property that N(f) is less than or equal to MF[f].

One goal of Nielsen theory is to calculate the Nielsen number in specific cases, either by finding explicit formulas where that is possible, or else by means of algorithms. For instance, if f: X --> X is a map of a compact Lie group, a theorem of Boju Jiang implies that N(f) is the order of the cokernel of the homomorphism I - f_#: \pi₁(X) --> \pi₁(X) where I is the identity and f_# is the fundamental group homomorphism induced by f. Another focus of attention is to establish what are called "Wecken"-type results in which the Nielsen number is shown to be equal to the minimum number of solutions to the equation among all the maps in the given homotopy class, rather than just a bound on it. The terminology refers to results of Franz Wecken in which he showed, for instance, that if f: X --> X is a map of an n-manifold where n is not 2, then N(f) = MF[f].

Some other areas of Nielsen theory are periodic point theory, relative fixed point theory, coincidence theory and root theory.

In periodic point theory, a map f: X --> X is iterated, that is, f²: X --> X is defined by f²(x) = f(f(x)) and, in general, f^k(x) = f(f^k-1(x)). A periodic point is a solution to the equation f^k(x) = x and Nielsen periodic point theory studies the minimum number of solutions to the equation g^k(x) = x among all maps g homotopic to f.

Relative fixed point theory is concerned with maps f: (X, A) --> (X,A) of compact ANR pairs and seeks the minimum number of fixed points among all maps of pairs homotopic to f through homotopies of pairs.

Concidence theory requires two maps f, g: X --> Y where, in the classical setting, X and Y are closed manifolds of the same dimension. The Nielsen coincidence number is a lower bound for the minimum number of solutions to the equation f'(x) = g'(x) among all maps f' homotopic to f and g' homotopic to g and, if n is not 2, it equals that minimum.

For root theory, there is just one map f: X --> Y between closed manifolds of the same dimension and c in Y is any point. The Nielsen root number is a lower bound for the number of solutions to g(x) = c for all maps g homotopic to f.

There is much more to Nielsen theory than the topics mentioned above. For a somewhat broader survey, along with historical background and references, see R. Brown, Fixed Point Theory in History of Topology (I. James, ed.), Elsevier, 1999, pages 271 - 299.

Nielsen theory has found applications within topology as well as in other areas of mathematics. Classical Nielsen fixed point theory has contributed to numerical analysis, see W. Forster, Some computational methods for systems of nonlinear equations and systems of polynomial equations, J. Global Optimization 2 (1992), pages 317 - 356. Although the classical setting of Nielsen fixed point theory is maps of compact ANRs, there are extensions of the theory that do not require that the spaces even be locally compact. This more general theory has been used in nonlinear analysis to establish the existence of multiple solutions to differential and integral equations, thus advancing a program for Nielsen theory first proposed by Jean Leray. Nielsen periodic point theory has been applied with considerable success in dynamics, see for instance B. Jiang, Applications of Nielsen theory to dynamics in Nielsen Theory and Reidemeister Torsion (J. Jezierski ed.), Banach Centre Publications 49 (1999), pages 203 - 221. Within topology, Nielsen root theory is used to extend the concept of topological degree, see R. Brown and H. Schirmer, Nielsen root theory and Hopf degree theory, Pacific J. Math., to appear.

Date published: August 26, 2000.