Nielsen theory is a branch of mathematical research with its origins in

topological Topology (from the Greek words , and ) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, wit ...

fixed-point theory. Its central ideas were developed by Danish mathematician Jakob Nielsen, and bear his name. The theory developed in the study of the so-called ''minimal number'' of a map ''f'' from a

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

space to itself, denoted ''MF'' 'f'' This is defined as: :

= \min \,

where ''~'' indicates

homotopy In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. ...

of mappings, and #Fix(''g'') indicates the number of fixed points of ''g''. The minimal number was very difficult to compute in Nielsen's time, and remains so today. Nielsen's approach is to group the fixed-point set into classes, which are judged "essential" or "nonessential" according to whether or not they can be "removed" by a homotopy. Nielsen's original formulation is equivalent to the following: We define an

equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...

on the set of fixed points of a self-map ''f'' on a space ''X''. We say that ''x'' is equivalent to ''y'' if and only if there exists a path ''c'' from ''x'' to ''y'' with ''f''(''c'') homotopic to ''c'' as paths. The equivalence classes with respect to this relation are called the Nielsen classes of ''f'', and the Nielsen number ''N''(''f'') is defined as the number of Nielsen classes having non-zero fixed-point index sum. Nielsen proved that :

N(f) \le \mathit

making his invariant a good tool for estimating the much more difficult ''MF'' 'f'' This leads immediately to what is now known as the Nielsen fixed-point theorem: ''Any map f has at least N(f) fixed points.'' Because of its definition in terms of the fixed-point index, the Nielsen number is closely related to the Lefschetz number. Indeed, shortly after Nielsen's initial work, the two invariants were combined into a single "generalized Lefschetz number" (more recently called the Reidemeister trace) by Wecken and Reidemeister.

Bibliography

*{{cite book , last=Fenchel , first=Werner , author-link=Werner Fenchel , author2=Nielsen, Jakob , author2-link=Jakob Nielsen (mathematician) , editor=Asmus L. Schmidt , title=Discontinuous groups of isometries in the hyperbolic plane , series=De Gruyter Studies in mathematics , volume=29 , publisher=Walter de Gruyter & Co. , location=Berlin , date=2003

External links

Survey article on Nielsen theory
by Robert F. Brown at Topology Atlas Fixed-point theorems Fixed points (mathematics) Topology