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

topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

fixed-point theory Fixed point may refer to: * Fixed point (mathematics), a value that does not change under a given transformation * Fixed-point arithmetic, a manner of doing arithmetic on computers * Fixed point, a benchmark (surveying) used by geodesists * Fixed p ...

. 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 A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...

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

= \min \,

where ''~'' indicates

homotopy In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defor ...

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 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 A path is a route for physical travel – see Trail. Path or PATH may also refer to: Physical paths of different types * Bicycle path * Bridle path, used by people on horseback * Course (navigation), the intended path of a vehicle * Desire p ...

''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 In mathematics, the fixed-point index is a concept in topological fixed point (mathematics), fixed-point theory, and in particular Nielsen theory. The fixed-point index can be thought of as a Multiplicity (mathematics), multiplicity measurement for ...

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

, 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