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 h ...

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 Jacob or Jakob Nielsen may refer to: * Jacob Nielsen, Count of Halland (died c. 1309), great grandson of Valdemar II of Denmark * , Norway (1768-1822) * Jakob Nielsen (mathematician) (1890–1959), Danish mathematician known for work on automorphi ...

, 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 Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

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 deforma ...

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. Each 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 theory, and in particular Nielsen theory. The fixed-point index can be thought of as a multiplicity measurement for fixed points. The index can be easily defined in the ...

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 In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is named ...

. Indeed, shortly after Nielsen's initial work, the two invariants were combined into a single "generalized Lefschetz number" (more recently called the

Reidemeister trace Kurt Werner Friedrich Reidemeister (13 October 1893 – 8 July 1971) was a mathematician born in Braunschweig (Brunswick), Germany. Life He was a brother of Marie Neurath. Beginning in 1912, he studied in Freiburg, Munich, Marburg, and Göttinge ...

) 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 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 h ...

Fixed-point theorems Fixed points (mathematics) Topology