- Nielsen theory
Nielsen theory is a branch of mathematical research with its origins in topological 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 space to itself, denoted MF[f]. This is defined as:
where ~ indicates homotopy 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 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
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.
- Fenchel, Werner; Nielsen, Jakob; edited by Asmus L. Schmidt (2003). Discontinuous groups of isometries in the hyperbolic plane. De Gruyter Studies in mathematics. 29. Berlin: Walter de Gruyter & Co..
- Survey article on Nielsen theory by Robert F. Brown at Topology Atlas
Wikimedia Foundation. 2010.
Look at other dictionaries:
Nielsen transformation — In mathematics, especially in the area of abstract algebra known as combinatorial group theory, Nielsen transformations, named after Jakob Nielsen, are certain automorphisms of a free group which are a non commutative analogue of row reduction… … Wikipedia
Nielsen–Schreier theorem — In group theory, a branch of mathematics, the Nielsen–Schreier theorem is the statement that every subgroup of a free group is itself free. It is named after Jakob Nielsen and Otto Schreier. Contents … Wikipedia
Nielsen–Olesen vortex — In theoretical physics, a Nielsen–Olesen vortex is a point like object localized in two spatial dimensions or, equivalently, a classical solution of field theory with the same property. This particular solution occurs if the configuration space… … Wikipedia
Nielsen-Ninomiya-Theorem — Das Nielsen Ninomiya Theorem ist ein Begriff aus der Gittereichtheorie und besagt, dass bei einer Diskretisierung einer Feldtheorie mit Fermionen, die Anzahl rechts und linkshändiger Teilchen gleich ist. Voraussetzung sind lediglich einige… … Deutsch Wikipedia
Nielsen-Olesen vortex — In theoretical physics, a Nielsen Olesen vortex is a point like object localized in two spatial dimensions or, equivalently, a classical solution of field theory with the same property. This particular solution occurs if the configuration space… … Wikipedia
Nielsen–Thurston classification — In mathematics, Thurston s classification theorem characterizes homeomorphisms of a compact surface. William Thurston s theorem completes the work initiated by Jakob Nielsen (1944). Given a homeomorphism f : S → S, there… … Wikipedia
Nielsen-Thurston classification — In mathematics, Thurston s classification theorem characterizes homeomorphisms of a compact surface. William Thurston s theorem completes the work initiated by Jakob Nielsen in the 1930s.Given a homeomorphism f : S rarr; S , there is a map g… … Wikipedia
Nielsen–Olsen string — In theoretical physics, Nielsen–Olsen string is a one dimensional object or equivalently a classical solution of certain equations of motion. The solution does not depend on the direction along the string; the dependence on the other two,… … Wikipedia
Jakob Nielsen (mathematician) — For other people with similar names see Jakob Nielsen. Jakob Nielsen (October 15, 1890 – August 3, 1959) was a Danish mathematician known for his work on automorphisms of surfaces. He was born in the village Mjels on the island of Als in North… … Wikipedia
String theory — This article is about the branch of theoretical physics. For other uses, see String theory (disambiguation). String theory … Wikipedia