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:
 * $$\mathit{MF}[f] = \min \{ \# \mathrm{Fix}(g) \, | \, g \sim f \},$$

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
 * $$N(f) \le \mathit{MF}[f],$$

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.