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**Equivariant Nielsen fixed point theory.**
*(English)*
Zbl 1222.55001

Classical Nielsen fixed point theory studies the fixed point sets of self-maps. This paper deals with a kind of equivariant version. Consider the \(G\)-ENR’s whose isotropy type sets are finite. The author defines two \(G\)-homotopy invariants which are respectively lower bounds for the number of fixed points and orbits of all \(G\)-maps in the \(G\)-homotopy class of a given \(G\)-map. A minimum theorem is also given, i.e. under some assumptions on the \(G\)-space the two new lower bounds can be realized by some \(G\)-map in the \(G\)-homotopy class of any given \(G\)-map. The key concept here is an abstract simplicial complex whose vertices are all essential fixed point classes of the restrictions of the given \(G\)-map on \(G\)-invariant subspaces, and simplexes are subset of vertices containing an (essential or inessential) fixed point class.

It should be mentioned that an equivariant Nielsen fixed point theory was introduced by P. Wong in a series of works, beginning with [Pac. J. Math. 150, No. 1, 179–200 (1991; Zbl 0691.55004)] and [ibid. 159, No. 1, 153–175 (1993; Zbl 0739.55001)]. The approach and corresponding invariants in the present paper are different from those of P. Wong. Furthermore, the author generalizes his invariants into those estimating fixed points and orbits of equivariant relative \(G\)-maps.

It should be mentioned that an equivariant Nielsen fixed point theory was introduced by P. Wong in a series of works, beginning with [Pac. J. Math. 150, No. 1, 179–200 (1991; Zbl 0691.55004)] and [ibid. 159, No. 1, 153–175 (1993; Zbl 0739.55001)]. The approach and corresponding invariants in the present paper are different from those of P. Wong. Furthermore, the author generalizes his invariants into those estimating fixed points and orbits of equivariant relative \(G\)-maps.

Reviewer: Xuezhi Zhao (Beijing)