Let f : V → X be a G-map
defined on an open invariant subset V of a G-ENR X where G is a compact Lie
group and the G-action on V is not necessarily free. In this paper, we introduce the
notion of an equivariant Nielsen number NGc(f,V ) which is an ordered
k-tuple that depends on the isotropy types (H1),…,(Hk) of V . When G is
finite, NGc(f,V ) gives a lower bound for the minimal number of fixed points
in the (restricted) G-homotopy class of f and this lower bound is sharp
when the G-action on V is free. We relate NGc(f,V ) to a local equivariant
obstruction to G-deforming a map to be fixed point free and we discuss the
relationship between the equivariant Nielsen number and the ordinary Nielsen
number.
