Fixed Point Theory for Homogeneous Spaces A Brief Survey

New directions in Nielsen–Reidemeister theory

“…This general type of result was first proved by B. Jiang (see [40] for more information on Nielsen fixed point theory) for a class of spaces now known as Jiang spaces. Jiang-type results hold for all selfmaps of a large class of spaces including simply-connected spaces, generalized lens spaces, H -spaces, topological groups, orientable coset spaces of compact connected Lie groups, nilmanifolds, certain C-nilpotent spaces where C denotes the class of finite groups, certain solvmanifolds and infra-homogeneous spaces [62]. For the equality N( f ) = R( f ) to hold, one must first determine the finiteness of the Reidemeister number R( f ).…”

“…Jiang-type results hold for all selfmaps of a large class of spaces including simply-connected spaces, generalized lens spaces, H-spaces, topological groups, orientable coset spaces of compact connected Lie groups, nilmanifolds, certain C-nilpotent spaces where C denotes the class of finite groups, certain solvmanifolds and infra-homogeneous spaces [96].…”

The R ∞ property for free groups, free nilpotent groups and free solvable groups

“…Under certain hypotheses the Reidemeister number R(ϕ) is then exactly the number of essential fixed point classes of f if R(ϕ) is finite and the number of essential fixed point classes of f is zero if R(ϕ) is infinite. See [5,9] and the references therein for more information.…”

The geometry of twisted conjugacy classes in wreath products

“…As the Reidemeister number is much easier to calculate than the Lefshetz number, this provides a valuable tool for computing the cardinality of the set of fixed points of the map f . Jiang's results have been extended to selfmaps of simply connected spaces, generalized lens spaces, topological groups, orientable coset spaces of compact connected Lie groups, nilmanifolds, certain C-nilpotent spaces where C denotes the class of finite groups, certain solvmanifolds and infra-homogeneous spaces (see, e.g., [Wo2], [Wo3]). Groups G which satisfy property R ∞ , that is, every automorphism ϕ has R(ϕ) = ∞, will never be the fundamental group of a manifold which satisfies the conditions above.…”