< Back to previous page

Project

An averaging formula for Nielsen numbers on infra-solvmanifolds

We prove an averaging formula for Nielsen numbers on any (compact) infra-solvmanifold. Our formula generalises the celebrated averaging formulas for Nielsen numbers on infra-nilmanifolds and infra-solvmanifolds of type (R).

A compact manifold X is said to satisfy the R-infinity property if every self homotopy equivalence of X has infinite Reidemeister number. If X satisfies the R-infinity property and X has a nice structure, like weakly Jiang, the Nielsen number of every self homotopy equivalence of X vanishes. Geometrically, this means that every self homotopy equivalence is homotopic to a fixed point free map. This inspires us to say that a compact manifold X satisfies the N-zero property if the Nielsen number of every self homotopy equivalence is zero. Using our averaging formula, we then classify all (compact) solvmanifolds up to dimension four satisfying the N_zero property.

Every infra-solvmanifold can be realised as a so-called polynomial manifold. A polynomial manifold is a quotient space of the form M=R^n/G where G is a virtually polycyclic group acting freely and properly discontinuously on R^n via polynomial diffeomorphims. In this setting, any self-map of M is homotopic to a polynomial map, by which we mean a map q induced by a polynomial p : R^n → R^n. We reformulate our averaging formula in terms of the polynomial lift p. Additionally, we study the number of fixed points of these polynomial maps and find that polynomial maps have either infinitely many fixed points or exactly N(q) fixed points. Polynomial maps in a sense thus realise the minimal number of fixed points prescribed by the Nielsen number.

Finally, we employ our averaging formula to study the rationality of the Nielsen zeta function on a (compact) solvmanifold. The Nielsen zeta function is known to be rational on solvmanifolds of type (E). We extend this result to the class of NR-solvmanifolds, which were introduced by Keppelmann and McCord as a class of solvmanifolds satisfying the Anosov relation. We next verify that Nielsen zeta functions on any solvmanifold up to dimension five are rational, and conjecture rationality in any dimension.

Date:15 Sep 2017 →  4 Oct 2021
Keywords:Nielsen number, Reidemeister number, infra-solvmanifolds, fixed-point theory
Disciplines:Analysis, Applied mathematics in specific fields, General mathematics, History and foundations, Other mathematical sciences and statistics
Project type:PhD project