Boju Jiang

Fixed Points and Braids. II

A basic problem in fixed point theory is to find the least number of fixed points for a homotopy class of maps. That is, given a map f : X ~ X of a connected compact polyhedron X into itself, to find M F [ f ] :=Min{ #Fix (g) lg~f :X,X} . The invariant central to this theory is the Nielson number N(f) , defined as the number of essential fixed point classes (see [1] ~)r [4]). N( f ) is always a lower bound to MF[f] and it has been a long-standing problem to prove or to disprove that N ( f ) = MFEf ]. We know [3] that if X has no local cut points and X is not a surface of negative Euler characteristic, then N ( f ) = M F [ f ] for every map f : X , X . The aim of the present paper is to generalize the result of [5] and to show

Twisted Topological Invariants Associated with Representations

The purpose of this note is to set up a framework for twisting the classical topological invariants via a matrix representation of the fundamental group, and to show how it works for two well known invariants — the Alexander polynomial and the Lefschetz number. As examples of knots with the same Alexander polynomial but different twisted Alexander polynomial have already been given by Lin, we supply some maps with zero Lefschetz number but non-zero twisted Lefschetz number.

The Wecken property of the projective plane

A proof is given of the fact that the real projective plane P 2 has the Wecken property, i.e. for every selfmap f : P 2 → P 2, the minimum number of fixed points among all selfmaps homotopic to f is equal to the Nielsen number N(f) of f . Let X be a compact connected polyhedron, and let f : X → X be a map. Let MF [f ] denote the minimum number of fixed points among all maps homotopic to f . The Nielsen number N(f) of f is always a lower bound to MF [f ]. A space X is said to have the Wecken property if N(f) = MF [f ] for all maps f : X → X. See [Br] for information about our current knowledge of such spaces. It is considered as a classical fact (cf. [J, §5]) that compact surfaces of non-negative Euler characteristic have the Wecken property. There are only seven such surfaces. The cases of the sphere, the disk, the annulus and the Mobius band are trivial. The Wecken property of the torus was first proved in [B1], later generalized to higher dimensional tori in [H1]. The Klein bottle was also treated in [B1], although the enumeration of homotopy classes of selfmaps was incomplete. A complete proof was given in the unpublished [Ha], see a sketch in [DHT, Theorem 5.8]. (The Wecken property of the Klein bottle is also a consequence of the result [HKW, Corollary 8.3] on solvmanifolds.) The case of the projective plane was only mentioned by Hopf at the end of [H1]. The purpose of this short note is to supply a proof for this case, to fill a gap in the literature. Let S be the unit sphere in the Euclidean 3-space. The map p : S → P 2 identifying antipodal pairs of points is the universal cover of the projective plane P . We know π1(P ) = H1(P ) = Z/2Z. 1991 Mathematics Subject Classification: Primary 55M20. Partially supported by NSFC. The paper is in final form and no version of it will be published elsewhere.

Applications of Nielsen theory to dynamics

In this talk, we shall look at the application of Nielsen theory to certain questions concerning the “homotopy minimum” or “homotopy stability” of periodic orbits under deformations of the dynamical system. These applications are mainly to the dynamics of surface homeomorphisms, where the geometry and algebra involved are both accessible.