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The Wecken property of the projective plane

100%

Jiang B.

tom 49

nr 1

223-225

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.

Applications of Nielsen theory to dynamics

100%

Jiang B.

tom 49

nr 1

203-221

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.

The twisted products of spheres that have the fixed point property

63%

Duan H. , Jiang B.

nr 2

157-168

By a twisted product of Sⁿ we mean a closed, 1-connected 2n-manifold M whose integral cohomology ring is isomorphic to that of Sⁿ × Sⁿ, n ≥ 3. We list all such spaces that have the fixed point property.

Reidemeister orbit sets

51%

Jiang B. , Lee S. , Woo M.

Fundamenta Mathematicae

2004

tom 183

nr 2

139-156

The Reidemeister orbit set plays a crucial role in the Nielsen type theory of periodic orbits, much as the Reidemeister set does in Nielsen fixed point theory. Extending Ferrario's work on Reidemeister sets, we obtain algebraic results such as addition formulae for Reidemeister orbit sets. Similar formulae for Nielsen type essential orbit numbers are also proved for fibre preserving maps.

On tame embeddings of solenoids into 3-space

51%

Jiang B. , Wang S. , Zheng H. , Zhou Q.

nr 1

57-75

Solenoids are inverse limits of the circle, and the classical knot theory is the theory of tame embeddings of the circle into 3-space. We make a general study, including certain classification results, of tame embeddings of solenoids into 3-space, seen as the "inverse limits" of tame embeddings of the circle. Some applications in topology and in dynamics are discussed. In particular, there are tamely embedded solenoids Σ ⊂ ℝ³ which are strictly achiral. Since solenoids are non-planar, this contrasts sharply with the known fact that if there is a strictly achiral embedding Y ⊂ ℝ³ of a compact polyhedron Y, then Y must be planar.