The local Nielsen fixed point theory of fibre preserving maps

The use of fibre techniques plays the important role in topological fixed point theory and its applications.In this book, we introduce recent developments in Nielsen fixed point theory of fibre preserving map.There is well known addition formulae expressing the Nielsen number of the fibre preserving map as a simple sum of Nielsen numbers on the fibres.We generalize these ideas to the setting of fibre preserving maps defined locally.We first extend the product formula for the index of a fixed point class of a fibre preserving map to the setting of fibre preserving maps defined locally.Then, we prove addition formula for Nielsen number of fibre preserving maps defined locally.Finally we introduce a Nielsen type number of fibre preserving maps defined locally, show that it is a lower bound for the least number of fixed points within the fibre homotopy class. Some calculations and examples are given.