| chapter i. the general problem. a particular case. a few historical remarks introduction. a. integral power mappings of the circle 1. integral power mappings. the lefschetz number and fixed points 2. exponential mappings. liftings of integral power mappings 3. fixed points of liftings. lifting classes and fixed point classes b. general self-mapping of the circle 4. index or algebraic counting of an isolated fixed point 5. liftings of a self-mapping. homotopic self-mapping classes. fixed points of liftings 6. theorem l for the circle 7. lifting classes. fixed point classes 8. index of a fixed point class. the nielsen number. theorem n for the circle c. a sketch of the theory of fixed point classes. a few historical remarks 9. from particular case to theory of fixed point classes 10. a few historical remarks (1) the problem of existence or non-existence of fixed points (2) the problem of the least number of fixed points chapter ii. the nielsen number 1. lifting class and fixed point class 2. nonempty fixed point class. equivalent definition. finiteness .3. correspondence between fixed point classes induced by homotopy 4. necessary and sufficient condition for the correspondence 5. index of fixed point class. the nielsen number 6. homotopy invariance: index of the fixed point class and the nielsen number 7. commutativity: index of the fixed point class and the nielsen number chapter iii. evaluation of the nielsen number 1. endomorphism fπ of π1 (x, x0). fπ class. algebraic definition of r(f) 2. a lower bound of r(f) 3. conditions for r(f)= # coker (id-f1*) 4. jiang group and three related lemmas 5. evaluation of the nielsen number in the case of maximal jiang group 6. applications of theorems 5.1 and 5.2 chapter iv. nielsen number and the least number of fixed points 1. point-tail homotopy and line-fence homotopy 2. moving and uniting of fixed points. #φ ([id]) of 2-dimensionally connected polyhedron 3. non-2-dimensionally-connected complex. welding set. good star motion 4. #φ([id]) of non-2-dimensionally-connected complex 5. a sufficient condition for #φ ([f])= n (f).. chapter v. the number n(f;h) and the root classes a. the number n(f;h) 1. definitions and theorems under basic assumption 2. examples b. the root classes 3. from fixed point classes of self-mapping to root classes of equation 4. the correspondence between root classes induced by a homotopy 5. another subgroup s(x, x*) of the fundamental group π1(x, x*) 6. reidemeister number of equation 7. index of root class. evaluation of n(f, x*) in the case of maximal s(x, x*) appendix a. homotopy and fundamental group 1. homotopy 2. path. product and inverse. subpath 3. two types of path classes 4. from path classes with fixed endpoints to fundamental group 5. basic properties of π1(x, x0) appendix b. covering spaces 1. formal or abstract definition of covering spaces. two basic theorems on liftings of paths 2. two basic theorems on liftings of self-mappings of x 3. homomorphisms and automorphisms of covering spaces 4. geometrical construction of covering spaces 5. geometrical formulas of liftings in the universal covering space appendix c. approximation theorems 1. short homotopy between self-mappings 2. approximation theorems bibliography epilogue list of symbols index... |
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