Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: COINCIDENCE THEORY FOR SPACES WHICH FIBER OVER A NILMANIFOLD | COINCIDENCE THEORY FOR SPACES WHICH FIBER OVER A NILMANIFOLD PETER WONG Received 20 August 2003 and in revised form 9 February 2004 Let Y be a finite connected complex and p Y N a fibration over a compact nilmanifold N. For any finite complex X and maps f g X Y we show that the Nielsen coincidence number N f g vanishes if the Reidemeister coincidence number R pf pg is infinite. If in addition Y is a compact manifold and g is the constant map at a point a e Y then f is deformable to a map f X Y such that f -1 a 0. 1. Introduction The celebrated Lefschetz-Hopf fixed point theorem states that if a selfmap f X X on a compact connected polyhedron X has nonvanishing Lefschetz number L f then every map homotopic to f must have a fixed point. On the other hand if L f 0 f need not be homotopic to a fixed point free map. A classical result ofWecken asserts that if X is a triangulable manifold of dimension at least three then the Nielsen number N f is the minimal number of fixed points of maps in the homotopy class of f. Thus in this case if N f 0 then f is deformable to be fixed point free. For coincidences of two maps f g X Y between closed oriented triangulable M-manifolds there is an analogous Lefschetz coincidence number L f g and L f g 0 implies x e X I f x g x 0 for all f f and g g. Schirmer 14 introduced a Nielsen coincidence number N f g and proved a Wecken-type theorem. While the theory of Nielsen fixed point coincidence classes is useful in obtaining multiplicity results in fixed point coincidence theory and in other applications the computation of the Nielsen number remains one of the most difficult and central issues. One of the major advances in recent development in computing the Nielsen number is a theorem of Anosov who proved that for any selfmap f N N of a compact nilmanifold N N f I L f I. By a nilmanifold we mean a coset space of a nilpotent Lie group by a closed subgroup. Thus the computation of N f reduces to that of the homological trace L f . Anosov s
