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: THE LEFSCHETZ-HOPF THEOREM AND AXIOMS FOR THE LEFSCHETZ NUMBER | THE LEFSCHETZ-HOPF THEOREM AND AXIOMS FOR THE LEFSCHETZ NUMBER MARTIN ARKOWITZ AND ROBERT F. BROWN Received Received 28 August 2003 The reduced Lefschetz number that is L - 1 where L denotes the Lefschetz number is proved to be the unique integer-valued function A on self-maps of compact poly-hedra which is constant on homotopy classes such that 1 A fg A gf for f X Y and g Y X 2 if f1 f2 f3 is a map of a cofiber sequence into itself then A f1 A f1 A f3 3 A f deg p1 fe1 deg pkfek where f is a self-map of a wedge of k circles er is the inclusion of a circle into the rth summand and pr is the projection onto the rth summand. If f X X is a self-map of a polyhedron and I f is the fixed-point index of f on all of X then we show that I 1 satisfies the above axioms. This gives a new proof of the normalization theorem if f X X is a self-map of a polyhedron then I f equals the Lefschetz number L f of f. This result is equivalent to the Lefschetz-Hopf theorem if f X X is a self-map of a finite simplicial complex with a finite number of fixed points each lying in a maximal simplex then the Lefschetz number of f is the sum of the indices of all the fixed points of f. 1. Introduction Let X be a finite polyhedron and denote by H X its reduced homology with rational coefficients. Then the reduced Euler characteristic of X denoted by X is defined by X Z -1 k dim Hk X . k Clearly X is just the Euler characteristic minus one. In 1962 Watts 13 characterized the reduced Euler characteristic as follows. Let e be a function from the set of finite poly-hedra with base points to the integers such that i e S0 1 where S0 is the 0-sphere and ii e X e A e XZA where A is a subpolyhedron of X. Then e X X . Let be the collection of spaces X of the homotopy type of a finite connected CW-complex. If X e we do not assume that X has a base point except when X is a sphere or a wedge of spheres. It is not assumed that maps between spaces with base points are based. A map f X X where X e induces .
