Lefschetz-Hopf定理和莱夫谢茨数的公理
The Lefschetz-Hopf theorem and axioms for the Lefschetz number
摘要：
本文证明了约化的莱夫谢茨数，即莱夫谢茨数减1，是紧致多面体自映射上唯一的整值函数L，它在同伦类上是常数，使得（1）L（fg）= L（gf），对f：X -->Y和g：Y -->X;（2）若（f_1，f_2，f_3）是上纤维序列到自身的映射，则L（f_2）= L（f_1）+ L（f_3）; + 度（p_k f e_k）），其中f是k个圆的楔形的映射，e_r是圆到第r个被加数中的包含，p_r是到第r个被加数上的投影。设f：X -->X是多面体的自映射，I（f）是f在所有X上的不动点指数，则证明了I减1满足上述公理.这给出了正规化定理的一个新证明：如果f：X -->X是多面体的自映射，则I（f）等于f的莱夫谢茨数.这个结果等价于Lefschetz-Hopf定理：如果f：X ->X是一个具有有限个不动点的有限单纯复形的自映射，每个不动点位于一个极大单纯形中，那么f的莱夫谢茨数是f的所有不动点的指数之和。
英文摘要：
The reduced Lefschetz number, that is, the Lefschetz number minus 1, is proved to be the unique integer-valued function L on selfmaps of compact polyhedra which is constant on homotopy classes such that (1) L(fg) = L(gf), for f:X -->Y and g:Y -->X; (2) if (f_1, f_2, f_3) is a map of a cofiber sequence into itself, then L(f_2) = L(f_1) + L(f_3); (3) L(f) = - (degree(p_1 f e_1) + ... + degree(p_k f e_k)), where f is a map of a wedge of k circles, e_r is the inclusion of a circle into the rth summand and p_r is the projection onto the rth summand. If f:X -->X is a selfmap of a polyhedron and I(f) is the fixed point index of f on all of X, then we show that I minus 1 satisfies the above axioms. This gives a new proof of the Normalization Theorem: If f:X -->X is a selfmap of a polyhedron, then I(f) equals the Lefschetz number of f. This result is equivalent to the Lefschetz-Hopf Theorem: If f: X -->X is a selfmap 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.
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