The Lefschetz-Hopf theorem and axioms for the Lefschetz number
| dc.creator | Arkowitz, Martin | |
| dc.creator | Brown, Robert F. | |
| dc.date | 2004-03-24 | |
| dc.date.accessioned | 2026-07-07T05:06:43Z | |
| dc.date.available | 2026-07-07T05:06:43Z | |
| dc.description | 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. | |
| dc.description | 9 pages | |
| dc.identifier | https://arxiv.org/abs/math/0403421 | |
| dc.identifier | http://arxiv.org/abs/math/0403421 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/70581 | |
| dc.subject | Algebraic Topology | |
| dc.subject | 55M20 | |
| dc.title | The Lefschetz-Hopf theorem and axioms for the Lefschetz number | |
| dc.type | text |