Izolacija negibnih točk preslikav na kompaktnem poliedru

V tem članku obravnavam izolacijo negibnih točk preslikav na kompaktnem poliedru. Najprej si bomo ogledali, kaj so poliedri, in izpeljali lastnosti, ki jih bomo kasneje potrebovali. Na poliedrih definiramo simplicialne preslikave, ki jih bomo potrebovali za deformiranje preslikave.

Pokazali bomo, da je mogoče vsako preslikavo “malo” deformirati v simplicialno preslikavo, ki ima izolirane negibne točke. S tem bomo dokazali, da je mogoče izolirati negibne točke vsake preslikave na kompaktnem poliedru.

Da je preslikave mogoče “malo” deformirati v preslikave z izoliranimi negibnimi točkami, je topološka invarianta in zato velja tudi za vse prostore, ki so homeomorfni poliedrom.

Isolating fixed points of maps on compact polyhedra

This article considers the isolation of fixed points of maps on compact polyhedra. First we have a look at what polyhedra are and then we show a few properties we are going to need later. We also define simplicial maps which we will need for deforming the map.

We show that every map can be slightly modified into a simplicial map which has isolated fixed points. That proves it is possible to isolate fixed points of an arbitrary map on a compact polyhedron.

The property that all maps can be slightly modified to a map with isolated fixed points is a topological invariant and therefore transfers to all spaces homeomorphic to polyhedra.