# Simplicial complexes and Lefschetz fixed-point theorem

 dc.contributor Helsingin yliopisto, Matemaattis-luonnontieteellinen tiedekunta fi dc.contributor University of Helsinki, Faculty of Science en dc.contributor Helsingfors universitet, Matematisk-naturvetenskapliga fakulteten sv dc.contributor.author Siekkinen, Aku dc.date.issued 2019 dc.identifier.uri URN:NBN:fi:hulib-201910303817 dc.identifier.uri http://hdl.handle.net/10138/306572 dc.description.abstract We study a subcategory of topological spaces called polyhedrons. In particular, the work focuses on simplicial complexes out of which polyhedrons are constructed. With simplicial complexes we can calculate the homology groups of polyhedrons. These are computationally easier to handle compared to singular homology groups. en We start by introducing simplicial complexes and simplicial maps. We show how polyhedrons and simplicial complexes are related. Simplicial maps are certain maps between simplicial complexes. These can be transformed to piecewise linear maps between polyhedrons. We prove the simplicial approximation theorem which states that for any continuous function between polyhedrons we can find a piecewise linear map which is homotopic to the continuous function. In section 4 we study simplicial homology groups. We prove that on polyhedrons the simplicial homology groups coincide with singular homology groups. Next we give an algorithm for calculating the homology groups from matrix presentations of boundary homomorphisms. Also examples of these calculations are given for some polyhedrons. In the last section, we assign an integer called the Lefschetz number for continuous maps from a polyhedron to itself. It is calculated using the induced map between homology groups of the polyhedron. With the help of Hopf trace theorem we can also calculate the Lefschetz number using the induced map between chain complexes of the polyhedron. We prove the Lefschetz fixed-point theorem which states that if the Lefschetz number is not zero, then the continuous function has a fixed-point. dc.language.iso eng dc.publisher Helsingin yliopisto fi dc.publisher University of Helsinki en dc.publisher Helsingfors universitet sv dc.subject algebraic topology en dc.subject simplicial complex en dc.subject simplicial homology en dc.subject Lefschetz fixed-point theorem en dc.title Simplicial complexes and Lefschetz fixed-point theorem en dc.type.ontasot pro gradu -tutkielmat fi dc.type.ontasot master's thesis en dc.type.ontasot pro gradu-avhandlingar sv dc.subject.discipline Matematiikka und dct.identifier.urn URN:NBN:fi:hulib-201910303817