Print Email Facebook Twitter on Cohomology and Ext-groups Title on Cohomology and Ext-groups Author Bakker, Hidde (TU Delft Electrical Engineering, Mathematics and Computer Science) Contributor de Jong, Robin (mentor) Hart, K.P. (mentor) Kraaij, R.C. (graduation committee) Degree granting institution Delft University of Technology Programme Electrical Engineering Date 2023-07-03 Abstract This thesis is about homological algebra and singular (co)homology. In the first chapter the notions of complexes of abelian groups, (co)homology of these complexes and injective resolutions will be introduced. Then Ext-groups will be defined and various properties dervied. A particularly interesting group, Ext(Q,Z), will be calculated which involves the p-adic integers. Lastly we will prove the universal coefficient theorem for complexes of free abelian groups.In the second chapter we will use the tools provided by the previous chapter to calculate the singular homology groups of topological spaces. First we will explicitely describe the zero'th and first singular homology groups for any topological space. For the spheres S^n and real projective n space P^n(R) we will calculate all singular homology and cohomology groups. For this we will use the universal coefficient and properties about Ext-groups which have been proven in chapter 1. We will also prove and use the long exact sequence of Mayer-Vietoris. This theorem proposes a way to calculate the singular homology groups of a space by using the singular homology groups of two subspaces. Subject Algebraic TopologyHomological AlgebraCategory TheorySingular HomologyUniversal Coefficient Theorem To reference this document use: http://resolver.tudelft.nl/uuid:a385741f-3723-4f96-b7e1-fa319e908e63 Part of collection Student theses Document type bachelor thesis Rights © 2023 Hidde Bakker Files PDF Cohomology_and_Ext_groups.pdf 410.45 KB Close viewer /islandora/object/uuid:a385741f-3723-4f96-b7e1-fa319e908e63/datastream/OBJ/view