on Cohomology and Ext-groups

on Cohomology and Ext-groups

Bakker, Hidde (TU Delft Electrical Engineering, Mathematics and Computer Science)

de Jong, Robin (mentor)
Hart, K.P. (mentor)
Kraaij, R.C. (graduation committee)

Delft University of Technology

Electrical Engineering

2023-07-03

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.

Algebraic Topology
Homological Algebra
Category Theory
Singular Homology
Universal Coefficient Theorem

http://resolver.tudelft.nl/uuid:a385741f-3723-4f96-b7e1-fa319e908e63

Student theses

bachelor thesis

© 2023 Hidde Bakker