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Lecture# Mayer-Vietoris Sequence

Description

This lecture covers the Mayer-Vietoris sequence, which deals with the exact sequence of homomorphisms between abelian groups. It explains the concept of complementary embedded spheres and the commutativity of rows in the homomorphism diagram. The lecture also discusses the isomorphisms between groups and the implications of the Barratt-Whitehead Lemma. Furthermore, it explores the application of the Mayer-Vietoris sequence in reduced homology and the van Kampen theorem. Through various examples, the instructor demonstrates how path-connected open subsets lead to path-connected spaces. The lecture concludes by emphasizing the importance of understanding the exact sequences and the abelianization process.

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In course

Related concepts (96)

MATH-323: Algebraic topology

Homology is one of the most important tools to study topological spaces and it plays an important role in many fields of mathematics. The aim of this course is to introduce this notion, understand it

Christians

Christians (ˈkɹɪstʃən,_-tiən) are people who follow or adhere to Christianity, a monotheistic Abrahamic religion based on the life and teachings of Jesus Christ. The words Christ and Christian derive from the Koine Greek title Christós (Χριστός), a translation of the Biblical Hebrew term mashiach (מָשִׁיחַ) (usually rendered as messiah in English). While there are diverse interpretations of Christianity which sometimes conflict, they are united in believing that Jesus has a unique significance.

Christianity

Christianity (ˌkrɪstɪˈænɪti) is an Abrahamic monotheistic religion based on the life and teachings of Jesus of Nazareth. It is the world's largest and most widespread religion with roughly 2.4 billion followers representing one-third of the global population. Its adherents, known as Christians, are estimated to make up a majority of the population in 157 countries and territories, and believe that Jesus is the Son of God, whose coming as the Messiah was prophesied in the Hebrew Bible (called the Old Testament in Christianity) and chronicled in the New Testament.

Exact sequence

An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an ) such that the of one morphism equals the kernel of the next. In the context of group theory, a sequence of groups and group homomorphisms is said to be exact at if . The sequence is called exact if it is exact at each for all , i.e., if the image of each homomorphism is equal to the kernel of the next. The sequence of groups and homomorphisms may be either finite or infinite.

Christianization

Christianization (or Christianisation) is a term for the specific type of change that occurs when someone or something has been or is being converted to Christianity. The term is applied to more than one type of conversion. For example, it can describe the changes that follow an individual's conversion, and it can also be used to designate the conversion of previously non-Christian practices, spaces and places to Christian uses and names.

Christian theology

Christian theology is the theology of Christian belief and practice. Such study concentrates primarily upon the texts of the Old Testament and of the New Testament, as well as on Christian tradition. Christian theologians use biblical exegesis, rational analysis and argument.

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