Lecture

Zig Zag Lemma

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

In course

MATH-323: Topology III - Homology

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 its properties and learn how to compute it. There will be m ...

Description

This lecture covers the Zig Zag Lemma, focusing on the long exact sequence of relative homology and induced homomorphisms on relative complexes. The instructor explains the proof of Theorem A and the concept of excision. The lecture emphasizes the importance of chain complexes and the construction of short exact sequences. Various diagrams are used to illustrate the concepts discussed.

Instructor

Leonid Monin

Teaching & PhD Courses Algebra III - rings and fields MATH-215 This is an introductory course in ring and field theory. Analysis I (English) MATH-101(en) We study the fundamental concepts of analysis, calculus and the integral of real-valued functions of a real variable. Topology III - Homology MATH-323 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

Official source

Ontological neighbourhood

Mathematics

Topology: Algebraic topology

Logic

Classical logic: First-order logic

Related concepts (31)

Excision theorem

In algebraic topology, a branch of mathematics, the excision theorem is a theorem about relative homology and one of the Eilenberg–Steenrod axioms. Given a topological space and subspaces and such that is also a subspace of , the theorem says that under certain circumstances, we can cut out (excise) from both spaces such that the relative homologies of the pairs into are isomorphic. This assists in computation of singular homology groups, as sometimes after excising an appropriately chosen subspace we obtain something easier to compute.

Relative homology

In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intuitively, it helps determine what part of an absolute homology group comes from which subspace. Given a subspace , one may form the short exact sequence where denotes the singular chains on the space X. The boundary map on descends to and therefore induces a boundary map on the quotient.

Homotopical algebra

In mathematics, homotopical algebra is a collection of concepts comprising the nonabelian aspects of homological algebra, and possibly the aspects as special cases. The homotopical nomenclature stems from the fact that a common approach to such generalizations is via abstract homotopy theory, as in nonabelian algebraic topology, and in particular the theory of . This subject has received much attention in recent years due to new foundational work of Vladimir Voevodsky, Eric Friedlander, Andrei Suslin, and others resulting in the A1 homotopy theory for quasiprojective varieties over a field.

Homomorphism

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word homomorphism comes from the Ancient Greek language: ὁμός () meaning "same" and μορφή () meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).

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.

Related lectures (64)

Relative Homology: Exact Sequence

Covers the long exact sequence of relative homology groups and chain complexes.

Group Cohomology

Covers the concept of group cohomology, focusing on chain complexes, cochain complexes, cup products, and group rings.

Simplicial and Singular Homology Equivalence

Demonstrates the equivalence between simplicial and singular homology, proving isomorphisms for finite s-complexes and discussing long exact sequences.

Algebraic Kunneth Theorem

Covers the Algebraic Kunneth Theorem, explaining chain complexes and cohomology computations.

Induced Homomorphisms on Relative Homology Groups