Lecture

Chain Maps: Homotopy Invariance

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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 concept of chain maps between chain complexes, including homotopy invariance, homology groups of quotients, and induced homomorphisms. It explains how chain maps induce homomorphisms between cycles and boundaries, ensuring the commutativity of certain diagrams.

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

Algebra: Abstract algebra

Related concepts (32)

Chain complex

In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the of each homomorphism is included in the kernel of the next. Associated to a chain complex is its homology, which describes how the images are included in the kernels. A cochain complex is similar to a chain complex, except that its homomorphisms are in the opposite direction. The homology of a cochain complex is called its cohomology.

Homology (mathematics)

In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry. The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes.

Induced homomorphism

In mathematics, especially in algebraic topology, an induced homomorphism is a homomorphism derived in a canonical way from another map. For example, a continuous map from a topological space X to a topological space Y induces a group homomorphism from the fundamental group of X to the fundamental group of Y. More generally, in , any functor by definition provides an induced morphism in the target for each morphism in the source category.

Quotient space (topology)

In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonical projection map (the function that maps points to their equivalence classes). In other words, a subset of a quotient space is open if and only if its under the canonical projection map is open in the original topological space.

Homotopy category of chain complexes

In homological algebra in mathematics, the homotopy category K(A) of chain complexes in an A is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of chain complexes Kom(A) of A and the D(A) of A when A is ; unlike the former it is a , and unlike the latter its formation does not require that A is abelian. Philosophically, while D(A) turns into isomorphisms any maps of complexes that are quasi-isomorphisms in Kom(A), K(A) does so only for those that are quasi-isomorphisms for a "good reason", namely actually having an inverse up to homotopy equivalence.

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