Lecture

Homotopy Lifting

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In course

MATH-225: Topology II - fundamental groups

On étudie des notions de topologie générale: unions et quotients d'espaces topologiques; on approfondit les notions de revêtements et de groupe fondamental,et d'attachements de cellules et on démontre le Théorème de Seifert-van Kampen. Des exemples de surf ...

Description

This lecture covers the concept of homotopy lifting, focusing on paths, proofs, uniqueness properties, constant paths, discrete spaces, and monomorphisms. It explains how to define constant paths, prove uniqueness, and establish properties of discrete spaces.

Instructors (2)

Jérôme Scherer

Teaching & PhD PhD Students Virgile Simon Constantin Past EPFL PhD Students Samuel Lavenir Past EPFL PhD Students as codirector Kay Remo Werndli, Lyne Moser Courses Topology IV.a - Cohomology rings MATH-506 Singular cohomology is defined by dualizing the singular chain complex for spaces. We will study its basic properties, see how it acquires a multiplicative structure and becomes a graded commutative algebra. We study an algebraic version, namely group cohomology, and compare both approa

Dimitri Stelio Wyss

I did my PhD under the supervision of Tamas Hausel at EPFL and the Institue of Science and Technology Austria . From 2017-2019 I was a postdoctoral researcher at the Sorbonne University Paris 6 with François Loeser. Since September 2019 I hold the chair of arithmetic geometry at EPFL as a tenure track assistant professor. For more information see my CV.I work in algebraic and arithmetic geometry. My research uses ideas from p-adic and motivic integration to study the interplay between geometric and arithmet

Official source

Related concepts (28)

Monomorphism

In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation . In the more general setting of , a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism. That is, an arrow f : X → Y such that for all objects Z and all morphisms g1, g2: Z → X, Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below.

Homotopy group

In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. To define the n-th homotopy group, the base-point-preserving maps from an n-dimensional sphere (with base point) into a given space (with base point) are collected into equivalence classes, called homotopy classes.

Morphism

In mathematics, particularly in , a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in analysis and topology, continuous functions, and so on.

Homotopy

In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from ὁμός "same, similar" and τόπος "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (həˈmɒtəpiː, ; ˈhoʊmoʊˌtoʊpiː, ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces.

Section (category theory)

In , a branch of mathematics, a section is a right inverse of some morphism. , a retraction is a left inverse of some morphism. In other words, if and are morphisms whose composition is the identity morphism on , then is a section of , and is a retraction of . Every section is a monomorphism (every morphism with a left inverse is left-cancellative), and every retraction is an epimorphism (every morphism with a right inverse is right-cancellative). In algebra, sections are also called split monomorphisms and retractions are also called split epimorphisms.

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