Lecture

The Fundamental Group of a Surface

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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 the fundamental group of a surface, starting with the associated polygonal presentation and the structure of rings and body. The proof of the cell structure of the surface is explained, along with the identification of vertices and the attachment application. The lecture also discusses the process of choosing vertices and summits, as well as walking through the surface to understand the attachment application.

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

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Related concepts (30)

Attachment theory

Attachment theory is a psychological, evolutionary and ethological theory concerning relationships between humans. The most important tenet is that young children need to develop a relationship with at least one primary caregiver for normal social and emotional development. The theory was formulated by psychiatrist and psychoanalyst John Bowlby. Within attachment theory, infant behaviour associated with attachment is primarily the seeking of proximity to an attachment figure in stressful situations.

Attachment therapy

Attachment therapy (also called "the Evergreen model", "holding time", "rage-reduction", "compression therapy", "rebirthing", "corrective attachment therapy", and "coercive restraint therapy") is a pseudoscientific child mental health intervention intended to treat attachment disorders. It is found primarily in the United States, and much of it is centered in about a dozen clinics in Evergreen, Colorado, where Foster Cline, one of the founders, established his clinic in the 1970s.

Attachment in children

Attachment in children is "a biological instinct in which proximity to an attachment figure is sought when the child senses or perceives threat or discomfort. Attachment behaviour anticipates a response by the attachment figure which will remove threat or discomfort". Attachment also describes the function of availability, which is the degree to which the authoritative figure is responsive to the child's needs and shares communication with them.

Category of rings

In mathematics, the category of rings, denoted by Ring, is the whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of rings is , meaning that the class of all rings is proper. The category Ring is a meaning that the objects are sets with additional structure (addition and multiplication) and the morphisms are functions that preserve this structure.

Product of rings

In mathematics, a product of rings or direct product of rings is a ring that is formed by the Cartesian product of the underlying sets of several rings (possibly an infinity), equipped with componentwise operations. It is a direct product in the . Since direct products are defined up to an isomorphism, one says colloquially that a ring is the product of some rings if it is isomorphic to the direct product of these rings.

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