Algebraic topology

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Below are some of the main areas studied in algebraic topology: 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, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. Homology In algebraic topology and abstract algebra, homology (in part from Greek ὁμός homos "identical") is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group. Cohomology In homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign 'quantities' to the chains of homology theory. Manifold A manifold is a topological space that near each point resembles Euclidean space. Examples include the plane, the sphere, and the torus, which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot be embedded in three dimensions, but can be embedded in four dimensions.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (8)

Related people (1)

Related concepts (193)

Related courses (29)

Related lectures (123)

Algebraic Homotopy Interleaving Distance

Nicolas Michel Berkouk

The theory of persistence, which arises from topological data analysis, has been intensively studied in the one-parameter case both theoretically and in its applications. However, its extension to the

SPRINGER INTERNATIONAL PUBLISHING AG2021

Numerical Methods for First and Second Order Fully Nonlinear Partial Differential Equations

Dimitrios Gourzoulidis

This thesis focuses on the numerical analysis of partial differential equations (PDEs) with an emphasis on first and second-order fully nonlinear PDEs. The main goal is the design of numerical methods

EPFL2021

Topological exploration of artificial neuronal network dynamics

Kathryn Hess Bellwald, Gard Spreemann

One of the paramount challenges in neuroscience is to understand the dynamics of individual neurons and how they give rise to network dynamics when interconnected. Historically, researchers have resor

MIT PRESS2019

MATH-506: Topology IV.b - cohomology rings

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 a

MATH-497: Homotopy theory

We propose an introduction to homotopy theory for topological spaces. We define higher homotopy groups and relate them to homology groups. We introduce (co)fibration sequences, loop spaces, and suspen

MATH-726(2): Working group in Topology II

The theme of the working group varies from year to year. Examples of recent topics studied include: Galois theory of ring spectra, duality in algebra and topology, topological algebraic geometry and t

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.

Algebraic topology

Michael Atiyah

Sir Michael Francis Atiyah (əˈtiːə; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded the Fields Medal in 1966 and the Abel Prize in 2004. Atiyah was born on 22 April 1929 in Hampstead, London, England, the son of Jean (née Levens) and Edward Atiyah. His mother was Scottish and his father was a Lebanese Orthodox Christian.

Homology of Riemann Surfaces MATH-410: Riemann surfaces

Explores the homology of Riemann surfaces, including singular homology and the standard n-simplex.

The Topological Künneth Theorem MATH-506: Topology IV.b - cohomology rings

Explores the topological Künneth Theorem, emphasizing commutativity and homotopy equivalence in chain complexes.

The Duplication of the Cube

Explores the historical significance of the duplication of the cube problem and the attribution of mathematical instruments.