Homotopy | EPFL Graph Search

Are you an EPFL student looking for a semester project?

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

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. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formally, a homotopy between two continuous functions f and g from a topological space X to a topological space Y is defined to be a continuous function from the product of the space X with the unit interval [0, 1] to Y such that and for all . If we think of the second parameter of H as time then H describes a continuous deformation of f into g: at time 0 we have the function f and at time 1 we have the function g. We can also think of the second parameter as a "slider control" that allows us to smoothly transition from f to g as the slider moves from 0 to 1, and vice versa. An alternative notation is to say that a homotopy between two continuous functions is a family of continuous functions for such that and , and the map is continuous from to . The two versions coincide by setting . It is not sufficient to require each map to be continuous. The animation that is looped above right provides an example of a homotopy between two embeddings, f and g, of the torus into R3. X is the torus, Y is R3, f is some continuous function from the torus to R3 that takes the torus to the embedded surface-of-a-doughnut shape with which the animation starts; g is some continuous function that takes the torus to the embedded surface-of-a-coffee-mug shape. The animation shows the image of ht(x) as a function of the parameter t, where t varies with time from 0 to 1 over each cycle of the animation loop.

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 (16)

Related people (2)

Related concepts (122)

Related courses (15)

Related lectures (189)

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

Homotopical relations between 2-dimensional categories and their infinity-analogues

Lyne Moser

In this thesis, we study the homotopical relations of 2-categories, double categories, and their infinity-analogues. For this, we construct homotopy theories for the objects of interest, and show that

EPFL2021

The cotangent complex of a map of commutative rings is a central object in deformation theory. Since the 1990s, it has been generalized to the homotopical setting of E-infinity-ring spectra in various

2021

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.

A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better properties than simplicial complexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex). The C stands for "closure-finite", and the W for "weak" topology.

This course will provide an introduction to model category theory, which is an abstract framework for generalizing homotopy theory beyond topological spaces and continuous maps. We will study numerous

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

This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex

Relative Homotopy Groups

Covers relative homotopy groups, establishing long exact sequences and defining boundary homomorphisms.

Categories and Functors

Explores building categories from graphs and the encoding of information by functors.

Natural Transformations in Algebra

Explores natural transformations in algebra, defining functors and isomorphisms.