0
This article is about topology. For chemistry, see Homotopic groups.
The two bold paths shown above are homotopic relative to their endpoints. Thin lines mark isocontours of one possible homotopy.
In topology, two continuous functions from one topological space to another are called homotopic (Greek homos = identical and topos = place) if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions. An outstanding 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 pathological spaces. Consequently most algebraic topologists work with compactly generated spaces, CW complexes, or spectra.
Still curious? Ask our experts.
Chat with our AI personalities
Ezra
Ross
BlakeAdd your answer:
Earn +20 pts
How many acute obtuse and right angles are there in word mathematics?
MATHEMATICS
What is the continuity in mathematics?
Continuity in mathematics is the first derivative equal to zero or the Boundary condition.
Who is grand grandfather of mathematics?
The "grandfather of mathematics" is often considered to be the ancient Greek mathematician Euclid. Euclid is known for his work "Elements," a mathematical and geometric treatise that has had a profound influence on the development of mathematics. His systematic approach to geometry and his emphasis on logical reasoning laid the foundation for much of modern mathematics.
What are his contribution in mathematics?
It depends on who "he" is (or was).
Can you major in geometry?
No, but you can major in mathematics
What has the author Mark Hovey written?
Mark Hovey has written: 'Model categories' -- subject(s): Model categories (Mathematics), Complexes, Homotopy theory 'Axiomatic stable homotopy theory' -- subject(s): Homotopy theory
What has the author Klaus Johannson written?
Klaus Johannson has written: 'Homotopy equivalences of 3-manifolds with boundaries' -- subject(s): Homotopy equivalences, Manifolds (Mathematics)
What has the author Myles Tierney written?
Myles Tierney has written: 'Categorical constructions in stable homotopy theory' -- subject(s): Categories (Mathematics), Complexes, Homotopy theory
When was Homotopy to Marie created?
Homotopy to Marie was created in 1982.
What has the author Hanno Ulrich written?
Hanno Ulrich has written: 'Fixed point theory of parametrized equivariant maps' -- subject(s): Fixed point theory, Homotopy theory, Mappings (Mathematics)
What has the author J Frank Adams written?
J. Frank Adams has written: 'Stable homotopy theory' -- subject(s): Homotopy theory
What has the author Donald M Davis written?
Donald M. Davis has written: 'From Representation Theory to Homotopy Groups' 'The nature and power of mathematics' -- subject(s): Cryptography, Fractals, Geometry, Non-Euclidean, Number theory
What has the author Richard M Hain written?
Richard M. Hain has written: 'Iterated integrals and homotopy periods' -- subject(s): Homotopy theory, Multiple integrals
What has the author James D Stasheff written?
James D. Stasheff has written: 'H-spaces from a homotopy point of view' -- subject(s): H-spaces, Homotopy theory
What has the author Michael Artin written?
Michael Artin has written: 'Etale homotopy' -- subject(s): Homotopy theory 'Algebraic spaces' -- subject(s): Algebraic functions, Algebraic spaces
What has the author Rosa Antolini written?
Rosa Antolini has written: 'Cubical structures and homotopy theory'
What has the author I M James written?
I. M. James has written: 'General topology and homotopy theory' -- subject(s): Topology 'Remarkable mathematicians' -- subject(s): Mathematicians, Biography 'The topology of Stiefel manifolds' -- subject(s): Stiefel manifolds 'Fibrewise topology' -- subject(s): Fiberings (Mathematics), Topology
