In the mathematical field of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another. The homotopy must be a 1-parameter family of immersions.
Similar to homotopy classes, one defines two immersions to be in the same regular homotopy class if there exists a regular homotopy between them. Regular homotopy for immersions is similar to isotopy of embeddings: they are both restricted types of homotopies. Stated another way, two continuous functions are homotopic if they represent points in the same path-components of the mapping space, given the compact-open topology. The space of immersions is the subspace of consisting of immersions, denote it by . Two immersions are regularly homotopic if they represent points in the same path-component of .
Read more about Regular Homotopy: Examples
Famous quotes containing the word regular:
““I couldn’t afford to learn it,” said the Mock Turtle with a sigh. “I only took the regular course.”
“What was that?” inquired Alice.
“Reeling and Writhing, of course, to begin with,” the Mock Turtle replied; “and then the different branches of Arithmetic—Ambition, Distraction, Uglification, and Derision.”
“I never heard of ‘Uglification,’” Alice ventured to say.”
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)