This result is very important, in that it underpins homotopy theory, allowing homotopic deformations to be understood as continuous paths in the space of functions.
32.
A closed curve is called "'essential "'if it is not homotopic to a point, a puncture, or a boundary component.
33.
The " cuff lengths " \ ell _ i are simply the lengths of the closed geodesics homotopic to the f ( \ gamma _ i ).
34.
By taking " k " big enough, we see that ? is homotopic, with respect to the base point, to the constant map . \ Box
35.
Recently, Discrete Morse theory has shown promise for computational homology because it can reduce a given simplicial complex to a much smaller cellular complex which is homotopic to the original one.
36.
If for some " i " all maps are null homotopic, then the group ? " i " consists of one element, and is called the trivial group.
37.
Among other things, he proved that, roughly speaking, any homotopy equivalence of Haken manifolds is homotopic to a homeomorphism, i . e . that closed Haken manifolds are topologically rigid.
38.
If c is not null-homotopic this mapping class is nontrivial, and more generally the Dehn twists defined by two non-homotopic curves are distinct elements in the mapping class group.
39.
If c is not null-homotopic this mapping class is nontrivial, and more generally the Dehn twists defined by two non-homotopic curves are distinct elements in the mapping class group.
40.
We can specify a homeomorphism of the boundary of a solid torus to " T " by having the meridian curve of the solid torus map to a curve homotopic to \ gamma.
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