Dehn proved that the filling area functions for compact Riemannian manifolds where the area of a minimal surface bounding a null-homotopic closed curve is bounded in terms of the length of that curve.
42.
If it is obtained from a polygon it is just the image of vertical lines, and the measure of an arc is just the euclidean length of the horizontal segment homotopic to the arc.
43.
:: If the closed Jordan curve ? separates the surface, it is homotopic to a smooth Jordan curve ? ( with non-vanishing derivative ) that separates the surface into two halves.
44.
Proved that closed Haken manifolds are topologically rigid : roughly, any homotopy equivalence of Haken manifolds is homotopic to a homeomorphism ( for the case of boundary, a condition on peripheral structure is needed ).
45.
A counterexample is given by the Warsaw circle, whose first cohomology group vanishes, but admits a map to " S " " 1 " which is not homotopic to a constant map
46.
In fact, since the Lefschetz number has been defined at the homology level, the conclusion can be extended to say that any map homotopic to " f " has a fixed point as well.
47.
No CTC can be continuously deformed as a CTC to a point ( that is, a CTC and a point are not timelike homotopic ), as the manifold would not be causally well behaved at that point.
48.
The converse is also true if one allows such things as " homotopy sections ", i . e . a map such that is homotopic ( as opposed to equal ) to the identity map on.
49.
When you say that " " you can't deform one into the other " ", that usually means that they are not homotopic, which is clearly false since they're both homeomorphic.
50.
Just as one point is distinguished, so one class is distinguished : all maps ( or curves ) homotopic to the constant map " S " 1 ?! " x " are called null homotopic.
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