| 31. | It is said that Hurewicz attended this conference, and his first work on higher homotopy groups appeared in 1935.
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| 32. | The suspension can be used to construct a homomorphism of homotopy groups, to which the Freudenthal suspension theorem applies.
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| 33. | On the other hand, cohomology groups are homotopy classes of maps to spaces with a single nonzero homotopy group.
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| 34. | The real line is also contractible, and as such all of its homotopy groups and reduced homology groups are zero.
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| 35. | Various types of defects in the medium can be characterized by elements of various homotopy groups of the order parameter space.
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| 36. | Thus a 2-dimensional van Kampen theorem which computes nonabelian second relative homotopy groups was given by Brown and Higgins.
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| 37. | In 1932, E . Cech submitted a paper on higher homotopy groups to the International Congress of Mathematics at Zurich.
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| 38. | In fact it generates the homotopy group ? 3 ( " S " 2 ) and has infinite order.
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| 39. | Most modern computations use spectral sequences, a technique first applied to homotopy groups of spheres by Jean-Pierre Serre.
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| 40. | The study of the homotopy groups of spheres has led to the development of many powerful tools used in algebraic topology.
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