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