純粋・応用数学（含むガロア理論）4

純粋・応用数学（含むガロア理論）4at MATH

純粋・応用数学（含むガロア理論）4 - 暇つぶし2ch146:to πk+2(Sn+2) is an isomorphism of p-components if k < p(n + 1) ? 3, and an epimorphism if equality holds (Serre 1952). The p-torsion of the intermediate group πk+1(Sn+1) can be strictly larger. Computational methods ・"The method of killing homotopy groups", due to Cartan and Serre (1952a, 1952b) involves repeatedly using the Hurewicz theorem to compute the first non-trivial homotopy group and then killing (eliminating) it with a fibration involving an Eilenberg?MacLane space. ・The Serre spectral sequence was used by Serre to prove some of the results mentioned previously. He used the fact that taking the loop space of a well behaved space shifts all the homotopy groups down by 1, so the nth homotopy group of a space X is the first homotopy group of its (n?1)-fold repeated loop space, which is equal to the first homology group of the (n?1)-fold loop space by the Hurewicz theorem. This reduces the calculation of homotopy groups of X to the calculation of homology groups of its repeated loop spaces. The Serre spectral sequence relates the homology of a space to that of its loop space, so can sometimes be used to calculate the homology of loop spaces. The Serre spectral sequence tends to have many non-zero differentials, which are hard to control, and too many ambiguities appear for higher homotopy groups. Consequently, it has been superseded by more powerful spectral sequences with fewer non-zero differentials, which give more information. （日本語ページ無いが、英文ページがある（＾＾；） https://en.wikipedia.org/wiki/Hiroshi_Toda Hiroshi Toda Hiroshi Toda (戸田宏, Toda Hiroshi, born 1928) is a Japanese mathematician, who specializes in stable and unstable homotopy theory. つづく

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