티스토리 뷰
The fixed point theorem for disks was first proved by Brouwer around 1910, quite early in the history of topology. Brouwer in fact proved the corresponding result for D^n, and we shall obtain this generalization in Corollary 2.15 using homology groups in place of first homotopy groups. One could also use the higher homotopy groups. Brouwer's original proof used neither homology nor homotopy groups, which had not been "invented" at the time. [Hatcher, Algebraic Topology, p32, 쌍따옴표는 내가 추가함]
In section 2.2 we used homology to distinguish different homotopy classes of maps from S^n -> S^n via the notion of degree. We will show here that cup product can be used to do something similar for maps S^{2n-1} -> S^n. Originally this was done by Hopf using more geometric constructions, before the "invention" of cohomology and cup groducts. [Hatcher, Algebraic Topology, p427, 쌍따옴표는 내가 추가함]
보통 수학적 개념은 발명된 것이라기보다는 발견된 것이라고 간주하는 경우가 많은데, Hatcher는 그렇지 않은가봄. 하긴 Inventiones mathematicae 라는 저널도 있는데...
신이 정수를 만들었고, 나머지는 모두 인간의 작업이다. --Leopold Kronecker