Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

380 colin mclartyThe key to the homology of the torus is that every 1-cycle on the torus ishomologous to a unique sum of C 1 and C 2 :a j C j + a k C k +···+a n C n ∼ a 1 C 1 + a 2 C 2for some unique a 1 , a 2 ∈ Z. Thepair{C 2 , C 3 } serves as wella j C j + a k C k +···+a n C n ∼ a 2 C 2 + a 3 C 3for some unique a 2 , a 3 ∈ Z.Thepair{C 1 , C 3 } also serves as do infinitely manyothers. No single cycle will do. So the first Betti number of the torus is 2. Thefirst Betti number of the sphere S 2 is 0, since all 1-cycles on it are homologousto 0.Higher Betti numbers are defined using higher dimensional figures in placeof circles. A good n-dimensional space M has an ith Betti number, counting thei-dimensional holes in M, for every i from 0 to n. The Betti numbers of a spacesay much about its topology but, in fact, Poincaré and all topologists of thetime knew that the numbers alone omit important relations between cycles.14.2.2 Brouwer to Noether to functorsBy 1910 the standard method in topology was algebraic calculation with cycles(Herreman, 2000). Everyone from Poincaré on knew that the cycles formedgroups. The 1-cycles on the torus add and subtract and they satisfy all theaxioms for an Abelian group, when the relation ∼ is taken as equality. Thesame holds for the i-dimensional cycles on any n-dimensional space M, foreach 0 ≤ i ≤ n. Only no one wanted to use group theoretic language intopology. Brouwer would not even calculate with cycles. But he was a friendof Noether’s and they shared some students.As Noether emphasized morphisms in algebra, so Brouwer organized histopology around maps or continuous functions f : M → N between topologicalspaces (van Dalen, 1999). His most famous theorem, the fixed pointtheorem, says:LetD n be the n-dimensional solid disk, that is all points on orinside the unit sphere in n-dimensional space R n ,theneverymapf : D n → D nhas a fixed point, a point x such that f (x) = x. Many of his theorems areexplicitly about maps, and essentially all of his proofs are based on findingsuitable maps.Noether’s homomorphism and isomorphism theorems unified her viewpointwith Brouwer’s. Given any topological space M, she would explicitly formthe group Z i (M) of i-cycles on M, as the formal sums of circles shown aboveform the group Z 1 (T ) of 1-cycles on the torus. Then she would form thesubgroup B i (M) ⊆ Z i (M) of n-boundaries on M, inotherwordsthei-cycles