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

‘there is no ontology here’ 379Intuitively the vertical circle on the left is not wrapped around any regionon the surface because it is wrapped around the ‘hole’ that runs throughthe interior of the torus. The larger horizontal circle wraps around the holethrough the center of the torus. The spiraling circle wraps around both holes.A hole is called 1-dimensional if a circle can wrap around it. The hole insidethe sphere is not 1-dimensional since any circle on the sphere can slip off to oneside and (for example) shrink down to the small bounding circle in Figure 14.1.That hole is 2-dimensional as the sphere surface wraps around it. The holes ineach dimension say a great deal about a topological space (Atiyah, 1976).To organize this information, Henri Poincaré called a circle C on a surfacehomologous to 0 if it bounds a region, and then wrote C ∼ 0 (Sarkaria, 1999).The two small circles in Figure 14.1 are both homologous to 0, while noneof the three on the torus in Figure 14.2 are. Name those three C 1 , C 2 , C 3respectively. Together they do bound a region.¹⁰ Poincaré saystheirsumishomologous to 0:C 1 + C 2 + C 3 ∼ 0He would use the usual rules of arithmetic to rewrite this asC 3 ∼−C 1 − C 2In fact he would consider all the circles C on the torus.¹¹ He called any formalsuma j C j + a k C k +···+a n C nof circles C i on the torus with integer coefficients a i a 1-cycle. If those circleswith those multiplicities form the boundary of some sum of regions then the1-cycle is homologous to 0:a j C j + a k C k +···+a n C n ∼ 0A pair of 1-cycles are homologous to each othera i C i +···+a k C k ∼ a n C n +···+a p C pif and only if their formal difference is homologous to 0:a i C i +···+a k C k − a n C n −···−a p C p ∼ 0¹⁰ Cut the torus along the small circle C 1 to form a cylinder, and then along the horizontal circleC 2 to get a flat rectangle with the spiral circle C 3 as diagonal. Either triangular half of the rectangle isbounded by one vertical side, one horizontal, and the diagonal.¹¹ Here a ‘circle’ in any space M is any continuous map C : S 1 → M from the isolated circle or1-sphere S 1 to M. It may be quite twisted and run many times around M in many ways.