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

‘there is no ontology here’ 383regions of a sphere or torus look like pieces of the plane R 2 .Aspherecanturnon an axis the way the earth does. A torus can turn around a central axis theway a bicycle tire turns around its axle:Fig. 14.3.A rotating sphere has two fixed points, call them the North and South poles.The rotating torus has none—obviously because the axis passes through ahole. Of course the matter is more complex with general continuous functionsrather than just rigid rotations. It is more complex yet for manifolds with moreholes or in higher dimensions. In general the theorem relates fixed points ofamapf : M → M to the way f acts on holes in all dimensions, that is to themorphisms H i (f ).On its face the fixed point theorem counts fixed points, which are solutionsto equations of the form f (x) = x. Weil saw that if he could apply it to suitablearithmetic spaces then he could use this plus Galois theory to count solutions tohis polynomials. There was one crying problem: it was nearly inconceivable thatarithmetic spaces could be defined so as to support such a topological theorem.14.2.4 CohomologyThe route to scheme theory ran through a variant of homology called cohomologyand the key to schemes is that they admit coverings analogous to the topologicalcase. For example, the torus can be covered by overlapping cylindrical sleeves,U 1 , U 2 , U 3 , drawn here in solid outline:U 1U 2U 3Fig. 14.4.TheroutefromU 1 to U 2 to U 3 travels around the hole in the center ofthe torus. The hole is revealed, very roughly, by the fact that every two