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

382 colin mclartycomplexities and some disorder over the next twenty years until topologistsfound a way to organize the subject by bypassing all the nuts and bolts:In order that these algebraic techniques not remain a special craft, the privatereserve of a few virtuosos, it was necessary to put them in a broad, coherent, andsupple conceptual setting. This was accomplished in the 1940sand1950s throughthe efforts of many mathematicians, notably Samuel Eilenberg at ColumbiaUniversity, Saunders Mac Lane of the University of Chicago, the late NormanSteenrod, and Henri Cartan. (Bass, 1978, p.505)They axiomatized homology as a correlation between patterns of continuousmaps and patterns of group morphisms. For each dimension i,thei-dimensionalhomology group became a functor H i . This means:• Homology preserves domain and codomain.f : M → N gives H i (f ) : H i (M) → H i (N)• Each identity map 1 M : M → M (which, intuitively, does not affect theholes of M) has identity homology.1 M : M → M gives 1 Hi (M) : H i (M) → H i (M)• The homology of a composite gf is the composite of the homologies.fT′ggivesH i ( f )H i (T′)H i ( g)Tg fT′′H i (T )H i (g f )H i (T′′)The axioms require more which we will not go into.¹³The structuralist point is that all the groups and morphisms are defined onlyup to isomorphism. Topologists still use nuts-and-bolts descriptions of cyclesand boundaries but textbooks use the axioms to define homology. The axiomsmake it easier to focus on geometry and they show how different nuts andbolts all yield the same calculations.14.2.3 The Lefschetz fixed point theoremThe Lefschetz fixed point theorem applies to especially nice spaces M, theorientable topological manifolds, where each small enough region of M looks like acontinuous piece of some Euclidean coordinate space R n as for example small¹³ See Eilenberg and Steenrod (1945), Hocking and Young (1961, Chapter7).