# Veblen O., Whitehead J. H.'s A Set of Axioms for Differential Geometry PDF

By Veblen O., Whitehead J. H.

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F ) 46 HOMOTOPY THEORY AND DUALITY we take c E TCr(f) represented by (u, v), S r-1 U X and set h(c) = the homology class of (ui, vK), where (c, K) is the fundamental r-cycle in Cr(i). For a differential triple F -'4 X - Y the transgression square in homology reads H r (Y) Hr Hr_1(F)4--S-Hr(j) Each homomorphism is induced by a "chain" map, a(z, x) = z, x s Cr(X ), z s C,,- 1 (F), E1 (z) = (Jz, 0), 82 (z, x) = px, µ(Y) = (0, Y), Y E Cr (Y) We see, for instance, that Elaz = (jaz, 0) and asl z = (- ajz, pjz) and so E1 az = - acl z since our chain complexes are normalized.

8 to show that 1 is a homotopy equivalence. 6. 6'. 6 requires the observation that all the homotopies entering into the proofs of Props. 9 are homotopies of pairs. 1. If p : X -+ Y is a fibre map with Y a 1-connected polyhedron and fibre F a K(G, n -1) space, then p is algebraically equivalent to the fibre map induced by some map f : Y-+ K(G, n). Before beginning the proof we note that it is possible to get rid of the hypothesis that Y be a polyhedron by using the functor which passes to the singular polyhedron.

Let IM -1P2 be the (m -1)-fold suspension of the projection plane P2. We have a cell decomposition Em -1P2 = Sm v 2 em + 1 where 2 denotes the map of degree 2. Clearly Sm v 2 em + 1 = K'(Z2, m). Any map of Sm u2 em + 1 to itself which is nullhomotopic on Sm induces the zero homomorphism of Z2. But the classes of maps of Sm u2 em + 1 to itself which are nullhomomotopic on Sm are in (1, 1) correspondence with the V 2 em + 1) = Z2. However it is clear that the homoelements of 7rm + 1(Sm topy class of a is uniquely determined by h provided that G1 is freeabelian.