By John Henry Constantine Whitehead
ISBN-10: 0080098711
ISBN-13: 9780080098715
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Additional info for Homotopy Theory. The Mathematical Works of J. H. C. Whitehead
Example text
This is valid for arbitrary (finite) complexes K, L, but the resulting co-chain map cannot, in general, be realized by a geometrical map K -+L. The step from the cochain map to the geo4 ) i. e. there is a map g: L-+K such t h a t gf&l, fgç^l, where ^ denotes the relation of homotopy and 1 denotes the identical map, both in K and in L. 5 ) Our μ includes and can be defined in terms of Bockstein's π and ω but our Δ is an additional element of algebraic structure. e ) i. e. homomorphisms of the groups of co-chains which commute with the co-boundary operator (cf.
Moreover the unit element has the same geometric inter pretation in R(K) as in R{K'). For let ueC°(K), ufeC°(X') be the co-chains with constant value 1. Then the unit element, e! ε R(K'), is the co-homology class of ur. Obviously i*ur = u, whence he', the unit element in R(K), is the co-homology class of u. Let / : K -> L be any map of K in a complex L. Let R (L) be defined in the same way as R(K), by means of a simplicial sub-division, V', of L. The map / determines a unique proper homomorphism R{Lf) -> R(K') and hence, in the obvious way, a unique proper homomorphism R(L)-+R(K).
The 3-spheres ÄJ , . . , S] may be triangulated and it follows from Lemma 6 that Kl has a simplicial sub-division. On applying a similar argument to the 4-cells it follows that Kx and hence K, is of the same homotopy type as a reduced complex, which has a simplicial sub division. Therefore the condition that each of our complexes is to have a simplicial sub-division does not restrict the homotopy type of a reduced complex. Lemma 8. Any simple, ^dimensional complex is of the same homotopy type 'as some reduced complex.