By H. Bass
ISBN-10: 0387064346
ISBN-13: 9780387064345
ISBN-10: 3540064346
ISBN-13: 9783540064343
Read or Download Higher K-Theories proceedings of the conference held at the Seattle Research Center of the Battelle Memorial Inst., from Aug. 28 to Sept. 3, 1972 PDF
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Additional info for Higher K-Theories proceedings of the conference held at the Seattle Research Center of the Battelle Memorial Inst., from Aug. 28 to Sept. 3, 1972
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.