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By Hans Joachim Baues

ISBN-10: 0521055318

ISBN-13: 9780521055314

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The cylinder IV has the structure maps V(D V `"i'+IV') V (3) with ix = x', i1x = x", p(x') = p(x") = x, p(sx) = 0. 9) Remark. Let fo, f 1: V-+ V' be chain maps. 2) can be identified with a chain homotopy a: fo -- f 1. This is a map a: V -> V' of degree + 1 with 1 da+ad=-fo+f1. (1) Now a yields a chain map G:IV- V' by G(x') = fox, G(x") = f1x and G(sx) = a(x). One easily checks that (1) is equivalent to Gd = dG. (2) In case G is given we get a by a(x) = G(sx). In particular, S: V -> I V, Sx = sx (3) is a chain homotopy, S:io = i1, with a = GS.

A (graded) algebra A is a positive (or negative) module A together with a map µ:A Qx A -* A of degree 0 and an element 1 eAo such that the multiplication x y = u(x (9 y) is associative and 1 is the neutral element, (1 - x = x- 1 = x). A (non-graded) algebra is a graded algebra A which is concentrated in degree 0, that is Ao = A. A map f :A -+ B between algebras is a map of degree 0 with f (1) = 1 and f (x y) = f (x) f (y). II The ring of coefficients R is a graded algebra which is concentrated in degree 0.

X in Top for which A and X are CW-spaces. ) - k*(X) is an isomorphism where the X,, run over the finite subcomplexes of the CW- complex X. Let CW-spaces be the full subcategory of Top consisting of CW-spaces. 10) Theorem. The category CW-spaces = C with the structure cof = maps in C which are cofibrations in Top, we = h*-equivalences = maps f :X -+ Y in C which induce an isomorphism f*:k*(X) = k*(Y) is a cofibration category. 13). The non-trivial part of the theorem is the existence of the fibrant models (C4).

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Algebraic Homotopy by Hans Joachim Baues

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