By Hans Joachim Baues

A primary challenge of algebraic topology is the category of homotopy forms and homotopy sessions of maps. during this paintings the writer extends result of rational homotopy conception to a subring of the reason. The tools of facts hire classical commutator calculus of nilpotent staff and Lie algebra idea and depend on an in depth and systematic learn of the algebraic homes of the classical homotopy operations (composition and addition of maps, wreck items, Whitehead items and better order James-Hopi invariants). The account is basically self-contained and will be available to non-specialists and graduate scholars with a few history in algebraic topology and homotopy concept.

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Similar remarks apply to Rm n(a' p). , It is easily seen that tp in (3. 6) satisfies the relations in (3. 8) [x®a, y 0 0] = (x u y) 0 [a, 0] for the Lie bracket in (3. 4). For this, it is important that Cn and Rm are in fact homogeneous terms. This is the advantage of Rm n over Qn in (2. 2). Theorem (3. 7) can be proved along the same lines as (5. 9) in chapter II. ¢4. n The general type of Zassenhaus terms and its characterization modulo a prime We first generalize the Zassenhaus formula (1. 1) Proposition.

Let L be a nilpotent Lie algebra over Q. Then there is an unique group multiplication on L satisfying (3. 3) x y = (x + y) TI c(x, y)/n! n=2 for x, y e L and this is the multiplication given by the Baker-CampbellHausdorff formula. Moreover we know from (2. 1) that the commutator in exp(L) satisfies the equation R (3. 3)' x n n>1 m>1 TI m, n! m! n(x, y) Since L is nilpotent only a finite number of factors are non trivial. For the special types of Lie algebras below we can deduce a new presentation of the exponential group.

X A and n = A ^ ... ^ A. From the unit interval I = [0, 1] we define the 1-sphere Sl = I/ 10, 1) and the n-sphere Sn = (Sl )^n. 2) with µ(t) _ (2t, *) for 0 <_ t 2 and µ(t) 2t-1) for i <_ t < 1. EX = S1 " X is called the suspension of X and the function space 62Y = { f : S1 - Y I f(*) = *) is called the loop space of Y. EX is a Co-H-space and OX an H-space by the induced map 5 JA =JA ^X:EX - ZX'sX, µ = Yµ : QY x 62Y - fly. A space X together with a map p : X - X - X is a Co-H-space when X -+ X - X c X x X is homotopic to the diagonal (1.