# Commutator calculus and groups of homotopy classes by Hans Joachim Baues

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 concept to a subring of the reason. The equipment of evidence hire classical commutator calculus of nilpotent staff and Lie algebra concept and depend on an intensive and systematic examine of the algebraic houses of the classical homotopy operations (composition and addition of maps, spoil items, Whitehead items and better order James-Hopi invariants). The account is basically self-contained and will be obtainable to non-specialists and graduate scholars with a few heritage in algebraic topology and homotopy concept.

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Let y-n be defined with respect to the lexicographical ordering from the left and let a, j3 E [EX, EA]. Then we have for n-1 n-1 Yn(a + Q) = Yna + Yn-i(a) i=1 U Yi(o) + Yn(R) Proof. Let µ be the multiplication on J(A). The proposition follows from the equation, f1, f2 E ['E (JA X JA), EA^n], n _ fi = gn(EN) _ = gigs a gn-iq2 i=0 =f 2 47 II where q1, q2 : E(JA x JA) EJA are the projections. Now let nN:AN=Ax... be given by nN(xl, ... , xN) = xl ... xN and let 71 = s(nN x It is enough to prove f1 ° n = f2 ° n for all N, M.

10). Dn and T are defined as in I, §1, and for a permutation a E S n let ... ^ xQ) Ta(t, xl ^ ... ^ xn) _ (t, xQ n 1 be the corresponding permutation of factors xi E A. More general than (2. 8) we have with the notation in I (4. 2), I (4. 3) and I (4. 4): (2. 9) Let X be finite dimensional and let A be a Co-H- Theorem. space. Then for a E [EX, TA] we have in [EX, EA ' ... ' EA] the equation Il a + ... + lka = (i1+... +ik)o +' ' n>2 cn(ii' ... , ik) o yn(a) where cn(i1 , lk) deDk 1'1' ... , ']0(d) 0 TT(d) n Proof of (2.

U (3 is the n-fold cup product and where ^n B^ n T : EA ^ - E (A ^ B) ^n is the shuffle map. Proof. This follows from the homotopy commutative diagram X^X a ^ 1 J(A)^X a^1 J(A ^ X) where p(aI... ar ^ x) _ (a1 x)... (ar ^ x). // (2. 15) Proposition. Let y-n be defined with respect to the lexicographical ordering from the left and let a, j3 E [EX, EA]. Then we have for n-1 n-1 Yn(a + Q) = Yna + Yn-i(a) i=1 U Yi(o) + Yn(R) Proof. Let µ be the multiplication on J(A). The proposition follows from the equation, f1, f2 E ['E (JA X JA), EA^n], n _ fi = gn(EN) _ = gigs a gn-iq2 i=0 =f 2 47 II where q1, q2 : E(JA x JA) EJA are the projections.