Differentiable Manifolds by Georges de Rham, F.R. Smith, S.S. Chern

By Georges de Rham, F.R. Smith, S.S. Chern

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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. 8). Let R : FQY - Y be the evaluation map with R (t, a) = Q(t). For the adjoint f : X - R Y of f : EX - Y we have f = R ° (Ef). (1) We consider the diagram 44 II v n>1 i1g + i2g where by use of (1) we have (2) (i19+i 2 i) - (Ea) = ila + i 2 a Since G is a homotopy equivalence, there exist mappings cn making the diagram homotopy commutative. We have to show (3) cn = c ri(i l i ), , 2 c =i +i , 1 1 2 as defined in (2.

Thus cn(a, (3) is needed only if I a l = 1,61 is even. In this case, we evaluate c n (a, p) in the graded Lie algebra 7r. 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.

4) T(ul) 0 T(u2) = H*(J(Sn) x J(Sm), Q). The embedding 0 is defined to be the Lie algebra homomorphism with $(x) = ill 0 U1, )(Y) = {a2 9) u2 where fit, f12 : T -+ Q map u1 9) 1 to 1 and 1 0 u2 to and map all other elements un g) um to zero. 1 respectively Looking at the images of basic commutators we see that 0 and are actually embeddings. We remark that (4. 5) L(u1, u2) = n*(S2(ESn , ESm)) 9) Q For each N we have the mapping (4. 6) IrN : PN = (Sn)N - J(Sn) which is the restriction of the identification map n in (2.

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