process

104 F. Muro and A. TonksProof of Theorem 4.2. In order to define a strong deformation retraction,˛ D C pj DC C,1 ˛' jp; pj D 1;the crucial step will be to define the homotopy ˛ W D 0 C ! D 1 C on the generators ŒAof D 0 C. Then one can use the equations1. ˛.c 0 C d 0 / D ˛.c 0 / d 0C ˛.d 0 /,2. jp.c 0 / D c 0 C @˛.c 0 /,3. jp.c 1 / D c 1 C ˛@.c 1 /,for c i ;d i 2 D i C to define the composite jp and the homotopy ˛ on all of D C.Itisa straightforward calculation to check that the map jp so defined is a homomorphismand that ˛ is well defined. In particular, Lemma 4.5 (a) implies that ˛ vanishes oncommutators of length 3. In order to define j we must show that jp factors throughthe projection p, that is,jp.ŒB i 2 A C B p 1 A/ D 0:If we also show thatp˛ D 0then equations (2) and (3) above say pjp.c i / D p.c i / C 0, and since p is surjective itfollows that the composite pj is the identity.The set of objects of C is the free monoid generated by objects S 2 S. Thereforewe can define inductively the homotopy ˛ by˛.ŒS C B/ D ŒB i 2 S C B p 1 SC ˛.ŒB/for B 2 C and S 2 S, with ˛.ŒS/ D 0. We claim that a similar relation then holds forall objects A, B of C,˛.ŒA C B/ D ŒB i 2 A C B p 1 A C ˛.ŒA C ŒB/ (4.1)D ŒB i 2 A C B p 1 A C ˛.ŒA/ ŒB C ˛.ŒB/:If A Dor A 2 S then (4.1) holds by definition, so we assume inductively it holds forgiven A, B and show it holds for S C A, B also:˛.ŒS C A C B/ D ŒA C B S C A C B SC ˛.ŒA C B/D ŒA C B S C A C B SC ŒB A C B A C ˛.ŒA/ ŒB C ˛.ŒB/D ŒB S C A C B S C A C ŒA S C A S ŒB C ˛.ŒA/ ŒB C ˛.ŒB/D ŒB S C A C B S C A C ˛.ŒS C A/ ŒB C ˛.ŒB/: