8 D. J. BENSON AND F. R. COHEN Let e2 denote ( £ ? ) + ( i o) + (01 ) + (1 o) i n t h e S rou P r i n g o f GL2(F2)] e2 is the second Steinberg idempotent (Mitchell and Priddy [20]). Observe that GL2(F2) acts on RP°° X RP°° and thus e2 may be regarded as a self-map of E(RP°° A RP°°) with addition taken with respect to the suspension coordinate. Let EM(2) denote the homotopy direct limit of e2 : E(RP°° A RP°°) -+ E(RP°° A KP°°). Next recall that E(A'(Z/3,1)) is homotopy equivalent to AVB where H2(A', Z) £ Z/3 and H4(B Z) £ Z/3 (Holzsager [15]). Theorem 1.6 The classifying space BTQ0 is stably equivalent to #£4 V SM(2) V E - 1 P . Theorem 1.7 If 1 q 2, then Hq(T%0 F3) = 0 and # 3 ( l t 0 F 3 ) = F 3 © JJi(E6 ^ 2 ( ^ 6 F 3 )). Furthermore, ^3+4^(^0 Z) contains a Z/3-summand. We remark that most of the calculations here are directed at T20 related results on rgo will te given in the sequel. We thank the Sonderforschnungsbereich »Geometrie und Analysis" for partial support while this paper was in preparation and C.-F. Bodigheimer for his suggestions. 2 Presentations A presentation for Artin's braid group on n strings is given as follows (see Artin [2], Birman [3]): (i) There are generators 0 1 , . . . , crn_i. (ii) A complete set of relations is given by Ti(Ti+\(Ji = (Ti+lTi(Ti+i for all I d(Tj = GjO~i if \i j \ 2. The symmetric group on n letters E n has a presentation given by (i) and (ii) above together with o\ 1 for all i. By work of Magnus [17] a presentation for TQ0 is given by (i) and (ii) above together with (iii) (7i72...an_i)(orn_1 ...a2ax) = 1 = (Tix2...7n-i)n- The group Ag is a central extension 1 - Z/2 - A, - Tl'+2 - 1 where Ag has a presentation given by (i) and (ii), together with (iv) the element 0 = {a\.. - cr2g+i)(cr2g+i.. .a\) is in the center and satisfies Q,2 = 1. Furthermore, {a\... J2g+i)2g*2 = 1. See Birman and Hilden [4].
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