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A-Simplicial Objects and A-Topological Groups by Smirnov V. A.

By Smirnov V. A.

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Maumary de Solt (ThQorie 1 tousles de , i # j , X r A oO (Eij) sont nuls saul celui termes est le groupe engendr~ = 1 - XEij = des c o m m u t a t e u r s i, dlstincts J, k, on a, en remar- . (1-XEij-pEjk+X~Eik) GL(A) . trois Indlces -I (1+XEij) I+XEij SL(A) l+XgEik = (I+XEij+~EjK+X~EiK). 9 -38- ce qul montre que : SL(A) Pour lation qua xy noterons m yx quels ~ l'incluslon solent par ; on contraire, ~,8 slmplement qua SL(A) , GL(A) ~il~xiste transformations nent ~L(A) v~rlfler d'~qulvalence nous des C r x m y x , y blocs 9 SL(A) .

B~ pose d del . ---k , , . g g 2 . . oO gk est un stationnaire C Ek - d' isomorphisme Lorsque E k r E~ -g d'el T' k ) {dim C. dim C ' } i saul Isomorphlsme C @ avec Ia p I u s car 0 0- De Isomorphlsme " " J , on a b o u t i t & la situation suivante : trlvlaux; dlmenslon -I simple. ~j[n /B nl n ~ Hn' ~ En' I B 'n ~ = T(g) g Isomorphisme c'est-&-dlre le d l a g r a m m e simple. ~ C' =E' dens :(f) p' T' n. commutetlf Alors & Iignes § = • T(g n) simple en t o u t e gn est un exactes >0 i~-1 B~n -1 = 0 B ~0 .

H([) p. HI(L) un 113). par On p o s e -58- Soit (suppos~e M(f) , Is cellulaire), cyllndre de c'est-~-dlre l'appllcatlon l'~qulvalence Is quotient r (K x I ) u L------+KuLu(K homotoplque de (KxI)~ L f, per x ~) donn~e p a r r = x,r = f(x),r = (x,t),@(y) x C'est f un CW-complexe )L : K fl@che ~tant l'InJection ~I(K) sclnde une r@trectlon. (resp. K ~quivalences est obtenue en (O,13 & Cn_ 1 ( L ) Dono que T(f) D~finition ple si C(M) e n) et : Une ~ q u i v a l e n c e dWhomotopie On note a l o r s K K et .

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