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352 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganLemma 2.1. H .V / Š H . yV/:Proof. We construct morphisms of complexes inducing the isomorphisms in cohomology.We will use the following notations. If f 2 yV n and W Œn ! we will denoteby f ./ 2 yV its component in yV .ForW Œn ! we denote by j Œjl W Œl j ! its truncation: j Œjl .i/ D .i C j/, j Œjl .i; k/ D ..i C j/ .k C j//. For 1 W Œn 1 ! and 2 W Œn 2 ! with 1 .n 1 / D 2 .0/ define their concatenationƒ D 1 2 W Œn 1 C n 2 ! by the following formulas.´ 1 .i/ if i n 1 ;ƒ.i/ D 2 .i n 1 / if i n 1 ;8ˆ< 1 .ik/ if i;k n 1 ;ƒ.ik/ D 2 ..i n 1 /.k n 1 // if i;k n 1 ;ˆ: 2 .0 .k n 1 // ı .in 1 / if i n 1 k:This operation is associative. Finally we will identify in our notations W Œ1 ! withthe morphism .01/ in .The morphism C .V / ! C . yV/is constructed as follows. Let W Œn ! be asimplex in and define k by .k/ D Œ k , k D 0;1;:::;n. Let ‡./W Œn ! .n/be a morphism in defined byThen define the map W V ! yV by the formula.‡.//.k/ D .kn/. k /: (2.3)..v//./ D ‡./ v for v 2 V n :This is a map of cosimplicial vector spaces, and therefore it induces a morphism ofcomplexes.The morphism W C . yV/! C .V / is defined by the formula.f / D . 1/ n.nC1/2X0i k kC1. 1/ i 0CCi n 1f.@ 0 i 0 @ 1 i 1@ n 1i n 1/ for f 2 yV nwhen n>0, and .f / is V 0 component of f if n D 0.The morphism ı is homotopic to Id with the homotopy hW C . yV/! C given by the formula1 . yV/hf ./ DXn 1Xj D0 0i k kC1. 1/ i 0CCi j 1f.@ 0 i 0@ j 1i j 1 ‡.j Œ0j / j Œj .n 1/ /for f 2 yV n when n>0, and h.f / D 0 if n D 0.The composition ı W C .V / ! C .V / preserves the normalized subcomplexxC .V / and acts as the identity on it. Therefore ı induces the identity map oncohomology. It follows that and are quasiisomorphisms inverse to each other.