# New PDF release: A basis for the non-crossing partition lattice top homology By Zoque E.

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Example text

Another proof has been given by [ C a t h e l i n e a u , 1998]. 11). 1 # ; ( 0 ( n ) , A^(R n )) = 0 if i + j < n, j > 0. Note t h a t this does not directly give the desired vanishing for n = 3 and j = 1 since the coefficients are untwisted (so t h a t in this case the homology vanishes trivially by "center kills"). Indeed we shall use the case n = 4 and j = 2 for our proof. The other ingredient is the calculation of the Hochschild homology of the real quaternions H considered as an algebra over Q together with the formulation of this homology in terms of non-commutative differential forms (due to M.

Hence, if q ^ n mod 2, the group H0(G, Vq(RnY) is annihilated by 2 and is therefore 0 since it is a rational vector space. e. by isometries of the form 37 38 Scissors congruences, group homology and characteristic classes Sp,p£ En, given by Sp(x) = p- (x -p) =2p- x, x G K". For example in the case of n — 2 we then have V{En,G) 9* H0(G,V2(R2)) = A|(R 2 ) 9* R and the isomorphism is induced by the area function. Before we proceed we need two simple but useful lemmas about group homology. 4. Let M be a (left) R[G]-module (R any commutative ring) and let 7 G G be a central element such that for some r G R, jx = rx,Mx G M.

N: THEOREM n PT(y)^®t,-i(^),A^)), 9= 1 and the splitting is independent of choice of integer a > 1. c) HP(T{V), A&(fl)) = 0 for p < n - q - 1, q > 0. R e m a r k : The spectral sequence gives rise to a filtration FPVT(V), p= - 1 , 0 , 1 , . . , n - 2, such that F p /F p _i 2 HP(T(V), A^(fl)) withp + g = n — 1 which corresponds to the a9-eigenspace for \ia. 11) X = Xq + Xq+i + ... such that Xi has "weight i", that is, is in the a J th eigenspace. We will now get a geometric interpretation for this filtration.