A Hopf algebra of spherical polytopes introduced more than 20 years ago by C.H. Sah, in his book on scissors congruences, is revisited through the light of shu e algebra. Oddly enough, these two topics where considered separately in the book of Sah, but not related. Their connection is based on previous work of Dupont. This new point of view is applied to the cohomology of Dehn complexes, considered recently by Goncharov. We get in particular a general procedure for linking this cohomology to the homology of some classical Lie groups, considered as discrete groups. A crucial role is played by certain bar and cobar constructions.
The late Chi-Han Sah wrote in his 1977 monograph (Sah, Hilbert's third problem: scissors congruence, Research Notes in Math 33, Pitman, 1979): "In my opinion, the third problem of Hilbert is of equal stature with the rest of the problems of Hilbert". The works of Dupont, Sah and others have amply justiΓΏed this vision. The following article would like to yet reinforce it. One can look at the recent book of Dupont (Scissors Congruences, Group Homology and Characteristic Classes, Nankai Tracts in Mathematics, Vol. 1, World ScientiΓΏc, Singapore, 2001) to get an idea of the richness and state of the problem.