▁used ▁here ▁to ▁represent ▁volume ▁( as ▁each ▁real ▁number ▁is ▁the ▁volume ▁of ▁exactly ▁one ▁element ▁of ▁this ▁group ). ▁The ▁map ▁from ▁the ▁group ▁of ▁poly hed ra ▁to ▁ ▁is ▁the ▁De hn ▁invariant . ▁ ▁is ▁the ▁E uclidean ▁point ▁rotation ▁group , ▁and ▁ ▁is ▁the ▁group ▁hom ology . ▁Syd ler ' s ▁theorem ▁that ▁volume ▁and ▁the ▁De hn ▁invariant ▁are ▁the ▁only ▁invari ants ▁for ▁E uclidean ▁dis section ▁is ▁represented ▁hom olog ically ▁by ▁the ▁statement ▁that ▁the ▁group ▁ ▁appearing ▁in ▁this ▁sequence ▁is ▁actually ▁zero . ▁If ▁it ▁were ▁non zero , ▁its ▁image ▁in ▁the ▁group ▁of ▁poly hed ra ▁would ▁give ▁a ▁family ▁of ▁poly hed ra ▁that ▁are ▁not ▁dis sect able ▁to ▁a ▁cube ▁of ▁the ▁same ▁volume ▁but ▁that ▁have ▁zero ▁De hn ▁invariant . ▁By ▁Syd ler ' s ▁theorem , ▁such ▁poly hed ra ▁do ▁not ▁exist . ▁ ▁The ▁group ▁ ▁appearing ▁towards ▁the ▁right ▁of ▁the ▁exact ▁sequence ▁is ▁is omorphic ▁to ▁the ▁group ▁ ▁of ▁K äh ler ▁differential s , ▁and ▁the ▁map ▁from ▁tensor ▁products ▁of ▁lengths ▁and ▁angles ▁to ▁K äh ler ▁differential s ▁is ▁given ▁by ▁ ▁where ▁ ▁is ▁the ▁universal ▁deriv ation ▁of ▁. ▁This ▁group ▁ ▁is ▁an ▁obst acle ▁to ▁realiz ability : ▁its ▁non zero ▁elements ▁come ▁from ▁elements ▁of ▁ ▁that ▁cannot ▁be ▁realized ▁as ▁De hn ▁invari ants . ▁ ▁Anal og ously , ▁in ▁hyper bol ic ▁or ▁spher ical