Page - 155 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Let jandqbetwopositive integerssuchthat j< q. Letξ= x1⊗x2...⊗xq.Todefinethe linearmap dj :Cq(B,V)→Cq−1(B,V) weput dj(ξ⊗v)=X∗j ·(∂jξ⊗v). Henceforthonedefines theboundaryoperator d :Cq(B,V)→Cq−1(B,V) bysetting d= q ∑ 1 (−1)jdj. Soweobtainachaincomplexwhosehomologyspaceofdegreeq isdenotedbyHq(B,V). Definition14. Thegradedvector space H∗(B,V)=∑ q Hq(B,V) is called the totalhomologyofBwithcoefficients inV. 3.2.6. TwoCochainComplexes We are going to define two cochain complexes. They are denoted by CKV(B,V) and by Cτ(B,V) respectively. Werecall that thevectorsubspace J(V)⊂V isdefinedby (s ·s∗) ·v−s ·(s∗·v)=0 ∀s s∗∈B. Letusset C0KV(B,V)= J(V), C0τ(B,V)=V, Cq(B,V)=HomR(Tq(B˜) ∀q≥1. Let (j,q)beapairofnonnegative integerssuchthat j< q.Wearegoingtodefinethe linearmap dj :Cq(B,V)→Cq+1(B,V). Given f ∈Cq(B,V)and ξ= x1⊗ ...⊗xq+1 weput djf(ξ)=X∗j · f(∂jξ)− f(djξ). The familyof linearmappingsdjhaspropertyS ·1,viz djdi=didj−1 ∀i, j with i< j. 155