Seite - 162 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 bysetting ∂KV(w)(X)=−X ·w+w ·X if w∈ J(W), (7a) ∂KVf=∑ [i<j] S[i,j](f) if q>0. (7b) ByLemma2weobtain the followingstatement Theorem3. Forevery twosidedKVmoduleWofaKValgebraA thepair (C∗KV,∂KV) is a cochaincomplex. 3.3.4. TheTotalCohomology LetWbeatwo-sidedmoduleofaKValgebraA.Ourconcern is theZ-gradedvectorspace Cτ =W+⊕q>0Cq(A,W). Forourpresentpurpose themapsSij arenotsubject therequirementas inStep2. Wedefinethecoboundaryoperator∂τ bysetting ∂τw(X)=−X ·w+w ·X ∀winW, ∂τ f(ξ)= ∑ 1≤i<j≤q+1 S[i,j](f)(ξ) ∀q>0. Thequantity (∂2τ f(ξ)depends linearlyontheKVanomaly functionsof thepair (A,W). Thus the pair (C∗τ,∂τ) isacochaincomplex. Its cohomologyiscalledtheW-valuedtotalKVcohomologyofA. Wedenote itbyH∗τ(A,W). 3.3.5. TheResidualCohomology,SomeExactSequences,RelatedTopics,DTO-HEG-IGE-ENT Inthenextsectionswewillseethatthelinksbetweentheinformationgeometryandthedifferential topology involve therealvaluedtotalKVcohomologyofKValgebroids.Manyrelevant relationships arebasedontheexact sequences →Hq−1KVres(A,R)→Hqe(A,R)→HqKV(A,R)→HqKVres(A,R)→ →Hq−1τres(A,R)→Hqe(A,R)→Hqτ(A,R)→Hqτres(A,R)→ Nowweareprovidedwithcohomological toolswhichwillbeusedin thenextsections. WeplantoperformKVcohomologicalmethodsforstudyingsomelinksbetweentheverticesof thesquare“DTO, IGE,ENTHGE”as inFigure1.Werecallbasicnotions. DTO IGE ENTHGE KVH Figure1.Federation. DTOstands forDifferentialTOpology. Thepurposes: Riemannian foliationsandRiemannianwebs. Symplectic foliationsandsymplecticwebs. Linearizationofwebs. Our aims: We use cohomologicalmethods for constructingRiemannian foliations, Riemannianwebs, linearizablewebs. 162