# Generalization of Tangential Complexes of Weight Three and Their Connections with Grassmannian Complex

Authors: Sadaqat Hussain; Nasreen Kausar; Sajida Kousar; Parameshwari Kattel; Tahir Shahzad
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Following earlier work by Gangl, Cathelineaue, and others, Siddiqui defined the Siegel’s cross-ratio identity and Goncharov’s triple ratios over the truncated polynomial ringFεν. They used these constructions to introduce both dialogarithmic and trilogarithmic tangential complexes of first order. They proposed various maps to relate first-order tangent complex to the Grassmannian complex. Later, we extended all the notions related to dialogarithmic complexes to a general order n. Now, this study is aimed to generalize all of the constructions associated to trilogarithmic tangential complexes to higher orders. We also propose morphisms between the tangent to Goncharov’s complex and Grassmannian subcomplex for general order. Moreover, we connect both of these complexes by demonstrating that the resulting diagrams are commutative. In this generalization process, the classical Newton’s identities are used. The results reveal that the tangent group TB3nF of a higher order and defining relations are feasible for all orders.


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## Body

## 1. Introduction

Polylogarithms have been known for almost three centuries in various disciplines of mathematics such as Feynman integrals, volume functions of hyperbolic tetrahedrons, quantum field theory, and Dedikend Zeta functions. Since the last two decades, it became important after Bloch’s work, in which he introduced a group (Bloch Group) and found connections of this group with algebraicK theory (see [1, 2]). Suslin introduced the well-known Grassmannian Complex and Bloch–Suslin complex (see [3, 4]). Later, Goncharov used geometric configurations in order to define the motivic complexes and to prove Zagier’s conjecture on polylogarithms and special L values for weight 2 and 3 (see [4]). He also introduced the triple ratio together with Zagier by antisymmetrization of his own formula f23. Moreover, he related the Grassmannian subcomplexes to his trilogarithmic motivic complexes by introducing several homomorphisms of the form fi3 (see [2, 5, 6]). Cathelineau studied variants (infinitesimal and tangential) of these motivic complexes and presented a tangent group Tℬ2F which is, in fact, a F vector space (see [7]). Siddiqui defined the tangent group Tℬ3F and its complexes for the first order. He used geometric configurations to construct cross-ratio, triple-ratio, and Siegel cross-ratio identity for dual numbers and then proposed various maps to relate Grassmannian complex and first-order tangent complex (see [8, 9]). He himself attempted to extend his constructions to the second-order tangent complex and gave some results for a special case.Hussain and Siddiqui [10] discussed the tangent group and its associated complexes for the second order by introducing second-order tangent group, denoted by Tℬ22F. They extended cross-ratio, Goncharov’s triple-ratio, and Siegel’s cross-ratio identity of the first order to second order. They also proposed morphisms such as τ0,ε22 and τ1,ε22 for weight two and τ0,ε23, τ1,ε23, and τ2,ε23 for weight three, in order to connect the second-order tangential complex with the Grassmannian complex. Recently, the degree of the tangent group of weight 2 is generalized by introducing a group Tℬ2nF [11]. This group is used to construct a generalized tangent complex:(1)Tℬ2nF⟶∂εnF⊗∧2F×⊕∧3F.This complex is further related to the Grassmannian complex by introducing mapsπ0,εn2 and π1,εn2. As the order of the tangent group is generalized only for weight two, it is still a matter of great concern and motivates to introduce and analyse higher order tangent groups of weight three which will eventually be used for the establishment of generalized tangent complexes of weight three. In this work, we construct higher order tangent to Goncharov’s complexes and its associated algebraic constructions (cross ratio, Siegel’s identity, triple ratio, etc.) of weight three and to obtain a generalized formula for the order n≥3. We also propose general formula for the maps which connect the trilogarithmic tangential complexes of order greater than two to the Grassmannian complexes. For this, we define nth-order tangent group Tℬ3nF of weight 3 along with its functional equations. We define a map ∂εn to construct the following tangent complex of general order:(2)Tℬ3nF⟶∂εnTℬ2nF⊗F×⊕F⊗ℬ2F⟶∂εnF⊗∧2F×⊕∧3F.Our next goal is to connect the above complex to the well-known Grassmannian complex. This will be achieved through an inductive approach and using Newton–Girard identities (see Section2.4). Using the results from Siddiqui [9], we will determine the coefficients of the cross-ratio and Siegel cross-ratio identity and determinants of order n. After these constructions, we will move to find morphisms of the connection π0,εn3, π1,εn3, and π2,εn3 so as to relate Grassmannian complex to Cathelineau’s trilogarithmic complex. The consequence of this connection is the formation of diagram D, and at the end, we prove that this diagram is commutative.


## 2. Materials and Methods

This section is devoted to give brief introduction to certain concepts and constructions related to this work. Many of the terms also found in [1, 4, 9, 10, 12, 13] can be consulted for more details.


### 2.1. Complex

Suppose that(3)Ai⟶fiAi−1⟶fi−1…⟶f0A0is a chain of abelian groups with corresponding maps fk; then, such a chain is said to be a complex if(4)fi−1∘fi=0.


### 2.2. Grassmannian Complex

LetX be any nonempty set. Consider CmX be any free Abelian group generated by the elements of G/Xm, where G is a group which acts on X; then, we define a differential map d:CmX⟶Cm−1X as(5)d:x1,…,xm↦∑i=0m−1mx1,…,x^i,…,xm.Also, if we denoteCmn to be a free Abelian group generated by the configurations of the elements of an n dimensional vector space Vn and xi|x1,…,x^i,…,xm be the projective configuration of the jth component xj along the ith component xi, where i≠j, j=1,…,m, then we define a projective differential map d′:Cm+1n+1⟶Cmn as(6)d′:x1,…,xm↦∑i=0m−1mxi|x1,…,x^i,…,xm.By using these differential maps and free Abelian groups generated by configurations, we have a bicomplex of the form called Grassmannian bicomplex.From this bicomplex, we can form many subcomplexes such as(7)Cm+2n+2⟶d′Cm+1n+1⟶d′Cmnor(8)Cm+2n⟶dCm+1n⟶dCmn.These subcomplexes are known as Grassmannian complexes.


### 2.3. Tensor Product

LetA and B be two free Abelian groups (ℤ—modules); then, the tensor product of A and B is denoted as A⊗B and defined as a free Abelian group with generators a⊗b for a∈A and b∈B, satisfying the relations(9)a1+a2⊗b=a1⊗b+a2⊗b,a⊗b1+b2=a⊗b1+a⊗b2,ra⊗b=ra⊗b=a⊗rb,where a,a1,a2∈A, b,b1,b2∈B, and r∈F.


### 2.4. Newton–Girard Identities

Girard introduced some identities which describe relationship between roots of a polynomial and its coefficients (see [14, 15]). Later, Newton generalized these identities and gave a recursive formula. Today these identities are known as Newton–Girard identities whose explanation is as follows. Suppose fy is a polynomial like(10)gy=yn+c1yn−1+c2yn−2+c3yn−3+……+cn−1y+cn,with n of its roots r1,r2,……rn. We use the notation δk for the sum of the kth powers of roots as(11)δk=r1k+r2k+……+rnk,where k∈ℤ+ and δk=0 for k〉n. And(12)δk+c1δk−1+c2δk−2+s3ck−3+……+ck−1δ1+kck=0.This identity allows us to deduce the relations:(13)δ1+c1=0,δ2+c1δ1+2c2=0,δ3+c1δ2+c2δ1+3c3=0,δ4+c1δ3+c2δ2+c3δ1+4c4=0.Now, considering the most generalized form of the polynomial,(14)fy=∑i=0ntiyi.And with the assumption thattk=0 for k0, we define δk, for k⩾0, as(15)δk=r1k+r2k+……+rnk.By interchanging “k” to “−k,” we obtain(16)δ−k=r1−k+r2−k+……+rn−k.Finally, we can conclude the general form of Newton’s identity as(17)tjn−j+tj+1δ−1+tj+2δ−2+tj+3δ−3+……+tnδj−n=0;j⩽n.Furthermore, we can deduct the following results:(18)M1=t1s,M2=2t2s−t12s2,M3=3t3s−3t1t2s2+t13s3,M4=4t4s−4t1t3s2−2t22s2+4t12t2s3−t14s4.In general notation,(19)Mn=ntns−∑r=1n−1tn−rsMr,here we used Mi=−δ−k,∀i=0,1,2,… and t0=s.When we consider the caset0=1−s, the above identities will become(20)N1=−t1s−1,N2=−2t2s−1−t12s−12,N3=−3t3s−1−3t1t2s−12−t13s−13,N4=−4t4s−1−4t1t3s−12−2t22s−12−4t12t2s−13−t14s−14.The general form will be(21)Nn=ntn1−s−∑r=1n−1tn−r1−sNr.In [10], we have extended the idea of trilogarithmic tangent group up to the second order by proposing a group Tℬ32F and its complex. It is quite natural to think for the generalization of the degree of this group and other constructions (complexes, morphisms, cross ratio, etc.) related to this group. Let us define the following.


### 2.5. First-Order Tangent Group of Weight 3

A first-order tangent group, denoted byTℬ2F, is a ℤ module generated by the elements of the form s;s′∈ZFε2, a quotient by the expression,(22)s;s′−t;t′+ts;ts′−1−t1−s;1−t1−s′+s1−tt1−s;s1−tt1−s′,where s,t≠0,1,s≠t, s;s′=s+s′ε−s;s,s′∈F,(23)ts′=st′−s′ts2,1−t1−s′=1−ts′−1−st′1−s2,and(24)s1−tt1−s′=t1−ts′−s1−st′t1−s2.As the second order is discussed in [10], we define the third order directly.


### 2.6. Third-Order Tangent Group of Weight 3

We denoteTℬ33F to be the trilogarithmic tangent group of order three which is a ℤ module over the truncated polynomial ring Fε4 whose generators are the elements of the form s;t1,t2,t3∈ℤFε4, where s;t1,t2,t3=s+t1ε+t2ε2+t3ε3−s,s,t1,t2,t3∈F and quotient by the kernel of map(25)∂:ℤFε4⟶Tℬ23F⊗F×⊕F⊗ℬ2F,such that(26)∂a;b1,b2,b32=s;t1,t2,t323⊗s+3t3s−3t1t2s2+t13s3⊗s2,where ⊕ and ⊗, respectively, represent the direct sum and tensor product of free Abelian groups or modules’ tensor product of the free Abelian groups.


### 2.7. Generalized Tangent Group of Weight 3

Let the characteristic of the fieldF be zero and Fεn+1 represent the ring of truncated polynomials. We call Tℬ3nF the tangent group of weight 3 and order n which can be defined as a ℤ module over Fεn+1 with generators of the form s;t1,t2,…,tn∈ℤFεn+, where s;t1,t2,…,tn=s+t1ε+t2ε2+⋯+tnεn−s,s,t1,t2,…,tn∈F and quotient by Ker∂:(27)∂:ℤFεn+1⟶Tℬ2nF⊗F×⊕F⊗ℬ2F,where ∂ can be written as(28)∂s;t1,t2,…,tn2=s;t1,t2,…,tn2n⊗s+Mn⊗s2.The factorMn is given in (8). For n=1,2, we obtain the groups Tℬ3F and Tℬ32F. The former is defined in [9] and later is in [10]. By using the group Tℬ3nF, the following tangential complex can be obtained:(29)Tℬ3nF⟶∂εnTℬ2nF⊗F×⊕F⊗ℬ2F⟶∂εnF⊗∧2F×⊕∧3F.


### 2.8. Triple Ratio

The triple ratioru0,…,u5 of six points is given in [12] as(30)r3u0,…,u5=u0u1u3u1u2u4u2u0u5u0u1u4u1u2u5u2u0u3,whose tangential version for ν=2 can be traced in [8, 10] as(31)r3,εl0∗,…,l5∗=l0∗l1∗l3∗l1∗l2∗l4∗l2∗l0∗l5∗εl0l1l4l1l2l5l2l0l3−l0l1l3l1l2l4l2l0l5l0l1l4l1l2l5l2l0l3l0∗l1∗l4∗l1∗l2∗l5∗l2∗l0∗l3∗εl0l1l4l1l2l5l2l0l3.We extend this notion forν=n+1 as(32)r3,εnu0∗,…,u5∗=Alt6u0∗u1∗u3∗u1∗u2∗u4∗u2∗u0∗u5∗εnu0u1u4u1u2u5u2u0u3−r3,εn−1u0∗,…,u5∗u0∗u1∗u4∗u1∗u2∗u5∗u2∗u0∗u3∗εu0u1u4u1u2u5u2u0u3−r3,εn−2u0∗,…,u5∗u0∗u1∗u4∗u1∗u2∗u5∗u2∗u0∗u3∗ε2u0u1u4u1u2u5u2u0u3−…−r3u0,…,u5u0∗u1∗u4∗u1∗u2∗u5∗u2∗u0∗u3∗εnu0u1u4u1u2u5u2u0u3.


## 2.1. Complex

Suppose that(3)Ai⟶fiAi−1⟶fi−1…⟶f0A0is a chain of abelian groups with corresponding maps fk; then, such a chain is said to be a complex if(4)fi−1∘fi=0.


## 2.2. Grassmannian Complex

LetX be any nonempty set. Consider CmX be any free Abelian group generated by the elements of G/Xm, where G is a group which acts on X; then, we define a differential map d:CmX⟶Cm−1X as(5)d:x1,…,xm↦∑i=0m−1mx1,…,x^i,…,xm.Also, if we denoteCmn to be a free Abelian group generated by the configurations of the elements of an n dimensional vector space Vn and xi|x1,…,x^i,…,xm be the projective configuration of the jth component xj along the ith component xi, where i≠j, j=1,…,m, then we define a projective differential map d′:Cm+1n+1⟶Cmn as(6)d′:x1,…,xm↦∑i=0m−1mxi|x1,…,x^i,…,xm.By using these differential maps and free Abelian groups generated by configurations, we have a bicomplex of the form called Grassmannian bicomplex.From this bicomplex, we can form many subcomplexes such as(7)Cm+2n+2⟶d′Cm+1n+1⟶d′Cmnor(8)Cm+2n⟶dCm+1n⟶dCmn.These subcomplexes are known as Grassmannian complexes.


## 2.3. Tensor Product

LetA and B be two free Abelian groups (ℤ—modules); then, the tensor product of A and B is denoted as A⊗B and defined as a free Abelian group with generators a⊗b for a∈A and b∈B, satisfying the relations(9)a1+a2⊗b=a1⊗b+a2⊗b,a⊗b1+b2=a⊗b1+a⊗b2,ra⊗b=ra⊗b=a⊗rb,where a,a1,a2∈A, b,b1,b2∈B, and r∈F.


## 2.4. Newton–Girard Identities

Girard introduced some identities which describe relationship between roots of a polynomial and its coefficients (see [14, 15]). Later, Newton generalized these identities and gave a recursive formula. Today these identities are known as Newton–Girard identities whose explanation is as follows. Suppose fy is a polynomial like(10)gy=yn+c1yn−1+c2yn−2+c3yn−3+……+cn−1y+cn,with n of its roots r1,r2,……rn. We use the notation δk for the sum of the kth powers of roots as(11)δk=r1k+r2k+……+rnk,where k∈ℤ+ and δk=0 for k〉n. And(12)δk+c1δk−1+c2δk−2+s3ck−3+……+ck−1δ1+kck=0.This identity allows us to deduce the relations:(13)δ1+c1=0,δ2+c1δ1+2c2=0,δ3+c1δ2+c2δ1+3c3=0,δ4+c1δ3+c2δ2+c3δ1+4c4=0.Now, considering the most generalized form of the polynomial,(14)fy=∑i=0ntiyi.And with the assumption thattk=0 for k0, we define δk, for k⩾0, as(15)δk=r1k+r2k+……+rnk.By interchanging “k” to “−k,” we obtain(16)δ−k=r1−k+r2−k+……+rn−k.Finally, we can conclude the general form of Newton’s identity as(17)tjn−j+tj+1δ−1+tj+2δ−2+tj+3δ−3+……+tnδj−n=0;j⩽n.Furthermore, we can deduct the following results:(18)M1=t1s,M2=2t2s−t12s2,M3=3t3s−3t1t2s2+t13s3,M4=4t4s−4t1t3s2−2t22s2+4t12t2s3−t14s4.In general notation,(19)Mn=ntns−∑r=1n−1tn−rsMr,here we used Mi=−δ−k,∀i=0,1,2,… and t0=s.When we consider the caset0=1−s, the above identities will become(20)N1=−t1s−1,N2=−2t2s−1−t12s−12,N3=−3t3s−1−3t1t2s−12−t13s−13,N4=−4t4s−1−4t1t3s−12−2t22s−12−4t12t2s−13−t14s−14.The general form will be(21)Nn=ntn1−s−∑r=1n−1tn−r1−sNr.In [10], we have extended the idea of trilogarithmic tangent group up to the second order by proposing a group Tℬ32F and its complex. It is quite natural to think for the generalization of the degree of this group and other constructions (complexes, morphisms, cross ratio, etc.) related to this group. Let us define the following.


## 2.5. First-Order Tangent Group of Weight 3

A first-order tangent group, denoted byTℬ2F, is a ℤ module generated by the elements of the form s;s′∈ZFε2, a quotient by the expression,(22)s;s′−t;t′+ts;ts′−1−t1−s;1−t1−s′+s1−tt1−s;s1−tt1−s′,where s,t≠0,1,s≠t, s;s′=s+s′ε−s;s,s′∈F,(23)ts′=st′−s′ts2,1−t1−s′=1−ts′−1−st′1−s2,and(24)s1−tt1−s′=t1−ts′−s1−st′t1−s2.As the second order is discussed in [10], we define the third order directly.


## 2.6. Third-Order Tangent Group of Weight 3

We denoteTℬ33F to be the trilogarithmic tangent group of order three which is a ℤ module over the truncated polynomial ring Fε4 whose generators are the elements of the form s;t1,t2,t3∈ℤFε4, where s;t1,t2,t3=s+t1ε+t2ε2+t3ε3−s,s,t1,t2,t3∈F and quotient by the kernel of map(25)∂:ℤFε4⟶Tℬ23F⊗F×⊕F⊗ℬ2F,such that(26)∂a;b1,b2,b32=s;t1,t2,t323⊗s+3t3s−3t1t2s2+t13s3⊗s2,where ⊕ and ⊗, respectively, represent the direct sum and tensor product of free Abelian groups or modules’ tensor product of the free Abelian groups.


## 2.7. Generalized Tangent Group of Weight 3

Let the characteristic of the fieldF be zero and Fεn+1 represent the ring of truncated polynomials. We call Tℬ3nF the tangent group of weight 3 and order n which can be defined as a ℤ module over Fεn+1 with generators of the form s;t1,t2,…,tn∈ℤFεn+, where s;t1,t2,…,tn=s+t1ε+t2ε2+⋯+tnεn−s,s,t1,t2,…,tn∈F and quotient by Ker∂:(27)∂:ℤFεn+1⟶Tℬ2nF⊗F×⊕F⊗ℬ2F,where ∂ can be written as(28)∂s;t1,t2,…,tn2=s;t1,t2,…,tn2n⊗s+Mn⊗s2.The factorMn is given in (8). For n=1,2, we obtain the groups Tℬ3F and Tℬ32F. The former is defined in [9] and later is in [10]. By using the group Tℬ3nF, the following tangential complex can be obtained:(29)Tℬ3nF⟶∂εnTℬ2nF⊗F×⊕F⊗ℬ2F⟶∂εnF⊗∧2F×⊕∧3F.


## 2.8. Triple Ratio

The triple ratioru0,…,u5 of six points is given in [12] as(30)r3u0,…,u5=u0u1u3u1u2u4u2u0u5u0u1u4u1u2u5u2u0u3,whose tangential version for ν=2 can be traced in [8, 10] as(31)r3,εl0∗,…,l5∗=l0∗l1∗l3∗l1∗l2∗l4∗l2∗l0∗l5∗εl0l1l4l1l2l5l2l0l3−l0l1l3l1l2l4l2l0l5l0l1l4l1l2l5l2l0l3l0∗l1∗l4∗l1∗l2∗l5∗l2∗l0∗l3∗εl0l1l4l1l2l5l2l0l3.We extend this notion forν=n+1 as(32)r3,εnu0∗,…,u5∗=Alt6u0∗u1∗u3∗u1∗u2∗u4∗u2∗u0∗u5∗εnu0u1u4u1u2u5u2u0u3−r3,εn−1u0∗,…,u5∗u0∗u1∗u4∗u1∗u2∗u5∗u2∗u0∗u3∗εu0u1u4u1u2u5u2u0u3−r3,εn−2u0∗,…,u5∗u0∗u1∗u4∗u1∗u2∗u5∗u2∗u0∗u3∗ε2u0u1u4u1u2u5u2u0u3−…−r3u0,…,u5u0∗u1∗u4∗u1∗u2∗u5∗u2∗u0∗u3∗εnu0u1u4u1u2u5u2u0u3.


## 3. Main Results and Discussion

### 3.1. Dilogarithmic Bicomplexes

In this section, we will connect the Grassmannian bicomplex to the tangent to the Bloch–Suslin complex. This connection can be displayed through the figure.The mapsπ0,εn3, π1,εn3, π2,εn3, and ∂εn are defined for n=1,2 in [9, 10], respectively. Here, we redefine these maps using Newton identities which will enable us to propose these maps for order “ n.” For this, we suppose the following notations. Δu0∗,…,u^i∗,…,u3∗εj=λij, Δu0∗,…,u^i+1∗,…,u3∗εj=λi+1j, Δu0∗,…,u^i+2∗,…,u3∗εj=λi+2j, and Δu0∗,…,u^i+3∗,…,u3∗εj=λi+3j for all i=0,1,2,3,j=0,1,2,3,….n with λi0=Δu0,…,u^i,…,u3 and λi+10=Δu0,…,u^i+1,…,u3λi+20=Δu0,…,u^i+2,…,u3λi+30=Δu0,…,u^i+3,…,u3. Note that we write Δu∗ε0=Δu.Newton’s theorem associates a power sumPij function for every polynomial with coefficients λij and gives the following relations (see Section 2.4):(33)Pi1=λi1λi0,Pi2=2λi2λi0−λi12λi02,Pi3=3λi3λi0−3λi1λi2λi02+λi13λi03,Pi4=4λi4λi0−4λi1λi3λi02−2λi22λi02+4λi12λi2λi03−λi14λi04,with the general term,(34)Pin=nλinλi0−∑r=1n−1λin−rλi0Pir.Above constructions enable us to rewrite the mapsπ0,ε3 and π0,ε23 in a precise form as(35)π0,ε3u0∗,…,u3∗=∑i=03−1iPi1⊗λi+10λi+20∧λi+30λi+20+∧k=0k≠i3Pk1,imod4,π0,ε23u0∗,…,u3∗=∑i=03−1iPi2⊗λi+10λi+20∧λi+30λi+20+∧k=0k≠i3Pk2,imod4.Forn=3, we propose(36)π0,ε33u0∗,…,u3∗=∑i=03−1iPi3⊗λi+10λi+20∧λi+30λi+20+∧k=0k≠i3Pk3,imod4.Using an inductive approach, one can write(37)π0,εn3u0∗,…,u3∗=∑i=03−1iPin⊗λi+10λi+20∧λi+30λi+20+∧k=0k≠i3Pkn,imod4.Next, we move to describeπ1,εn3u0∗,…,u3∗. We follow an inductive approach to generalize this map because already we have the definitions of π1,εn3u0∗,…,u3∗, for n=1,2,3 (see [8, 10]). There, we observe that the complexity of this map increases when its order become higher and higher. So, to overcome this situation, we introduce some notations as follows. Let Δu0∗,…,u^i∗,…,u^j∗,…,u4∗εr=δijr with Δu0,…,u^i,…,u^j,…,u4=δij0 and r=0,….,n; then, for a polynomial whose coefficients are δijr, there should be an rth power sum say Sijr satisfying the identities below:(38)Sij1=δij1δij0,Sij2=2δij2δij0−δij12δij02,Sij3=3δij3δij0−3δij1δij2δij02+δij13δij03,Sij4=4δij4δij0−4δij1δij3δij02−2δij22δij02+4δij12δij2δij03−δij14δij04.And in general,(39)Sijn=nδijnδij0−∑m=1n−1δijn−mδij0Sijm.Again, we rewrite the mapsπ1,ε3 and π1,ε23, which are defined in [9, 10], respectively, in current settings as(40)π1,ε3u0∗,…,u4∗=−13∑i,j=0j≠i4−1irui|u0,…,u^i,…,u4;rεui∗|u0∗,…,u∗^i,…,u4∗2⊗∏i≠jΔu^i,u^j+Sij1⊗rui|u0,…,u^i,…,u42,π1,ε23u0∗,…,u4∗=−13∑i,j=0j≠i4−1irui|u0,…,u^i,…,u4;rεui∗|u0∗,…,u∗^i,…,u4∗,rε2ui∗|u0∗,…,u∗^i,…,u4∗22⊗∏i≠jΔu^i,u^j+Sij2⊗rui|l0,…,u^i,…,l42.Forn=3, we also propose(41)π1,ε33u0∗,…,u4∗=−13∑i,j=0j≠i4−1irui|u0,…,u^i,…,u4;rεui∗|u0∗,…,u∗^i,…,u4∗,…,rε3ui∗|u0∗,…,u∗^i,…,u4∗23⊗∏i≠jΔu^i,u^j+Sij3⊗rui|u0,…,u^i,…,u42.It is now easy to expressπ1,εn3 as(42)π1,εn3u0∗,…,u4∗=−13∑i,j=0j≠i4−1irui|u0,…,u^i,…,u4;rεui∗|u0∗,…,u∗^i,…,u4∗,…,rεnui∗|u0∗,…,u∗^i,…,u4∗2n⊗∏i≠jΔu^i,u^j+Sijn⊗rui|u0,…,u^i,…,u42.The mapπ2,εn3 can simply be defined as the triple ratio which is given in (17). For ν=2,3, it is given in [8, 10] as(43)π2,ε3u0∗,…,u5∗=245Alt6r3u0,…,u5;r3,εu0∗,…,u5∗3,π2,ε23u0∗,…,u5∗=245Alt6r3u0,…,u5;r3,εu0∗,…,u5∗,r3,ε2u0∗,…,u5∗3.And, forn=3, we propose(44)π2,ε33u0∗,…,u5∗=245Alt6r3u0,…,u5;r3,εu0∗,…,u5∗,r3,ε2u0∗,…,u5∗,r3,ε3u0∗,…,u5∗3.Therefore, in general, it may be(45)π2,εn3u0∗,…,u5∗=245Alt6r3u0,…,u5;r3,εu0∗,…,u5∗,r3,ε2u0∗,…,u5∗,…,r3,εnu0∗,…,u5∗3.The horizontal map∂εn of the diagram D works in two ways. First is when it sends the members of Tℬ3nF to Tℬ2nF⊗F×⊕F⊗ℬ2F and second way is when it sends elements of Tℬ2nF⊗F×⊕F⊗ℬ2F into F⊗∧2F×⊕∧3F. This map is defined in [9, 10], for n=1,2, as(46)∂εs;t2⊗c+x⊗y2=ts⊗1−s∧c−t1−s⊗s∧c+x⊗1−y∧y+ts∧t1−s∧x,∂ε2s;t1,t22⊗c+x⊗y2=2t2s−t12s2⊗1−s∧c−2t21−s+t121−s2⊗s∧c+x⊗1−y∧y+2t2s−t12s2∧2t21−s+t121−s2∧x.And, forn=3, we propose this map as(47)∂ε3s;t1,t2,t323⊗c+x⊗y2,=3t3a−3t1t2s2−t13s3⊗1−s∧c−3t31−s−3t1t21−s2−t131−s3⊗s∧c,+x⊗1−y∧y+3t3s−3t1t2s2−t13s3∧3t31−s−3t1t21−s2−t131−s3∧x.We can make these definitions more precise using identities of Newton, i.e.,(48)∂ε2s;t1,t22⊗c+a⊗b2=M1⊗1−s∧c−N1⊗s∧c+a⊗1−b∧b+M1∧N1∧a,∂ε3s;t1,t2,t32⊗c+a⊗b2=M3⊗1−s∧c−N3⊗s∧c+a⊗1−b∧b+M3∧N3∧a.Therefore, we can write(49)∂εns;t1,t2,…,tn2⊗c+a⊗b2,=Mn⊗1−s∧c−Nn⊗s∧c+a⊗1−b∧b+Mn∧Nn∧a.The map∂εn of right square of the diagram (D) has already been defined in [9, 10] for order 1 and 2 as(50)∂εs;t3=s;t2⊗s+ts⊗s2,∂ε2s;t1,t232=s;t1,t222⊗s+2t2s−t12s2⊗s2.And, for order 3, we propose(51)∂ε3s;t1,t2,t333=s;t1,t2,t323⊗s+3t3s−3t1t2s2−t13s3⊗s2.If we consider a polynomial having coefficientsti together with a kth power sum function Tk, then we can express the above maps in a precise form like(52)∂εt0;t132=t0;t12⊗t0+T1⊗t02,∂ε2t0;t1,t232=t0;t1,t222⊗t0+T2⊗t02,∂ε3t0;t1,t2,t333=t0;t1,t2,t323⊗t0+T3⊗t02.Therefore,(53)∂εnt0;t1,t2,t3,…,tn3n=t0;t1,t2,t3,…,tn2n⊗t0+Tn⊗t02,where ti=r3,εiu0∗,…,u5∗,i=0,…,n, and(54)Tn=ntnt−∑r=1n−1tn−rtTr.Theorem 1.
The right square of (D) commutes, that is,(55)∂εn∘π1,εn3=π0,εn3∘d.Proof.
We split the map,(56)π0,εn3:C4AFεn+13⟶F⊗∧2F×⊕∧3F,into the sum of two maps:(57)ϕ1n:C4AFεn+13⟶F⊗∧2F×and(58)ϕ2n:C4AFεn+13⟶∧3F.
Then, we write(59)π0,εn3=ϕ1n+ϕ2n.
The definitions ofd and π0,εn3 allow us to write(60)ϕ1n∘du0∗,…,u4∗=ϕ1n∑i=04−1iu0∗,…,u^i∗,…,u4∗=Alt˜01234∑i=03−1inλinλi0−∑r=1n−1λin−rλi0Pir⊗λi+10λi+20∧λi+30λi+20,ϕ2n∘du0∗,…,u4∗=ϕ2n∑i=04−1iu0∗,…,u^i∗,…,u4∗=Alt˜01234∑i=03−1i∧k=0k≠i3nλknλk0−∑r=1n−1λkn−rλk0Pkr.
The inner expression of (37) will gives us terms such as a1εn/a1⊗b∧c, a1εia2εj/a12⊗b∧c,i+j=n, and a1εia2εja3εk/a13⊗b∧c,i+j+k=n. Moreover, the simplification process vanishes the terms with xα≠xβ≠xγ… and remains only those terms whose xi′s are the same. Next, we expand the outer sum and use some arithmetic to simplify the result. We can use the same algorithm on the other part ϕ2n∘du0∗,…,u4∗ and get the result in precise form and hence will get the value of π0,εn3∘d. Next, we move to evaluate ∂εn∘π1,εn3. For this, we take u0∗,…,u4∗∈C5AFεn+13 so that(61)∂εn∘π1,εn3u0∗,…,u4∗,=∂εn−13∑i,j=0j≠i4−1ia;b1,b2,…,bn2n⊗∏i≠jΔu^i,u^j+Sijn⊗rui|u0,…,u^i,…,u42,where s=rui|u0,…,u^i,…,u4, t1=rεui∗|u0∗,…,u∗^i,…,u4∗, t2=rε2ui∗|su0∗,…,u∗^i,…,u4∗, and tn=rεnui∗|u0∗,…,u∗^i,…,u4∗ and Sijn is defined in (22). Using definition of ∂εn and settings of (8) and (10), we can write as(62)=−13∑i,j=0j≠i4−1intns−∑r=1n−1tn−rsMr⊗1−s∧∏i≠jΔu^i,u^j−ntn1−s−∑r=1n−1tn−r1−sNr⊗s∧∏i≠jΔu^i,u^j+Sijn⊗1−y∧y+nbns−∑r=1n−1tn−rsMr∧ntn1−s−∑r=1n−1tn−r1−sNr∧Sijn,where y=rui|u0,…,u^i,…,u42. This expression can further be transformed into a simpler form by using first the axioms of tensor and wedge product and then expanding through summation. This will give us a precise value of ∂εn∘π1,εn3u0∗,…,u4∗ which will be exactly equal to the value π1,εn3∘du0∗,…,u4∗.Theorem 2.
The left part of the diagram (D) commutes, that is,(63)∂εn∘π2,εn3=π1,εn3∘d.Proof.
The mapsπ1,εn3, ∂εn, and π2,εn3 are explained in (25), (35), and (28), respectively. Taking u0∗,…,u5∗∈C6AFεn+13 and applying (28), we have(64)∂εn∘π2,εn3u0∗,…,u5∗=∂εn245Alt6r3u0,…,u5;r3,εu0∗,…,u5∗,r3,ε2u0∗,…,u5∗,…,r3,εnu0∗,…,u5∗3n.
Applying definition (35) and using (36), we obtain(65)=245Alt6c;c1,c2,…,cn2n⊗c+ncnc−∑r=1n−1cn−rcTr⊗c2,where ci=r3,εiu0∗,…,u5∗,i=0,1,2,…,n, and T1=c1/c Here, we can use the combinatorial techniques which are used in the proof of Theorem 5.6 of [9] and can write equation (42) as(66)=13Alt6c;c1,c2,…,cn2n⊗u0u1u3+nu0∗u1∗u3∗εnl0l1l3−∑r=1n−1u0∗u1∗u3∗εn−ru0u1u3Vr⊗c2,where Vr represents the rth sum of powers of the polynomial having coefficients u0∗u1∗u3∗εi and V1=u0∗u1∗u3∗ε/u0u1u3. To attain RHS, we express the map π1,εn3 as an alternation sum:(67)π1,εn3u0∗,…,u5∗=13Alt6ru0|u1u2u3u4;rεu0∗|u1∗u2∗u3∗u4∗,…,rεnu0∗|u1∗u2∗u3∗u4∗22⊗u0u1u2+nu0∗u1∗u3∗εnu0u1u3−∑r=1n−1u0∗u1∗u3∗εn−ru0u1u3Vr⊗c2.
Now, it becomes easy to attain the value ofπ1,εn3∘du0∗,…,u5∗ with the help of (44). For this, we proceed by enforcing d and then applying π1,εn3. Now, we are going to follow the procedure of Theorem 5.6 of [8] which will provide us a result exactly the same as (43).


## 3.1. Dilogarithmic Bicomplexes

In this section, we will connect the Grassmannian bicomplex to the tangent to the Bloch–Suslin complex. This connection can be displayed through the figure.The mapsπ0,εn3, π1,εn3, π2,εn3, and ∂εn are defined for n=1,2 in [9, 10], respectively. Here, we redefine these maps using Newton identities which will enable us to propose these maps for order “ n.” For this, we suppose the following notations. Δu0∗,…,u^i∗,…,u3∗εj=λij, Δu0∗,…,u^i+1∗,…,u3∗εj=λi+1j, Δu0∗,…,u^i+2∗,…,u3∗εj=λi+2j, and Δu0∗,…,u^i+3∗,…,u3∗εj=λi+3j for all i=0,1,2,3,j=0,1,2,3,….n with λi0=Δu0,…,u^i,…,u3 and λi+10=Δu0,…,u^i+1,…,u3λi+20=Δu0,…,u^i+2,…,u3λi+30=Δu0,…,u^i+3,…,u3. Note that we write Δu∗ε0=Δu.Newton’s theorem associates a power sumPij function for every polynomial with coefficients λij and gives the following relations (see Section 2.4):(33)Pi1=λi1λi0,Pi2=2λi2λi0−λi12λi02,Pi3=3λi3λi0−3λi1λi2λi02+λi13λi03,Pi4=4λi4λi0−4λi1λi3λi02−2λi22λi02+4λi12λi2λi03−λi14λi04,with the general term,(34)Pin=nλinλi0−∑r=1n−1λin−rλi0Pir.Above constructions enable us to rewrite the mapsπ0,ε3 and π0,ε23 in a precise form as(35)π0,ε3u0∗,…,u3∗=∑i=03−1iPi1⊗λi+10λi+20∧λi+30λi+20+∧k=0k≠i3Pk1,imod4,π0,ε23u0∗,…,u3∗=∑i=03−1iPi2⊗λi+10λi+20∧λi+30λi+20+∧k=0k≠i3Pk2,imod4.Forn=3, we propose(36)π0,ε33u0∗,…,u3∗=∑i=03−1iPi3⊗λi+10λi+20∧λi+30λi+20+∧k=0k≠i3Pk3,imod4.Using an inductive approach, one can write(37)π0,εn3u0∗,…,u3∗=∑i=03−1iPin⊗λi+10λi+20∧λi+30λi+20+∧k=0k≠i3Pkn,imod4.Next, we move to describeπ1,εn3u0∗,…,u3∗. We follow an inductive approach to generalize this map because already we have the definitions of π1,εn3u0∗,…,u3∗, for n=1,2,3 (see [8, 10]). There, we observe that the complexity of this map increases when its order become higher and higher. So, to overcome this situation, we introduce some notations as follows. Let Δu0∗,…,u^i∗,…,u^j∗,…,u4∗εr=δijr with Δu0,…,u^i,…,u^j,…,u4=δij0 and r=0,….,n; then, for a polynomial whose coefficients are δijr, there should be an rth power sum say Sijr satisfying the identities below:(38)Sij1=δij1δij0,Sij2=2δij2δij0−δij12δij02,Sij3=3δij3δij0−3δij1δij2δij02+δij13δij03,Sij4=4δij4δij0−4δij1δij3δij02−2δij22δij02+4δij12δij2δij03−δij14δij04.And in general,(39)Sijn=nδijnδij0−∑m=1n−1δijn−mδij0Sijm.Again, we rewrite the mapsπ1,ε3 and π1,ε23, which are defined in [9, 10], respectively, in current settings as(40)π1,ε3u0∗,…,u4∗=−13∑i,j=0j≠i4−1irui|u0,…,u^i,…,u4;rεui∗|u0∗,…,u∗^i,…,u4∗2⊗∏i≠jΔu^i,u^j+Sij1⊗rui|u0,…,u^i,…,u42,π1,ε23u0∗,…,u4∗=−13∑i,j=0j≠i4−1irui|u0,…,u^i,…,u4;rεui∗|u0∗,…,u∗^i,…,u4∗,rε2ui∗|u0∗,…,u∗^i,…,u4∗22⊗∏i≠jΔu^i,u^j+Sij2⊗rui|l0,…,u^i,…,l42.Forn=3, we also propose(41)π1,ε33u0∗,…,u4∗=−13∑i,j=0j≠i4−1irui|u0,…,u^i,…,u4;rεui∗|u0∗,…,u∗^i,…,u4∗,…,rε3ui∗|u0∗,…,u∗^i,…,u4∗23⊗∏i≠jΔu^i,u^j+Sij3⊗rui|u0,…,u^i,…,u42.It is now easy to expressπ1,εn3 as(42)π1,εn3u0∗,…,u4∗=−13∑i,j=0j≠i4−1irui|u0,…,u^i,…,u4;rεui∗|u0∗,…,u∗^i,…,u4∗,…,rεnui∗|u0∗,…,u∗^i,…,u4∗2n⊗∏i≠jΔu^i,u^j+Sijn⊗rui|u0,…,u^i,…,u42.The mapπ2,εn3 can simply be defined as the triple ratio which is given in (17). For ν=2,3, it is given in [8, 10] as(43)π2,ε3u0∗,…,u5∗=245Alt6r3u0,…,u5;r3,εu0∗,…,u5∗3,π2,ε23u0∗,…,u5∗=245Alt6r3u0,…,u5;r3,εu0∗,…,u5∗,r3,ε2u0∗,…,u5∗3.And, forn=3, we propose(44)π2,ε33u0∗,…,u5∗=245Alt6r3u0,…,u5;r3,εu0∗,…,u5∗,r3,ε2u0∗,…,u5∗,r3,ε3u0∗,…,u5∗3.Therefore, in general, it may be(45)π2,εn3u0∗,…,u5∗=245Alt6r3u0,…,u5;r3,εu0∗,…,u5∗,r3,ε2u0∗,…,u5∗,…,r3,εnu0∗,…,u5∗3.The horizontal map∂εn of the diagram D works in two ways. First is when it sends the members of Tℬ3nF to Tℬ2nF⊗F×⊕F⊗ℬ2F and second way is when it sends elements of Tℬ2nF⊗F×⊕F⊗ℬ2F into F⊗∧2F×⊕∧3F. This map is defined in [9, 10], for n=1,2, as(46)∂εs;t2⊗c+x⊗y2=ts⊗1−s∧c−t1−s⊗s∧c+x⊗1−y∧y+ts∧t1−s∧x,∂ε2s;t1,t22⊗c+x⊗y2=2t2s−t12s2⊗1−s∧c−2t21−s+t121−s2⊗s∧c+x⊗1−y∧y+2t2s−t12s2∧2t21−s+t121−s2∧x.And, forn=3, we propose this map as(47)∂ε3s;t1,t2,t323⊗c+x⊗y2,=3t3a−3t1t2s2−t13s3⊗1−s∧c−3t31−s−3t1t21−s2−t131−s3⊗s∧c,+x⊗1−y∧y+3t3s−3t1t2s2−t13s3∧3t31−s−3t1t21−s2−t131−s3∧x.We can make these definitions more precise using identities of Newton, i.e.,(48)∂ε2s;t1,t22⊗c+a⊗b2=M1⊗1−s∧c−N1⊗s∧c+a⊗1−b∧b+M1∧N1∧a,∂ε3s;t1,t2,t32⊗c+a⊗b2=M3⊗1−s∧c−N3⊗s∧c+a⊗1−b∧b+M3∧N3∧a.Therefore, we can write(49)∂εns;t1,t2,…,tn2⊗c+a⊗b2,=Mn⊗1−s∧c−Nn⊗s∧c+a⊗1−b∧b+Mn∧Nn∧a.The map∂εn of right square of the diagram (D) has already been defined in [9, 10] for order 1 and 2 as(50)∂εs;t3=s;t2⊗s+ts⊗s2,∂ε2s;t1,t232=s;t1,t222⊗s+2t2s−t12s2⊗s2.And, for order 3, we propose(51)∂ε3s;t1,t2,t333=s;t1,t2,t323⊗s+3t3s−3t1t2s2−t13s3⊗s2.If we consider a polynomial having coefficientsti together with a kth power sum function Tk, then we can express the above maps in a precise form like(52)∂εt0;t132=t0;t12⊗t0+T1⊗t02,∂ε2t0;t1,t232=t0;t1,t222⊗t0+T2⊗t02,∂ε3t0;t1,t2,t333=t0;t1,t2,t323⊗t0+T3⊗t02.Therefore,(53)∂εnt0;t1,t2,t3,…,tn3n=t0;t1,t2,t3,…,tn2n⊗t0+Tn⊗t02,where ti=r3,εiu0∗,…,u5∗,i=0,…,n, and(54)Tn=ntnt−∑r=1n−1tn−rtTr.Theorem 1.
The right square of (D) commutes, that is,(55)∂εn∘π1,εn3=π0,εn3∘d.Proof.
We split the map,(56)π0,εn3:C4AFεn+13⟶F⊗∧2F×⊕∧3F,into the sum of two maps:(57)ϕ1n:C4AFεn+13⟶F⊗∧2F×and(58)ϕ2n:C4AFεn+13⟶∧3F.
Then, we write(59)π0,εn3=ϕ1n+ϕ2n.
The definitions ofd and π0,εn3 allow us to write(60)ϕ1n∘du0∗,…,u4∗=ϕ1n∑i=04−1iu0∗,…,u^i∗,…,u4∗=Alt˜01234∑i=03−1inλinλi0−∑r=1n−1λin−rλi0Pir⊗λi+10λi+20∧λi+30λi+20,ϕ2n∘du0∗,…,u4∗=ϕ2n∑i=04−1iu0∗,…,u^i∗,…,u4∗=Alt˜01234∑i=03−1i∧k=0k≠i3nλknλk0−∑r=1n−1λkn−rλk0Pkr.
The inner expression of (37) will gives us terms such as a1εn/a1⊗b∧c, a1εia2εj/a12⊗b∧c,i+j=n, and a1εia2εja3εk/a13⊗b∧c,i+j+k=n. Moreover, the simplification process vanishes the terms with xα≠xβ≠xγ… and remains only those terms whose xi′s are the same. Next, we expand the outer sum and use some arithmetic to simplify the result. We can use the same algorithm on the other part ϕ2n∘du0∗,…,u4∗ and get the result in precise form and hence will get the value of π0,εn3∘d. Next, we move to evaluate ∂εn∘π1,εn3. For this, we take u0∗,…,u4∗∈C5AFεn+13 so that(61)∂εn∘π1,εn3u0∗,…,u4∗,=∂εn−13∑i,j=0j≠i4−1ia;b1,b2,…,bn2n⊗∏i≠jΔu^i,u^j+Sijn⊗rui|u0,…,u^i,…,u42,where s=rui|u0,…,u^i,…,u4, t1=rεui∗|u0∗,…,u∗^i,…,u4∗, t2=rε2ui∗|su0∗,…,u∗^i,…,u4∗, and tn=rεnui∗|u0∗,…,u∗^i,…,u4∗ and Sijn is defined in (22). Using definition of ∂εn and settings of (8) and (10), we can write as(62)=−13∑i,j=0j≠i4−1intns−∑r=1n−1tn−rsMr⊗1−s∧∏i≠jΔu^i,u^j−ntn1−s−∑r=1n−1tn−r1−sNr⊗s∧∏i≠jΔu^i,u^j+Sijn⊗1−y∧y+nbns−∑r=1n−1tn−rsMr∧ntn1−s−∑r=1n−1tn−r1−sNr∧Sijn,where y=rui|u0,…,u^i,…,u42. This expression can further be transformed into a simpler form by using first the axioms of tensor and wedge product and then expanding through summation. This will give us a precise value of ∂εn∘π1,εn3u0∗,…,u4∗ which will be exactly equal to the value π1,εn3∘du0∗,…,u4∗.Theorem 2.
The left part of the diagram (D) commutes, that is,(63)∂εn∘π2,εn3=π1,εn3∘d.Proof.
The mapsπ1,εn3, ∂εn, and π2,εn3 are explained in (25), (35), and (28), respectively. Taking u0∗,…,u5∗∈C6AFεn+13 and applying (28), we have(64)∂εn∘π2,εn3u0∗,…,u5∗=∂εn245Alt6r3u0,…,u5;r3,εu0∗,…,u5∗,r3,ε2u0∗,…,u5∗,…,r3,εnu0∗,…,u5∗3n.
Applying definition (35) and using (36), we obtain(65)=245Alt6c;c1,c2,…,cn2n⊗c+ncnc−∑r=1n−1cn−rcTr⊗c2,where ci=r3,εiu0∗,…,u5∗,i=0,1,2,…,n, and T1=c1/c Here, we can use the combinatorial techniques which are used in the proof of Theorem 5.6 of [9] and can write equation (42) as(66)=13Alt6c;c1,c2,…,cn2n⊗u0u1u3+nu0∗u1∗u3∗εnl0l1l3−∑r=1n−1u0∗u1∗u3∗εn−ru0u1u3Vr⊗c2,where Vr represents the rth sum of powers of the polynomial having coefficients u0∗u1∗u3∗εi and V1=u0∗u1∗u3∗ε/u0u1u3. To attain RHS, we express the map π1,εn3 as an alternation sum:(67)π1,εn3u0∗,…,u5∗=13Alt6ru0|u1u2u3u4;rεu0∗|u1∗u2∗u3∗u4∗,…,rεnu0∗|u1∗u2∗u3∗u4∗22⊗u0u1u2+nu0∗u1∗u3∗εnu0u1u3−∑r=1n−1u0∗u1∗u3∗εn−ru0u1u3Vr⊗c2.
Now, it becomes easy to attain the value ofπ1,εn3∘du0∗,…,u5∗ with the help of (44). For this, we proceed by enforcing d and then applying π1,εn3. Now, we are going to follow the procedure of Theorem 5.6 of [8] which will provide us a result exactly the same as (43).


## 4. Conclusion

Polylogarithmic groups have been studied by several renowned mathematicians such as Goncharove, Zagier, Bloch, Suslin, and Cathelineaue. A tangential version of these groups of weight two and three is studied by Siddiqui for the first order. Recently, the study of tangential groups of weight two and their associated maps are extended to a general ordern. In this work, we have shown that the notions associated to trilogarithmic tangential groups Tℬ3nF such as cross ratio, triple ratio, Siegel’s identity and other relations are valid for higher orders. All these notions are being constructed for the trilogarithmic tangential groups of higher order. Using the groups Tℬ3nF and a map ∂εn, we formed the following complexes:(68)Tℬ3nF⟶∂εnTℬ2nF⊗F×⊕F⊗ℬ2F⟶∂εnF⊗∧2F×⊕∧3F.Moreover, we have proposed morphisms,π0,εn3, π1,εn3, and π2,εn3 in order to connect Grassmannian complexes to the trilogarithmic tangential complexes for higher orders. This generalization process has been carried out with the help of Newton’s identities. Lastly, we proved that the resulting diagrams of connectivity are commutative.The above results motivate us to compute higher order tangent groups for weightn≥4 and use it to construct the higher order tangent to Goncharov’s complex for weight 4 or even higher [16].



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