Parameterizing the Simplest Grassmann-Gaussian Relations for Pachner Move 3-3

We consider relations in Grassmann algebra corresponding to the four-dimensional Pachner move 3-3, assuming that there is just one Grassmann variable on each 3-face, and a 4-simplex weight is a Grassmann-Gaussian exponent depending on these variables on its five 3-faces. We show that there exists a large family of such relations; the problem is in finding their algebraic-topologically meaningful parameterization. We solve this problem in part, providing two nicely parameterized subfamilies of such relations. For the second of them, we further investigate the nature of some of its parameters: they turn out to correspond to an exotic analogue of middle homologies. In passing, we also provide the 2-4 Pachner move relation for this second case.


Introduction
Discrete topological field theories -specifically, field theories on piecewise linear (PL) manifolds -are definitely a challenging research subject. As there are now many interesting topological quantum field theories (TQFT's) in three dimensions, it looks reasonable to concentrate on the four-dimensional case. Such a theory is expected to bring about interesting results both by itself and when compared with the existing theories on smooth manifolds.
As explained, for instance, in [10, Section 1], it makes sense first to construct algebraic relations corresponding to Pachner moves. And the simplest nontrivial relations of such kind arise, as we believe, in Grassmann algebras. In three dimensions, a relation corresponding to Pachner move 2-3 is often called pentagon relation, and there are some Grassmann-algebraic constructions for pentagon relation, presented, in particular, in paper [9]. As we hope to demonstrate here, the four-dimensional case has its own specific beauty; it is more complicated but also yields to systematic investigation.
If we consider an ansatz -a (tentative) specific form of quantities or expressions entering in our relations, and consider the relations as equations for the ansatz parameters, and if our ansatz is simple enough, then it may happen that the existence of many solutions for such equations follows already from parameter counting.
In this paper, we take the simplest possible form (6) of Grassmann-algebraic relation corresponding to Pachner move 3-3 -with just one Grassmann variable on each 3-face, and further assume that the Grassmann weight of a 4-simplex has the form of a Grassmann-Gaussian exponent, depending on the five variables on the 3-faces. A heuristic parameter count shows that there exists a large -and intriguing -family of relations of such form. We prefer to go further and prove the rigorous Theorem 4, formulated in terms of isotropic linear spaces of Grassmann differential operators annihilating our Grassmann-Gaussian exponents. In doing so, we not only prove the existence of the 4-simplex weights satisfying the 3-3 relations, but discover some interesting operators (namely, (16) and (17)) that may deserve further investigation; at least, they have an elegant form (namely, (28) and (29)) in one specific case.
Having proved our Theorem 4, we are naturally led to the problem of finding an algebraic-topologically meaningful parameterization of our Grassmann weights, which would enable us to move further and construct topological field theories. In the present paper, we make two steps in this direction by presenting two explicitly -and nicely -parameterized subfamilies of such weights, found largely by guess-and-try method. The first subfamily resembles the (more cumbersome) constructions in [6,7] -both are related to exotic homologies. The striking new fact is, however, that this is now only a subfamily of something mysterious, on whose nature only our parameterized second family sheds some additional light.
Some of the results of this paper have first appeared, in a preliminary form, in the preprint [8]. Below, • in Section 2, we recall the basic definitions from the theory of Grassmann algebras and Berezin integral, • in Section 3, we recall the four-dimensional Pachner moves, mainly moves 3-3 and 2-4 with which we will be dealing in this paper, and introduce some notational conventions, • in Section 4, we introduce a 3-3 relation for Grassmann 4-simplex weights. First, we do it in a general form, then we specialize the weights to be Grassmann-Gaussian exponents and explain their connection with isotropic spaces of Grassmann differential operators, • in Section 5, based on these isotropic spaces, we show the existence of a vast family of 4-simplex weights satisfying the 3-3 relation. The way we do it is constructive; what lacks in it is a parameterization for this whole family relevant for algebraic-topological applications, • in Section 6, we present two subfamilies of Grassmann 4-simplex weights satisfying the 3-3 relation where such parameterization has been obtained, • in Section 7, we do some preparational work in order to expose some exotichomological structures lying behind the second of the mentioned subfamilies. Namely, we introduce, for a given triangulated four-manifold, a sequence of two linear mappings -supposedly a fragment of an exotic chain complex, prove their chain property (their composition vanish), and present computational evidence showing that they provide an exotic analogue of usual middle (i.e., second) homologies, and • in Section 8, guided by the fact that the mentioned exotic-homological structures manifest themselves more clearly for the Pachner move 2-4, we present the relations corresponding to this move, study a new factor -edge weight -appearing in these relations, and then formulate the relations for both moves 3-3 and 2-4 using these exotic-homological terms.

Grassmann algebras and Berezin integral
In this paper, a Grassmann algebra is an associative algebra over the field C of complex numbers, with unity, generators x i -also called Grassmann variables -and relations This implies that, in particular, x 2 i = 0, so each element of a Grassmann algebra is a polynomial of degree ≤ 1 in each x i .
The degree of a Grassmann monomial is its total degree in all Grassmann variables. If an algebra element consists of monomials of only odd or only even degrees, it is called odd or, respectively, even. If all the monomials have degree 2, we call such element a Grassmannian quadratic form.
The exponent is defined by its usual Taylor series. We call the exponent of a quadratic form Grassmann-Gaussian exponent. Here is an example of it: There are two kinds of derivations in a Grassmann algebra: left derivative ∂ ∂x i and right derivative ← − ∂ ∂x i , with respect to a Grassmann variable x i . These are C-linear operations in Grassmann algebra defined as follows. Let f be an element not containing variable x i , then and From (1) and (2), the following Leibniz rules follow: if f is either even or odd, then where ǫ = 1 for an even f and ǫ = −1 for an odd f . The Grassmann-Berezin calculus of anticommuting variables is in many respects parallel to the usual calculus, see [1] and especially [2]. Still, there are some peculiarities, and one of them is that the integral in a Grassmann algebra is the same operation as derivative; more specifically, Berezin integral in a variable x i is defined, traditionally, as the right derivative w.r.t. x i . Independently, Berezin integral is defined as follows: it is a C-linear operator in Grassmann algebra satisfying where g does not contain x i . Multiple integral is defined according to the following Fubini rule: In "differential" notations, integral (4) is

Pachner moves in four dimensions
Pachner moves [12] are elementary local rebuildings of a manifold triangulation. A triangulation of a PL manifold can be transformed into any other triangulation using a finite sequence of Pachner moves. In four dimensions, each Pachner move replaces a cluster of 4-simplices with a cluster of some other 4-simplices, occupying the same place in the triangulation and having the same boundary. There are five (types of) Pachner moves in four dimensions: 3 → 3, 2 ↔ 4, and 1 ↔ 5, where the numbers indicate how many 4-simplices have been withdrawn and how many have replaced them. As the withdrawn and the replacing clusters of 4-simplices have the same common boundary, we can glue them together in a natural way (forgetting for a moment about the rest of the manifold); then, for all Pachner moves, they must form together a sphere S 4 triangulated in five 4-simplices as the boundary of a 5simplex, which we denote ∂∆ 5 . More details and a pedagogical introduction can be found in [10].
Move 3-3 is, in some informal sense, central: experience shows that if we have managed to find an algebraic formula whose structure can be regarded as reflecting the structure of the move, then we can also find (usually more complicated) formulas corresponding to the other Pachner moves. This may be compared to the three-dimensional case, where the popular "pentagon relation" often corresponds to the "central" three-dimensional Pachner move 2-3, while it is believed that, having done something interesting with this pentagon equation, one will be also able to work with the move 1-4.
We call the initial cluster of 4-simplices on a move the left-hand side (l.h.s.) of that move, and the final cluster -its right-hand side (r.h.s.). All moves in this paper will involve six vertices denoted i = 1, . . . , 6. Below are some more details. There are no inner edges (1-faces) or vertices (0-faces) in either side of this move.

Moves 2 ↔ 4
We describe move 2 → 4; move 4 → 2 is its inverse. Move 2 → 4 replaces, in the notations of this paper, the cluster of two 4-simplices 12345  There are no inner vertices in either side of this move.

Moves 1 ↔ 5
We don't work with these moves in this paper, so we only indicate that the move 1 → 5 adds a new vertex 6 inside the 4-simplex 12345, thus dividing it into five new 4-simplices. Move 5 → 1 is, of course, its inverse.

A few conventions
Any side of a Pachner move, as well as a single 4-simplex, is a triangulated fourmanifold with boundary. In Section 7, we also consider an arbitrary orientable triangulated four-manifold. Here are some our conventions concerning manifolds, their simplices, and also some complex parameters appearing in our theory, like vertex coordinates (see Section 6).
Convention 1. All manifolds in this paper are assumed to be oriented. In the case of Pachner moves, the orientation is defined so that, for the 4-simplex 12345, it is given by this order of its vertices.

Convention 2.
We denote by N k the number of k-simplices in a triangulation, and by N ′ k the number of inner k-simplices. Vertices are numbered from 1 through N 0 (as we have already done for Pachner moves, where N 0 = 6).

Convention 3.
When simplices are written in terms of their vertices, these go in the increasing order of their numbers (again, as we have already done).

Convention 4.
If the order of vertices is unknown, we use the following notation. Let there be, for instance, a 4-simplex whose set of vertices is {i, j, k, l, m}, then we denote it, omitting the commas for brevity, as {ijklm}.
Convention 5. The complex parameters appearing in our theory -to be exact, the eighteen parameters in Section 5 and vertex coordinates introduced in Section 6 -lie in the general position with respect to any considered algebraic formula, unless the opposite is explicitly stated. For instance, there is no division by zero in formula (26). Moreover, concerning vertex coordinates, all functions of them in this paper are rational, so the reader can assume that the coordinates are indeterminates over C (and we extended C to the relevant field of rational functions).

Relation 3-3 with Grassmann-Gaussian exponents: generalities 4.1 The form of relation 3-3
The Grassmann-algebraic Pachner move relations for move 3-3, considered in this paper, have the following general form: Here Grassmann variables x ijkl are attached to all 3-faces, i.e., tetrahedra t = ijkl; the Grassmann weight W ijklm of a 4-simplex u = ijklm depends on (i.e., contains) the variables on its 3-faces, e.g., W 12345 depends on x 1234 , x 1235 , x 1245 , x 1345 and x 2345 . Also, W u may depend on parameters attached to the 4-simplex u or/and its subsimplices. The integration goes in variables on inner three-faces in the corresponding side of Pachner move, while the result depends on the variables on boundary faces. Finally, const in the right-hand side is a numeric factor. Formula (6) appears to give the simplest possible form for a Grassmannalgebraic relation imitating the 3-3 move. Remark 1. As an example of a more heavyweight relation corresponding to the same Pachner move, we can cite [6, formula (38)], where, in particular, two Grassmann variables live on each tetrahedron.
Further simplification is achieved by using Grassmann-Gaussian exponents (which corresponds to free fermions in physical language) and assuming that We hope to demonstrate in this paper that the relation (6) is interesting already in the case of such exponents.

Isotropic subspaces of operators
The exponent (7) is characterized, up to a factor that does not depend on those x t that enter in it, by the equations where we denote ∂ t = ∂/∂x t . Generalizing the operators in the big parentheses in (9), we consider C-linear combinations of operators of left differentiations and multiplying by Grassmann generators: where t runs over all 3-faces in a given triangulated manifold.
We regard the anticommutator of two operators (10) (defined as [A, B] + = AB + BA for operators A and B) as their scalar product: (1) , d (2) ] + = With this scalar product, operators (10) form a complex Euclidean space, while all polynomials of these operators form a Clifford algebra.
Recall that an isotropic, or totally singular, subspace of a complex Euclidean space C 2n is such linear subspace where the scalar product identically vanish. We will need some basic facts about isotropic subspaces; for the reader's convenience, we formulate them as the following Theorem 1 and give it a simple proof. Much more interesting facts about Clifford algebras and isotropic subspaces in Euclidean spaces can be found, e.g., in [3]. Theorem 1. Maximal isotropic spaces in complex Euclidean space C 2n have dimension n. The manifold of these maximal isotropic spaces -isotropic Grassmannian -splits up in two connected components.
Proof. The first statement is an easy exercise. The second can be proved as follows. Let V ⊂ C 2n be a maximal isotropic subspace. For a generic orthonormal basis e 1 , . . . , e 2n , the orthogonal projection of V onto the space W spanned by the first half of basis vectors, i.e., e 1 , . . . , e n , coincides with the whole W . Also, considering the manifold B of orthonormal bases, it has two connected components (the determinant of transition matrix is 1 within a component, and −1 between the components), and the same components remain in B \ S -the result of taking away the set S of non-generic (in the sense indicated above) bases: the components cannot split any further, because S has complex codimension ≥ 1 and thus real codimension ≥ 2. Taking some liberty, we call, in this proof, the components of B orientations of C 2n We prefer to arrange basis vectors in a column, and vector coordinates in a row; so, an arbitrary vector in C 2n is written like    , and vectors in C 2n are identified with row vectors if a basis is given. For a space V and basis e 1 , . . . , e 2n such as in the previous paragraph, we can represent V as the linear span of the rows of the following matrix: where 1 n is the identity matrix and O is an orthogonal matrix, both of sizes n×n, and i = √ −1. The determinant of O is either 1 or −1, and its sign obviously cannot change within one component of B \S. So, given a fixed orientation of C 2n and a maximal isotropic space V , we get either 1 or −1 as det O.
It remains to prove that there is no further splitting between maximal isotropic spaces. Consider two such spaces, V 1 and V 2 . There exists a generic, in the above sense, basis e 1 , . . . , e 2n for both of them. Using this basis, V 1 and V 2 can be written in terms of matrices (12), and the corresponding orthogonal matrices O 1 and O 2 belong to the same connected component in the space of orthogonal matrices. Theorem 2. (i) For a given weight W u of the form (7), the operators d satisfying equation form a five-dimensional isotropic linear space.
(ii) For the set of equations (13) corresponding to a five-dimensional isotropic space V of operators (10) with t running over the five 3-faces of a 4simplex u, there exists a nonzero Grassmann algebra element W u , containing only the Grassmann generators x t and satisfying all these equations. This W u is determined by these equations uniquely up to a numeric factor.
(iii) The element W u from item (ii) is even for one connected component of the set of five-dimensional isotropic spaces V , and odd for the other. In the first case, it is, for a generic V , a Grassmann-Gaussian exponent (7).
Proof. (i) Five such linearly independent equations are already written in (9). It follows from the antisymmetry (8) of coefficients α tt ′ that any two operators d written in the big parentheses in (9) anticommute (including the case where they coincide). This means that the scalar product (11) vanishes.
(ii) We denote the ten-dimensional Euclidean space of all operators (10), where t ⊂ u, simply as C 10 . There exists an orthogonal transform O of C 10 sending V into the subspace generated by the five ∂ ∂xt , and to O corresponds, according to the general theory, a C-linear automorphism B of the Grassmann algebra such that Oy = ByB −1 for y ∈ C 10 .
As ∂ ∂xt 1 = 0 for all five t, it follows that W u = B −1 1 is annihilated by all operators in V . On the other hand, if there were two linearly independent W u annihilated by all operators in V , it would follow that the two corresponding algebra elements BW u would be annihilated by all five ∂ ∂xt , but this holds only for the one-dimensional space of constants.
(iii) A Zariski open set of even elements W u is already provided, and it consists of Grassmann-Gaussian exponents (7). Similar Zariski open set of odd elements consists of those W u that satisfy equations (9) with ∂ t and x t interchanged, i.e., where in both cases the relevant operators d form a 9-dimensional isotropic space.
(ii) Such 9-dimensional space of isotropic equations determines the l.h.s. or r.h.s. of (6) up to a numeric factor, if it is also assumed that this l.h.s. or r.h.s. depends only on Grassmann variables on the boundary 3-faces, as explained after formula (6).
for each boundary or inner tetrahedron t; these equations follow from equations (9) for individual weights and the first Leibniz rule in (3). Next, if t is an inner tetrahedron and if some operator t ′ (β t ′ ∂ t ′ + γ t ′ x t ′ ) anticommutes with the differentiation ∂ t -that is, in the sum -and if also then ∂ t W satisfies a similar equation, from which ∂ t and x t are absent, namely and it is not hard to see that W dx t -the right derivative -satisfies the same equation.
Due to condition (14), now there remains, at least, an 11-dimensional isotropic space of equations instead of the 12-dimensional. Proceeding this way further with the two remaining inner tetrahedra t, we get, at least, a 9-dimensional space of equations. As an isotropic subspace in a 18-dimensional complex Euclidean space (nine boundary tetrahedra t, operators ∂ t and x t for each of them) cannot be more than 9-dimensional, it is exactly 9-dimensional.
(ii) This is proved in full analogy with similar statement in item (ii) in Theorem 2.
Remark 2. This time, each side of (6) is easily shown to be an odd elementnamely, of Grassmann degree 3, and this determines the connected component in the manifold of maximal isotropic subspaces where our subspaces belong. This will be important for the construction in Section 5, see Remark 3.

A large family of Grassmann-Gaussian weights satisfying relation 3-3
In this Section, we construct a 18-parameter family of Grassmann weights depending on the variables x ijkl on the boundary tetrahedra of either l.h.s. or r.h.s. of Pachner move 3-3 and such that a weight in this family can be represented as both the l.h.s. and r.h.s. of (6), with all the 4-simplex weights W ijklm having the form (7). Although the search for an algebraic-topologically meaningful parameterization for these weights is still in progress, the very existence of such family is already of interest; moreover, some properties of these weights can be seen already from the parameterization given below.

Heuristic parameter count
Before presenting our construction below in Subsection 5.2, we would like to explain it heuristically, using parameter counting. For a single 4-simplex, the corresponding isotropic space of operators, spanned by the operators in big parentheses in (9), depends on 10 parameters. When we compose the l.h.s. or r.h.s. of (6) (not yet demanding that l.h.s. be equal to r.h.s.), there are thus 30 parameters. Three of them are, however, redundant, because of the possible scalings of variables x t on three inner tetrahedra t -it is easily seen that such scalings may only multiply the considered integrals by a numeric factor. So, we have 3 × 10 − 3 = 27 essential parameters in each side of (6). On the other hand, a 9-dimensional isotropic subspace in a 18-dimensional complex Euclidean space is determined by 36 parameters. So, requiring this equalness, we subtract 36 parameters and are left with 2 × 27 − 36 = 18 parameters.

Rigorous construction
There are nine boundary tetrahedra in the l.h.s. or r.h.s. of Pachner move 3-3. We see it convenient to arrange them in the following Thus, the tetrahedra in every row correspond to a 4-simplex in the l.h.s., and the tetrahedra in every column correspond to a 4-simplex in the r.h.s. of the move. First, we introduce nine nonvanishing parameters κ t ∈ C for all tetrahedra t = ijkl in the table, and then eighteen orthonormal vectors-operators -a pair Then, we introduce six more parameters: λ u for each table row, and µ u for each table column, where u = ijklm is the corresponding 4-simplex. With these parameters, we construct the following six isotropic and mutually orthogonal vectors: for the table rows, and for the table columns. Next, we bring into consideration six more unit vectors, orthogonal to each other and to all g u and h u : for each row and for each column, and an orthogonal 3 × 3 matrix where ψ, ψ ′ and ψ ′′ -Euler angles for A -are our three remaining parameters. With these vectors and matrix, we construct isotropic, and orthogonal to each other as well as to all g u and h u , vectors r, s and t. It is convenient for us to arrange these vectors in a column, and we define them as follows: Remark 3. The plus sign before the second term in (20) cannot be changed to minus without making change(s) elsewhere in our construction. As a direct calculation shows, this sign ensures that the isotropic space spanned by vectors g u , h u , r, s and t (see item (i) below in Theorem 4) belongs to the desired connected component, according to Remark 2.

Theorem 4. (i)
The linear space V spanned by vectors g 12345 , g 12346 , g 12356 , h 12456 , h 13456 , h 23456 , r, s and t is 9-dimensional isotropic -a maximal isotropic subspace in the 18-dimensional space of operators (10) for tetrahedra t in the table (15).
(ii) The 18 parameters κ t , λ u , µ u , ψ, ψ ′ and ψ ′′ , used in our construction, are independent: the Jacobian matrix of the mapping from the space of these parameters to the Grassmannian (which consists of 9-dimensional subspaces in the mentioned 18-dimensional linear space) has rank 18 in a generic point.
(iii) For generic parameters, V is such that there exist such weights W u of the form (7) for all 4-simplices in the l.h.s. and r.h.s. of (6) that both sides of (6) are turned into zero by all operators in V.
Proof. (i) This follows directly from our construction.
(ii) This is shown by a direct calculation (enough to find that rank is 18 for some specific values of parameters).
(iii) We begin with considering the four vectors g 12346 , g 12356 , s and t, see (16) and ( (where the 4-simplex 12356 corresponds to the third row, and the inner tetrahedron 1236 is common for it and the "second-row" 4-simplex 12346), satisfy and g 12346 (23) span a three-dimensional isotropic subspace in the space of operators (10) for which t ∈ ∂(12346).
To move further, we note that our space V depends on the vectors p u and q u only modulo the six vectors g u and h u : the definitions (18) and (19) can be changed by adding any linear combinations of g u 's and h u 's to their right-hand sides, and this does not affect V. In particular, this means that each q u can be changed to a linear combination of itself and h u (17) (with the same u = ijklm) in such two ways that all the new q u 's will fit into the pattern  (the signs before x (u) t must be different for the two 4-simplices u containing the tetrahedron t).
Of course, the five-dimensional isotropic space of operators can be found in a similar way also for the 4-simplices in the r.h.s. of our Pachner move. Then we define the weights W u for all the six 4-simplices as satisfying each the corresponding five equations, according to item (ii) in Theorem 2, and (see the proof of item (ii) in Theorem 3) the relation (6) does hold.
What remains is to show that the above construction can be performed in such way that we get even Grassmann elements -exponents (7) (see item (iii) in Theorem 2) as our weights W u . We think that the easiest way to do this is to refer to the nontrivial example given below in Subsection 6.2, where the weights have the form (7), and the construction of isotropic spaces mentioned in this proof works well; then it is extended to the general case by continuity. So, to within this example, Theorem 4 is proven.
6 Two explicit constructions of parameterized weights satisfying relation 3-3 The first construction, given in Subsection 6.2, is a particular case of the construction in the previous Section 5. It depends on five (six, one of which is redundant) parameters -so, the nature of remaining 18 − 5 = 13 parameters is still mysterious. Note that the rank of matrix (α (u) (7) with (25), (26) and (27)) is 4 for this construction, as is the rank of a generic antisymmetric 5 ×5 matrix, hence this rank is also 4 in the general case of Section 5.
The second construction, given in Subsection 6.3, is probably a limiting case of the construction in Section 5, because here rank(α (u) t 1 t 2 ) = 2. Despite this kind of degeneracy, the second construction exhibits extremely interesting relations to exotic homologies, studied in Sections 7 and 8.

Some formulas common for the two constructions
We are going to present, in Subsections 6.2 and 6.3, two explicit constructions of nicely parameterized Grassmann four-simplex weights W ijklm of the form (7), satisfying the 3-3 algebraic relation (6). In this subsection, we write out some formulas that belong to both of them.
First, in both cases a quantity ϕ ijk is introduced for each 2-face ijk. These ϕijk enter both in the expressions for weights and in the multiplier const in (6). To be more exact, our relations here look as follows: Second, the following Grassmannian quadratic form is used in both cases. For a 4-simplex u = ijklm, let abc be its 2-face, and d 1 < d 2 -two remaining vertices. We put where ǫ d 1 abcd 2 = 1 if the order d 1 abcd 2 of vertices determines the orientation of ijklm induced by the fixed orientation of the manifold -l.h.s. or r.h.s. of a Pachner move in our case -and ǫ d 1 abcd 2 = −1 otherwise. Recall also Convention 4 concerning the curly brackets in the subscripts in (25).
Remark 4. In practical calculations, we use formula where p ijklm reflects the consistent orientation of 4-simplices. Namely, for the simplices in the l.h.s. of move 3-3, p 12345 = −p 12346 = p 23456 = 1, and for the simplices in the r.h.s. p 12456 = −p 13456 = p 23456 = 1. As for ǫ ijklm d 1 abcd 2 , it is the sign of permutation between the sequences of its subscripts and superscripts.

First family of weights
Let an complex number ξ i be put in correspondence to every vertex i = 1, . . . , 6. We call these numbers vertex coordinates. Then we define ϕ ijk as follows: then Φ ijklm according to (25), and then the weight W ijklm as the following Grassmann-Gaussian exponent: These formulas for weights first appeared in [8,Appendix].
Proof. Direct computer calculation.
Direct calculations show that the isotropic spaces of operators for the weights introduced in this Subsection fit well into the scheme of Section 5. We think that this subject of isotropic spaces for 4-simplex weights deserves a detailed study in further works; right here we write out, just for illustration, the elegant explicit formulas for one g u and one h u (and, looking at them, it will not be hard to guess the formulas for other g u and h u ). First, introduce the following auxiliary quantities: Then, g 12345 is proportional to the following vector:

Second family of weights
This time, let each vertex i have three complex coordinates ξ i , η i , ζ i over field C. We define ϕ ijk as the following determinant: Then we define the quantity where α, β, γ ∈ C, and n ∈ {1 . . . 6} is the number missing in the set {i, j, k, l, m}. Finally, we define the 4-simplex weight W ijklm as follows: Theorem 6. The weight (32) is a Grassmann-Gaussian exponent: Proof. A direct calculation shows that the form Φ ijklm has now rank 2. So, the Grassmann exponent in (33) cannot include terms of degree > 2, and the terms of degree ≤ 2 are exactly as in (32).
Proof. Direct calculation. We used our package PL [5] for manipulations in Grassmann algebra.
Remark 5. Formula (31) is simple and works well, but conceals the real nature of quantities h ijklm . This will be explained below in Sections 7 and 8, and we will rephrase the statement of Theorem 7 in new terms, as part of Theorem 11.

An exotic analogue of middle homologies
The terms h ijklm of zero Grassmann degree in weights (32) have actually an exotic homological nature. This becomes especially clear if we consider not only move 3-3, but also move 2-4, and this we are going to do in Section 8. Right here, we are presenting the sequence (34) of two linear mappings and some related notions and statements. We explain that what matters in a set of terms h ijklm corresponds to an element in Ker g 4 / Im g 3 . Sequence (34) is expected to be part of a longer chain complex, but we don't need here that complex in full.
Let there be an oriented triangulated PL manifold M with boundary. We introduce C-linear spaces C N ′ 1 and C N 4 whose bases are inner edges and 4-simplices of M, respectively (notations in accordance with Convention 2), and two C-linear mappings between them as follows: We use notations g 3 and g 4 because these mappings have a clear analogy with mappings g 3 and g 4 in [7, formula(13)]; we leave the definition and discussion of other g i (namely, g 1 , g 2 , g 5 and g 6 ) for further papers. A set of admissible values for h ijklm will correspond to an element of Ker g 4 , while such Grassmann weights as the right-hand side of formula (37) below do not change when an element of Im g 3 is added to it.
By definition, the matrix element of mapping g 3 between an edge b = ij and a 4-simplex u vanishes unless b ⊂ u. Assuming b ⊂ u, we can write u = {ijklm}, which means, according to Convention 4, that u has vertices i, i, k, l and m but they all don't necessarily go in the increasing order. In this case, the matrix element of g 3 is 1 Recall that ϕ ijk is defined in (30). Similarly, by definition, the matrix element of mapping g 4 between a 4simplex u and an edge b is nonzero only if b ⊂ u. We write again u = {ijklm} and b = ij, and define this matrix element as where ǫ ijklm = 1 if the sequence ijklm, in this order, gives the consistent (with the fixed orientation of manifold M) orientation of u, and ǫ ijklm = −1. There are three cases; in all three ijklmn is a permutation of the set of vertices 1 . . . 6.
Case 1: Edges a and b coincide, a = b = ij. The matrix element is The numerator, after the reducing to a common denominator, vanishes: this is a Plücker relation. Case 2: Edges a and b have one vertex in common, a = ij, b = ik. The matrix element is Here the numerator is obtained from the numerator in our Case 1 by letting k = j.
Case 3: Edges a and b do not intersect, a = ij, b = kl. The matrix element is Next, we prove the theorem again for M = S 4 , but triangulated in an arbitrary way. This arbitrary triangulation can be achieved by a sequence of Pachner moves performed on the initial triangulation considered above. Each Pachner move replaces some 4-simplices with some other ones, in such way that the withdrawn and the replacing 4-simplices will form together a sphere ∂∆ 5 if we change the orientation of, say, the withdrawn 4-simplices. It follows then from what we have proved for ∂∆ 5 that the contribution of all the replacing 4-simplices into any matrix element of g 4 • g 3 is the same as the contribution of all the withdrawn 4-simplices, including the cases where an edge appears or disappears during the move.
Finally, it remains to say that any manifold M has locally the same structure as S 4 , and any matrix element of g 4 • g 3 consists only of obviously local contributions.
Experimental result. For a closed oriented 4-dimensional PL manifold M, the vector space Ker g 4 / Im g 3 is six times (i.e., isomorphic to the direct sum of six copies of ) usual second homologies H 2 (M; C).

The edge weight
For any edge a = ij in a triangulation of a 4-manifold with boundary, we define a Grassmann differential operator ∂ a = ∂ ij as the following sum over all tetrahedra t = {ijkl} containing this edge: Lemma. The weight W u (32) of a 4-simplex u satisfies "edge equations" for any edge a. vanishes, it vanishes for all the product: It is an easy exercise (just break X into the even and odd parts) to see that the right analogue of (41) also holds: where ← − ∂ 56 is the same linear combination (39), but with left derivatives replaced with the right ones. Finally, recall that the integration means, according to Section 2, the right differentiation, and specifically in (40) this can be represented as follows (compare (5)): Due to Theorem 9, the following definition of w 56 is not surprising: by which we understand any Grassmann algebra element w 56 such that ∂ 56 w 56 = 1. For instance, we can take w 56 = ϕ 156 ϕ 256 x 1256 .

The quantities h ijklm
The quantities h ijklm for both moves 3-3 and 2-4 considered in this paper can be obtained as follows. First, glue together both sides of a Pachner move in the natural way -identifying like-named boundary simplices. This gives S 4 = ∂∆ 5 (recall that this means a 4-sphere triangulated as the boundary of a 5-simplex). Remark 6. This gluing implies that we have changed the orientation in one of the sides. Nevertheless, our mapping g 3 is defined in such way that the orientation issues do not affect the definition of allowable set of values for h ijklm given below.
Proof. Taking into account Theorem 8, it is enough to show that rank g 3 + rank g 4 = 6, where 6 is the number of 4-simplices in the triangulation and thus the dimension of the middle space in (34). This is done by a direct calculation; both ranks prove to be 3.
Remark 7. Compare this also with the Experimental result on page 21. We now take an arbitrary chain c edges on edges of our S 4 -element of the first linear space in (34), and then its image under g 3 -a chain on 4-simplices. We call the resulting coefficients at 4-simplices u = ijklm, both in the l.h.s. and r.h.s. of our Pachner move, an allowable set of values for h ijklm .
Theorem 11. Let there be an allowable set of values h u = h ijklm for the six 4-simplices in both sides of either Pachner move 3-3 or 2-4, and, in case of move 2-4, let there be chosen any edge weight w 56 according to (42). Let also the 4-simplex weights be defined according to (32), with the quadratic forms Φ u defined according to (25) and (30). Then, the relation (24) holds for move 3-3 or, respectively, (37) holds for move 2-4.
Remark 8. It is an easy exercise to show that our allowable sets of values h u = h ijklm for Pachner moves can be represented in the form (31) (although this form may disguise their nature). Hence, the part of Theorem 11 dealing with move 3-3 says the same as Theorem 7, as was promised in Remark 5. Now consider the r.h.s. of the 2-4 relation (37). As it is equal to the l.h.s., it does not depend on the coefficient in c edges at the edge 56, the latter being absent from the l.h.s. of the Pachner move, while being the only inner edge for its r.h.s. Consider the sequence (34) for the r.h.s. of move 2-4. The first and third spaces in this sequence are one-dimensional, the only basis vector being the edge 56. Any allowable set of h u , with the u's in this r.h.s., obviously makes a cycle in the sense that it is annihilated by the mapping g 4 . So, the r.h.s. of (37) is determined by this cycle modulo a boundary -image of g 3 .
Especially interesting question for future research is to uncover the analogue(s) of such exotic-homological structures for more general 4-simplex weights described in Section 5.