Odd-gon relations and their cohomology

A triangulation of a closed piecewise linear (PL) manifold can be transformed into any other triangulation of the same manifold using a finite sequence of elementary re-buildings—Pachner moves [15, 13]. For a manifold $M$ of dimension $\nu$, these moves can be described as follows.

The boundary $\partial\Delta^{\nu+1}$ of a simplex $\Delta^{\nu+1}$ of the next dimension consists of $\nu+2$ simplices $\Delta^{\nu}$. Represent $\partial\Delta^{\nu+1}$ as the union of two parts, the first containing $\kappa$ simplices $\Delta^{\nu}$ and called initial cluster, and the other containing the remaining $(\nu+2-\kappa)$ simplices $\Delta^{\nu}$ and called final cluster, where $\kappa$ is any of the numbers $1,\ldots,\nu+1$. Suppose that a triangulation of $M$ contains a part isomorphic to the initial cluster, then “Pachner move $\kappa\mapsto\nu+2-\kappa$” consists in replacing this part with the final cluster. This is possible because the two clusters have the same boundary, and either of them can be glued to the rest of the triangulation of $M$ by this boundary.

There are thus $\nu+1$ kinds of Pachner moves in $\nu$ dimensions:

and any two equidistant from the ends of the list (1) are inverse to each other.

In this paper, we will be interested mostly in the case of an odd dimension

A special role will be played by the ‘central’ moves $n+1\leftrightarrow n$, although we will consider all other moves as well.

A polygon relation is an algebraic imitation of a Pachner move.

Before making this definition more specific, we note that, with algebraic formulas of this kind, there is a hope to construct some kind of topological field theories for piecewise linear manifolds, yielding, in particular, PL manifold invariants.

Besides, polygon relations are beautiful algebraic and combinatorial structures that will definitely find connections and applications in many branches of mathematics. Note especially that they arise within the higher Tamari orders approach [2].

In this paper, we combine the ‘higher Tamari’ and ‘topological’ approaches. Our terminology will largely be in accordance with the mentioned paper [2] and especially [3] (where we show how the ideas of [2] can be developed both combinatorially and algebraically). In particular, this implies that we call “odd polygon”, or simply “odd-gon”, the relation corresponding to the move $n+1\mapsto n$, in the way we are going to explain. As for all other moves, we treat them within the “full polygon” framework of Section 3.

We number the simplices $\Delta^{\nu}$ taking part in the mentioned move from 1 through $\nu+2=2n+1$. The initial cluster consists, by definition, of the ‘odd’ simplices $1,3,\ldots,2n+1$, while the final cluster—of the ‘even’ simplices $2,4,\ldots,2n$. Each $\Delta^{\nu}$ has $2n$ faces (of dimension $\nu-1$), each of these belongs also to one of the other $\Delta^{\nu}$. For $\Delta^{\nu}$ number $q$, the other $\Delta^{\nu}$, and hence the faces of the $q$-th $\Delta^{\nu}$, form the ordered set

where ‘hat’ means omission. We declare ‘input’ the $n$ faces corresponding to the odd positions in $L_{q}$, and ‘output’ those corresponding to the even positions in $L_{q}$.

We introduce a set $X$ whose elements are called colors. These colors are assigned to $(\nu-1)$-dimensional faces, and for each $\Delta^{\nu}$, by definition, the $n$ ‘output’ colors are functions of the $n$ ‘input’ colors. For each of the initial and final cluster, there are $\frac{n(n+1)}{2}$ ‘global input’ faces for the whole cluster, whose colors can be taken arbitrary, and determine the colors of all other faces. Among these, there are $\frac{n(n+1)}{2}$ ‘global output’ faces, that is, such faces that are input for none of the simplices $\Delta^{\nu}$. Moreover, both the ‘global input’ and ‘global output’ faces are the same for the two clusters.

So, each of the clusters provides dependences of the ‘global output’ colors on ‘global input’ colors, and we say that the odd-gon relation holds if these dependences are the same. Combinatorial details can be found in [3, Section II] (the fact that only linear dependences are considered there does not affect the combinatorics) and in Section 2 below.

The ‘global input’ and ‘global output’ faces form together the common boundary of the initial and final clusters. Our odd-gon relation means thus that the dependences between the boundary colors are the same for the two clusters.

$\Delta^{m}$ denotes an $m$-dimensional simplex, that is, a simplex with $(m+1)$ vertices. Four kinds of simplices will be of most importance in this paper:

• $\bullet$

  $\Delta^{2n}$ —its boundary $\partial\Delta^{2n}$ is the union of the l.h.s. and r.h.s. of Pachner move $(n+1)\mapsto n$ or any other move in $2n-1$ dimensions. We call such simplex P-simplex, from the word ‘Pachner’;

• $\bullet$

  $\Delta^{2n-1}$ —a simplex of the maximal dimension in a $(2n-1)$-dimensional PL manifold. We call such simplex d-simplex, from the word ‘dimension’;

• $\bullet$

  $\Delta^{2n-2}$ —a face of $\Delta^{2n-1}$. These faces will be “colored” by elements $x$ of some “set of colors” $X$. We call such simplex simply face, if there is no risk of confusion;

• $\bullet$

  $\Delta^{n-2}$ —to such simplices special ‘generating’ colorings will correspond. We call such simplex g-simplex, from the word ‘generating’.

For brevity, we often use the following “complemental” notations. A d-simplex $\Delta^{2n-1}$ considered in the context of a Pachner move is determined by pointing out the only vertex $q$ from the set $\{1,\ldots,2n+1\}$ that is does not contain. So, in this paper it can be denoted simply as $q$. Similarly, a face $\Delta^{2n-2}$ not containing vertices $q$ and $r$ can be denoted simply $qr$ or, equivalently, $rq$; this is of course exactly the face common for d-simplices $q$ and $r$.

Below,

• $\bullet$

  in Section 2, we construct a specific algebraic odd-gon relation. We start with introducing objects related to simplices of different dimensions, and arrive at a relation between products of matrices, linking together the ‘combinatorial topological’ and ‘higher Tamari’ approaches;

• $\bullet$

  in Section 3, we extend the mentioned relation from move $(n+1)\mapsto n$ onto all Pachner moves—what we call “full polygon”;

• $\bullet$

  in Section 4, we describe a cohomology theory for our polygon relations;

• $\bullet$

  in Section 5, we explain how that cohomology theory can be used for constructing invariants of piecewise linear manifolds, using nontrivial polygon $(2n-1)$-cocycles;

• $\bullet$

  in Section 6, we show one way of how to construct actual manifold invariants. Namely, there exists a $(2n-2)$-cocycle in characteristic zero from which a desired $(2n-1)$-cocycle can be obtained in a finite characteristic (using a procedure reminiscent remotely of a Bockstein homomorphism);

• $\bullet$

  in Section 7, we show the existence and present calculations related to a nontrivial heptagon $(2n-1)$-cocycle in characteristic zero. It probably does not have immediate topological applications but demonstrates an interesting algebraic structure;

• $\bullet$

  in Section 8, we discuss the results and directions of future research.

Let $n\geq 2$ be an integer, and consider a $2n$-simplex $\Delta^{2n}$ whose vertices $i$ we would like to denote by numbers from $i=1$ through $i=2n+1$. In this section, its boundary $\partial\Delta^{2n}$ will serve us as the union of the two sides of our Pachner move $(n+1)\mapsto n$.

Let $X$ be any fixed set, called set of colors.

All such colorings form the $n(2n+1)$-th Cartesian degree $X^{\times n(2n+1)}$ of set $X$.

Let

be a $3\times(2n+1)$ matrix whose entries are indeterminates (algebraically independent elements) over a field $F$. We do not specify this $F$ at the moment, but the reader may think of $F=\mathbb{Q}$ —the field of rational numbers, or $F=\mathbb{F}_{q}$ —a finite field.

We now choose the color set to be the field

of rational functions of the entries of matrix $\mathcal{M}$ over $F$.

We denote

the determinant made of its $i$-th, $j$-th and $k$-th columns.

Recall (Subsection 1.3) that a g-simplex is any simplex of dimension $(n-2)$.

Definition (6) generalizes the edge vectors introduced in [10] for the heptagon relation, see [10, (16)]. Note that

There are $n+2$ vertices involved in linear relation (7): $i_{1},\ldots,i_{n-2}$ plus $j,k,l,m$. The following proposition shows that there are no linear relations in which not more than $n+1$ vertices are involved, or, in other words, no linear relations within any simplex $\Delta^{n}$.

One $\Delta^{n}$ has $\binom{n+1}{n-1}=\frac{n(n+1)}{2}$ faces of dimension $n-2$, so it generates a $\frac{n(n+1)}{2}$-dimensional linear space of g-colorings for $\Delta^{2n}$.

On the other hand, it is an easy exercise to show, using linear relations (7), that $e_{b}$ for any $b\subset\Delta^{2n}$ can be expressed through the vectors $e_{b}$ taken for $b\subset\Delta^{n}$ with one chosen $\Delta^{n}$. Hence, the following proposition holds.

In the previous subsection, we considered colorings of the whole $2n$-simplex, which we also call P-simplex (see Subsection 1.3), and whose boundary is the union of the l.h.s. and r.h.s. of a Pachner move. Now, we are going to consider colorings of just one $(2n-1)$-simplex $\Delta^{2n-1}$, which we also call d-simplex.

Take $\Delta^{2n-1}=m$ —remember (see again Subsection 1.3) that in our “complemental” notations this means the simplex without vertex $m$. We denote $e_{b}|_{m}$ the restriction of $e_{b}$ onto the faces of $m$. There is the following analogue of Proposition 1.

One $\Delta^{n-1}$ has $\binom{n}{n-1}=n$ faces of dimension $n-2$, so it generates an $n$-dimensional linear space of g-colorings for $\Delta^{2n-1}$.

On the other hand, it is an easy exercise to show, using linear relations (10), that $e_{b}|_{m}$ for any $b\subset\Delta^{2n-1}\not\ni m$ can be expressed through the vectors $e_{b}|_{m}$ taken for $b\subset\Delta^{n-1}$ with one chosen $\Delta^{n-1}$ not containing $m$. Hence, the following proposition holds.

Figure 1

depicts d-simplex $i_{1}\ldots i_{n}j_{1}\ldots j_{n}$ in a ‘dual’ way: the vertex—or actually the circle with the letter $q$—symbolizes the simplex itself, while the edges, or ‘legs’, correspond to its faces. We use ‘complemental’ notations (recall Subsection 1.3): numbers $i_{1},\ldots,i_{n},j_{1},\ldots,j_{n},q$ are supposed to make a permutation of numbers $1,\ldots,2n+1$, so our simplex is the $(2n-1)$-simplex without vertex $q$; similarly, each leg corresponds to the $(2n-2)$-face not having the two vertices with which the leg is labeled.

Proposition 6 suggests that we can take arbitrary colors for $n$ of $2n$ faces of our simplex $q$, let these be $i_{1}q,\ldots,i_{n}q$, then the colors of $j_{1}q,\ldots,j_{n}q$ are expected to be determined uniquely. The following reasoning shows that it is indeed so.

Consider $(n-2)$-simplex vector $e_{i_{2}\ldots i_{n}}$. As $(2n-2)$-faces denoted $i_{2}q$, …, $i_{n}q$ do not contain $i_{2}$, …, $i_{n}$, respectively, the corresponding components of $e_{i_{2}\ldots i_{n}}$ vanish. Other components are given by formula (6), so, the components of $e_{i_{2}\ldots i_{n}}$ on the $n$ lower legs of Figure 1—call them together, for a moment, $e_{i_{2}\ldots i_{n}}|_{\mathrm{lower}}$—are

while the components on the $n$ upper legs are

Remembering that

we get immediately the first row of matrix $A^{(q)}$. Similarly, we get other rows, and the final answer for $A^{(q)}$ is

Proposition 6 guarantees that there will arise no contradiction if we use other g-colorings for determining $A^{(q)}$.

In such context, we call legs $i_{1}q,\ldots,i_{n}q$ input, and legs $j_{1}q,\ldots,j_{n}q$ output. In our pictures (as well as in [3]), input legs lie below the corresponding vertex, and output legs—above. It is clear from our reasoning that any $n$ legs—that is, faces of a d-simplex—can be declared input; the corresponding matrix will be given by (15) with the relevant permutation of indices.

Our odd polygon relation is the equality of two products of $\frac{n(n+1)}{2}\times\frac{n(n+1)}{2}$ matrices called $A_{\mathcal{B}_{q}}^{(q)}$, each being a direct sum of matrix (15) and an identity matrix. More specifically: think of each $A_{\mathcal{B}_{q}}^{(q)}$ as acting on $\frac{n(n+1)}{2}$-rows, then by definition it acts nontrivially only on the entries with numbers in the set $\mathcal{B}_{q}$ (which we are going to specify), while all the other entries remain intact under $A_{\mathcal{B}_{q}}^{(q)}$. This implies of course that the cardinality of each set $\mathcal{B}_{q}$ equals $n$.

The definition of $\mathcal{B}_{q}$ is as follows.

It is an easy exercise to check that Definition 4 gives the same as the definition of $\mathcal{B}_{q}$ given in [3, Subsection II.B].

The following proposition states a fundamental property of sets $\mathcal{B}_{q}$.

Before proving (18), we study its combinatorial structure. We think of both sides of (18) as acting on $\frac{n(n+1)}{2}$-rows. Let there be an arbitrary $\frac{n(n+1)}{2}$-row, call it initial row. The action of each $A_{\mathcal{B}_{q}}^{(q)}$ is as follows: take the $\frac{n(n+1)}{2}$-row resulting after the action of all $A_{\mathcal{B}_{i}}^{(i)}$ staying to the left of $A_{\mathcal{B}_{q}}^{(q)}$ on the initial row, then take its $n$-subrow with entry numbers (positions) in $\mathcal{B}_{q}$—we call these inputs for $A^{(q)}$—and act with $A^{(q)}$ upon it; we call the entries of the resulting $n$-row outputs for $A^{(q)}$. Entries with positions outside $\mathcal{B}_{q}$ stay intact; they remain thus either initial or outputs of a previous $A^{(i)}$.

We call the results of action of the l.h.s. and r.h.s. of (18) on the initial row final rows; we will show soon, in Theorem 1, that these two rows coincide.

Proposition 8 shows that the structure of (18) corresponds to $\partial\Delta^{2n}$ divided in two parts. To make this statement more visual, it makes sense to think of (18) in terms of its diagram.

A leg linking $A^{(q)}$ and $A^{(r)}$ (in the sense of Definition 6) corresponds to $(2n-2)$-face $qr$. This face is boundary provided $A_{\mathcal{B}_{q}}^{(q)}$ and $A_{\mathcal{B}_{r}}^{(r)}$ are in the different sides of (18).

Relation (18) guarantees that the boundary colorings mentioned in Definition 10 are the same for one specific Pachner move, namely, $(n+1)\mapsto n$ and such the there are d-simplices $1,3,\ldots,2n+1$ in its initial cluster, and $2,4,\ldots,2n$ in its final cluster; d-simplex $q$ means here, as we remember, the simplex with all vertices $1,\ldots,2n+1$ except $q$. If there are other d-simplices in both sides of a move $(n+1)\mapsto n$, there is still no big problem to get a corresponding odd-gon rela

First, a small preparatory work.

Odd-gon relation (18) remains valid if we renormalize colors $x_{u}$ of faces $u$, that is, make a substitution

where $\lambda_{u}$ are nonzero elements of the field $\mathcal{F}$.

For a matrix $A^{(q)}$ with input faces $u_{1},\ldots,u_{n}$ and output faces $v_{1},\ldots,v_{n}$, this means the following transformation with two diagonal matrices:

(remember that our matrices act on the rows and from the right!)

In this subsection, we define permitted colorings of a d-simplex to be the same as its g-colorings (Definition 3). We denote the vertices of simplices involved in any Pachner move by numbers $1,\ldots,2n+1$. tion: we just make a relevant permutation of numbers $1,\ldots,2n+1$. One small thing to be taken into account is that, with this permutation, colors of some faces may change their signs with respect to definition (6), because the condition $l<m$ (just below (6)) may change to its reverse. The corresponding change of matrices $A^{(q)}$ with respect to (15) is, however, just a gauge transformation, with some lambdas in (21) equal to unity and some to minus unity.

For a Pachner move $(n+1+k)\mapsto(n-k)$, with $k>0$, move $k$ matrices $A^{(q)}$ from the r.h.s. of (18) to the l.h.s., by multiplying both sides by their inverses $B^{(q)}=\left(A^{(q)}\right)^{-1}$, for instance, from the right:

then make a relevant permutation of vertices, and the gauge transformation if necessary, as we just did for move $(n+1)\mapsto n$.

Any matrix $A_{\mathcal{B}_{q}}^{(q)}$ or $B_{\mathcal{B}_{q}}^{(q)}$ in (22) is still linked to any other matrix in the sense of Definition 6, so its structure corresponds indeed to the move $(n+1+k)\mapsto(n-k)$. Note also that matrices $B_{\mathcal{B}_{q}}^{(q)}$ are in fact of the very same form (15) as $A_{\mathcal{B}_{q}}^{(q)}$, with only input and output legs interchanged, exactly as the structure of (22) requires.

Finally, for a Pachner move having less d-simplices in the initial cluster than in the final cluster, just take the relation for its inverse, and interchange the l.h.s. and r.h.s.

This gives us the following theorem.

Note that the r.h.s. of (22) may have less inputs—and outputs—than the l.h.s. (and it actually always happens when $k>0$). The ‘superfluous’ output values in the l.h.s. coincide with the corresponding input values (because they obviously coincide in the r.h.s). Topologically, any ‘superfluous’ input and its corresponding output belong to the same inner face of the initial cluster, while the inputs and outputs of the r.h.s. form the boundary of either initial or final cluster.

As a permitted coloring of either l.h.s. or r.h.s. of (22) is determined by its (arbitrary) inputs, and the l.h.s. has, generally, more inputs than the r.h.s., it follows that to a permitted coloring of r.h.s. corresponds a whole linear space of permitted colorings of l.h.s., of dimension equal to the number of ‘superfluous’ inputs.

In this subsection, we define “nonconstant polygon cohomology” in a general context. It works equally well for both even and odd polygons, so, in this section, we speak of a “$(\nu+2)$-gon”, where meaningful values of $\nu$ can be $\nu=3,4,\ldots$, although, as we already stated (2), we deal mostly with the odd $\nu=2n-1$ in this paper. Our $(\nu+2)$-gon corresponds to a Pachner move in a $\nu$-dimensional PL manifold.

Our definition will depend on a chosen simplicial complex $K$. In principle, $K$ can be of any dimension, although the main work in this paper will take place in the standard $(\nu+1)$-simplex $K=\Delta^{\nu+1}$, whose boundary is the union of the l.h.s. and r.h.s. of any $\nu$-dimensional Pachner move.

Suppose that every $(\nu-1)$-face $u\subset K$ is colored by some element $\mathsf{x}_{u}\in X$ of a set $X$ of colors (for instance, field $X=\mathcal{F}$ considered in the preceding sections), and that a subset $R_{v}$ of permitted colorings is defined in the set of all colorings of every $\nu$-simplex $v$.

We define also the set of ‘permitted’ colorings for any simplex $\Delta\subset K$ of any dimension.

This agrees, of course, with our earlier Definition 9.

For other complexes, there may be more permitted colorings than g-colorings, see Examples 4 and 5 below.

It is implied in Definition 11 that all colorings $\mathsf{x}\in X$ are permitted for an individual $(\nu-1)$-face. Also, Definition 11 is void of course for $\dim\Delta<\nu-1$; it has a nontrivial meaning for $\dim\Delta>\nu$.

The set of all permitted colorings of an $m$-simplex $i_{0}\dots i_{m}$ will be denoted $\mathfrak{C}_{i_{0}\dots i_{m}}$. We assume here that the vertices of any simplex are ordered: $i_{0}<\ldots<i_{m}$.

It turns out that there are some interesting variations of the cochain definition (23). For instance, preprint [9] (although devoted to constant hexagon cohomology) suggests that polynomials may be used instead of general functions (23)—of course, in a situation where the notion of polynomial in the variables determining a permitted coloring makes sense.

We can also “double” the colorings and permitted colorings, introducing the Cartesian square $X\times X$ as a new color set, and Cartesian squares $R_{v}\times R_{v}$ as new sets of permitted colorings for each $\nu$-simplex $v$. This construction will be of special interest for us in the symmetric bilinear and bipolynomial cases.

In a characteristic $\neq 2$, symmetric bilinear cochains are of course just polarizations of quadratic homogeneous polynomial cochains. Preprint [9] suggests, however, that characteristic $2$ is interesting and deserves attention.

In any characteristic, symmetric bilinear mapping (26) can be treated as a scalar product of two permitted colorings of simplex $i_{0}\dots i_{m}$; to define a symmetric bilinear cochain is the same as to define scalar products for all corresponding simplices. This language of scalar products will be used extensively in what follows.

In Section 2, we parameterized g-colorings for a Pachner move by entries of matrix $\mathcal{M}$ (3). In order to consider more general PL manifolds, it makes sense to switch to a different—‘dual’—parameterization. As a first step, we introduce here these dual parameters for that same P-simplex whose boundary is the union of the two sides of the Pachner move.

The three rows of matrix $\mathcal{M}$ span a three-dimensional linear subspace $L$ in the $(2n+1)$-dimensional space $\mathcal{L}$ of $(2n+1)$-rows (over the field $\mathcal{F}$ (4)). Its orthogonal, $L^{\perp}$, is $(2n-2)$-dimensional, lies in the dual space $\mathcal{L}^{*}$, and can be represented as spanned by the $(2n-2)$ columns of a matrix

Denote $p_{i_{1}\dots i_{2n-2}}$ the determinant made of rows $i_{1},\dots,i_{2n-2}$ of $\mathcal{N}$. A well-known fact about the duality between such determinants and the determinants $d_{jkl}$ (5) is that $\mathcal{N}$ can be normalized so that the following relation will hold:

where

We are going to give an alternative definition of $(n-2)$-simplex vectors, and formulate it in such way that it will work not only for a Pachner move (taking place in the boundary of a P-simplex), but also for more general simplicial complexes. We introduce notation $\mathcal{N}_{i}$ which will mean, in this subsection, simply the $i$-th row of matrix $\mathcal{N}$ (28) —because here we are still in the situation of one Pachner move or, in other words, P-simplex $\Delta^{2n}$ with its $(2n+1)$ vertices—but $\mathcal{N}_{i}$ will be a more general row in the following sections.

Let $M$ be a $(2n-1)$-dimensional closed triangulated PL manifold, and $F$ a prime field $\mathbb{F}_{p}$ or the field $\mathbb{Q}$ of rational numbers. If $\operatorname{char}F\neq 2$, let also $M$ be oriented.