The Height of a Class in the Cohomology Ring of Polygon Spaces

LetM n,r be the configuration space of planar n-gons having side lengths 1, . . . , 1 and rmodulo isometry group. For generic r, the cohomology ringH(M n,r ;Z 2 ) has a formH(M n,r ;Z 2 ) = Z 2 [R(n, r), V 1 , . . . , V n−1 ]/I n,r , where R(n, r) is the first Stiefel-Whitney class of a certain regular 2-cover π : M n,r 󳨀→ M n,r and the ideal I n,r is in general big. For generic r, we determine the number h(n, r) such that R(n, r)h(n,r) ̸ = 0 but R(n, r)h(n,r)+1 = 0.


Introduction
Given a string r = ( 1 , . . .,   ) of  positive real numbers, one considers the configuration space  r of planar polygon linkages having side lengths   modulo isometry group.Starting in [1][2][3], the topology of  r has been considered by many authors.A notable achievement is [4] which determined the mod 2 cohomology ring  * ( r ; Z 2 ) for generic r.The study culminated in the proof by [5] of a conjecture by Kevin Walker which states that one can recover relative lengths of edges from the mod 2 cohomology ring of the configuration space.
By [4], the cohomology ring  * ( r ; Z 2 ) has a form where  r is the first Stiefel-Whitney class of a certain regular 2-cover  :  r →  r and the ideal I r is in general big.For example, when r = (1, . . ., 1), I r is generated by polynomials whose number is approximately 2 −2 .While [5] made clever arguments to distinguish the cohomology rings, it is difficult to extract more concrete information from the ring.The reason is that the polynomials do not form a Gröbner basis.
In this direction, one natural problem is to compute the height of  r , that is, the unique ℎ(r) such that  ℎ(r) r ̸ = 0 but  ℎ(r)+1 r = 0.The purpose of this paper is to determine ℎ(r) for the case r = (1, . . ., 1, ).Note that such r is a string which comes next to the equilateral case.For example, the homology groups  * ( r ; Z) were determined in [6].
Finally, we note that the height of an element in the cohomology ring of a space  has been studied in order to give nice lower bounds on the Lusternik-Schnirelmann category of .For example, [7] studied the problem for the case that  =   (R + ) and the element is the first Stiefel-Whitney class of the universal bundle.
This paper is organized as follows.In Section 2 we state our main results.Theorem A determines the height of  r .Theorem B determines which element represents the mod 2 fundamental class of  r .In Section 3 we prove auxiliary results.In Section 4 we prove Theorem B and in Section 5 we prove Theorem A.
( The following examples are well known: The following theorem is crucial in this paper. where (, ) and  1 , . . .,  −1 are of degree 1 and I , is the ideal generated by the three families of elements: (R3) The symbol  in (3) runs over all subsets of  including the empty set.By (2) a term of the sum in (3) The class (, ) coincides with the first Stiefel-Whitney class of the regular 2-cover  :  , →  , .We define the height of (, ) is the unique ℎ(, ) such that (, ) ℎ(,) ̸ = 0 but (, ) ℎ(,)+1 = 0.
In order to state our main results, we prepare notations.Throughout this paper, the notation  ≡  means that  ≡  (mod 2).We set Moreover, we set Now our first result is the following.
We study the generator of then we have in We set Our second result is the following.

Theorem B.
Let  and  be as in Theorem A. Then for 0 ≤  ≤ (, ), the following equality holds in  −3 ( , ; Z 2 ): Theorem A implies that (, ) −3 is a generator if and only if ( () (,) ) ≡ 1.This is in agreement with Theorem B for  = 0.
For the case of almost equilateral polygons, that is, the case for  = 1 or 2, we can write Theorem A more explicitly.
Proposition C. (i) About  ,1 for odd , one has the following.
(ii) About  ,2 for even , one has the following.
One deduces the proposition from Theorem A in Section 5.
We give two examples of Theorem A. The first example is about the case when  is small.Example 2. We consider Table 1 for the case ℎ(, ) =  − 3. Then (, , ) = 0 (where (, , ) is defined in (15)) in  −3 ( , ; Z 2 ) if and only if (, , ) satisfies the case which is given in Table 2.
Our second example is about the case when  is large.
In fact, (i) for odd  and (ii) correspond to the case ( () (,) ) ≡ 1 in Theorem A. But we know a stronger result as in (6).

Auxiliary Results
Proposition 4. Let  be indeterminate and  ∈ N. Let   () be an  ×  matrix over Z 2 [] defined as follows: Then one has the following results.
(ii) One has rank   () =  if and only if (iii) When rank   () =  − 1, the following results hold.Proof.We define 4 matrices as follows: Since det P = det Q () = 1, they are regular matrices.We write   and   () as follows: ) . ( Using the Chu-Vandermonde identity we have (i) Since rank 1q = 1 and P Q () is a regular matrix, we have rank   () ≥  − 1.
The following property of (, ) will be used in Section 5.
Proof.(i) We need to consider two cases according to We shall prove that  By Kummer, ( +  ) is odd if and only if there are no carries when adding  and  in base 2. Hence if we write  = 2 −1 +  (where 1 ≤  ≤ 2 −1 ), the greatest  (where  ≤  − 2) such that ( +  ) ≡ 1 is Hence (, ) = 2 −1 − 1 −  and Modifying Proposition 4 slightly, we have the following.
Lemma 7. Let Ǎ () be an ×(+1) matrix over Z 2 [] defined as follows: Then one has the following results.

Proof of Theorem A
Theorem A is proved by showing the upper bound for those  with (, )  ̸ = 0 coinciding with the lower bound.About the upper bound, we have the following.Proposition 8.In  * ( , ; Z 2 ), one has (, ) (,)+1 = 0.

)
Moreover, if  is even and   ∈ (0, 1), then we have  ,  ≅  −1,1 ×   1 , where  acts on  1 by complex conjugate.On the other hand, if  > 4 and  ∈ N has the different parity as , then  , has singular points.Hereafter we assume that  is a natural number which satisfies  ≤  − 2 and has the same parity as .In this case,  , is a connected closed manifold of dimension  − 3.Moreover,  , is orientable if and only if  is even.