TThe Stable Limit of Moduli Spaces of Polygons
Jack LoveSeptember 17, 2020
Keywords
Polygons, moduli spaces, configuration spaces
Mathematical Subject Classification (2020)
Primary: 58D29; Secondary: 58A35,52C25
Abstract
Polygon spaces have been studied extensively ([KM95], [HK96], [KM96], [HK97],[Kin99], [KM02], [Man09], [HMM11], [FF13], [GM13], [Kou14], [CS16]), and yet miss-ing from the literature is a simple property that every polygon has: dimension. This isdistinct (possibly) from the dimension of the ambient space in which the polygon lives.A square, in the usual sense of the word, is 2-dimensional no matter the dimensionof the ambient space in which it is embedded. If the ambient space has dimensiongreater than or equal to 3 we may bend the square along a diagonal to produce a3-dimensional polygon with the same edge-lengths. And yet even if the dimension ofthe ambient space is large, no amount of bending of the square will produce a poly-gon of dimension larger than 3. We generalize this idea to show that there are onlyfinitely many moduli spaces of polygons with given edge-lengths, even as the ambientdimension increases without bound. A polygon is a chain of line segments, called edges, that begins and ends at the origin insome Euclidean space R d . A polygon space is the collection of all polygons in R d with agiven vector of edge-lengths (cid:96) = ( l , . . . , l n ). A moduli space of polygons is the quotientof a polygon space by rotations in R d . For context and a point of contrast, we mentionthe work of Michael Farber and Viktor Fromm in [FF13]. It is well-known that polygonspaces are smooth manifolds if (cid:96) is sufficiently generic. Farber and Fromm showed thatupon fixing the dimension d of the ambient space and allowing (cid:96) to vary generically,the diffeomorphism types of polygon spaces are in bijection with the components ofa discrete geometric object. Rather than looking at diffeomorphism types of polygonspaces with fixed d and varying (cid:96) , we look at homeomorphism types of moduli spacesof polygons with fixed (cid:96) and varying d . Since any polygon in R d can be embedded in R d +1 , we define a directed system of moduli spaces of polygons by fixing (cid:96) = ( l , . . . , l n ) a r X i v : . [ m a t h . GN ] S e p v v v v v v v v v v v Figure 1: Pictures of 1-, 2-, and 3-dimensional (cid:96) -gons in R , for (cid:96) =(1 , , , and letting d increase without bound. Our main theorem is presented in Section 5, butwe state a version of it here: Theorem 1.1.
Given a vector (cid:96) = ( l , . . . , l n ) of edge-lengths, the directed system ofmoduli spaces of (cid:96) -gons stabilizes when the ambient dimension is equal to n . At the crux of this result is the notion of polygon dimension. We show that thedimension of a polygon with n edges is bounded by a function of n , and also plays a rolein the relationship between reflections and rotations of the polygon. These intermediateresults give conditions for when the arrows in the directed system mentioned above areeither injective or surjective and ultimately tell us that they are homeomorphisms when d ≥ n . Taken together with the result of Farber and Fromm, this implies that uponfixing the number n of edges, there are only finitely many homeomorphism types ofpolygon spaces even as (cid:96) varies generically throughout R n and d goes to infinity.In Section 2 we define the directed system of moduli spaces of polygons and intro-duce polygon dimension. In Section 3 we look at the possible dimensions for polygonsin a given polygon space, and Section 4 looks at the interplay between polygon dimen-sion, ambient space dimension, and orbits of polygons under rotations and reflections.The results of these sections will allow us to state and prove our main theorem inSection 5. Lastly, in Section 6 we illustrate our result with an example and use theexample to motivate new questions. This work is based on the author’s Ph.D. thesis at George Mason University, begununder Chris Manon and directed by Sean Lawton. The author is grateful to themboth for their guidance, as well as to Jim Lawrence, Neil Epstein, and Rebecca Goldinfor their helpful comments. The author also acknowledges the work of John Millsonand Michael Kapovich, and Michael Farber and Viktor Fromm, whose papers providedmuch of the background and context for what is presented here. Definitions
The goal of this section is to define the directed system of moduli spaces of (cid:96) -gons, andintroduce polygon dimension which will be our main tool in proving that these systemsstabilize. We begin with the definition of polygon space.
Definition 2.1.
Given n ≥ , (cid:96) = ( l , . . . , l n ) ∈ R n > , and d ≥ , the polygon space V d ( (cid:96) ) is the topological subspace of R d ( n − defined as V d ( (cid:96) ) = (cid:110) P = ( v , . . . , v n − ) ∈ R d ( n − : (cid:107) v i − v i − (cid:107) = l i , i = 1 , . . . , n (cid:111) , where v = v n = . Elements of V d ( (cid:96) ) are called polygons or (cid:96) -gons . Given an ( l , . . . , l n ) -gon ( v , . . . , v n − ) , the v i are its vertices and the l i are its edge-lengths . The polygon space V d ( (cid:96) ) admits natural SO ( d ) and O ( d ) actions by the map V d ( (cid:96) ) × G → V d ( (cid:96) ) , (( v , . . . , v n − ) , T ) (cid:55)→ ( T ( v ) , . . . , T ( v n − )) , where G is SO ( d ) or O ( d ). We write T ( P ) to mean the image of ( P, T ) under thismap and we let G ( P ) = { T ( P ) : T ∈ G } denote the orbit of P under the action of G .Thus two polygons in V d ( (cid:96) ) belong to the same SO ( d ) orbit if they are rotations of oneanother, and they belong to the same O ( d ) orbit if they are rotations or reflections ofone another. Definition 2.2.
The moduli space of (cid:96) -gons in R d , denoted M d ( (cid:96) ) , is the quotientspace V d ( (cid:96) ) /SO ( d ) . We let π : V d ( (cid:96) ) → M d ( (cid:96) ) denote the canonical projection, andgiven P = ( v , . . . , v n − ) ∈ V d ( (cid:96) ) we let [ P ] = [ v , . . . , v n − ] denote π ( P ) . Since any (cid:96) -gon in R d can be thought of as an (cid:96) -gon in R d +1 by embedding it ina codimension-1 hyperplane, there is a natural map M d ( (cid:96) ) → M d +1 ( (cid:96) ). In this way,we are led to a directed system of moduli spaces of (cid:96) -gons for fixed (cid:96) . We make thisconcrete in the following proposition/definition. Proposition/Definition 2.3.
Let R f −→ R f −→ R f −→ · · · be any directed systemof linear Euclidean isometries. Fix (cid:96) ∈ R n > and for d ≥ , define the maps F d : V d ( (cid:96) ) → V d +1 ( (cid:96) )( v , . . . v n − ) (cid:55)→ ( f d ( v ) , . . . , f d ( v n − )) and ϕ d : M d ( (cid:96) ) → M d +1 ( (cid:96) )[ P ] (cid:55)→ [ F d ( P )] . Then ϕ d is continuous and the directed system (cid:104) M d ( (cid:96) ) , ϕ d (cid:105) d ≥ is independent of thechoice of maps f d . Thus given (cid:96) ∈ R n > and any directed system (cid:104) R d , f d (cid:105) d ≥ of linearEuclidean isometries, we call (cid:104) M d ( (cid:96) ) , ϕ d (cid:105) d ≥ the directed system of moduli spaces of (cid:96) -gons . roof. Let (cid:96) ∈ R n > , fix a directed system (cid:104) R d , f d (cid:105) d ≥ of linear Euclidean isometries,and consider the diagram V d ( (cid:96) ) V d +1 ( (cid:96) ) M d ( (cid:96) ) M d +1 ( (cid:96) ) F d π d π d +1 ϕ d where π k : V k ( (cid:96) ) → M k ( (cid:96) ) is the canonical projection map. As with any topologicaltransformation group, these projection maps are open and continuous. The map F d iscontinuous as well, and thus if U ⊂ M d +1 ( (cid:96) ) is open, then π d ( F − d ( π − d +1 ( U ))) is open.But this is precisely ϕ − d ( U ), and thus ϕ d is continuous.To show that (cid:104) M d ( (cid:96) ) , ϕ d (cid:105) d ≥ is independent of our choice of f d ’s, let (cid:104) R d , f d (cid:105) d ≥ and (cid:104) R d , g d (cid:105) d ≥ be two directed systems of linear Euclidean isometries and define themaps F d : V d ( (cid:96) ) → V d +1 ( (cid:96) )( v , . . . v n − ) (cid:55)→ ( f d ( v ) , . . . , f d ( v n − ))and G d : V d ( (cid:96) ) → V d +1 ( (cid:96) )( v , . . . v n − ) (cid:55)→ ( g d ( v ) , . . . , g d ( v n − )) . Since SO ( d + 1) acts transitively on orthonormal bases of proper linear subspaces of R d +1 , for every d ≥ P = ( v , . . . v n − ) ∈ V d ( (cid:96) ) there exists T d ∈ SO ( d + 1)such that g d ( v i ) = T d ( f d ( v i )), and thus [ G d ( P )] = [ F d ( P )] = ϕ d ([ P ]).At each step M d ( (cid:96) ) ϕ d −−→ M d +1 ( (cid:96) ) in this system, potentially two things are happen-ing. First, there are new (cid:96) -gons appearing in M d +1 ( (cid:96) ) that are not in the image of ϕ d ,in which case ϕ d is not surjective; second, polygons P and Q that were not identified in M d ( (cid:96) ) become identified in M d +1 ( (cid:96) ), in which case ϕ d is not injective. Our main resultis that neither of these possibilities occurs when, and only when, d ≥ n , and thus thesystem stabilizes there. All of these results rely on the notion of polygon dimension,and we conclude this section with its definition. Definition 2.4.
Let P = ( v , . . . , v n − ) ∈ V d ( (cid:96) ) . The dimension of P , denoted dim( P ) , is the dimension of Span( { v , . . . , v n − } ) as a linear subspace of R d , andwe say that P is a dim( P ) -dimensional polygon. Given [ P ] ∈ M d ( (cid:96) ) the dimension of [ P ] is the dimension of P . V d ( (cid:96) ) In this section, we show that, as long as (cid:96) is not too degenerate, the polygon space V d ( (cid:96) )contains polygons of every dimension from 2 up through an upper bound dependenton d and (cid:96) . Let us first consider the conditions on a vector (cid:96) ∈ R n > for which thepolygon space V d ( (cid:96) ) is nonempty. Just as the triangle inequalities give necessary and ufficient conditions for three positive real numbers to be the edge-lengths of a triangle,the polygon inequalities l i ≤ (cid:88) j (cid:54) = i l j , l i > i = 1 , . . . , n (1)give necessary and sufficient conditions for ( l , . . . , l n ) to be the edge-lengths of apolygon. (See Lemma 1 in [KM95], keeping in mind that their polygons have beennormalized so that (cid:80) ni =1 l i = 1 and edges of length 0 are allowed.) In other words,letting C n denote the solution space of the inequalities (1), the polygon space V d ( (cid:96) )is nonempty if and only if (cid:96) ∈ C n . It may be helpful to think of C n as “almost apolyhedron”, in that the topological closure C n = (cid:96) ∈ R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l i ≤ (cid:88) j (cid:54) = i l j , l i ≥ i = 1 , . . . , n is a polyhedral cone pointed at the origin. Now we give an upper bound on thedimension of polygons in V d ( (cid:96) ) that holds for all (cid:96) ∈ C n . Proposition 3.1.
Let (cid:96) ∈ C n and let P ∈ V d ( (cid:96) ) . Then dim( P ) ≤ min { n − , d } .Proof. This is immediate since Span( v , . . . , v n − ) is a linear subspace of R d spannedby n − (cid:96) bounds the dimension of (cid:96) -gons—iscentral to our main theorem. Together with the upcoming results, it implies boththe injectivity and surjectivity of the maps ϕ d for large d . Before moving forwardwe characterize a subset of C n for which the corresponding moduli spaces are rathertrivial. The border of C n —the intersection of C n with its boundary—consists of those (cid:96) such that l i = (cid:80) j (cid:54) = i l j for some i = 1 , . . . , n . If (cid:96) lies on the border of C n then V d ( (cid:96) )consists only of 1-dimensional polygons, since the condition l i = (cid:80) j (cid:54) = i l j forces v i and v i − to be the endpoints of a line segment containing the rest of the vertices. In thiscase, SO ( d ) acts transitively on V d ( (cid:96) ) so M d ( (cid:96) ) is a singleton. Thus from here forward,we will assume (cid:96) ∈ int( C n ).It is well-known that if (cid:96) ∈ int( C n ) then V d ( (cid:96) ) contains 2-dimensional polygons.Proposition 3.3 below says that V d ( (cid:96) ) contains k -dimensional polygons for all values of k from 2 to min { d, n − } . In the proof we construct a ( k + 1)-dimensional (cid:96) -gon froma k -dimensional (cid:96) -gon for any 2 ≤ k < min { d, n − } . In the construction, we use avertex with a special property, and Lemma 3.2 shows that such a vertex exists. Referto Figures 2 and 3 when reading Lemma 3.2 and Proposition 3.3, respectively. Lemma 3.2.
Let P = ( v , . . . , v n − ) ∈ V d ( (cid:96) ) with ≤ dim( P ) < n − . For each i = 1 , . . . , n − , let L i denote the line through v i − and v i +1 , and let U i denote Span( { v j } j (cid:54) = i ) . There exists i ∈ { , . . . , n } such that v i ∈ U i \ L i .Proof. Let P ∈ V d ( (cid:96) ) with 2 ≤ dim( P ) < n −
1. Since dim( P ) < n − { v , . . . , v n − } , and thus v t ∈ U t for some t = 1 , . . . , n − v t (cid:54)∈ L t we are done by taking i = t , so suppose v t ∈ L t . Then v t − , v t , v t +1 arecolinear and in particular v t +1 ∈ Span( { v t − , v t } ), so v t +1 ∈ U t +1 . Now if v t +1 (cid:54)∈ L t +1 e are done by taking i = t +1 so suppose v t +1 ∈ L t +1 . Then by the same argument asabove v t +2 ∈ U t +2 , and we also have v t − , v t , v t +1 , v t +2 colinear. Continuing in thisway, if v i (cid:54)∈ U i \ L i for any i ∈ { , . . . , n − } , then all of v , v , . . . , v n − are co-linear,contradicting our assumption that dim( P ) ≥ L i v i − v i +1 v i U i Figure 2: If the dimension of a polygon is small enough, there exists avertex ( v i in the figure) that lies in the span of the other vertices and isnot co-linear with its neighbors. Proposition 3.3. If (cid:96) ∈ int( C n ) , then V d ( (cid:96) ) contains k -dimensional polygons for all k ∈ { , . . . , min { d, n − }} .Proof. Let (cid:96) ∈ int( C n ) and let P = ( v , . . . , v n − ) ∈ V d ( (cid:96) ) with 2 ≤ dim( P ) < min { d, n − } . For i = 1 , . . . , n − L i and U i as in Lemma 3.2, and choose i sothat v i ∈ U i \ L i . Let c be the orthogonal projection of v i onto L i and let u (cid:54)∈ U i bea unit vector. Define w := c + (cid:107) v i − c (cid:107) u and Q = ( v , . . . , v i − , w , v i +1 , . . . , v n − ).Since v i − , v i , v i +1 are not colinear, v i (cid:54) = c and thus w (cid:54)∈ U i , so dim( Q ) = dim( P )+1.It remains to show that (cid:107) w − v i − (cid:107) = l i − and (cid:107) w − v i +1 (cid:107) = l i +1 so that Q ∈ V d ( (cid:96) ).Let y be v i − or v i +1 . We have (cid:107) w − y (cid:107) = (cid:107) w − c (cid:107) + (cid:107) c − y (cid:107) (2)= (cid:107) v i − c (cid:107) + (cid:107) c − y (cid:107) (3)= (cid:107) v i − y (cid:107) , (4)where Equations (2) and (4) are the Pythagorean theorem, and Equation (3) followsfrom the definition of w . The result follows.In the following remark, we observe two implications of the results in this section onthe directed system (cid:104) M d ( (cid:96) ) , ϕ d (cid:105) —we will make these observations concrete in Section5, Proposition 5.1. Remark 3.4.
The map ϕ d is not surjective if d < n − since new polygons of higherdimensions continue to appear until we reach M n − ( (cid:96) ) , and ϕ d is surjective for d = n − (and thereafter) since every polygon in M n ( (cid:96) ) has dimension less than n and is thusthe image of a polygon in M n − ( (cid:96) ) . Given these observations about the surjectivity of ϕ d , it remains to make similarobservations about the injectivity of ϕ d . We make these observations in the followingsection. i v i − w v i +1 u v i U i c Figure 3: The vertex v i , guaranteed by Lemma 3.2, of a k -dimensional (cid:96) -gon for sufficiently small k , allows the (cid:96) -gon to bend along the line L i into a ( k + 1)-dimensional (cid:96) -gon, whose vertices are identical with thoseof the original (cid:96) -gon except that w has replaced v i (here, u is orthogonalto U i to make the picture easy to read, but the construction above doesnot require this). (cid:104) M d ( (cid:96) ) , ϕ d (cid:105) We begin with some preliminary remarks about O ( d ) and SO ( d ) and their actions onpolygons. The orthogonal group O ( d ) has two connected components: one is SO ( d )and the other, consisting of reflections through codimension-1 hyperplanes, we will call SO ( d ) − . Though SO ( d ) − is not a group we may none-the-less consider its action on V d ( (cid:96) ) by defining SO ( d ) − ( P ) = { T ( P ) : T ∈ SO ( d ) − } for every P ∈ V d ( (cid:96) ). Thus forany P ∈ V d ( (cid:96) ) we have O ( d )( P ) = SO ( d )( P ) ∪ SO ( d ) − ( P ) . (5)Moreover, SO ( d ) acts transitively by left (or right) multiplication on SO ( d ) − , so thatgiven any T ∈ SO ( d ) − we may recover all of SO ( d ) − by composing T with elementsof SO ( d ). Thus given any P ∈ V d ( (cid:96) ) and any T ∈ SO ( d ) − we have O ( d )( P ) = SO ( d )( P ) ∪ SO ( d )( T ( P )) . (6)We also note that SO ( d ) acts transitively on orthonormal bases of k -dimensional linearsubspaces of R d if k < d . Thus if X and Y are proper linear subspaces of R d ofdimension k , there exists T ∈ SO ( d ) so that T ( X ) = Y . Our final preliminary is thefollowing lemma. Lemma 4.1. If X and Y are proper linear subspaces of R d with X ⊂ Y and T ∈ SO ( d ) is such that T ( X ) ⊂ Y , then there exists S ∈ SO ( d ) such that S ( Y ) = Y and S | X = T | X .Proof. Let X , Y , and T be as above. Let { u , . . . , u k } be an orthonormal basis for X .Then { T ( u ) , . . . , T ( u k ) } is an orthonormal basis for T ( X ). If X and T ( X ) are bothsubspaces of Y we may extend these to orthonormal bases { u , . . . , u k , v k +1 . . . , v l } and { T ( u ) , . . . , T ( u k ) , w k +1 , . . . , w l } for Y . Since Y is a proper linear subspace of R d here exists S ∈ SO ( d ) such that S ( u i ) = T ( u i ) for i = 1 , . . . , k and S ( v i ) = w i for i = k + 1 , . . . , l . Thus S ( Y ) = Y and S | X = T | X .The main results of this section are two interpretations of the informal statement“reflections in R d are rotations in R d +1 ”. One interpretation, Proposition 4.4, is that if P and Q belong to the same SO ( d + 1) orbit in V d +1 ( (cid:96) ), then they come from polygonsin the same O ( d ) orbit of V d ( (cid:96) ). The second interpretation, Proposition 4.5, is that ifthe dimension of a polygon is less than the dimension of the ambient space, then everyreflection of that polygon may be obtained by rotation. We first state two lemmas thatlead to Proposition 4.4. Lemma 4.2.
Let f d : R d → R d +1 be a linear Euclidean isometry. For every T ∈ O ( d ) there exists S ∈ SO ( d + 1) such that S ( f d ( x )) = f d ( T ( x )) for all x ∈ R d . Conversely,for every S ∈ SO ( d + 1) such that S (im( f d )) = im( f d ) , there exists T ∈ O ( d ) such that S ( f d ( x )) = f d ( T ( x )) for all x ∈ R d .Proof. Let { a , . . . , a d } be a basis for R d and let f d : R d → R d +1 be a linear Euclideanisometry. There is a natural embedding (cid:15) : O ( d ) → SO ( d + 1) T (cid:55)→ (cid:40) T ⊕ if T ∈ SO ( d ) T ⊕ − if T ∈ SO ( d ) − where is the 1 × (cid:15) ( T ) is written with respect toa basis for R d +1 whose first d elements are { f d ( a ) , . . . , f d ( a d ) } . Thus if T ∈ O ( d )then (cid:15) ( T ) ∈ SO ( d + 1) and (cid:15) ( T )( f d ( x )) = f d ( T ( x )) for all x ∈ R d . Conversely,let S ∈ SO ( d + 1) such that S (im( f d )) = im( f d ). Then S ∈ im( (cid:15) ) and we have S ( f d ( x )) = f d ( (cid:15) − ( S )( x )) for all x ∈ R d .Lemma 4.2 says that rotating a point in the image of f d is the same as rotatingor reflecting its preimage in R d and then pushing it forward. Lemma 4.3 extends thisidea from points to polygons. Lemma 4.3.
Given P ∈ V d ( (cid:96) ) and T ∈ O ( d ) , there exists S ∈ SO ( d + 1) such that S ( F d ( P )) = F d ( T ( P )) . Conversely, given F d ( P ) ∈ im( F d ) and S ∈ SO ( d + 1) suchthat S ( F d ( P )) ∈ im( F d ) , there exists T ∈ O ( d ) such that S ( F d ( P )) = F d ( T ( P )) .Proof. Let P = ( v , . . . , v n − ) ∈ V d ( (cid:96) ) and T ∈ O ( d ). By Lemma 4.2 there exists S ∈ SO ( d + 1) such that S ( f d ( v i )) = f d ( T ( v i )) for all i = 1 , . . . , n −
1, and thus S ( F d ( P )) = F d ( T ( P )).Conversely, let F d ( P ) ∈ im( F d ) and S ∈ SO ( d + 1) such that S ( F d ( P )) ∈ im( F d ).Let X = Span( { f d ( v ) , . . . , f d ( v n − ) } ) and Y = im( f d ). Then X and Y are properlinear subspaces of R d +1 and X and S ( X ) are subsets of Y . Thus by Lemma 4.1 thereexists S (cid:48) ∈ SO ( d + 1) such that S (cid:48) ( Y ) = Y and S (cid:48) | X = S | X . Since S (cid:48) (im( f d )) = im( f d ) , Lemma 4.2 says there exists T ∈ O ( d ) such that S (cid:48) ( f d ( x )) = f d ( T ( x )) for all x ∈ R d ,and thus S (cid:48) ( F d ( P )) = F d ( T ( P )). Since S (cid:48) | Span( { f d ( v ) ,...,f d ( v n − ) } ) = S | Span( { f d ( v ) ,...,f d ( v n − ) } ) , e have S ( F d ( P )) = F d ( T ( P )). Proposition 4.4.
Let P ∈ V d ( (cid:96) ) . The preimage under F d of the SO ( d + 1) orbit of F d ( P ) is the O ( d ) orbit of P : F − d ( SO ( d + 1)( F d ( P ))) = O ( d )( P ) . Proof.
Let P ∈ V d ( (cid:96) ). We begin with the inclusion F − d ( SO ( d +1)( F d ( P ))) ⊂ O ( d )( P ).Let Q ∈ F − d ( SO ( d + 1)( F d ( P ))). Then F d ( Q ) = S ( F d ( P )) for some S ∈ SO ( d + 1).Since F d ( P ) and S ( F d ( P )) are both in the image of F d , by Lemma 4.3 there exists T ∈ O ( d ) such that F d ( T ( P )) = S ( F d ( P )), and thus F d ( T ( P )) = F d ( Q ). Since F d isinjective, T ( P ) = Q , and so Q ∈ O ( d )( P ).For the inclusion O ( d )( P ) ⊂ F − d ( SO ( d + 1)( F d ( P ))), let Q ∈ O ( d )( P ) and let T ∈ O ( d ) such that Q = T ( P ). By Lemma 4.3 there exists S ∈ SO ( d + 1) such that S ( F d ( P )) = F d ( T ( P )). Thus S ( F d ( P )) = F d ( Q ), so Q ∈ F − d ( SO ( d + 1)( F d ( P ))).In the context of the map ϕ d : M d ( (cid:96) ) → M d +1 ( (cid:96) ), Proposition 4.4 says that ϕ d isnot a priori injective. It says the preimage of [ F d ( P )] ∈ M d +1 ( (cid:96) ) in M d ( (cid:96) ) is the π d projection of the O ( d ) orbit of P , which may be strictly larger than its SO ( d ) orbit.However, as the next proposition shows, this is only the case if the dimension of P matches the dimension d of the ambient space. Proposition 4.5.
Let P ∈ V d ( (cid:96) ) . Then O ( d )( P ) = SO ( d )( P ) if and only if dim( P ) 1, so S ( P ) = T ( P ), and thus Q ∈ SO ( d )( P ).Now suppose dim( P ) = d , and suppose for a contradiction that SO ( d ) − ( P ) ⊂ SO ( d )( P ). Then given T ∈ SO ( d ) − there exists S ∈ SO ( d ) such that T ( P ) = S ( P ),and thus T ( v i ) = S ( v i ) for all i = 1 , . . . , n − 1. But since dim( P ) = d , some subset ofvertices of P form a basis for R d , and thus T = S , a contradiction. Remark 4.6. Propositions . and . imply that ϕ d is injective on the set of polygonsof dimension less than d . In Proposition . in the following section, we state a strongerresult and describe explicitly the preimage under ϕ d of ϕ d ([ P ]) for any [ P ] ∈ M d ( (cid:96) ) . Stabilization theorem In this section we give our main result, stated as Theorem 5.3, which says that asidefrom trivial cases the directed system (cid:104) M d ( (cid:96) ) , ϕ d (cid:105) d ≥ of moduli spaces of (cid:96) -gons sta-bilizes when d is equal to the length of (cid:96) . Remarks 3.4 and 4.6 summarize our resultsso far: the map ϕ d is surjective if and only if d ≥ n − 1, and injective on the set ofpolygons of dimension less than d . We begin by formalizing Remarks 3.4 and 4.6 withPropositions 5.1 and 5.2, respectively. Proposition 5.1. Let (cid:96) ∈ int( C n ) . The map ϕ d : M d ( (cid:96) ) → M d +1 ( (cid:96) ) is surjective ifand only if d ≥ n − .Proof. Let d < n − 1. By Proposition 3.3, M d +1 ( (cid:96) ) contains ( d + 1)-dimensionalpolygons. Since M d ( (cid:96) ) does not contain ( d + 1)-dimensional polygons, and sincedim( ϕ d ([ P ])) = dim([ P ]), ϕ d is not surjective. Now let d ≥ n − 1, and let [ P ] =[ v , . . . , v n − ] ∈ M d +1 ( (cid:96) ). We will show that [ P ] ∈ im( ϕ d ). By Proposition 3.1 wehave dim( P ) < d + 1, so there is a d -dimensional linear subspace X ⊂ R d +1 such that v i ∈ X for all i = 1 , . . . , n − 1. Let T ∈ SO ( d + 1) such that T ( X ) = im( f d ). Then T ( v i ) ∈ im( f d ) for all i = 1 , . . . , n − 1, so we may write T ( P ) = ( T ( v ) , . . . , T ( v n − )) = ( f d ( w ) , . . . , f d ( w n − ))for some w , . . . , w n − ∈ R d . Since ( f d ( w ) , . . . , f d ( w n − )) ∈ V d +1 ( (cid:96) ) and f d is alinear isometry, ( w , . . . , w n − ) ∈ V d ( (cid:96) ). Thus T ( P ) = F d (( w , . . . , w n − )) ∈ im( F d )and [ T ( P )] ∈ im( ϕ d ). Finally, since T ∈ SO ( d + 1), [ T ( P )] = [ P ]. Proposition 5.2. The map ϕ d : M d ( (cid:96) ) → M d +1 ( (cid:96) ) is -to- on the set of d -dimensionalpolygons and -to- elsewhere.Proof. Recall the diagram V d ( (cid:96) ) V d +1 ( (cid:96) ) M d ( (cid:96) ) M d +1 ( (cid:96) ) F d π d π d +1 ϕ d from Section 2. Tracing backwards from M d +1 ( (cid:96) ) we have ϕ − d ◦ ϕ d ([ P ]) = π d ◦ F − d ◦ π − d +1 ( ϕ d ([ P ])) . Since π − d +1 ( ϕ d ([ P ])) = SO ( d + 1)( F d ( P )), and since Proposition 4.4 says F − d ( SO ( d + 1)( F d ( P ))) = O ( d )( P )we have ϕ − d ◦ ϕ d ([ P ]) = π d ( O ( d )( P )) . Thus by the partition O ( d )( P ) = SO ( d )( P ) ∪ SO ( d )( T ( P )) in (6) we have ϕ − d ◦ ϕ d ([ P ]) = π d ( SO ( d )( P ) ∪ SO ( d )( T ( P ))) = { [ P ] , [ T ( P )] } where T is any element of SO ( d ) − . The result now follows from Proposition 4.5 whichimplies [ P ] = [ T ( P )] if and only if dim( P ) < d . e will use Proposition 5.2 in the proof of Theorem 5.3 to show that ϕ d is injectivewhen d ≥ n , but it tells us more. The proposition along with its proof tells us exactlyhow ϕ d is not injective when d < n , namely, that reflections of d -dimensional polygonsin R d become identified in R d +1 . We will see this exemplified in Section 6. For now,we are finally ready to state our main theorem, and its proof is easy since we have doneall the heavy lifting. Theorem 5.3. Let (cid:96) ∈ int( C n ) . Then the directed system M ( (cid:96) ) ϕ −→ M ( (cid:96) ) ϕ −→ M ( (cid:96) ) ϕ −→ · · · ϕ d − −−−→ M d ( (cid:96) ) ϕ d −−→ · · · of moduli spaces of (cid:96) -gons stabilizes at d = n , that is, M d ( (cid:96) ) ≈ M n ( (cid:96) ) if and only if d ≥ n .Proof. We show that for (cid:96) ∈ int( C n ) the map ϕ d : M d ( (cid:96) ) → M d +1 ( (cid:96) ) is a homeomor-phism if and only if d ≥ n . Let (cid:96) ∈ int( C n ). By Proposition 5.1 ϕ d is surjective ifand only if d ≥ n − 1, so it remains to show that ϕ d is injective if and only if d ≥ n .By Proposition 5.2, ϕ d is injective if and only if M d ( (cid:96) ) does not contain d -dimensionalpolygons. By Proposition 3.3 this happens if and only if d ≥ n . -gons We end with an example of a directed system of moduli spaces of (cid:96) -gons, in the hopethat it aids the reader’s intuition for these systems. Let (cid:96) = (1 , , , ∈ int( C ).Theorem 5.3 says that the directed system of moduli spaces of (cid:96) -gons stabilizes at d = 4, and thus we have M ( (cid:96) ) ϕ −→ M ( (cid:96) ) ϕ −→ M ( (cid:96) ) ≈ M ( (cid:96) ) ≈ M ( (cid:96) ) ≈ · · · . Subsections 6.1, 6.2, and 6.3 give descriptions of the moduli spaces M ( (cid:96) ) , M ( (cid:96) ),and the stable limit M ( (cid:96) ), respectively, and Figure 4 provides a picture of each. Inthe figure, several points in the moduli spaces are marked with their correspondingpolygons drawn next to them. A given picture should not be thought of as a staticimage, but as flows of polygons smoothly transforming into one another (see [KM96]for more on Hamiltonian flows in polygon spaces). Also, the pictures should not bethought of as wholly distinct from one another, but instead as being related to eachother by the identification of some polygons and the arising of others at each step inthe directed system. M ( (cid:96) ) The moduli space M ( (cid:96) ) of (cid:96) -gons in R is homeomorphic to three circles, every two ofwhich meet at a single point ([KM95], p15). The points of intersection correspond to 1-dimensional polygons. A given circle is partitioned by these points of intersection intotwo arcs of 2-dimensional polygons, where the polygons along one arc are reflectionsof the polygons along the other arc. See the polygons next to the marked points in M ( (cid:96) ) in the top left image of Figure 4 and try to imagine the polygons that are not rawn. For example, the curves going from the square at the top of the picture to eitherendpoint of the arc it is sitting on, those consist of parallelograms that get narrowerand narrower as you approach the 1-dimensional degenerate parallelograms at the arc’sendpoints. M ( (cid:96) ) The moduli space M ( (cid:96) ) is homeomorphic to a sphere. As we move from polygonsin R to polygons in R , two things happen. First, every 2-dimensional polygon in M ( (cid:96) ) becomes identified with its reflection, and thus each pair of arcs comprising acircle in M ( (cid:96) ) is projected onto a single arc in M ( (cid:96) ); these three arcs make up theequator in M ( (cid:96) ). Second, 3-dimensional polygons appear that did not exist in M ( (cid:96) );they comprise the two hemispheres. Analogous to the arc pairs consisting of reflectedpolygons in R , the hemisphere pairs consist of reflected polygons in R —every polygonin the southern hemisphere has its reflection in the northern hemisphere. M d ( (cid:96) ) , d ≥ Next, we arrive at the moduli space M ( (cid:96) ) which is homeomorphic to a disc. As wemove from R to R , no new polygons appear because the maximum dimension ofa 4-gon is 3—in other words ϕ is surjective. However, ϕ is not injective, since 3-dimensional polygons and their reflections in M ( (cid:96) ) get identified in M ( (cid:96) ). Thus thenorthern and southern hemispheres in M ( (cid:96) ) collapse to form the interior of the disk.This disk is the stable limit. As we move forward to M ( (cid:96) ) no new polygons appearbecause they all appeared back in M ( (cid:96) ). Moreover, no new identifications are made:if ϕ ([ P ]) = ϕ ([ Q ]) then P and Q are rotations or reflections of each other in R ,but since their dimension is less than 4 every reflection is obtained by rotation, so[ P ] = [ Q ]. We conclude with some questions inspired by this example. First, is the stable limitalways contractible and connected? Does it always have boundary, and if so doesthe interior consist of polygons of maximum dimension? We also notice that the1-dimensional polygons partition M ( (cid:96) ) into connected components of 2-dimensionalpolygons, and the 1- and 2-dimensional polygons partition M ( (cid:96) ) into connected com-ponents of 3-dimensional polygons. Can this be generalized? The answers to all thesequestions may come from an algebro-geometric view of polygon spaces. It is not men-tioned earlier, but V d ( (cid:96) ) is a real algebraic variety, and thus M d ( (cid:96) ) is real semi-algebraicspace. Is there an analogous directed system of semi-algebraic spaces? Is there a corre-spondence between polygon dimension and the dimension of the (pieces of) subvarietiescomprising M d ( (cid:96) )? v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v Figure 4: Clockwise from top-left, the moduli spaces M ( (cid:96) ), M ( (cid:96) ), and M d ( (cid:96) ) , d ≥ 4, for (cid:96) = (1 , , , References [CS16] J. Cantarella and C. Shonkwiler. The symplectic geometry of closed equi-lateral random walks in 3-space. Annals of Applied Probability , 26:549–596,2016.[FF13] Michael Farber and Viktor Fromm. The topology of spaces of polygons. Trans.Amer. Math. Soc. , 365(6):3097–3114, 2013.[GM13] Leonor Godinho and Alessia Mandini. Hyperpolygon spaces and modulispaces of parabolic Higgs bundles. Adv. Math. , 244:465–532, 2013. HK96] J. Hausmann and A. Knutson. 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