Abstract
Let a be a real number satisfying . We denote by the configuration space of regular spherical n-gons with side lengths a. In our previous paper, we determined for odd n. In this paper, we determine it for even n. The main difference from our previous paper is given as follows. When n is odd, we prove that is obtained from by successive Morse surgeries. On the other hand, when n is even, we show that is obtained from by successive Morse surgeries. Here, denotes the configuration space of equilateral n-gons in , which has singular points when n is even.
1. Introduction and Statement of the Main Result
Recently, the topology of polygon spaces in the Euclidean space of dimension two or three has been considered by many authors. The study of planar polygon spaces started in [1–3]. For example, the homology groups were determined in [4]. On the other hand, the study of spatial polygon spaces started in [5].
Morse theory plays a key role in the study of polygon spaces. [6] is an excellent exposition about polygon spaces with emphasis on Morse theory. In [7], Milgram and Trinkle obtained results by making excellent use of Morse surgery.
Later, Kapovich and Millson [8] studied spherical polygon spaces. They proved a remarkable and very useful theorem about a Morse function. We first recall their result.
Let be an n-tuple of real numbers satisfying . We set and definewhere d is the spherical distance. Let act on diagonally and we set . Kapovich and Millson defined the function by
We will restrict to u’s such that so that is differentiable. Note thatis the moduli space of closed polygonal linkages in with side lengths .
It is proved in [8] that an element of is a critical point of if and only if it is degenerate, i.e., it lies in a great circle γ in . In order to describe the signature, we give the following.
Definition 1 (orienting γ). Suppose is a closed degenerate linkage contained in a great circle γ. Orient γ so that the arc joining to is positively directed. Thus, an edge is a back-track if it has the same direction as .
Theorem 1 ([8], Main Theorem). Let be a degenerate free linkage and P be the associated degenerate closed n-gon linkage. Then, the signature of is given bywhere denotes the number of forward-tracks and denotes the number of back-tracks so . Moreover, denotes the winding number of the degenerate configuration P, which is given by the formula , where is defined as follows: if is the forward-track, the we set , and if is the back-track, then we set .
Let a be a real number satisfying . We consider the case that , that is, for . We set
Note that is the moduli space of regular spherical polygons with side lengths a.
We realize as a level set of a Morse function, which is different from the above . We first set
Let act on bywhere . Then, we set
We define the function by
Note that for all , we have
In [9], we studied the case of odd n and proved that is a Morse function. Since a level set is obtained by successive Morse surgeries, we could determine for all a.
The purpose of this paper is to study the case of even n. Hereafter, we will always assume n to be even and set . When n is even, has the following two characteristics, which do not hold for odd n. One is that there is a homeomorphism for all a (see Theorem 5 (i)). The other is that has singular points for all a. In fact, we setwhere is defined in Theorem 1. Then, an element of is a singular point of .
Avoiding , we give the following definitions. First, we set
Second, we set
Third, we denote to be the restriction of to .
Theorems 2 and 3, which are main theorems of this paper, assert that is a Morse function.
Theorem 2. An element is a critical point of if and only if P is degenerate.
Theorem 2 implies the following two results:(i)In contrast to the fact that the critical points of are the degenerate linkages, the critical points of are the degenerate linkages P such that .(ii)All a’s which are critical values for are rational multiples of π. Moreover, the smallest critical value is and the largest one is .
From the definition of f, b, and in Theorem 1, for a degenerate linkage, the following relations hold:
Then, we have the following:
Theorem 3. Let be a degenerate polygon. Then, the following hold:(i)When , the signature of is given by (ii)When , the signature of is given by
This paper is organized as follows. In Section 2, we first combine Theorems 2 and 3 into Theorem 4. Then, using the theorem, we determine in Theorem 6. In Section 3, we prove Theorems 2 and 3. Theorem 1 is a key to proving Theorem 3. In Section 4, we give several examples about .
2. Conclusions from Theorems 2 and 3
In Theorem 3, we set
Then, we obtain the following theorem. We denote by the set of natural numbers.
Theorem 4. We setThen, the following assertions hold:(i)To each , there corresponds a certain number of critical points of . All critical points are nondegenerate such that their information is given by Table 1.(ii)Conversely, a critical point of is attained by a unique .
Remark 1. (i)It is not true that critical points of the same critical value have the same index. For example, contains elements and such that their critical values are . On the other hand, the index of the former is 4 but that of the latter is 3.(ii)The number of critical points does not depend on t. Moreover, the index of has the same parity as , which also does not depend on t.
Proof of Theorem 4. (i)Let be the set of critical points of . We define the map by making correspond with in (15). We check that F is certainly a map to . First, we check that . In fact, this is clear from (15) for positive or negative. Second, we check that . In fact, we have from the second equation of (14) that Since , (18) implies that . Third, we check that . In fact, we have from the second equation of (15) that if , then , and if , then . Next, we check Table 1. First, (18) tells us that the critical value in Table 1 is true. Second, we compute the index of at . From the first equation of (14) and (15), we have Then, using Theorem 3, (15), and (19), we have the index of at is for positive or negative. Hence, the index of in Table 1 is true. Third, we compute the number of critical points. (15) tells us that critical points and satisfy if and only if or . The description of f in terms of s is computed in (19). Since is always back-track, we need to choose f elements from . The total number of such choices is Hence, the number of critical points in Table 1 is true. This completes the proof of Theorem 4 (i).(ii)The item is clear from (15).
Theorem 5. (i)For , we set , where we put for . Then, there is a homeomorphism , where the space is defined in (3). In particular, there is a homeomorphism for all .(ii)Let be the configuration space of equilateral n-gons in . Then, if or , there is a homeomorphism .
Proof. (i)The item is proved in ([10], Theorem 2.2) and ([8], Section 4). More precisely, we define the map by where we set . Then, G is a homeomorphism.(ii)It is clear that if ε is a sufficiently small positive real number, then is homeomorphic to . Moreover, since any element of is a regular value of by Theorem 4, we have for . Using the above item (i), we also have for .
Theorem 6. Let be the set of critical values of (note that we can write the elements of using Table 1). Then, the following assertions hold:(i)Assume that . Then, we have(ii)Assume that . Let be the unique element of , which satisfies the following two conditions:Then, we haveNote that the right-hand side of (24) does not depend on .
In order to prove Theorem 6, we need the following.
Theorem 7. Let be a smooth function on a d-dimensional manifold M. For numbers , we assume that is compact and contains a unique nondegenerate critical point p of index r. Then, the following results hold:(i)The level set is obtained from by removing and attaching along the boundary. We call this construction a surgery of type .(ii)If d is even, then we have(iii)We set . Then, level set is obtained from by removing and attaching along the boundary, where C denotes the cone. In particular, if d is even, then we have
Proof. The theorem is well known in Morse surgery (see [11]).
Proof of Theorem 6. We apply Theorem 7 to the function μ in (9). Every level set contains the singular point set and the information on is determined in Theorem 4.(i)If , then Table 1 tells us that a is a regular value of . Combining Theorem 7 (ii) and Theorem 5 (ii), we have . Recall that was determined in ([12], Theorem A). Hence, (i) follows.(ii)If , then Table 1 tells us that the elements of whose critical value equals to a are given by for . Then, combining Table 1 and Theorem 7 (iii), we obtain (ii).
3. Proofs of Theorems 2 and 3
Proof of Theorem 2. We prove the theorem along the lines of ([8], Theorem 2.9) (see also the proof of ([9], Theorem 1.3)). Combining the following three assertions, we obtain Theorem 2:(i)An analogue of ([8], Lemma 2.7 (ii)): by (10), the Zariski tangent space is given by(ii)An analogue of ([8], Corollary 2.8): we see from (i) that a point P is a singular point of if and only if is a critical point of .(iii)By ([13], Theorem 1.1), P is a singular point of if and only if P is degenerate.In order to prove Theorem 3, we need the following.
Lemma 1. We consider the function in (2) for the case that , that is, for . Let O be an open neighborhood of in such that O contains no other degenerate polygons than . We fix a sufficiently small positive real number ε. Then, the following assertions hold:(i)Assume that . Then, for all , is diffeomorphic to an open set of (ii)Assume that . Then, for all , is diffeomorphic to an open set of
Proof of Lemma 1. (i)We write the side lengths of a spherical polygon as . For , we deform the side lengths of an element of by We need to check that this deformation is indeed possible. To see this, it will suffice to see that (28) does not cross a wall for any δ, where a wall is defined in ([8], p. 311). Recall that is always back-track (see Definition 1). Hence, we have about (28) that for some integer u. We claim that . In fact, the assumption tells us that at least of the first components of (28) are forward-track. Hence, we have and the claim follows. Now, since the term in (2) is near zero but not zero for all , (28) does not cross a wall for any δ. Finally, setting or ε in (28), we complete the proof of (i).(ii)Instead of (28), we deform the side lengths of an element of byWe have about (30) thatfor some integer . We claim that . In fact, the assumption tells us that at most of the first components of (30) are forward-track. Hence, we have and the claim follows.
Now, since the term in (9) is near zero but not zero for all , (30) does not cross a wall for any δ. Finally, setting or ε in (30), we complete the proof of (ii). This completes the proof of Lemma 1.
Proof of Theorem 3. We shall deduce Theorem 3 from Theorem 1. We apply Theorem 7 (i) to the map . Note that . We denote by r the index of P. When the level set descends to , the critical point P gives a surgery of typeWe set .(i)If , then Lemma 1 (i) tells us that the above descent is equivalent to the ascent from to . Combining Theorem 7 (i) and Theorem 1, when we cross through P, a surgery of type occurs. Comparing (32) and (33), we have(ii)If , then the descent from to is equivalent to the descent from to . When we cross through P, a surgery of type occurs. Comparing (32) and (35), we have This completes the proof of Theorem 3.
4. Examples
Proposition 1. We have the following examples:(i)(ii)(iii)(iv)
Proof. Computing the right-hand side of (24) explicitly, we obtain the proposition.
Remark 2. Proposition 1 (i) was obtained in ([14], Theorem 2) by a different method.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by JSPS KAKENHI (grant no. 15K04877).