Abstract
For , let be an -dimensional smooth closed manifold and a smooth function. We set and assume that is attained by unique point such that is a nondegenerate critical point. Then the Morse lemma tells us that if is slightly bigger than , is diffeomorphic to . In this paper, we relax the condition on from being nondegenerate to being an isolated critical point and obtain the same consequence. Some application to the topology of polygon spaces is also included.
1. Introduction and Statement of the Result
Throughout this paper, denotes the standard topological sphere equipped with the standard differential structure. For , let be an -dimensional smooth closed manifold and a smooth function. We set and assume that is attained by unique point . Then the following result is a consequence of the Morse lemma (see, e.g., [1]): If is a nondegenerate critical point and there are no critical points in (where ), then there is a diffeomorphism .
The purpose of this paper is to study the question of whether the same result holds if we relax the condition on from being nondegenerate to being an isolated critical point. We also give an application of our result to the topology of polygon spaces.
Now our main result is the following.
Theorem A. For , let be an -dimensional smooth closed manifold and a smooth function. One sets and assumes that is attained by unique point . If is an isolated critical point and there are no critical points in where , then the following results hold: (i)If , there is a diffeomorphism .(ii)If , there is a homeomorphism .
Corollary B. For , let be an -dimensional smooth closed manifold and a smooth function. One sets and assumes that is attained by unique point . If is an isolated critical point and there are no critical points in where , then the following results hold: (i)If , there is a diffeomorphism .(ii)If , there is a homeomorphism .
This paper is organized as follows. In Section 2 we prove Theorem A. In Section 3 we study an application of it. Theorem 4 is the main result in this section. Remark 7(ii) states the essential difference between the known map and ours.
2. Proof of Theorem A
We keep the notations of Theorem A. For , we set
Lemma 1. There is a diffeomorphism .
Proof. By [2, Lemma ], it will suffice to prove that, for any compact set , there exists an open set of such that and .
We fix an open set which satisfies that (Such indeed exists because .) Note that and is compact. Note also that is compact. Hence there exist and (where ) such that We fix such and .
For a compact set , since holds, there exists such that . This implies that .
By assumption, there are no critical points in . Hence there is a diffeomorphism We set . Then is an open set of , and .
Lemma 2. The manifold is homotopy equivalent to .
Proof. Since is a manifold, the assertion is clear for . We assume . We claim that To prove (5), note that is a manifold with boundary such that . Lefschetz duality implies that Recall that the inclusion is a homotopy equivalence. Since by Lemma 1, has the cohomology of a point. In the homology long exact sequence of the pair , we apply (6). Then we obtain (5).
Since the fundamental group at infinity of is trivial by Lemma 1, is simply connected. (See, e.g., [3, p. 389] or [4, Examples and ] for this kind of argument.) Hence is a homotopy sphere.
Proof of Theorem A. For or , Lemma 2 immediately implies that is diffeomorphic to .
For , Lemma 2 tells us that is contractible with simply connected boundary. Using the -cobordism theorem (see [5, p. 108, Proposition ]), we have . Hence .
For , combining Freedman’s resolution of the -dimensional Poincaré conjecture [6] and Lemma 2, we have .
For , combining Perelman’s resolution of the -dimensional Poincaré conjecture [7], Lemma 2, and the fact [8] that the differential structure on is unique, we have .
Proof of Corollary B. If satisfies the assumption of Corollary B, the function satisfies the assumption of in Theorem A. Hence Corollary B follows.
3. An Application
Starting in [9–11], the topology of the configuration space of planar polygons has been considered by many authors. We refer to [12] for an excellent exposition.
For simplicity, we consider the case that the edge lengths are and . Let act on the -dimensional torus diagonally. For , we setHere denotes the unit vectors in the direction of the sides of a polygon.
It is clear that for and It is also known that there is a diffeomorphism
Recall that (8) can be understood Morse-theoretically. The following arguments are particularly well described in [12]: Using the -action in the definition of , we normalize to be and write as (See Figure 1.)
We identify and define a function by Then (9) gives an identification . It is elementary to prove that an element is a critical point of if and only if for all . All these critical points are nondegenerate and their indices are known. (See, e.g., [9, 12–14].) In particular, attains its maximum value at and this is a nondegenerate critical point. Hence, using the Morse lemma, (8) follows.
The configuration space of spatial polygons has also been studied by many authors. (See, e.g., [12, 14, 15].) In this case, our object is defined by
Recently, motivated by chemistry, Crippen [16], Goto and Komatsu [17], and O’Hara [18] studied certain subspaces of . Namely, they studied the configuration space of equilateral polygons with restriction on the splay angle of each vertex.
First, we define the angle to be and as and , respectively. Goto and Komatsu [17] chose the angle with molecular model in mind. Then they studied the space defined by where denotes the standard inner product on . The closed chains in are equilateral polygons in with vertices such that the interior angles are all equal to except for the two angles at the successive vertices and .
The main result in [17] states that when , is a manifold homeomorphic to . Since they use Reeb’s theorem, they state their result as a homeomorphism. But actually and are diffeomorphic because the differential structure on is unique for . But what is more important is that it is not known whether is a manifold for .
Second, for all , we define a space by Here we understand to be . The closed chains in are equilateral polygons in with vertices such that the interior angles are all equal to . Crippen [16] studied the topological type of for various when . The result is that is either , one point, or two points depending on . Next, O’Hara [18] studied for various . The result is that is disjoint union of a certain number of ’s and points. It is to be noted that since , it is natural to expect that . But the above results imply that the defining equations for do not intersect transversally when is small.
Note that the above results in [16–18] concentrate on the case for small . This is understandable because imposing some conditions on the interior angles causes difficulties in computations. Nevertheless, we would like to prove some assertion which holds for general . Modifying the definition of , we define a space as follows: The closed chains in are polygons in with edge lengths and such that the interior angles are all equal to except for the two angles at the endpoints of the edge of length .
Let us obtain a similar description to (9). We set and . By the -action in the definition of , we can normalize and to be and , respectively. Thus we have the following description of : Hereafter we use description (15). (See Figure 2.)
We recall the following.
Theorem 3 (see [19, Theorem ], [20, Theorem ]). One sets Then the following results hold: (i) if .(ii). We write the element of by in the notation of (15). Then is given as follows: For , is given by As a consequence, is given by Note that is a planar -gon. (See Figure 3.)
Now we state the main result in this section (cf. (8)).
Theorem 4. Let be slightly smaller than . Then, for all , there is a diffeomorphism
For the rest of this paper, we prove Theorem 4. For that purpose, we define Moreover, similarly to in (10), we define a map byThen (15) gives an identification and Theorem 3 tells us that is attained by . In order to compute the Hessian matrix of at , we construct the commutative diagram shown in Figure 4.
First, we set Namely, is the universal cover.
Second, we construct by induction on . We define to be the unique map between one-point spaces.
Assuming that is constructed, we construct . We write Since , there are two choices for which satisfies that . Among these ’s, we choose the one which satisfies the condition that Using this, we defineAnd we set
From the construction, induces a map such that . It is easy to see that is a diffeomorphism. (See Remark 7(ii).)
Third, we set , where is defined in (20). Thus we have completed the construction of the diagram in Figure 4.
Note that , where is defined in Theorem 3.
Lemma 5. Let be the Hessian matrix of at . Note that this is an matrix. Let be the th entry of . Then, for all , the following result holds:
Proof. The lemma is proved by direct computations.
Proof of Theorem 4 for Even . We set It is easy to see that when is even, Thus we have ; hence is a nondegenerate critical point of . Using the Morse lemma, we complete the proof of Theorem 4 for even .
Proof of Theorem 4 for Odd . Lemma 5 tells us that when is odd, . Hence we need to use Corollary B. For that purpose, it will suffice to prove that is an isolated critical point of .
We define a map by where the right-hand side denotes the Jacobian matrix of at
First we prove the following.
Lemma 6. For each , there exists such that for all .
Proof. We fix . In order to prove the lemma by contradiction, assume that were an accumulation point of . For , let be the th element of . Since is a polynomial in , we can define for and this is a holomorphic function. If were an accumulation point of , then the identity theorem would tell us that is identically for all and for all .
We write the Maclaurin expansion of as where and are polynomials in of degrees and , respectively. Since , we haveThen standard computations show that . This contradicts the assumption that .
This completes the proof of Lemma 6.
Now using Lemma 6 and the continuity of the function , there exists an open neighborhood of in such that if , then Thus is an isolated critical point of .
This completes the proof of Theorem 4 for odd .
Remark 7. (i) Recall that Corollary B for just gives a topological assertion. In order to prove Theorem 4, we have studied the map . Since , implies that . But we have seen in Lemma 5 that when is even, is a nondegenerate critical point and deduced Theorem 4 directly from the Morse lemma, which gives a differential assertion.
(ii) Although we have not used the diffeomorphism in the above arguments, it is to be noted that the map is much more difficult than the known map in (10).
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The author is grateful to the referee for valuable comments.