# On the Level Set of a Function with Degenerate Minimum Point

Authors: Yasuhiko Kamiyama
Journal: International Journal of Mathematics and Mathematical Sciences
 (2015)

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## Abstract

Forn≥2, let M be an n-dimensional smooth closed manifold and f:M→R a smooth function. We set minf(M)=m and assume that m is attained by unique point p∈M such that p is a nondegenerate critical point. Then the Morse lemma tells us that if a is slightly bigger than m, f-1(a) is diffeomorphic to Sn-1. In this paper, we relax the condition on p 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.


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## Body

## 1. Introduction and Statement of the Result

Throughout this paper,Sk denotes the standard topological sphere equipped with the standard differential structure. For n≥2, let M be an n-dimensional smooth closed manifold and f:M→R a smooth function. We set minf(M)=m and assume that m is attained by unique point p∈M. Then the following result is a consequence of the Morse lemma (see, e.g., [1]):Ifpis a nondegenerate critical point and there are no critical points inf-1((m,a]) (wherem<a),then there is a diffeomorphismf-1(a)≅Sn-1.The purpose of this paper is to study the question of whether the same result holds if we relax the condition onp 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.
Forn≥2, let M be an n-dimensional smooth closed manifold and f:M→R a smooth function. One sets minf(M)=m and assumes that m is attained by unique point p∈M. If p is an isolated critical point and there are no critical points in f-1((m,a])  (where m<a), then the following results hold: (i)
Ifn≠5, there is a diffeomorphism f-1(a)≅Sn-1.(ii)
Ifn=5, there is a homeomorphism f-1(a)≈S4.Corollary B.
Forn≥2, let M be an n-dimensional smooth closed manifold and f:M→R a smooth function. One sets maxf(M)=m and assumes that m is attained by unique point p∈M. If p is an isolated critical point and there are no critical points in f-1([a,m))  (where a<m), then the following results hold: (i)
Ifn≠5, there is a diffeomorphism f-1(a)≅Sn-1.(ii)
Ifn=5, there is a homeomorphism f-1(a)≈S4.This paper is organized as follows. In Section2 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. Forr∈R, we set (1)Mr≔f-1m,r,Mr∘≔f-1m,r.Lemma 1.
There is a diffeomorphismMa∘≅Rn.Proof.
By [2, Lemma 3], it will suffice to prove that, for any compact set K⊂Ma∘, there exists an open set V of Ma∘ such that K⊂V and V≅Rn.
We fix an open setU which satisfies that (2)p∈U⊂U-⊂Ma∘,U≅Rn.(Such U indeed exists because p∈Ma∘.) Note that p∉Ma∖U and Ma∖U is compact. Note also that U- is compact. Hence there exist ε0 and ε1∈R (where m<ε0<ε1) such that (3)Mε0⊂U⊂U-⊂Mε1∘⊂Ma∘.We fix such ε0 and ε1.
For a compact setK⊂Ma∘, since maxf(K)<a holds, there exists b such that maxf(K)<b<a. This implies that K⊂Mb∘.
By assumption, there are no critical points inf-1((m,a]). Hence there is a diffeomorphism (4)φ:Mε1,Mε0⟶≅Ma,Mb.We set V:=φ(U). Then V is an open set of Ma∘, K⊂V and V≅Rn.Lemma 2.
The manifoldf-1(a) is homotopy equivalent to Sn-1.Proof.
Sincef-1(a) is a manifold, the assertion is clear for n=2. We assume n≥3. We claim that (5)Hqf-1a;Z≅HqSn-1;Z∀q.To prove (5), note that Ma is a manifold with boundary such that ∂Ma=f-1(a). Lefschetz duality implies that (6)HqMa,∂Ma;Z≅Hn-qMa;Z∀q.Recall that the inclusion i:Ma∘→Ma is a homotopy equivalence. Since Ma∘≅Rn by Lemma 1, Ma has the cohomology of a point. In the homology long exact sequence of the pair (Ma,∂Ma), we apply (6). Then we obtain (5).
Since the fundamental group at infinity ofMa∘ is trivial by Lemma 1, f-1(a) is simply connected. (See, e.g., [3, p. 389] or [4, Examples 5 and 6] for this kind of argument.) Hence f-1(a) is a homotopy sphere.Proof of Theorem A.
Forn=2 or 3, Lemma 2 immediately implies that f-1(a) is diffeomorphic to Sn-1.
Forn≥6, Lemma 2 tells us that Ma is contractible with simply connected boundary. Using the h-cobordism theorem (see [5, p. 108, Proposition A]), we have Ma≅Dn. Hence f-1(a)≅Sn-1.
Forn=5, combining Freedman’s resolution of the 4-dimensional Poincaré conjecture [6] and Lemma 2, we have f-1(a)≈S4.
Forn=4, combining Perelman’s resolution of the 3-dimensional Poincaré conjecture [7], Lemma 2, and the fact [8] that the differential structure on S3 is unique, we have f-1(a)≅S3.Proof of Corollary B.
Iff satisfies the assumption of Corollary B, the function g≔-f satisfies the assumption of f 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 are1,…,1 and l. Let SO(2) act on the n-dimensional torus Tn=(S1)n diagonally. For l>0, we set(7)Xn,l=z1,…,zn∈Tn∣∑i=1n-1zi+lzn=0SO2.Here zi∈S1⊂C denotes the unit vectors in the direction of the sides of a polygon.It is clear thatXn,l=⌀ for l>n-1 and Xn,n-1={onepoint}. It is also known that there is a diffeomorphism (8)Xn,l≅Sn-3forn-3<l<n-1.Recall that (8) can be understood Morse-theoretically. The following arguments are particularly well described in [12]: Using the SO(2)-action in the definition of Xn,l, we normalize z1 to be 1 and write Xn,l as (9)Xn,l=z1,…,zn∈Tn∣z1=1,∑i=1n-1zi+lzn=0.(See Figure 1.)Figure 1
An element ofX5,l.We identifyTn-2=(z1,…,zn-1)∈Tn-1∣z1=1 and define a function fn:Tn-2→R by (10)fnz1,…,zn-1=∑i=1n-1zi2.Then (9) gives an identification Xn,l=fn-1(l2). It is elementary to prove that an element (z1,…,zn-1)∈Tn-2∖fn-1(0) is a critical point of fn if and only if zi∈{-1,1} for all 2≤i≤n-1. All these critical points are nondegenerate and their indices are known. (See, e.g., [9, 12–14].) In particular, fn attains its maximum value at (z1,…,zn-1)=(1,…,1) 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 (11)Yn,l=a1,…,an∈S2n∣∑i=1n-1ai+lan=0SO3.Recently, motivated by chemistry, Crippen [16], Goto and Komatsu [17], and O’Hara [18] studied certain subspaces of Yn,1. Namely, they studied the configuration space ofequilateral polygons with restriction on the splay angle of each vertex.First, we define the angleαn to be (7/12)π and arccos(-1/3) as n=5 and n≥6, respectively. Goto and Komatsu [17] chose the angle αn with molecular model in mind. Then they studied the space defined by (12)Zn=a1,…,an∈Yn,1∣-ai,ai+1=cos⁡αnfor1≤i≤n-2,where , denotes the standard inner product on R3. The closed chains in Zn are equilateral polygons in R3 with n vertices such that the interior angles are all equal to αn except for the two angles at the successive vertices O and ∑i=1n-1‍ai.The main result in [17] states that when n=5,6,7, Zn is a manifold homeomorphic to Sn-4. Since they use Reeb’s theorem, they state their result as a homeomorphism. But actually Zn and Sn-4 are diffeomorphic because the differential structure on Sk is unique for k≤3. But what is more important is that it is not known whether Zn is a manifold for n≥8.Second, for allβ∈R, we define a space by (13)Wnβ=a1,…,an∈Yn,1∣-ai,ai+1=cos⁡βfor1≤i≤n.Here we understand an+1 to be a1. The closed chains in Wn(β) are equilateral polygons in R3 with n vertices such that the interior angles are all equal to β. Crippen [16] studied the topological type of Wn(β) for various β when n=3,4,5. The result is that Wn(β) is either ⌀, one point, or two points depending on β. Next, O’Hara [18] studied W6(β) for various β. The result is that W6(β) is disjoint union of a certain number of S1’s and points. It is to be noted that since dimYn,1=2n-6, it is natural to expect that dimWn(β)=2n-6-n=n-6. But the above results imply that the defining equations for Wn(β) do not intersect transversally when n is small.Note that the above results in [16–18] concentrate on the case for small n. 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 n. Modifying the definition of Zn, we define a space as follows: (14)An,l=a1,…,an∈Yn,l∣ai,ai+1=0for1≤i≤n-2.The closed chains in An,l are polygons in R3 with edge lengths 1,…,1︸n-1 and l such that the interior angles are all equal to π/2 except for the two angles at the endpoints of the edge of length l.Let us obtain a similar description to (9). We set e1=(1,0,0) and e2=(0,1,0). By the SO(3)-action in the definition of Yn,l, we can normalize a1 and a2 to be e1 and e2, respectively. Thus we have the following description of An,l: (15)An,l=a1,…,an∈S2n∣a1=e1,a2=e2,∑i=1n-1ai+lan=0,ai,ai+1=02≤i≤n-2.Hereafter we use description (15). (See Figure 2.)Figure 2
An element ofA5,l.We recall the following.Theorem 3 (see [19, Theorem 5.6], [20, Theorem 4]).
One sets(16)rn≔2mn=2m+1,2m2-2m+1n=2m.Then the following results hold: (i)
An,l=⌀ if l>rn.(ii)
An,rn={onepoint}. We write the element of An,rn by Pn=a1,…,an in the notation of (15). Then Pn is given as follows: For 1≤i≤n-1, ai is given by(17)ai=e1iisodd,e2iiseven.As a consequence,an is given by (18)an=-mrne1+e2n=2m+1,-1rnme1+m-1e2n=2m.Note thatPn is a planar n-gon. (See Figure 3.)Figure 3
P
5 and P6.Now we state the main result in this section (cf. (8)).Theorem 4.
Letl be slightly smaller than rn. Then, for all n≥5, there is a diffeomorphism An,l≅Sn-4.For the rest of this paper, we prove Theorem4. For that purpose, we define (19)Bn=a1,…,an-1∈S2n-1∣a1=e1,a2=e2,ai,ai+1=02≤i≤n-2.Moreover, similarly to fn in (10), we define a map gn:Bn→R by(20)gna1,…,an-1=∑i=1n-1ai2.Then (15) gives an identification An,l=gn-1(l2) and Theorem 3 tells us that maxgn(Bn) is attained by Pn. In order to compute the Hessian matrix of gn at Pn, we construct the commutative diagram shown in Figure 4.Figure 4First, we set(21)πnθ1,…,θn-3=eiθ1,…,eiθn-3.Namely, πn is the universal cover.Second, we constructψn by induction on n. We define ψ3 to be the unique map between one-point spaces.Assuming thatψn is constructed, we construct ψn+1. We write (22)ψnθ1,…,θn-3=a1,…,an-1.Since an-2,an-1=0, there are two choices for ξ∈S2 which satisfies that ξ,an-2=ξ,an-1=0. Among these ξ’s, we choose the one which satisfies the condition that det⁡(an-2,an-1,ξ)=(-1)n+1. Using this, we define(23)η≔cos⁡θn-2an-2+sin⁡θn-2ξ.And we set (24)ψn+1θ1,…,θn-3,θn-2≔a1,…,an-1,η.From the construction,ψn induces a map φn:Tn-3→Bn such that ψn=φn∘πn. It is easy to see that φn is a diffeomorphism. (See Remark 7(ii).)Third, we sethn≔gn∘ψn, where gn is defined in (20). Thus we have completed the construction of the diagram in Figure 4.Note thatψn(0,…,0)=Pn, where Pn is defined in Theorem 3.Lemma 5.
LetH(hn)(0,…,0) be the Hessian matrix of hn at (θ1,…,θn-3)=(0,…,0). Note that this is an (n-3)×(n-3) matrix. Let γ(n,i,j) be the (i,j)th entry of H(hn)(0,…,0). Then, for all n≥5, the following result holds: (25)γn,i,j=-2n-1-i2j+12ifi≥j,-2n-1-j2i+12ifi<j.Proof.
The lemma is proved by direct computations.Proof of Theorem4 for Even n.
We set(26)Γn,i≔γn,i,i+1-γn,i,1γn,n-3,1γn,n-3,i+1.It is easy to see that when n is even, (27)det⁡Hhn0,…,0=-2∏i=1n-4Γn,i.Thus we have det⁡(H(hn)(0,…,0))≠0; hence (0,…,0) is a nondegenerate critical point of hn. Using the Morse lemma, we complete the proof of Theorem 4 for even n.Proof of Theorem4 for Odd
n. Lemma 5 tells us that when n is odd, det⁡(H(hn)(0,…,0))=0. Hence we need to use Corollary B. For that purpose, it will suffice to prove that (0,…,0) is an isolated critical point of hn.We define a mapFn:Sn-4×[0,∞)→Rn-3 by (28)Fnu1,…,un-3,t=grad⁡hntu1,…,tun-3,where the right-hand side denotes the Jacobian matrix of hn at (29)θ1,…,θn-3=tu1,…,tun-3.First we prove the following.Lemma 6.
For each(u1,…,un-3)∈Sn-4, there exists ε>0 such that Fn(u1,…,un-3,t)≠0 for all t∈(0,ε).Proof.
We fix(u1,…,un-3)∈Sn-4. In order to prove the lemma by contradiction, assume that t=0 were an accumulation point of Fn(u1,…,un-3,t). For 1≤k≤n-3, let Fn,k(u1,…,un-3,t) be the kth element of Fn(u1,…,un-3,t). Since Fn,k(u1,…,un-3,t) is a polynomial in sin⁡tui,cos⁡tui∣1≤i≤n-3, we can define Fn,k(u1,…,un-3,t) for t∈C and this is a holomorphic function. If t=0 were an accumulation point of Fn(u1,…,un-3,t), then the identity theorem would tell us that Fn,k(u1,…,un-3,t) is identically 0 for all t∈C and for all 1≤k≤n-3.
We write the Maclaurin expansion ofFn,k(u1,…,un-3,t) as (30)Fn,ku1,…,un-3,t=σku1,…,un-3t+τku1,…,un-3t3+Higher  Order  Terms  int,where σk(u1,…,un-3) and τk(u1,…,un-3) are polynomials in u1,…,un-3 of degrees 1 and 3, respectively. Since Fn,k(u1,…,un-3,t)≡0, we have(31)σku1,…,un-3=τku1,…,un-3=0∀1≤k≤n-3.Then standard computations show that u1=⋯=un-3=0. This contradicts the assumption that (u1,…,un-3)∈Sn-4.
This completes the proof of Lemma6.Now using Lemma6 and the continuity of the function Fn, there exists an open neighborhood U of O in Rn-3 such that if (a1,…,an-3)∈U∖O, then (32)grad⁡hna1,…,an-3≠0,…,0.Thus (0,…,0) is an isolated critical point of hn.This completes the proof of Theorem4 for odd n.Remark 7.
(i) Recall that Corollary B fordimM=5 just gives a topological assertion. In order to prove Theorem 4, we have studied the map gn:Bn→R. Since dimBn=n-3, dimBn=5 implies that n=8. But we have seen in Lemma 5 that when n is even, Pn 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 diffeomorphismφn in the above arguments, it is to be noted that the map gn∘φn:Tn-3→R is much more difficult than the known map fn:Tn-2→R in (10).



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