Abstract

A Morse function \(f\) on a connected closed manifold \(M\) is called a self-indexing function if the index of \(f\) at \(p\) is equal to \(f(p)\) for any critical point \(p\) of \(f\). We consider the following question: Does there exist an embedding $$\varepsilon_n: (S^1)^n \to \R^{n+1}$$

such that \(p_1\circ\varepsilon_n\) is a self-indexing Morse function? Here \(p_1\:\R^{n+1}\to \R$ is the first projection.

In this paper, we give an affirmative answer to the question for the case $n=1$ or $2$. For that purpose, we construct embeddings $\varepsilon_n$ explicitly. On the other hand, for $n\geq 3$, we prove a weaker version of the question. That is, we construct an embedding $$\delta_n:(S^1)^n \to \R^{n+2}$$

such that $p_1\circ \delta_n$ is a self-indexing Morse function. Here $p_1:\R^{n+2}\to \R$ is the first projection.

Introduction

Self-indexing Morse function

A Morse function \(f\) on a connected closed manifold \(M\) is called a self-indexing function if the index of \(f\) at \(p\) is equal to \(f(p)\) for any critical point \(p\) of \(f\). The existence of a self-indexing function was proved by Smale [5] and successfully used by him in solving the Poincaré conjecture. (See also [2] and [3].)

Height function

Let \(f\) be a Morse function on a connected closed manifold \(M\). Assume that there is an embedding \(\mu: M \to {\mathbb R}^d\) such that \(f=p_1\circ \mu\), where \(p_1: {\mathbb R}^d\to {\mathbb R}\) denotes the first projection. In this case, we call \(f\) a height function.

A question

In [4], it is asked whether \((S^1)^2\) admits a self-indexing Morse function which is also a height function. More precisely, does there exist an embedding \[\varepsilon_2: (S^1)^2 \to {\mathbb R}^3\] such that \(p_1\circ \varepsilon_2\) is a self-indexing Morse function?

In [4], an answer is given by showing an embedding by a picture. It is desirable to provide an explicit formula for the embedding.

Generalization and summary of the result

Generalizing the question in §1.3, we pose the following:

Question 1. Does there exist an embedding \[\varepsilon_n: (S^1)^n \to {\mathbb R}^{n+1}\] such that \(p_1 \circ \varepsilon_n\) is a self-indexing Morse function? Here \(p_1:{\mathbb R}^{n+1}\to {\mathbb R}\) is the first projection.

The purpose of this paper is to study Question 1. We summarize the main results in §2 as follows:

Theorem 2. (i)For \(n\geq 1\), we define \[f_n: (S^1)^n \to {\mathbb R}\] by \[\label{jul28a} f_n(e^{i\theta_1},\dots,e^{i \theta_n})= \sum_{i=1}^n \log_3(2+\cos \theta_i)--------------------(1).\] Then \(f_n\) is a self-indexing Morse function.

(ii) We define \(\varepsilon_2: (S^1)^2 \to {\mathbb R}^3\) by \[\varepsilon_2 (e^{i \theta_1},e^{i \theta_2}):= (f_2(e^{i \theta_1},e^{i \theta_2}), (2+\sin \theta_1)\cos \theta_2,(2+\sin \theta_1)\sin \theta_2).\] Then \(\varepsilon_2\) is an embedding. Thus we have obtained an explicit answer to the question in §1.3.

The organization of this paper

This paper is organized as follows. In §2 we state our main results and in §3 we prove them.

Main results

Theorem A . For \(n\geq 1\), we define \(f_n\) as in (1). Then \(f_n\) is a self-indexing Morse function.

Remark 3. The function \(f_n\) is also a perfect Morse function. That is, the Morse inequalities are in fact equalities.

Theorem B. (i) We define the map \(\varepsilon_1:S^1 \to {\mathbb R}^2\) by \[\varepsilon_1(e^{i\theta_1})=(f_1(e^{i \theta_1}), \sin \theta_1).\] Then \(\varepsilon_1\) is an embedding.

(ii) We define the map \(\varepsilon_2:(S^1)^2 \to {\mathbb R}^3\) by \[\varepsilon_2(e^{i\theta_1},e^{i \theta_2})=(f_2(e^{i \theta_1},e^{i \theta_2}), (2+\sin \theta_1)\cos \theta_2,(2+\sin \theta_1)\sin \theta_2).\] Then \(\varepsilon_2\) is an embedding.

Remark 4. Combining Theorems A and B, we obtain an affirmative answer to Question 1 for the case \(n=1\) or \(2\).

We do not know whether Question 1 is true for \(n\geq 3\). But the following weaker version holds:

Theorem C. For \(n\geq 3\), there is an embedding \[\delta_n:(S^1)^n\to {\mathbb R}^{n+2}\] such that \(p_1\circ \delta_n=f_n\) holds. Here \(p_1:{\mathbb R}^{n+2}\to {\mathbb R}\) is the first projection.

Proofs of the main results

Proof of Theorem A. We obtain from (1) that \[\text{grad}\, f_n=-\frac{1}{\log 3}\left(\frac{\sin \theta_1}{2+\cos \theta_1}, \frac{\sin \theta_2}{2+\cos \theta_2},\dots,\frac{\sin \theta_n}{2+\cos \theta_n}\right).\] Hence \((e^{i\theta_1},\dots,e^{i\theta_n})\) is a critical point of \(f_n\) if and only if \(\theta_i \in \{0,\,\pi\}\) for \(1\leq i \leq n\).

To see that \(f_n\) is a Morse function, the critical points are clearly non-degenerate and the index of the critical point \((e^{i\theta_1},\dots,e^{i\theta_n})\) is equal to the number of \(\theta_i\)’s with \(\theta_i=0\). (See, for example, .)

The fact that \(f_n\) is self-indexing is proved as follows: If \((e^{i\theta_1},\dots,e^{i\theta_n})\) is a critical point of \(f_n\) of index \(k\), then we have \[f_n(e^{i\theta_1},\dots,e^{i\theta_n})=k.\hfill \qedhere\]

Proof of Theorem B. Since the proofs of (i) and (ii) are similar, we prove only (ii). We set \[\label{may12e} \varepsilon_2 (e^{i\theta_1},e^{i\theta_2})=(\xi_1,\xi_2,\xi_3).----------------(2)\] Hereafter, to say “\(\theta_i\) is determined uniquely" precisely means that there is at most one solution in \(\theta_i \;(\text{mod}\; 2\pi)\).

We need to prove that if \(\xi_1,\xi_2\) and \(\xi_3\) are given in (2), then \(\theta_1\) and \(\theta_2\) are determined uniquely. From the second and third components of \(\varepsilon_2\), we have \[\begin{aligned} &\sin \theta_1=\sqrt{\xi_2^2+\xi_3^2}-2-------------------------(3) \label{may12a}\\ {and} &\begin{dcases} \cos \theta_2=\frac{\xi_2}{\sqrt{\xi_2^2+\xi_3^2}}\\ \sin \theta_2=\frac{\xi_3}{\sqrt{\xi_2^2+\xi_3^2}}---------------------------(4)\label{may12b} \end{dcases} \end{aligned}\]

Firstly, from (4), \(\theta_2\) is determined uniquely. Secondly, from \(f_2\) in (1), we have \[\label{may12d} \large{ \cos \theta_1=\frac{3^{\xi_1}}{2+\cos \theta_2}-2.}-----------------------(5)\] Now combining (3) and (5), \(\theta_1\) is determined uniquely.

Proof of Theorem C. We first prove the following:

Lemma 5. For \(n\geq 1\), there is an embedding \[\label{sep2e} \lambda_n: (S^1)^n \to {\mathbb R}^{n+1}.-------------------------(6)\]

Proof. We prove the lemma by induction on \(n\).

Base case: We define \(\lambda_1\) to be the inclusion \(i: S^1 \to {\mathbb R}^2\).

Induction step: We show the implication ‘‘ (6) for \(n=k\)” \(\Rightarrow\) ‘‘ (6) for \(n=k+1\)”. Assume that there is an embedding \[\label{sep2d} \lambda_{k}: (S^1)^{k} \to {\mathbb R}^{k+1}.-----------------------(7)\] We define the map \[\label{sep2j} \phi: S^1 \times {\mathbb R}\to {\mathbb R}^2---------------------(8)\]by \[\phi (e^{i\theta},t) := e^t e^{i\theta}.\] Then it is clear that \(\phi\) is an embedding.

Now we define \(\lambda_{k+1}\) to be the composition of the following inclusions: \[(S^1)^{k+1} =S^1 \times (S^1)^{k}\subset S^1 \times {\mathbb R}^{k+1} =(S^1\times {\mathbb R})\times {\mathbb R}^{k} \subset {\mathbb R}^2\times {\mathbb R}^{k}={\mathbb R}^{k+2}.\] Here the left inclusion is a consequence of (7). On the other hand, the right inclusion is a consequence of (8).

Now we complete the proof of Theorem C. Using Lemma 5, we define \(\delta_n\) by \[\delta_n(e^{i\theta_1}, \dots, e^{i \theta_n}) =(f_n(e^{i\theta_1}, \dots, e^{i \theta_n}),\; \lambda_n (e^{i\theta_1}, \dots, e^{i \theta_n})).\] Then it is clear that \(\delta_n\) is an embedding such that \(p_1\circ \delta_n=f_n\).

Conclusions

We have obtained an affirmative answer to Question 1 for the case \(n=1\) or \(2\). On the other hand, for \(n\geq 3\), we obtained a weaker version of Question 1. That is, we constructed an embedding \(\delta_n: (S^1)^n \to {\mathbb R}^{n+2}\) such that \(p_1 \circ \delta_n\) is a self-indexing Morse function. To study whether Question 1 itself holds for \(n\geq 3\) is our further task.

Acknowledgments

The author would like to thank the reviewer for carefully reading the manuscript and for valuable comments to improve it.

Conflicts of Interest:

The author declares no competing interests.

References

1. Matsumoto, Y. (2002). An Introduction to Morse Theory, Translations of Mathematical Monographs, American Mathematical Society, 208, Providence, RI. View

2. Milnor, J. (1965). Lectures on the h-cobordism Theorem, Princeton Univ. Press, Princeton, NJ. View

3. Nicolaescu, L. (2011). An invitation to Morse Theory, 2nd ed., Universitext, Springer, New York. View

4. PVAL-inactive (2016). Self-indexing Morse function on a torus which is a height function, Math-Stack-Exchange. View

5. Smale, S. (1961). Generalized Poincar´e conjecture in dimensions greater than four, Ann. of Math. 74 , 391–406. View