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Cancellations of periodic orbits for non-singular Morse–Smale flows

Published online by Cambridge University Press:  25 May 2023

D. V. S. LIMA
Affiliation:
Federal University of the ABC, Center of Mathematics Computing and Cognition, Santo Andre, Brazil (e-mail: dahisy.lima@ufabc.edu.br)
K. A. DE REZENDE
Affiliation:
University of Campinas, Institute of Mathematics, Statistics and Scientific Computing, Campinas, Brazil e-mail: (ketty@ime.unicamp.br)
M. R. DA SILVEIRA*
Affiliation:
Federal University of the ABC, Center of Mathematics Computing and Cognition, Santo Andre, Brazil (e-mail: dahisy.lima@ufabc.edu.br)
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Abstract

In this work, we explore the dynamical implications of a spectral sequence analysis of a filtered chain complex associated to a non-singular Morse–Smale (NMS) flow $\varphi $ on a closed orientable $3$-manifold $M^3$ with no heteroclinic trajectories connecting saddle periodic orbits. We introduce the novel concepts of cancellations and reductions of pairs of periodic orbits based on Franks’ morsification and Smale’s cancellation theorems. The main goal is to establish an algebraic-dynamical correspondence between the unfolding of this spectral sequence associated to $\varphi $ and a family of flows obtained by cancelling and reducing pairs of periodic orbits of $\varphi $ on $M^3$. This correspondence is achieved through a spectral sequence sweeping algorithm (SSSA), which determines the order in which these cancellations and reductions of periodic orbits occur, producing a family of NMS flows that reaches a core flow when the spectral sequence converges.

Type
Original Article
Copyright
© The Author(s), 2023. Published by Cambridge University Press

1 Introduction

In this article, we adopt a homotopical approach in the investigation of non-singular Morse–Smale (NMS) flows on closed 3-manifolds by considering the unfolding of a spectral sequence of an NMS chain complex associated to it. Similar questions have been considered within other classes of dynamical systems, such as Morse–Smale flows with no periodic orbits [Reference Bertolim, Lima, Mello, de Rezende and da Silveira3, Reference Bertolim, Lima, Mello, de Rezende and da Silveira4, Reference Lima, Mazoli Neto, de Rezende and da Silveira15], Morse–Bott flows [Reference Lima and de Rezende14], gradient flows associated to circle-valued Morse functions [Reference Lima, Mazoli Neto, de Rezende and da Silveira15] and Gutierrez–Sotomayor flows [Reference Lima, Raminelli and de Rezende16].

Based on results of Franks established in [Reference Franks9], we prove algebraic-dynamical correspondence theorems between the cancellations and reductions of periodic orbits in an NMS flow $\varphi $ and the differentials of a spectral sequence of a filtered NMS chain complex associated to $\varphi $ .

Let $\gamma _k$ and $\gamma _{k-1}$ be two periodic orbits of an NMS flow $\varphi $ , of indices k and $k-1$ , respectively. The orbits $\gamma _k$ and $\gamma _{k-1}$ can be cancelled if there exists an NMS flow ${\varphi }'$ in M that coincides with $\varphi $ outside a neighborhood U of $W^u(\gamma _k) \cap W^s(\gamma _{k-1})$ and ${\varphi }'$ has empty chain recurrent set in U. The orbits $\gamma _k$ and $\gamma _{k-1}$ can be reduced if there exists an NMS flow ${\varphi }'$ on M that coincides with $\varphi $ outside a neighborhood U of $W^u(\gamma _k) \cap W^s(\gamma _{k-1})$ and the chain recurrent set of ${\varphi }'$ restricted to U is a unique periodic orbit $\gamma '$ . More specifically, if $k=2$ , $\gamma '$ is a repelling periodic orbit and if $k=1$ , $\gamma '$ is an attracting periodic orbit. Local cancellations and reductions are described in more detail in §3.2. See Figure 1.

Figure 1 Flow cancellation and reduction in $S^3$ .

Let $ \Sigma (M^3)$ be the set of NMS flows on an orientable closed $3$ -manifold $M^3$ without heteroclinic trajectories between saddle periodic orbits. It follows that flows in $\Sigma (M^3)$ that undergo cancellations and reductions remain in $\Sigma (M^3)$ .

We say that $\varphi \in \Sigma (M^3)$ is a core flow if each inessential connection between two consecutive periodic orbits is part of a double inessential connection. Equivalently, $\varphi \in \Sigma (M^3)$ is a core flow if pairs of consecutive periodic orbits in $\varphi $ cannot be cancelled or reduced. See §3 for more details.

Given $\varphi \in \Sigma (M^3)$ , one can associate it to a chain complex $(C^{\mathrm {NMS}},\Delta ^{\mathrm {NMS}})$ called an NMS chain complex, which will be defined in §3. In this work, we consider flows in $\Sigma (M^3)$ whose differentials $\Delta ^{\mathrm {NMS}}$ are totally unimodular (TU) matrices, which is an essential technical hypothesis for obtaining cancellation results. Also, as a consequence of this hypothesis, $M^3$ is torsion free. See §3. Let

$$ \begin{align*} { \mathcal{S}}= \{ \phi\in \Sigma(M^3) \,|\, \Delta^{\mathrm{NMS}} \mbox{ is a TU matrix}\}. \end{align*} $$

One defines in $\Sigma (M^3)$ a partial order relation $<$ as follows: given $\varphi ,\phi \in \Sigma (M^3)$ , $\varphi <\phi $ if $\phi $ is obtained from $\varphi $ by a cancellation or a reduction of a pair of periodic orbits and is extended by transitivity . Every totally ordered subset of $\mathcal {S}$ has a maximal element, which is a core flow in $\Sigma (M^3)$ . The set $\mathcal {S}_{\varphi }=\{\phi \in \mathcal {S} \mid \varphi < \phi \}$ is the set of flows in $\Sigma (M^3)$ that can be obtained from $\varphi $ by performing cancellations and reductions of periodic orbits.

Given $\varphi \in \mathcal {S}$ , the goal of this paper is to construct a family of NMS flows in $\mathcal {S}_{\varphi }$ that takes $\varphi $ to a core flow through an algebraic procedure based on a spectral sequence analysis of the chain complex $(C^{\mathrm {NMS}},\Delta ^{\mathrm {NMS}})$ that is realized by using a spectral sequence algorithm. Spectral sequence algorithms are always filtration sensitive. Thus, the results obtained in §5 that use a spectral sequence algorithm will explore different types of filtrations as well as subclasses of $\mathcal {S}$ . Moreover, by requiring certain properties on the filtration and possibly on $M^3$ , the core flow $\varphi ^{\omega }$ reached by the algorithm minimizes the number of periodic orbits in $\mathcal {S}_{\varphi }$ .

In §2, we present background material on NMS flows and an outline of the spectral sequence sweeping algorithm (SSSA).

In §3.1, we characterize topological-dynamical properties of the connections between periodic orbits for flows in $\Sigma (M^3)$ . These properties are essential for defining an NMS chain complex through a morsification process. In §3.2, this characterization is used to establish Theorems 3.11 and 3.12, which determine the conditions under which local cancellations and reductions of periodic orbits occur.

In §4.1, we provide a proof that the SSSA applied to $\Delta ^{\tiny {NMS}}$ determines the modules $\mathcal {E}^r_p$ and induces the differentials $d^r_p$ of the spectral sequence. This will be crucial in demonstrating the correspondence of the algebraic behavior of the modules and of the differentials in the spectral sequence with the dynamical cancellations and reductions of the periodic orbits of the flow. In §4.2, we prove Theorems 4.12 and 4.13 which characterize properties of the differentials of the spectral sequence under the hypothesis that the filtration is NMS-compatible.

In §5, several algebraic-dynamical correspondence theorems are proved under different hypotheses either on the filtration or on the flow. In §5.1, by imposing restrictions on the filtration, we prove Theorems 5.1 and 5.2 which ensure that the unfolding of the spectral sequence provides a corresponding list of cancellations and reductions of periodic orbits that lead to a core flow. In §5.2, we consider flows in $\Sigma (M^3)$ that do not have pairs of cancelling periodic orbits. By requiring that the filtration satisfies the reduction ordering condition, Theorem 5.7 guarantees that the algebraic-dynamical correspondence obtained in Theorem 5.1 provides a corresponding list of reductions leading to a core flow that has the lowest number of attracting, saddle and repelling orbits in $\mathcal {S}_{\varphi }$ . By removing the reduction ordering condition restriction, Theorem 5.8 still arrives at a core flow that does not need to have the minimal number of periodic orbits. In §5.3, we prove Theorem 5.15, which asserts that, given any flow $\varphi \in \Sigma (S^3)$ whose differential is a TU matrix, all algebraic-dynamical correspondence theorems will lead to a core flow in $S_{\varphi }$ with exactly one repelling and one attracting periodic orbit.

Finally, in §6, we investigate NMS flows with heteroclinic connections between saddle periodic orbits. In Theorem 6.1, we prove a local reduction result between two saddle orbits.

2 Background

NMS flows have been studied by several authors who at first sought to answer the question of determining which manifolds admit these flows. Asimov proved in [Reference Asimov1] that, for a closed manifold $M^n$ , $n\neq 3$ , a necessary and sufficient condition is $\chi (M)=0$ . This is not true in dimension three. Morgan proved in [Reference Morgan19] that an orientable closed prime 3-manifold admits an NMS flow if and only if it is a graph manifold. Moreover, as a result of Asimov’s and Morgan’s work, a manifold admits an NMS flow if and only if it admits a round handle decomposition. Later, Wada [Reference Wada23] characterized which classes of links could be realized as NMS flows on $S^3$ . Based on this work, Campos and Vindel [Reference Campos and Vindel5] found topological conditions for the existence of NMS flows on $S^3$ without heteroclinic trajectories connecting saddle orbits. A nice survey on the classification of Morse–Smale systems, which includes NMS flows on 3-manifolds, can be found in [Reference Grines, Gurevich, Pochinka and Zhuzhoma12]. In [Reference Pochinka and Shubin20], Pochinka and Shubin characterize, up to topological equivalence, NMS flows on n-manifolds which admit exactly two periodic orbits. The aforementioned authors are mainly interested in the realization of NMS flows on closed manifolds.

Franks [Reference Franks8Reference Franks11] adds a new perspective to this investigation by applying homotopical tools to the study of the structure of Morse–Smale flows on closed manifolds M by considering an associated CW-complex as well as the distinct attaching maps between cells and the connecting manifolds. Our work is inspired by this approach and will further bridge the gap between the topological-algebraic information of an NMS chain complex and the dynamics of an NMS flow.

In this section, we present some background material on NMS flows and provide a review of results on spectral sequences and on the sweeping algorithm introduced in [Reference Cornea, de Rezende and da Silveira6], which is used to recover the spectral sequence associated to a filtered chain complex.

2.1 NMS flows

A an NMS flow on an orientable smooth 3-manifold $M^3$ is a smooth flow with a chain recurrent set consisting of a finite number of hyperbolic periodic orbits that satisfies the transversality condition.

The index of a periodic orbit $\gamma $ is defined as the fiber dimension of the unstable bundle $E^u$ over $\gamma $ . We say that a periodic orbit $\gamma $ is untwisted if its unstable manifold $W^u(\gamma )$ is orientable, that is, $W^u(\gamma )$ is diffeomorphic to $S^1 \times \mathbb {R}^k$ , where k is the index of $\gamma $ . The periodic orbit $\gamma $ of index k is twisted if its unstable manifold is non-orientable, that is, $W^u(\gamma )$ is diffeomorphic to $\mathcal {M}\times \mathbb {R}^{k-1}$ , where $\mathcal {M}$ is an open Möebius strip.

In this work, we assume that the flow $\varphi $ has no twisted periodic orbits. Indeed, the results herein have the assumption that the differential of the chain complex associated to the flow is represented by a TU matrix, which is not the case when the flow has a twisted periodic orbit. This fact follows from results in [Reference Franks11, Reference Lima and de Rezende14].

An untwisted periodic orbit $\gamma $ of index k is in standard form if there is a tubular neighborhood V of $\gamma $ with coordinates $(\theta ,x,y) \in S^1\times \mathbb {R}^{k}\times \mathbb {R}^{n-k-1}$ such that the vector field has the form

$$ \begin{align*} X= \dfrac{\partial}{\partial \theta} + \sum_{i=1}^k x_i\dfrac{\partial}{\partial x_i} - \sum_{j=1}^{n-k-1} y_j\dfrac{\partial}{\partial y_j}. \end{align*} $$

In [Reference Franks9], Franks shows that any Morse–Smale flow is topologically conjugate to a Morse–Smale flow whose periodic orbits are in standard form. See Figure 2.

Figure 2 Standard form of untwisted periodic orbits in dimension three.

The next result shows that a periodic orbit is, in a certain sense, interchangeable with a pair of rest points. See Figure 3.

Figure 3 Interchanging a periodic orbit with a pair of rest points.

Theorem 2.1. (Asimov [Reference Asimov1], Franks [Reference Franks10])

Suppose $\varphi $ is a Morse–Smale flow with rest points p and q of index $k+1$ and k, respectively, such that $W^u(p) \cap W^s(q)$ consists of two trajectories. Then $\varphi $ can be changed on any given neighborhood of the two trajectories from p to q in such a way as to eliminate the two critical points and replace them with:

  1. (a) an untwisted periodic orbit of index k if $W^u(p) \cap W^s(q)$ consists of two trajectories with opposite signs; or

  2. (b) a twisted periodic orbit if $W^u(p) \cap W^s(q)$ consists of two trajectories with the same sign.

Franks’ next result shows the reverse construction, that is, the replacement of a periodic orbit of index k by two rest points of indices k and $k+1$ . See Figure 3.

Theorem 2.2. (Franks [Reference Franks9])

Suppose $\varphi $ is a Morse–Smale flow on an orientable manifold M with a periodic orbit $\gamma $ of index k in standard form. Then, given a neighborhood U of $\gamma $ , there exists a Morse–Smale flow $\widetilde {\varphi }$ whose vector field agrees with that of $\varphi $ outside U and that has rest points p, q of indices $k+1$ and k in U but no other chain recurrent points in U. For $\widetilde {\varphi }$ , $W^u(p) \cap W^s(q)$ will consist of two trajectories with the same sign if $\gamma $ is twisted and opposite signs if $\gamma $ is untwisted. Moreover, the unstable manifold for $\gamma $ will be equal to $W^u(p)\cup W^u(q)$ .

2.2 Spectral sequences and the sweeping algorithm for Morse complexes

In this subsection, we introduce a method that allows us to extract the modules and differentials of a spectral sequence of a filtered chain complex from its differential. This computation makes use of the sweeping algorithm that was introduced in [Reference Cornea, de Rezende and da Silveira6] and which will be described in this subsection.

In order to state a major theorem on spectral sequences, we make use of the following definitions.

Definition 2.3. Let $(C,\partial )$ be a chain complex and let $F=\{F_{p}C\}$ be a filtration in $(C,\partial )$ .

  1. (1) F is said to be an increasing filtration if there is a sequence of submodules $F_pC$ , for $p \in \mathbb {Z}$ , such that $F_pC\subset F_{p+1}C$ . Moreover, the filtration must be compatible with the gradation of C, that is, $F_pC$ is graded by $\{F_pC_q\}$ .

  2. (2) F is said to be convergent if $\bigcap _p F_pC=0$ and $\cup F_{p}C=C$ .

  3. (3) F is finite if $F_pC=0$ for some p and $F_{p^{\prime }}C=C$ for some $p^{\prime }$ .

  4. (4) F is bounded below if, given q, there exists $p(q)$ such that $F_{p(q)}C_q=0$ .

The following theorem associates a spectral sequence to a filtered chain complex and can be found in [Reference Spanier22].

Theorem 2.4. Given a filtration F for the chain complex $(C,\partial )$ that is convergent and bounded below, there exists a convergent spectral sequence with

$$ \begin{align*} E^{0}_{p,q}=F_pC_{p+q}/F_{p-1}C_{p+q}=G(C)_{p,q}, \end{align*} $$
$$ \begin{align*} E^1_{p,q}\thickapprox H_{(p+q)}(F_pC_{p+q}/F_{p-1}C_{p+q}) \end{align*} $$

and $E^{\infty }$ is isomorphic to the module $GH_*(C)$ . The algebraic formulas for the modules are

$$ \begin{align*} E^{r}_{p,q}=Z^{r}_{p,q}/(Z^{r-1}_{p-1,q+1}+\partial Z^{r-1}_{p+r-1,q-r+2}), \end{align*} $$

where

$$ \begin{align*} Z^r_{p,q}=\{c\in F_pC_{p+q}\,|\, \partial c\in F_{p-r}C_{p+q-1}\}. \end{align*} $$

In [Reference Bertolim, Lima, Mello, de Rezende and da Silveira3, Reference Bertolim, Lima, Mello, de Rezende and da Silveira4, Reference Cornea, de Rezende and da Silveira6], a spectral sequence analysis for a Morse chain complex $(C_{\ast }(f),\partial _{\ast })$ with a TU matrix $\Delta $ associated to $ \partial _{\ast }$ and a specific filtration induced by the flow $\varphi _{f}$ was developed. The Morse chain complex describes the dynamics of a flow associated to a Morse–Smale function with no periodic orbits. Recall that $C(f)= \{C_{k}(f)\}$ is the $\mathbb {Z}$ -module generated by the critical points of f and graded by their Morse index: that is,

$$ \begin{align*} C_{k}(f) := \bigoplus_{x \in \mathrm{Crit}_k(f)} \mathbb{Z} \langle x\rangle. \end{align*} $$

The Morse boundary operator $\partial _{k}(x) : \mathcal {C}_{k}(f) \longrightarrow \mathcal {C}_{k-1}(f)$ is given on a generator x of $\mathcal {C}_{k}(f)$ by

(1) $$ \begin{align} \partial_k\langle x\rangle := \displaystyle\sum_{y \in \mathrm{Crit}_{k-1}(f)} n(x,y) \langle y\rangle , \end{align} $$

and it is extended to general chains by linearity, where $n(x,y)$ is the intersection number of x and y. The intersection number is defined for a pair $x,y \in \mathrm {Crit}(f)$ such that $ind_{f} (x) - ind_{f} (y) = 1$ , as follows: a choice of orientations for the unstable manifolds $W^{u}(x)$ and $W^{u}(y)$ induces orientations on the connecting manifold ${\mathcal M}_ {xy}$ through the isomorphism

(2) $$ \begin{align} T_{{\mathcal M}_{xy}}W^{u}(x) \simeq T{\mathcal M}_{xy} \oplus {\mathcal V}_{{\mathcal M}_{xy}}W^{s}(y), \end{align} $$

where ${\mathcal V}_{{\mathcal M}_ {xy}}W^{s}(y)$ is the normal bundle of $W^{s}(y)$ restricted to ${\mathcal M}_{xy}$ . The orientation on ${\mathcal V}_{{\mathcal M}_{xy}}W^{s}(y)$ is determined by the orientation on a fiber ${\mathcal V}_{y}W^{s}(y)$ , which is given by the isomorphism

$$ \begin{align*}T_{y}W^{u}(y) \oplus T_{y}W^{s}(y) \simeq T_{y}M \simeq {\mathcal V}_{y}W^{s}(y) \oplus T_{y}W^{s}(y). \end{align*} $$

The moduli space between x and y is defined by ${{\mathcal M}}^{x}_{y}={\mathcal M}_{xy}\cap f^{-1}(a)$ , where a is a regular value of f such that $f(y)<a<f(x)$ . Given $u\in {{\mathcal M}}^{x}_{y}$ , the characteristic sign $n_{u} $ of the orbit ${\mathcal O}(u)$ through u is defined by $[{\mathcal O}(u)]_{\mathrm {ind}} = n_u [\dot {u}]$ , where $[\dot {u} ]$ and $[{\mathcal O}(u)]_{\mathrm {ind}}$ denote the orientations on $\mathcal {O}(u)$ induced by the flow and by $\mathcal {M}_{xy}$ . The intersection number of x and y is defined by

$$ \begin{align*} n(x,y) = \displaystyle\sum_{u \in {{\mathcal M}}^{x}_{y}} n_u. \end{align*} $$

The pair $(C_{\ast }(f),\partial _{\ast })$ is a chain complex, known as the Morse chain complex, and its homology coincides with the singular homology of the surface M. For more details see [Reference Weber24].

The filtration on $(C_{\ast }(f),\partial _{\ast })$ considered in [Reference Bertolim, Lima, Mello, de Rezende and da Silveira3, Reference Bertolim, Lima, Mello, de Rezende and da Silveira4, Reference Cornea, de Rezende and da Silveira6] is induced by the flow $\varphi _{f}$ . More specifically, given a finest Morse decomposition $\{ M(p) \mid p\in P=\{1,\ldots , m \}, \ m = \# \mathrm {Crit}(f) \}$ such that there are distinct critical values $c_p$ with $f^{-1}(c_p)\supset M(p)$ , we define a finest filtration on M by

$$ \begin{align*} \{F_{p-1}\}_{p=1}^{m}=\{f^{-1}(-\infty ,c_p+\epsilon)\}_{p=1}^{m}, \end{align*} $$

since, for each $p\in P$ , there is only one singularity in $F_p\setminus F_{p-1}$ .

Whenever F is a finest filtration and the singularity in $F_p\setminus F_{p-1}$ has index k, then the only q such that $E^r_{p,q}$ is non-zero is $q=k-p$ . Hence, in this case, we omit reference to q, that is, $E^r_p$ is, in fact, $E^r_{p,k-p}$ .

Note that $E^{\infty }$ does not determine $H_*(C)$ completely, but

$$ \begin{align*} E^{\infty}_{p,q}\approx GH_*(C)_{p,q}=\frac{F_pH_{p+q}(C)}{F_{p-1}H_{p+q}(C)}. \end{align*} $$

However, it is a well-known fact that whenever $GH_*(C)_{p,q}$ is free and the filtration is bounded,

(3) $$ \begin{align} \displaystyle\bigoplus_{p+q=k}GH_*(C)_{p,q}\approx H_{p+q}(C). \end{align} $$

See [Reference Davis and Kirk7].

In [Reference Cornea, de Rezende and da Silveira6], where the spectral sequence associated to a Morse complex was considered in the setting of an n-dimensional manifold, for $n\geq 1$ , and computed over $\mathbb {Z}$ , (3) is not necessarily true. However, a characterization theorem for primary pivots for TU differentials in dimension n, which states that the primary pivots are $\pm 1$ (Theorem 2.1) is proved in [Reference Bertolim, Lima, Mello, de Rezende and da Silveira4]. As a consequence of this theorem and the fact that the differentials of the spectral sequence are induced by the primary pivots, whenever the differential is TU, $GH_*(C)_{p,q}$ is free for all p and q and thus (3) holds in the setting of a filtered Morse flow with a finest filtration.

2.2.1 Spectral Sequence Sweeping Algorithm—SSSA

The SSSA was introduced in [Reference Cornea, de Rezende and da Silveira6] for a chain complex with a finest filtration. It constructs recursively a family of matrices $\{\Delta ^r\}$ for $r\geq 0$ , where $\Delta ^0=\Delta $ , by considering at each stage the rth diagonal. It is shown in [Reference Cornea, de Rezende and da Silveira6] that this family of matrices, with marked entries called primary pivots and change of basis pivots, determines the spectral sequence $(E^r, d^r)$ .

For each fixed r and each non-zero entry $\Delta ^r_{i,j}$ in the rth diagonal of $\Delta ^r$ , the mark-up is carried out as follows.

  • If there is a primary pivot below $\Delta ^r_{i,j}$ , then it is left unmarked.

  • If there is no primary pivot below and to the left of $\Delta ^r_{i,j}$ , then this entry is marked as a primary pivot.

  • If there is no primary pivot below $\Delta ^r_{i,j}$ but there is a primary pivot to the left of it, then the entry is marked as a change of basis pivot.

Whenever $\Delta ^r_{i,j}$ is marked as a change of basis pivot on the rth diagonal, then there exists a column, namely, the tth column $(t<j)$ , associated to a k-chain such that $\Delta ^r_{k_{i,t}}$ is a primary pivot. To construct the next matrix $\Delta ^{r+1}$ , a change of basis is performed on $\Delta ^{r}$ which zeroes out the entry $\Delta _{i,j}^{r}$ without introducing non-zero entries below the ith row, that is, $\Delta ^{r}_{s,j}=0$ for $s>i$ . Since $\Delta $ is a TU matrix, the SSSA has primary pivots always equal to $\pm 1$ as proved in [Reference Bertolim, Lima, Mello, de Rezende and da Silveira4]. Hence, it is always possible to choose the particular change of basis which uses only the tth and the jth column of $\Delta ^r$ . In other words, we zero out the entry $\Delta _{k_{i,j}}^{r}$ by adding or subtracting the tth column to or from the jth column of $\Delta ^{r}$ .

Let $k_1, k_2,\ldots $ be the columns of $\Delta ^{r}$ that are associated to k chains. We denote by $\sigma _k^{k_{\ell },r}$ the k chain represented in the $k_{\ell }$ th column of $\Delta ^r$ . Hence, the $k_j$ th column of $\Delta ^{r+1}$ is

(4) $$ \begin{align} \sigma_k^{k_j,{r+1}}&=\underbrace{\sum_{\ell=1}^{j}c^{k_j,r}_{\ell}h_k^{k_{\ell}}}_{\sigma_k^{k_j,r}}\pm \underbrace{\sum_{\ell=1}^{t}c^{k_t,r}_{\ell}h_k^{k_{\ell}}}_{\sigma_k^{k_t,r}}\nonumber\\ &=c^{k_j,r+1}_{1}h_k^{k_1} +c^{k_j,r+1}_{2}h_k^{k_2}+\cdots +c^{k_j,r+1}_{j-1}h_k^{k_{j-1}}+ c^{k_j,r+1}_{j}h_k^{k_j},\end{align} $$

where $c^{k_{\ell },r}\in \mathbb {Z}$ .

Therefore, the matrix $\Delta ^{r+1}$ has numerical values determined by the change of basis over $\mathbb {Z}$ of $\Delta ^r$ . In particular, all the changes of basis pivots on the rth diagonal $\Delta ^r$ are zero in $\Delta ^{r+1}$ .

It is easy to see that all $\Delta ^r$ are upper triangular and $\Delta ^r\circ \Delta ^r=0$ since they are recursively obtained from the initial connection matrix $\Delta $ by specific changes of basis over $\mathbb {Z}$ . Note that if the entry $\Delta ^{r}_{p-r+1,p+1}$ has been identified by the SSSA as a primary pivot or a change of basis pivot, then the entries below it are all zero, that is, $\Delta ^{r}_{s,p+1}=0$ for all $s>p-r+1$ .

In [Reference Cornea, de Rezende and da Silveira6], it is shown how the $\mathbb {Z}$ -modules $E^{r}_{p}$ are determined by the connection matrix $\Delta $ . To do this, a formula for the module $Z^{r}_{p,k-p}$ was established in terms of the $\sigma _k^{r,j}$ determined by the SSSA. Let $k_{\ell _{p}}$ be the rightmost $h_k$ column such that $k_{\ell _{p}}\leq p+1$ , that is, the rightmost $h_k$ column in $F_pC$ . Then

$$ \begin{align*}Z^{r}_{p,k-p}= \mathbb{Z}&[\mu^{k_{\ell_{p}},{r-p-1+k_{\ell_{p}}}}\sigma_k^{k_{\ell_{p}},{r-p-1+k_{\ell_{p}}}},\mu^{k_{\ell_{p}-1},{r-p-1+k_{\ell_{p}-1}}}\sigma_k^{k_{\ell_{p}-1},{r-p-1+k_{\ell_{p}-1}}},\ldots,\\&\quad\mu^{k_1,{r-p-1+k_1}}\sigma_k^{k_1,{r-p-1+k_1}}],\end{align*} $$

where $\mu ^{j,\zeta }=0$ whenever the primary pivot of the jth column is below the $(p-r+1)$ th row and $\mu ^{j,\zeta }=1$ otherwise. (Whenever $j<0$ , we consider $\sigma ^{i,j}_{k}=\sigma ^{i,0}_{k}$ .) Moreover, throughout the SSSA, $\Delta $ induces the differentials $d^r_p$ in the spectral sequence. In fact, whenever $E^{r}_{p}$ and $E^{r}_{p-r}$ are both non-zero, the map $d^r_p:E^{r}_{p}\to E^{r}_{p-r}$ is multiplication by the entry $\Delta ^r_{p-r+1,p+1}$ which is either a primary pivot or a zero with a column of zero entries below it. Otherwise, $d^r_p$ is zero. Since, in this case, the primary pivots are always equal to $\pm 1$ , the non-zero $d^r$ are always isomorphisms induced by primary pivots.

For a more detailed explanation of the SSSA as well as other sweeping algorithms, see [Reference Bertolim, Lima, Mello, de Rezende and da Silveira4].

3 Local behaviour of NMS flows

3.1 NMS chain complex

In this section, we consider a chain complex associated to an NMS flow $\varphi $ on a closed orientable 3-manifold and prove properties of its differential. We assume that $\varphi $ has no heteroclinic trajectories connecting saddle orbits, that is, given two saddle periodic orbits $ \gamma _1^i $ and $\gamma _1^j$ , $W^u(\gamma _1^i)\cap W^s(\gamma _1^j)$ is empty. The set of NMS flows satisfying this property is denoted by $\Sigma (M^3)$ . The next result shows that, for flows in $\Sigma (M^3)$ , the connecting manifold between periodic orbits has a nice behavior in the sense that, whenever $W^u(\gamma _1)\cap W^s(\gamma _0)\neq \emptyset $ , each of its connected components is homeomorphic to a cylinder $S^1 \times (0,1)$ . The dynamical behavior of these cylinders characterizes the differentials of the spectral sequence. As the spectral sequence unfolds, these differentials will play an important role in the construction of a continuation to a core NMS flow.

Proposition 3.1. Let $\varphi $ be an NMS flow in $\Sigma (M^3)$ , where $M^3$ is an orientable closed $3$ -manifold, $\gamma _1$ is a saddle periodic orbit and $\gamma _0$ (respectively, $\gamma _2$ ) is an attracting (respectively, repelling) periodic orbit, such that $W^u(\gamma _1)\cap W^s(\gamma _0)$ (respectively, $W^u(\gamma _2)\cap W^s(\gamma _1)$ ) is non-empty. Then each connected component of $W^u(\gamma _1)\cap W^s(\gamma _0)$ (respectively, $W^u(\gamma _2)\cap W^s(\gamma _1)$ ) is homeomorphic to $S^1 \times (0,1)$ .

Proof. Assume that $W^u(\gamma _1)\cap W^s(\gamma _0)$ is non-empty and let C be one of the connected components of $W^{u}(\gamma _1)\setminus \{\gamma _1\}$ such that $ W^s(\gamma _0)\cap C$ is also non-empty. Note that $W^s(\gamma _0)\cap C$ is an open set in C, since $W^s(\gamma _0)$ is an open set in M. Also, $W^s(\gamma _0)\cap C$ is a closed set in C. By the connectedness of C, it follows that $W^s(\gamma _0)\cap C$ must be equal to C.

Even though a non-empty connected component C of $W^u(\gamma _1)\cap W^s(\gamma _0)$ (respectively, $W^u(\gamma _2)\cap W^s(\gamma _1)$ ) is always a cylinder, there are distinct embeddings of C into $W^s(\gamma _0)$ (respectively, $W^s(\gamma _2)$ ).

A compressing disc for a surface F in a $3$ -manifold M is an embedded disc $D \subset M$ that meets F along its boundary, that is, $D\cap F = \partial D$ . The compressing disc D is inessential if the curve $\partial D$ is trivial on F, that is, bounds a disc on F. Otherwise, D is essential. A surface $F \subset M$ is called incompressible if it admits no essential compressing discs. Otherwise, it is called compressible. See [Reference Jaco13, Reference Matveev17].

Definition 3.2. Let $\gamma _1$ be a saddle periodic orbit and let $\gamma _0$ (respectively, $\gamma _2$ ) be an attracting (respectively, repelling) periodic orbit of $\varphi $ . Given C a connected component of $W^u(\gamma _1)\cap W^s(\gamma _0)$ ( $W^u(\gamma _2)\cap W^s(\gamma _1)$ ), one says that C is an inessential connection if C is an incompressible surface in $W^s(\gamma _0)$ (respectively, $W ^u(\gamma _2)$ ). Otherwise, one says that C is an essential connection.

In other words, if C is an incompressible surface in $W^s(\gamma _0)$ (respectively, $W^s(\gamma _2)$ ), then there are only inessential compressing discs. If C is a compressible surface in $W^s(\gamma _0)$ (respectively, $W^s(\gamma _2)$ ), then there exists an essential compressing disc. See Figure 4.

Figure 4 Inessential cylinder of connections (left) and essential cylinder of connections (right).

Let U be a neighborhood of a periodic orbit $\gamma $ in a Morse–Smale flow $\varphi $ such that U is disjoint from every other connected component of the recurrent set of $\varphi $ . A flow $\widetilde {\varphi }$ is a one step morsification of $\varphi $ on U if:

  1. (1) $\widetilde {\varphi }$ is a Morse–Smale flow;

  2. (2) $\widetilde {\varphi }$ agrees with $\varphi $ outside of U; and

  3. (3) U has two rest points $ p$ and q of index $k+1$ and k, where k is the index of the periodic orbit $\gamma $ . There are exactly two orbits connecting p and q and there are no other rest points or periodic orbits in U. Finally, $W^u(\gamma )$ is equal to $W^u(p) \cup W^u(q)$ and $W^s(\gamma )$ is equal to $W^s(p) \cup W^s(q)$ . See Franks [Reference Franks9, Reference Franks10].

Let $\mathcal {U} = \{U_1, U_2, \ldots , U_m\}$ be a collection of pairwise disjoint neighborhoods of the periodic orbits $\gamma ^1, \ldots , \gamma ^m$ in the Morse–Smale flow $\varphi $ such that $\gamma ^i \subset U_i$ for $i = 1, \ldots , m$ and each $U_i$ is disjoint from every other periodic orbit and every rest point of $\varphi $ . A flow $\widetilde {\varphi }$ is a morsification of $\varphi $ on $\mathcal {U}$ if:

  1. (1) $\widetilde {\varphi }$ is a Morse-Smale flow;

  2. (2) $\widetilde {\varphi }$ agrees with $\varphi $ outside of $\bigcup _{i=1}^{m} U_i$ ; and

  3. (3) in each $U_i$ , $\widetilde {\varphi }$ agrees with $\widetilde {\varphi }_i$ , where $\widetilde {\varphi }_i$ is a one step morsification of $\varphi $ on $U_i$ .

Whenever $\mathcal {U}$ contains a neighborhood of each periodic orbit of $\varphi $ , for simplicity, we refer to $\widetilde {\varphi }$ as a morsification of $\varphi $ .

Given a periodic orbit $\gamma _{k}$ of a Morse–Smale flow $\varphi $ , denote by $h_{k}(\gamma _{k})$ and $h_{k+1}(\gamma _{k})$ the index k and index $k+1$ rest points corresponding to $\gamma _{k}$ in a morsification $\widetilde {\varphi }$ of $\varphi $ . Denote by $\Gamma _{k}(\varphi )$ the set of all periodic orbits of index k of the flow $\varphi $ and denote by $\Gamma (\varphi ) = \Gamma _0(\varphi ) \cup \Gamma _1(\varphi ) \cup \Gamma _2(\varphi )$ the set of all periodic orbits.

A natural chain complex that describes the dynamics of a Morse–Smale flow without periodic orbits is the Morse chain complex. In order to study the dynamics of an NMS flow $\varphi $ , we consider the Morse chain complex of a morsification $\widetilde {\varphi }$ of $\varphi $ . More specifically, given periodic orbits $\gamma $ , $\beta $ , denote by $n(h_k(\gamma ), h_{k-1}(\beta );\widetilde {\varphi })$ the intersection number between the unstable manifold of $h_k(\gamma )$ and the stable manifold of $h_{k-1}(\beta )$ with respect to $\widetilde {\varphi }$ . If it is clear from the context, we omit reference to $\widetilde {\varphi }$ . The Morse group is the free $\mathbb {Z}$ -module $C=C(\widetilde {\varphi })$ generated by the critical points of $\widetilde {\varphi }$ and graded by the Morse index, that is, $C_{k}(\widetilde {\varphi }) = \mathbb {Z}[\mathrm {Crit}_k(\widetilde {\varphi })]$ . The boundary operator $\partial $ of $\varphi $ on a generator $h_k(\gamma )$ is given by

$$ \begin{align*} \partial_k : C_{k}(\widetilde{\varphi}) & \longrightarrow C_{k-1}(\widetilde{\varphi}) \nonumber \\ h_k(\gamma) & \longmapsto \sum_{\beta \in \Gamma_{k\!-\!1}(\varphi) \cup \Gamma_{k\!-\!2}(\varphi) }n(h_k(\gamma), h_{k-1}(\beta);\widetilde{\varphi}) h_{k-1}(\beta). \nonumber \end{align*} $$

Note that $(C_{\ast }(\widetilde {\varphi }),\partial _{\ast })$ is a particular case of a connection matrix for a Morse decomposition of $(M,\varphi )$ , where each Morse set is a periodic orbit and the partial order is the flow order. This fact follows from the results in [Reference Lima and de Rezende14, Reference Salamon21]. Let $\Delta $ be the matrix corresponding to the map $\partial _{\ast }$ with respect to the basis of $C_{\ast }(\widetilde {\varphi })$ given by $h_{k}(\gamma )$ , for $\gamma \in \Gamma (\varphi )$ and $k=0,1,2$ . The chain complex $(C_{\ast }(\widetilde {\varphi }),\partial _{\ast })$ is said to be an NMS-chain complex for $(M,\varphi )$ . In this paper, we use the notation $(C^{\mathrm {NMS}}(\widetilde {\varphi }),\Delta ^{\mathrm {NMS}})$ or $(C^{\mathrm {NMS}},\Delta ^{\mathrm {NMS}})$ whenever the morsification $\widetilde \varphi $ is clear from the context.

Recall that, in order for Smale’s cancellation theorem to hold, the intersection number between two singularities must be $\pm 1$ . Hence, it is natural to consider differentials $\Delta ^{\mathrm {NMS}}$ that are TU matrices and thus all entries are $0$ , $\pm 1$ . Later, this hypothesis will be instrumental in proving the cancellation theorems.

Let $TUM(\varphi )$ be the class of morsifications $\widetilde {\varphi }$ of $\varphi $ such that the differential $\Delta ^{\mathrm {NMS}}$ of the Morse chain complex associated to $\widetilde {\varphi }$ is a TU matrix. In this paper, we consider flows $\varphi $ such that $TUM(\varphi )\neq \emptyset $ .

The next theorem gives a correspondence between an inessential connection of an NMS flow $\varphi $ and the associated connections between the singularities of a morsification of $\varphi $ .

Theorem 3.3. Let $\varphi $ be an NMS flow in $\Sigma (M^3)$ , where $M^3$ is an orientable closed $3$ -manifold. Let $\gamma _0$ and $\gamma _1$ (respectively, $\gamma _1$ and $\gamma _2$ ) be two periodic orbits of $\varphi $ and let $\widetilde {\varphi }\in TUM(\varphi )$ be a morsification of $\varphi $ on $\mathcal {U}=\{U_1,U_2\}$ , where $U_1$ and $U_2$ are two disjoint neighborhoods of $\gamma _0$ and $\gamma _1$ (respectively, $\gamma _1$ and $\gamma _2$ ).

  1. (1) If $W^u(\gamma _1)\cap W^s(\gamma _0) $ (respectively, $W^u(\gamma _2)\cap W^s(\gamma _1) $ ) has one connected component only, which is inessential, then there is a unique connection between $h_2(\gamma _1)$ and $h_1(\gamma _0)$ (respectively, $h_2(\gamma _2)$ and $h_1(\gamma _1)$ ) in $\widetilde {\varphi }$ .

  2. (2) If $W^u(\gamma _1)\cap W^s(\gamma _0) $ (respectively, $W^u(\gamma _2)\cap W^s(\gamma _1) $ ) has two connected components where both are inessential connections, then the algebraic intersection number between $h_2(\gamma _1)$ and $h_1(\gamma _0)$ (respectively, $h_2(\gamma _2)$ and $h_1(\gamma _1)$ ) in $\widetilde {\varphi }$ is zero.

Proof. Let $\widetilde {\varphi }$ be a morsification of $\varphi $ and let C be a connected component of $W^u(\gamma _1)\cap W^s(\gamma _0)$ . The next two claims determine properties of the connecting orbits of $\widetilde {\varphi }$ in C.

Claim 1. If C is inessential, then $W^u(h_2(\gamma _1))\cap W^s(h_1(\gamma _0))\cap C\neq \emptyset $ in $\widetilde {\varphi }$ .

Suppose $W^u(h_2(\gamma _1))\cap W^s(h_1(\gamma _0))\cap C=\emptyset $ . Hence, $[W^u(h_1(\gamma _1))\cup W^u( h_2(\gamma _1))]\cap W^s(h_0(\gamma _0))\cap C = S^1\times (0,1)$ . There is a contractible neighborhood U of $h_0(\gamma _0)$ such that $U\subset W^s(\gamma _0)$ and

$$ \begin{align*} F:=W^u(h_1(\gamma_1))\cup W^u( h_2(\gamma_1))\cap C\cap U \cong S^1\times (0,1).\end{align*} $$

Note that, given a non-contractible closed curve $\alpha $ in F, there is a disc $D $ in U such that $\alpha $ is the boundary of D, that is, $\partial {D}=\alpha $ . Hence, D is an essential compressing disc for F and the connection between $\gamma _1$ and $\gamma _0$ is an essential connection.

Claim 2. If C is inessential, then all connections between $h_2(\gamma _1)$ and $h_1(\gamma _0)$ of $\widetilde {\varphi }$ in C have the same characteristic sign.

In fact, the characteristic sign of an orbit connecting $h_2(\gamma _1)$ and $h_1(\gamma _0)$ is determined by two orientations: the orientation induced by the flow and the orientation on the connecting manifold $\mathcal {M}_{h_2(\gamma _1)h_1(\gamma _0)}$ induced by the isomorphism (2). Either the orientation induced by the isomorphism (2) is compatible with the flow orientation for all orbits in $\mathcal {M}_{h_2(\gamma _1)h_1(\gamma _0)}\cap C$ or they are all not compatible. See Figure 5. Hence, all connecting orbits between $h_2(\gamma _1)$ and $h_1(\gamma _0)$ in C have the same characteristic sign.

Figure 5 Orientation of the connecting manifolds.

Case (1) follows from the claims above and the fact that $\Delta ^{\mathrm {NMS}}$ is a TU matrix.

In order to prove (2), let $C_1$ and $C_2$ be the connected components of $W^u(\gamma _1)\cap W^s(\gamma _0)$ . The numbers of connecting orbits in $W^u(h_2(\gamma _1))\cap W^s(h_1(\gamma _0))\cap C_1$ and $W^u(h_2(\gamma _1))\cap W^s(h_1(\gamma _0))\cap C_2$ must be equal since, otherwise, there would be self-intersection in $W^u(h_2(\gamma _1))$ . It follows that the intersection number $n(h_2(\gamma _1),h_1(\gamma _0);\widetilde {\varphi })$ is a sum of an even number of $+1$ and $-1$ and hence it must be an even number. Since $\widetilde {\varphi }\in TUM(\varphi )$ , then $n(h_2(\gamma _1),h_1(\gamma _0);\widetilde {\varphi })=0$ .

The proof with respect to the connections between a periodic orbits $\gamma _2$ and $\gamma _1$ is completely analogous.

Definition 3.4. Let C be an essential connected component of $W^u(\gamma _{i+1})\cap W^s(\gamma _i)$ , for $i=0,1$ . We say that C is $h_2$ - $h_1$ free if there exists a morsification $\widetilde {\varphi }\in TUM(\varphi )$ of $\varphi $ such that $W^u(h_2(\gamma _{i+1}))\cap W^s(h_1(\gamma _i))\cap C=\emptyset $ in $\widetilde {\varphi }$ . In this case, $\widetilde {\varphi }$ is $h_2$ - $h_1$ free in C. If $\widetilde {\varphi }\in TUM(\varphi )$ is $h_2$ - $h_1$ free in all essential connections of $\varphi $ , then $\widetilde {\varphi }$ is said to be a $h_2$ - $h_1$ free morsification. The class of $h_2$ - $h_1$ free morsifications of $\varphi $ is denoted by $FM(\varphi )$ .

Corollary 3.5. Let $\varphi $ be an NMS flow in $\Sigma (M^3)$ , where $M^3$ is an orientable closed $3$ -manifold. Let $\gamma _0$ and $\gamma _1$ (respectively, $\gamma _1$ and $\gamma _2$ ) be periodic orbits of $\varphi $ and let $\widetilde {\varphi }\in TUM(\varphi )$ be a morsification of $\varphi $ on $\mathcal {U}=\{U_1,U_2\}$ , where $U_1$ and $U_2$ are disjoint neighborhoods of $\gamma _0$ and $\gamma _1$ (respectively, $\gamma _1$ and $\gamma _2$ ). If $W^u(\gamma _1)\cap W^s(\gamma _0) $ (respectively, $W^u(\gamma _2)\cap W^s(\gamma _1) $ ) has two connected components, where one is an inessential connection and the other is a $h_2$ - $h_1$ free essential connection, then there is a unique connection between $h_2(\gamma _1)$ and $h_1(\gamma _0)$ (respectively, $h_2(\gamma _2)$ and $h_1(\gamma _1)$ ) in $\widetilde {\varphi }$ .

Given periodic orbits $\gamma \in \Gamma _k(\varphi )$ and $\beta \in \Gamma _{k-1}(\varphi )$ , we denote by $\Delta (\gamma ,\beta )$ the following submatrix of $\Delta ^{\mathrm {NMS}}$ .

$$ \begin{align*} \Delta(\gamma,\beta) = \left[ \begin{array}{@{}cc@{}} n(h_k(\gamma), h_{k-1}(\beta)) & n(h_{k+1}(\gamma), h_{k-1}(\beta)) \\ n(h_k(\gamma), h_{k}(\beta)) & n(h_{k+1}(\gamma), h_{k}(\beta)) \end{array} \right]\hspace{-2pt}. \end{align*} $$

We denote by $E_{ij}$ the $2\times 2$ matrix having one non-zero entry only, namely, the entry $(i,j)$ , which is equal to one. Let I be the identity $2\times 2$ matrix and let $\widetilde {I}$ be the matrix

$$ \begin{align*} \left[ \begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array} \right].\end{align*} $$

Proposition 3.6. Let $\varphi $ be an NMS flow in $\Sigma (M^3)$ , where $M^3$ is an orientable closed $3$ -manifold, and let $(C^{\mathrm {NMS}}(\widetilde {\varphi }),\Delta ^{\mathrm {NMS}})$ be the chain complex associated to a morsification $\widetilde {\varphi }\in TUM(\varphi )$ of $\varphi $ . Let $\gamma _0$ and $\gamma _1$ (respectively, $\gamma _1$ and $\gamma _2$ ) be periodic orbits of $\varphi $ such that $\widetilde {\varphi }$ is $h_2$ - $h_1$ free in all essential connections between $\gamma _0$ and $\gamma _1$ (respectively, $\gamma _1$ and $\gamma _2$ ). Then the following hold.

  1. (1) $\Delta (\gamma _1,\gamma _0)$ is an isomorphism if and only if there is a unique inessential cylinder of connections between the periodic orbits $\gamma _1$ and $\gamma _{0}$ . Analogously, $\Delta (\gamma _2,\gamma _1)$ is an isomorphism if and only if there is a unique inessential cylinder of connections between the periodic orbits $\gamma _2$ and $\gamma _{1}$ .

  2. (2) $ \Delta (\gamma _1,\gamma _{0})= \pm E_{11}$ if and only if there is a unique essential cylinder of connections between $\gamma _1$ and $\gamma _{0}$ and there is an attracting periodic orbit $\gamma _0^{\prime }\neq \gamma _{0}$ such that $\Delta (\gamma _1,\gamma _{0}^{\prime }) $ is non-zero. Analogously, $ \Delta (\gamma _2,\gamma _{1})= \pm E_{22}$ if and only if there is a unique essential cylinder of connections between $\gamma _1$ and $\gamma _{2}$ and there is a repelling periodic orbit $\gamma _2^{\prime }\neq \gamma _{2}$ such that $\Delta (\gamma _2,\gamma _{1}^{\prime }) $ is non-zero.

  3. (3) $ \Delta (\gamma _1,\gamma _{0})= \pm E_{22}$ if and only if $\gamma _1$ double connects with $\gamma _0$ through an inessential and an essential cylinder of connections. Analogously, $ \Delta (\gamma _2,\gamma _{1})= \pm E_{11}$ if and only if $\gamma _1$ double connects with $\gamma _2$ through an inessential and an essential cylinder of connections.

Proof. This follows from the proof of Theorem 3.3 and from Corollary 3.5.

Note that if $ \Delta (\gamma _k,\gamma _{k\!-\!1})$ is zero, for $k=1$ or $k=2$ , it does not necessarily mean that $W^u(\gamma _k)\cap W^s(\gamma _{k\!-\!1})$ is empty. For instance, this is the case when there are two inessential cylinders of connections between $\gamma _k$ and $ \gamma _{k\!-\!1}$ . Also, if $\gamma ^1_k$ and $\gamma ^2_k$ are both attracting periodic orbits or are both repelling periodic orbits, then the associated map in the connection matrix is the null map.

Theorems 3.7 and 3.8 provide a characterization for the columns and rows of the differential $\Delta $ associated to an $h_2$ - $h_1$ free morsification $\widetilde {\varphi }$ of $\varphi $ . This characterization is done in terms of the matrices $E_{ij}$ , I and $\widetilde {I}$ .

Theorem 3.7. (Column characterization of the differential matrix)

Let $\varphi $ be an NMS flow in $\Sigma (M^3)$ , where $M^3$ is an orientable closed $3$ -manifold, and let $(C^{\mathrm {NMS}}(\widetilde {\varphi }),\Delta ^{\mathrm {NMS}})$ be an NMS-chain complex such that $\widetilde {\varphi }\in TUM(\varphi )$ . Let $\{\gamma _{0}^{\ell }\}_{\ell }$ be attracting periodic orbits of $\varphi $ . Given a saddle periodic orbit $\gamma _1$ , the maps $\Delta (\gamma _1,\gamma _0^{\ell })$ satisfy one of the following conditions.

  1. (1) $\Delta (\gamma _1,\gamma _0^{\ell })$ is the null map for all $\ell $ .

  2. (2) There is ${\ell }_1$ such that $\Delta (\gamma _1,\gamma _0^{\ell _1})$ is the map $\pm E_{22}$ and $\Delta (\gamma _1,\gamma _0^{\ell })$ is the zero map for all ${\ell }\neq {\ell }_{1}$ .

  3. (3) There are ${\ell }_1,{\ell }_2$ such that the map $\Delta (\gamma _1,\gamma _0^{{\ell }_1})$ is either I, $\widetilde {I}$ or $ E_{11}$ , the map $\Delta (\gamma _1,\gamma _0^{{\ell }_2})$ is either $- I$ , $- \widetilde {I}$ or $- E_{11}$ and $\Delta (\gamma _1,\gamma _0^{\ell })$ is the zero map for all ${\ell }\neq {\ell }_{1},{\ell }_{2}$ .

Proof. By Proposition 3.1, $\gamma _1$ connects with either one or two attracting periodic orbits. Hence, at most two maps $\Delta (\gamma _1,\gamma _0^{\ell })$ are non-zero.

Suppose that $\gamma _1$ connects with only one attracting periodic orbit $\gamma _{0}^{{\ell }_1}$ (see Figure 6). Then $h_{1}(\gamma _{1})$ double connects to an index $0$ critical point of $\widetilde {\varphi }$ , which means that the column of $h_{1}(\gamma _{1})$ has only zero entries. On the other hand, the column corresponding to $h_{2}(\gamma _{1})$ is either a zero column or it has a unique non-zero entry in the row corresponding to $h_{1}(\gamma _{0}^{{\ell }_1})$ , which is $\pm 1$ . Hence, $\Delta (\gamma _1,\gamma _0^{{\ell }_1})$ is the null map or $\pm E_{22}$ .

Figure 6 Double connection through an inessential and an essential cylinder of connections.

Suppose that $\gamma _1$ connects with two attracting periodic orbits $\gamma _{0}^{{\ell }_1}$ and $\gamma _{0}^{{\ell }_2}$ . Then $h_{1}(\gamma _{1})$ connects with two index $0$ critical points, namely, $h_0(\gamma _0^{{\ell }_1})$ and $h_0(\gamma _0^{{\ell }_2})$ of $\widetilde {\varphi }$ , and hence the column of $h_{1}(\gamma _{1})$ has exactly two non-zero entries, $+1$ and $-1$ , corresponding to these connections. On the other hand, the column corresponding to $h_{2}(\gamma _{1})$ is either a zero column or it has non-zero entries in the rows corresponding to $h_{1}(\gamma _{0}^{{\ell }_1})$ or $h_{1}(\gamma _{0}^{{\ell }_2})$ . Hence, $\Delta (\gamma _1,\gamma _0^{{\ell }_i})$ is the map $\pm I$ , $\pm \widetilde {I}$ or $\pm E_{11}$ for $i=1,2$ and $\Delta (\gamma _1,\gamma _0^{{\ell }})$ is the null map for $\ell \neq {\ell }_1,{\ell }_2$ .

Theorem 3.8. (Row characterization of the differential matrix)

Let $\varphi $ be an NMS flow in $\Sigma (M^3)$ , where $M^3$ is an orientable closed $3$ -manifold, and let $(C^{\mathrm {NMS}}(\widetilde {\varphi }),\Delta ^{\mathrm {NMS}})$ be an NMS-chain complex such that $\widetilde {\varphi }\in TUM(\varphi )$ . Let $\{\gamma _{2}^{\ell }\}_{\ell }$ be the repelling periodic orbits of $\varphi $ . Given a saddle periodic orbit $\gamma _1$ , the maps $\Delta (\gamma _2^{{\ell }},\gamma _1)$ satisfy one of the following conditions.

  1. (1) $\Delta (\gamma _2^{{\ell }},\gamma _1)$ is the null map for all ${\ell }$ .

  2. (2) There is ${\ell }_1$ such that $\Delta (\gamma _2^{{\ell }_1},\gamma _1)$ is the map $\pm E_{11}$ and $\Delta (\gamma _2^{{\ell }},\gamma _1)$ is the zero map for all ${\ell }\neq {\ell }_{1}$ ;

  3. (3) There are ${\ell }_1,{\ell }_2$ such that the map $\Delta (\gamma _2^{{\ell }_1},\gamma _1)$ is either $ I$ , $ \widetilde {I}$ or $ E_{22}$ , the map $\Delta (\gamma _2^{{\ell }_2},\gamma _1)$ is either $- I$ , $- \widetilde {I}$ or $- E_{22}$ and $\Delta (\gamma _2^{{\ell }},\gamma _1)$ is the zero map for all ${\ell }\neq {\ell }_{1},{\ell }_{2}$ .

Proof. The proof follows from the previous theorem by considering the reverse flow.

Note that, if $\gamma _2 \in \Gamma _2(\varphi )$ is a repelling periodic orbit and $\gamma _0 \in \Gamma _{0}(\varphi )$ is an attracting periodic orbit, the corresponding submatrix of $\Delta $ is

$$ \begin{align*} \Delta(\gamma_2,\gamma_0) = \left[ \begin{array}{@{}cc@{}} n(h_2(\gamma_2), h_{0}(\gamma_0)) & n(h_{3}(\gamma_2), h_{0}(\gamma_0)) \\ n(h_2(\gamma_2), h_{1}(\gamma_0)) & n(h_{3}(\gamma_2), h_{1}(\gamma_0)) \end{array} \right]\hspace{-2pt},\end{align*} $$

which is either a zero matrix or $\pm E_{21}$ .

Example 3.9. Let $H_1$ and $H_2$ be two handlebodies. Consider $H_i\setminus D^3_i$ a handlebody minus a $3$ -ball in the interior of $H_i$ , for $i=1,2$ . Consider an NMS flow $\varphi _1$ on the attractor $H_1\setminus D^3_1$ transverse to the boundaries with two attracting periodic orbits $\gamma _0^1$ and $\gamma _0^2$ and two saddle periodic orbits $\gamma _1^3$ and $\gamma _1^4$ . Similarly, consider an NMS flow $\varphi _2$ on the repeller $H_2\setminus D^3_2$ transverse to the boundaries with one repelling periodic orbit $\gamma _2^6$ and one saddle periodic orbit $\gamma _1^5$ . By gluing $H_1\setminus D^3_1$ to $H_2\setminus D^3_2$ via the identification of $m_1$ with $\ell _2$ and $m_2$ with $\ell _1$ , one obtains $S^3\setminus \{D^3_1\cup D^3_2\}$ (see Figure 7). Now, by identifying the boundaries of $D^3_1$ and $D^3_2$ , one obtains an NMS flow $\varphi $ on $S^1\times S^2$ .

Figure 7 NMS flow on $S^1\times S^2$ .

Figure 8 shows the differential of an NMS-chain complex $(C^{\mathrm {NMS}}(\widetilde {\varphi }),\Delta ^{\mathrm {NMS}})$ associated to ${\varphi }$ , where $\widetilde {\varphi }$ is a $h_2$ - $h_1$ free morsification of $\varphi $ .

Figure 8 Differential $\Delta ^{\mathrm {NMS}}$ associated to ${\varphi }$ .

3.2 Local cancellations and reductions of periodic orbits

The goal of this section is to determine sufficient conditions under which two periodic orbits in $\varphi \in \Sigma (M^3)$ can be cancelled or reduced.

Definition 3.10. Let $\gamma _k$ and $\gamma _{k-1}$ be periodic orbits of an NMS flow $\varphi $ , of indices k and $k-1$ , respectively. The orbits $\gamma _k$ and $\gamma _{k-1}$ can be cancelled if there exists an NMS flow ${\varphi }'$ in M that coincides with $\varphi $ outside a neighborhood U of $W^u(\gamma _k) \cap W^s(\gamma _{k-1})$ and ${\varphi }'$ has empty chain recurrent set in U. The orbits $\gamma _k$ and $\gamma _{k-1}$ can be reduced if there exists an NMS flow ${\varphi }'$ on M that coincides with $\varphi $ outside a neighborhood U of $W^u(\gamma _k) \cap W^s(\gamma _{k-1})$ and the chain recurrent set of ${\varphi }'$ restricted to U is a unique periodic orbit $\gamma '$ .

Note that, if $k=2$ , $\gamma '$ is a repelling periodic orbit, and if $k=1$ , $\gamma '$ is an attracting periodic orbit.

Theorem 3.11. (Local cancellation of periodic orbits)

Let $\varphi $ be an NMS flow in $\Sigma (M^3)$ , where $M^3$ is an orientable closed $3$ -manifold, and let $(C^{\mathrm {NMS}}(\widetilde {\varphi }),\Delta ^{\mathrm {NMS}})$ be the NMS-chain complex such that $\widetilde {\varphi }\in TUM(\varphi )$ . Let $\gamma _k$ and $\gamma _{k-1}$ be two periodic orbits of indices k and $k-1$ , respectively, such that $\widetilde {\varphi }$ is $h_2$ - $h_1$ free in an essential connection between $\gamma _k$ and $\gamma _{k-1}$ . If the map $\Delta (\gamma _k,\gamma _{k\!-\!1})$ is an isomorphism, then there exists an NMS flow ${\varphi }'$ in M that coincides with $\varphi $ outside a neighborhood U of $W^u(\gamma _k) \cap W^s(\gamma _{k-1})$ and ${\varphi }'$ has empty chain recurrent set in U.

Proof. Since $\Delta (\gamma _k,\gamma _{k\!-\!1})$ is an isomorphism, by Proposition 3.6, there is a unique inessential cylinder of connections between $\gamma _k$ and $\gamma _{k-1}$ .

Let U be a neighborhood of $W^u(\gamma _k) \cap W^s(\gamma _{k-1})$ that does not intersect any other periodic orbit of $\varphi $ .

By the proof of Theorem 3.3, there is a unique orbit $u_1$ between $h_{k+1}(\gamma _{k})$ and $h_{k}(\gamma _{k\!-\!1})$ , as well as, a unique orbit $u_2$ between $h_{k}(\gamma _{k})$ and $h_{k-1}(\gamma _{k\!-\!1})$ in the morsification $\widetilde {\varphi }$ of $\varphi $ . Consider disjoint neighborhoods $V_i$ of $u_i$ , for $i=1,2$ , such that $V_i\subset U$ . By Smale’s first cancellation theorem [Reference Milnor18], one may cancel $h_{k+1}(\gamma _{k})$ and $h_{k}(\gamma _{k\!-\!1})$ , resulting in a flow $\widetilde {\varphi }_1$ that coincides with $\widetilde {\varphi }$ outside $V_1$ and has no critical points in $V_1$ . Note that, during this process of cancellation, the flow is not altered in $V_2$ . Hence, there remains a unique orbit $u_2$ between $h_{k}(\gamma _{k})$ and $h_{k-1}(\gamma _{k\!-\!1})$ in $\widetilde {\varphi }_1$ . Applying Smale’s first cancellation theorem once more to cancel $h_{k}(\gamma _{k})$ and $h_{k-1}(\gamma _{k\!-\!1})$ , one obtains a flow $\widetilde {\varphi }_2$ that coincides with $\widetilde {\varphi }_1$ outside $V_2$ and has no critical points in $V_2$ . By Theorem 2.1, one obtains a new NMS flow $\varphi '$ with two fewer periodic orbits, which coincides with $\varphi $ outside U and has empty chain recurrent set in U.

It follows from Theorem 3.11 that, given periodic orbits of $\varphi $ , $\gamma _k$ and $\gamma _{k-1}$ , such that $W^u(\gamma _k)\cap W^s(\gamma _{k-1})$ has only one connected component that is inessential, one can always perturb $\varphi $ in order to obtain a flow with two fewer periodic orbits. In this case, one says that the pair ( $\gamma _k,\gamma _{k-1}$ ) can be cancelled. See Figure 9.

Figure 9 Inessential cylinder of connections between an attracting and a saddle orbit (left) and their cancellation (right).

Theorem 3.12. (Local reduction of periodic orbits)

Let $\varphi $ be an NMS flow in $\Sigma (M^3)$ , where $M^3$ is an orientable closed $3$ -manifold and let $(C^{\mathrm {NMS}}(\widetilde {\varphi }),\Delta ^{\mathrm {NMS}})$ be an NMS-chain complex such that $\widetilde {\varphi }\in TUM(\varphi )$ . Let $\gamma _k$ and $\gamma _{k-1}$ be periodic orbits of indices k and $k-1$ , respectively, such that $\widetilde {\varphi }$ is $h_2$ - $h_1$ free in an essential connection between $\gamma _k$ and $\gamma _{k-1}$ . If the map $ \Delta (\gamma _{1},\gamma _{0}) = \pm E_{22}$ (respectively, $ \Delta (\gamma _{2},\gamma _{1})=\pm E_{11}$ ), then there exists an NMS flow ${\varphi }'$ on M that coincides with $\varphi $ outside a neighborhood U of $W^u(\gamma _k) \cap W^s(\gamma _{k-1})$ and the chain recurrent set of ${\varphi }'$ restricted to U is a unique attracting (respectively, repelling) periodic orbit.

Proof. Consider $k=1$ and assume that $ \Delta (\gamma _{1},\gamma _{0}) = \pm E_{22}$ . By Proposition 3.6, $\gamma _1$ double connects with $\gamma _0$ and there is a unique inessential cylinder of connections. By Theorem 3.3, there is a unique orbit u between $h_{2}(\gamma _{1})$ and $h_{1}(\gamma _{0})$ as well as two orbits $v_1$ and $v_2$ between $h_{1}(\gamma _{1})$ and $h_{0}(\gamma _{0})$ in the morsification $\widetilde {\varphi }$ of $\varphi $ . Let U be a neighborhood of $W^u(\gamma _1) \cap W^s(\gamma _{0})$ that does not intersect any other periodic orbit of