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

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.

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.

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.

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].

3.1 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.

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].

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 $ .

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.

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

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

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 $\widetilde {I}$ .

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 $\ell \neq {\ell }_1,{\ell }_2$ .

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.

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

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).