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).

It follows from Theorem 3.12 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 two connected components, where one is an inessential connection and the other is a $h_2$ - $h_1$ free essential connection, one can always perturb $\varphi $ in order to obtain a flow with one fewer periodic orbit. In this case, one says that the pair ( $\gamma _k,\gamma _{k-1}$ ) can be reduced. See Figure 10.

Figure 10

Double connection between a saddle and an attracting orbit (left) and reduction to an attracting orbit (right).

3.3 Birth and death of connections

The first step towards understand the global behavior of the flow $\varphi '$ obtained from $\varphi $ via the cancellation of the periodic orbits $\gamma _{1}$ and $\gamma _{0}$ is to study the effect of the cancellation in terms of essential and inessential connections with the remaining periodic orbits such as $\overline \gamma _{1}$ and $\overline \gamma _{0}$ .

Whenever the connection between $\overline \gamma _{1}$ and $\gamma _{0}$ and the connection between $\gamma _{1}$ and $\overline \gamma _{0}$ are both inessential connections in $\varphi $ , then the corresponding connection between $\overline \gamma _{1}$ and $\overline \gamma _{0}$ in $\varphi '$ is an inessential connection. In fact, considering a morsification $\widetilde \varphi \in TUM(\varphi )$ (see Figure 11(a)), the cancellations of corresponding pairs of critical points $(h_2(\gamma _1), h_1(\gamma _0))$ and $(h_1(\gamma _1),h_0(\gamma _0))$ give rise to the flow $\widetilde \varphi '$ (see Figure 11(b)). The new connections from $h_2(\overline \gamma _1)$ to $h_1(\overline \gamma _0)$ and from $h_1(\overline \gamma _1)$ to $h_0(\overline \gamma _0)$ characterize an inessential connection between $\overline \gamma _{1}$ and $\overline \gamma _{0}$ in the corresponding NMS flow $\varphi '$ .

Figure 11

Birth and death of flow lines within $\overline \gamma _{1}$ - $\gamma _{0}$ inessential and $\gamma _{1}$ - $\overline \gamma _{0}$ inessential connections.

On the other hand, whenever either the connection between $\overline \gamma _{1}$ and $\gamma _{0}$ or the connection between $\gamma _{1}$ and $\overline \gamma _{0}$ is an $h_2$ - $h_1$ free essential connection in $\varphi $ , then a similar analysis shows that the corresponding connection in $\varphi '$ is an $h_2$ - $h_1$ free essential connection. See Figures 12, 13, and 14.

Figure 12

Birth and death of flow lines within $\overline \gamma _{1}$ - $\gamma _{0}$ essential and $\gamma _{1}$ - $\overline \gamma _{0}$ inessential connections.

Figure 13

Birth and death of flow lines within $\overline \gamma _{1}$ - $\gamma _{0}$ inessential and $\gamma _{1}$ - $\overline \gamma _{0}$ essential connections.

Figure 14

Birth and death of flow lines within $\overline \gamma _{1}$ - $\gamma _{0}$ essential and $\gamma _{1}$ - $\overline \gamma _{0}$ essential connections.

Note that the flow $\widetilde \varphi '$ can be chosen such that the transversality condition is satisfied and

$$ \begin{align*} n(h_{2}(\overline\gamma_{1}),h_{1}(\overline\gamma_{0});\widetilde\varphi') &= n(h_{2}(\overline\gamma_{1}),h_{1}(\overline\gamma_{0});\widetilde\varphi)\\ &\quad- \dfrac{n(h_{2}(\overline\gamma_{1}),h_{1}(\gamma_{0});\widetilde\varphi) }{n(h_{2}(\gamma_{1}),h_{1}(\gamma_{0});\widetilde\varphi) } n(h_{2}(\gamma_{1}),h_{1}(\overline\gamma_{0});\widetilde\varphi). \end{align*} $$

Moreover, if C is an essential connected component of $W^u(\gamma _{1})\cap W^s(\gamma _0)$ in $\varphi '$ , then the morsification $\widetilde \varphi '$ of $\varphi '$ is such that $W^u(h_2(\gamma _{1}))\cap W^s(h_1(\gamma _0))\cap C=\emptyset $ in $\widetilde \varphi '$ .

The same analysis can be carried out to understand the global behavior of the flow obtained from $\varphi $ via the cancellation of the periodic orbits $\gamma _{2}$ and $\gamma _{1}$ .

The following proposition is a straightforward consequence of the analysis above.

4.1 SSSA for NMS complexes

Consider an NMS flow $\varphi $ on M. Let $m_k$ be the cardinality of the set of all periodic orbits of index k of the flow $\varphi $ , that is, $m_k=\# \Gamma _{k}(\varphi )$ . Also, let $m=m_0+m_1+m_2$ . Then, by Proposition 1.14 of [Reference Franks10], there exists a Lyapunov function $f:M\rightarrow \mathbb {R}$ for $\varphi $ such that:

1. (1) given $\gamma ^s$ , $\gamma ^{t}$ in $\Gamma (\varphi )$ , then $f(\gamma ^{s})\neq f(\gamma ^{t})$ , for all $s\neq t$ , where $s,t\in \{1, \ldots , m\}$ ; and

2. (2) $f(\gamma _{k})<f(\gamma _{k+1})$ for all $k=0,1$ .

For each $p\in \{1, \ldots , m\}$ , let $c_p:=f(\gamma ^p)$ . One can define a filtration on M by

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

where $\epsilon $ is sufficiently small so as to guarantee that there is only one periodic orbit $\gamma ^{p}$ in $\mathcal {F}_{p-1}\setminus \mathcal {F}_{p-2}$ , for each $p\in \{1, \ldots , m\}$ (assume that $\mathcal {F}_{-1}=\emptyset $ ).

Throughout this paper, we consider the filtration $\mathcal {F}=\{\mathcal {F}_pC\}$ induced by $\{\mathcal {F}_p\}$ on the NMS chain complex $(C^{\mathrm {NMS}},\Delta ^{\mathrm {NMS}})$ associated to $\varphi $ .

For each p, the chains generating $\mathcal {F}_pC$ that are not in $ \mathcal {F}_{p-1}C$ are represented in the matrix $\Delta $ in columns $2p+1$ and $2p+2$ and are denoted by $h^{2p+1}_{k}(\gamma _k^{p+1})$ and $h^{2p+2}_{k+1}(\gamma _k^{p+1})$ , respectively (see Figure 15). To make reference to these generators, one uses the notation $\mathcal {F}_{p}C\setminus \mathcal {F}_{p-1}C=\{h^{2p+1}_{k}(\gamma _k^{p+1}) ,h^{2p+2}_{k+1}(\gamma _k^{p+1})\}$ .

Figure 15

Filtration seen in the matrix $\Delta $ .

Let $(\mathcal {E}^r_p,d^r_p)$ be the spectral sequence for $(C^{\mathrm {NMS}},\Delta ^{\mathrm {NMS}})$ and a filtration $\mathcal {F}$ as defined above. The modules of the spectral sequence are given by

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

where

$$ \begin{align*} \mathcal{Z}^r_p=\{c\in \mathcal{F}_pC\,|\, \partial c\in \mathcal{F}_{p-r}C\}. \end{align*} $$

The proof of this theorem follows from Theorems 4.3 and 4.6 in §4.1.1.

This algorithm produces a sequence of matrices $\{\Delta ^r\}$ . A consequence of Theorem 4.1 is that each differential $d^{r}_{p}$ is induced by a submatrix of $\Delta ^r$ , as in Corollary 4.7.

As an important consequence of Theorem 4.1 and of a result proved in [Reference Bertolim, Lima, Mello, de Rezende and da Silveira4], which states that primary pivots for TU matrices are $\pm 1$ , we have that, for NMS flows in $ TUM (\varphi )$ , the non-zero differentials $d^r_{p,q}$ of the associated spectral sequence are all isomorphisms and the modules $E^r_{p,q}$ are all torsion free. In particular, the following result holds.

Thus, in this case, $GH_*(C)_{p,q}$ is free for all p and q and (3) holds: that is, the spectral sequence strongly converges to the homology of the NMS chain complex.

4.1.1 Proof of Theorem 4.1

In this section, we show how the modules $\mathcal {E}^r_p$ and the differentials $d^r_p$ of the spectral sequence associated to a filtered NMS chain complex can be determined by the SSSA. It was shown in [Reference Cornea, de Rezende and da Silveira6] that the modules of the spectral sequence $(E^r_p,d^r_p)$ associated to the finest filtration are determined by the SSSA. We use this fact to prove the following more general theorem.

Theorem 4.3. The SSSA determines the modules $\mathcal {E}^r_p$ .

Proof. Note that the module $\mathcal {Z}^r_p$ consists of chains in $\mathcal {F}_pC$ with boundary in $\mathcal {F}_{p-r}C$ . These chains are associated to all the columns of $\Delta $ to the left of and including the column $2p+2$ . Furthermore, since the boundary of the chains must be in $\mathcal {F}_{p-r}C$ , we must consider columns or their linear combinations that have the property that the entries in rows $i>2(p-r)+2$ are all zeros. Hence, in terms of the finest filtration $\{F_p\}$ ,

(5) $$ \begin{align}\mathcal{Z}^r_p=\{c\in F_{2p+1}C\,|\,\partial c\in F_{2(p-r)+1}C\}=Z^{2r}_{2p+1}. \end{align} $$

Analogously,

(6) $$ \begin{align}\mathcal{Z}^{r-1}_{p-1}=\{c\in F_{2p-1}C\,|\,\partial c\in F_{2(p-r)+1}C\}=Z^{2r-2}_{2p-1}\end{align} $$

and

$$ \begin{align*}\partial \mathcal{Z}^{r-1}_{p+r-1}=\partial\{c\in F_{2(p+r)-1}C\,|\,\partial c\in F_{2p+1}C\}=\partial Z^{2r-2}_{2(p+r)-1}.\end{align*} $$

If there is a k-chain in $\mathcal {F}_i\setminus \mathcal {F}_{i-1}$ , then the column of $\Delta $ associated to this chain, denoted by $k_{i}$ , is $2i+1$ or $2i+2$ . Then, using the formula given in [Reference Cornea, de Rezende and da Silveira6], (5) and (6) we describe the modules $\mathcal {Z}^r_{p,k-p}$ and $\mathcal {Z}^{r-1}_{p-1,k-(p-1)}$ in terms of the basis determined by $\Delta $ in the SSSA: that is,

(7) $$ \begin{align} \mathcal{Z}^r_{p,k-p}=\mathbb{Z}[\mu^{k_{{p}},{k_{{p}}-a}}\sigma_k^{k_{{p}},{k_{{p}}-a}}, \mu^{k_{p-1},{k_{p-1}-a}}\sigma_k^{k_{p-1},{k_{p-1}-a}},\ldots,\mu^{k_0,{k_0-a}}\sigma_k^{k_0,{k_0-a}}], \end{align} $$

(8) $$ \begin{align} \mathcal{Z}^{r-1}_{p-1,k-(p-1)}=\mathbb{Z}[\mu^{k_{{p-1}},{k_{{p-1}}-a}}\sigma_k^{k_{{p-1}},{k_{{p-1}}-a}}, \ldots,\mu^{k_0,{k_0-a}}\sigma_k^{k_0,{k_0-a}}], \end{align} $$

where $a=2(p-r)+2$ and, whenever there is no k-chain in $\mathcal {F}_i\setminus \mathcal {F}_{i-1}$ , $i=0,\ldots , P$ , the term $\mu ^{k_{{i}},{k_{{i}}-a}}\sigma _k^{k_{{i}},{k_{{i}}-a}}$ is supressed. Note that, if there is no k-chain in $\mathcal {F}_p\setminus \mathcal {F}_{p-1}$ , then $\mathcal {Z}^r_{p,k-p}=\mathcal {Z}^{r-1}_{p-1,k-(p-1)}$ and $\mathcal {E}^r_{p,k-p}=0$ . Now suppose that there is a k-chain in $\mathcal {F}_p\setminus \mathcal {F}_{p-1}$ . Consequently, $\mathcal {E}^{r}_{p,k-p}$ is isomorphic to a submodule of $\mathbb {Z}[\mu ^{k_{p},{k_{p}-a}}\sigma _k^{k_{p},{k_{p}-a}}].$

To determine $\mathcal {E}^{r}_{p,k-p}$ , we need to understand the module $\partial \mathcal {Z}^{r-1}_{p+r-1}$ . As we have seen, the module $\mathcal {Z}^{r-1}_{p+r-1,(k+1)-(p+r-1)}$ is

$$ \begin{align*} \mathbb{Z}&[\mu^{(k+1)_{{p+r-1}},(k+1)_{{p+r-1}}-(2p+2)}\sigma_{k+1}^{(k+1)_{{p+r-1}},{(k+1)_{{p+r-1}}-(2p+2)}}, \ldots,\\ &\quad\mu^{(k+1)_0,{(k+1)_0-(2p+2)}}\sigma_{k+1}^{(k+1)_0,{(k+1)_0-(2p+2)}}]. \end{align*} $$

Since there is a unique k-chain in $\mathcal {F}_p\setminus \mathcal {F}_{p-1}$ that is in row $k_p$ , this formula is equivalent to

$$ \begin{align*}\mathbb{Z}[\mu^{(k+ 1)_{{p+ r- 1}},(k+ 1)_{{p+ r- 1}}- k_p}\sigma_{k+ 1}^{(k+ 1)_{{p+ r- 1}},{(k+ 1)_{{p+ r- 1}}- k_p}}, \ldots,\mu^{(k+ 1)_0,{(k+ 1)_0- k_p}}\sigma_{k+ 1}^{(k+ 1)_0,{(k+ 1)_0- k_p}}]\end{align*} $$

and hence its boundary is

$$ \begin{align*} \mathbb{Z}&[\mu^{(k+1)_{{p+r-1}},{(k+1)_{{p+r-1}}-k_p}}\partial\sigma_{k+1}^{(k+1)_{{p+r-1}},{(k+1)_{{p+r-1}}-k_p}}, \ldots,\\ &\quad \mu^{(k+1)_0,{(k+1)_0-k_p}}\partial\sigma_{k+1}^{(k+1)_0,{(k+1)_0-k_p}}]. \end{align*} $$

If the column associated to $\sigma _{k+1}^{(k+1)_{\xi },{(k+1)_{\xi }-k_p}}$ has non-zero entries below the row $k_{p}$ , then $\mu ^{(k+1)_{\xi },{(k+1)_{\xi }-k_p}}=0$ . Hence, for each $\xi =0,\ldots , p+r-1$ , if $\mu ^{(k+1)_{\xi },{(k+1)_{\xi }-k_p}}=1$ , then

$$ \begin{align*} \partial\sigma_{k+1}^{(k+1)_{\xi},{(k+1)_{\xi}-k_p}}\ \ = \ \ \Delta^{(k+1)_{\xi}-k_p}_{k_{p},(k+1)_{\xi}}\sigma_k^{k_{p},(k+1)_{\xi}-k_p}+\underbrace{\ldots}_{\displaystyle\in \mathcal{Z}_{p-1}^{r-1}}.\end{align*} $$

Since $\Delta ^{(k\!+\!1)_{\xi }\!-\!k_p}_{k_{{p}},(k\!+\!1)_{\xi }}=0$ whenever $\xi <p$ , it follows that $\partial \mathcal {Z}^{r-1}_{p+r-1,(k+1)-(p+r-1)}$ is

(9) $$ \begin{align} [\mu^{(k\!+\!1)_{\xi},(k\!+\!1)_{\xi}\!-\!k_p} (\Delta^{(k\!+\!1)_{\xi}\!-\!k_p}_{k_{{p}},(k\!+\!1)_{\xi}} \sigma_k^{k_{p},(k\!+\!1)_{\xi}\!-\!k_p}), {\xi=p,\ldots, p+r-1}]+ \mathcal{Z}_{p-1}^{r-1}. \end{align} $$

Using equations (7), (8) and (9), we obtain a formula for $\mathcal {E}^{r}_{p}$ in terms of the $\sigma $ determined by the SSSA.

Corollary 4.4. Each module $\mathcal {E}^{r}_{p,k-p}$ of the spectral sequence is either $0$ or $\mathbb {Z}[\sigma _k^{k_p,k_p-2(p-r)-2}]$ .

Corollary 4.5. Each non-zero module $\mathcal {E}^{r}_{p}$ of the spectral sequence is either

$$ \begin{align*}\mathbb{Z}[\sigma_k^{k_{p},k_p-2(p-r)-2}] \quad\text{or} \quad \mathbb{Z}[\sigma_k^{(k-1)_{p},(k-1)_p-2(p-r)-2}, \sigma_k^{k_{p},k_p-2(p-r)-2}].\end{align*} $$

We now want to prove that the differentials of the spectral sequence are induced by the matrices produced in the SSSA.

Theorem 4.6. Whenever the modules $\mathcal {E}^{r}_{p,k-p}$ and $\mathcal {E}^{r}_{p-r,k-1-(p-r)}$ are non-zero, the differential $d^{r}_{p,k-p}$ is induced by entry $\Delta ^{k_p - (k-1)_{p-r}}_{(k-1)_{p-r},k_p}$ .

Proof. By equations (7) and (9) in the proof of Theorem 4.3,

$$ \begin{align*}\mathcal{E}^{r}_{p,k-p}&= \dfrac{\mathbb{Z}[\mu^{k_{{p}},{k_{{p}}-(k-1)_{p-r}}}\sigma_k^{k_{{p}},{k_{{p}}-(k-1)_{p-r}}}]+\mathcal{Z}_{p-1}^{r-1}}{\mathbb{Z}[\mu^{(k\!+\!1)_{\xi},(k\!+\!1)_{\xi}\!-\!k_p} \Delta^{(k\!+\!1)_{\xi}\!-\!k_p}_{k_{{p}},(k\!+\!1)_{\xi}} \sigma_k^{k_{p},(k\!+\!1)_{\xi}\!-\!k_p} ]_{{\xi=p,\ldots, p\!+\!r\!-\!1}}+ \mathcal{Z}_{p-1}^{r-1}}\\ & = \mathbb{Z}[\sigma_k^{k_{p},k_p-(k-1)_{p-r}}], \end{align*} $$

and hence $[\mu ^{(k\!+\!1)_{\xi },(k\!+\!1)_{\xi }\!-\!k_p} \Delta ^{(k\!+\!1)_{\xi }\!-\!k_p}_{k_{{p}},(k\!+\!1)_{\xi }} \sigma _k^{k_{p},(k\!+\!1)_{\xi }\!-\!k_p} ]_{{\xi =p,\ldots , p\!+\!r\!-\!1}}=0$ . It follows that

$$ \begin{align*} \mathcal{E}^{r+1}_{p,k-p}= \dfrac{\mathbb{Z}[\mu^{k_{{p}},{k_{{p}}-(k-1)_{p-r-1}}}\sigma_k^{k_{{p}},{k_{{p}}-(k-1)_{p-r-1}}}]}{\mathbb{Z}[\mu^{(k\!+\!1)_{p+r},(k\!+\!1)_{p+r}\!-\!k_p} \Delta^{(k\!+\!1)_{p+r}\!-\!k_p}_{k_{{p}},(k\!+\!1)_{p+r}} \sigma_k^{k_{p},(k\!+\!1)_{p+r}\!-\!k_p} ]}. \end{align*} $$

Consider the map $\delta ^{r}_{p,k-p}: \mathcal {E}^{r}_{p,k-p}\to \mathcal {E}^{r}_{p-r,k-1-(p-r)}$ induced by multiplication by the entry $\Delta ^{k_p - (k-1)_{p-r}}_{(k-1)_{p-r},k_p}$ . We show that

$$ \begin{align*} \mathcal{E}^{r+1}_{p,k-p}=\displaystyle\frac{\mathrm{Ker} \ \delta^{r}_{p,k-p}}{\mathrm{Im} \ \delta^{r}_{p+r,k+1-(p+r)}}. \end{align*} $$

Note that, for column $k_p$ of $\Delta ^{k_{p}-2(p-r)}$ , one of the following two situations may occur.

(1) The non-zero entries of column $k_p$ are above row $2(p-r)$ : that is, $\sigma _k^{k_{p},k_{p}-2(p+r)}$ is a generator of $\mathcal {Z}^{r+1}_{p,k-p}$ (that is, $\mu ^{k_{p},k_{p}-2(p-r)}=1$ ). In this case, $\Delta ^{k_p - (k-1)_{p-r}}_{(k-1)_{p-r},k_p}=0$ and hence $\mathrm {Ker} \ \delta ^{r}_{p,k-p}=0$ .

(2) The non-zero entries of column $k_p$ are above row $2(p-r)+2$ and below row $2(p-r)$ : that is, it has a primary pivot in row $(k-1)_{p-r}$ and hence $\sigma _k^{k_{p},k_{p}-(k-1)_{p-r}}$ is a generator of $\mathcal {Z}^r_{p,k-p}$ ( $\mu ^{k_{p},k_{p}-(k-1)_{p-r}}=1$ ) but $\sigma _k^{k_{p},k_{p}-2(p-r)}$ is not a generator of $\mathcal {Z}^{r+1}_{p,k-p}$ (that is, $\mu ^{k_{p},k_{p}-2(p-r)}=0$ ). In this case, $\Delta ^{k_p - (k-1)_{p-r}}_{(k-1)_{p-r},k_p}=\pm 1$ and hence $\mathrm {Ker} \ \delta ^{r}_{p,k-p}=\mathbb {Z}[\sigma _k^{k_{p},k_{p}-2(p+r)} ]$ .

It follows that $\mathrm {Ker} \ \delta ^{r}_{p,k-p}=\mathbb {Z}[\mu ^{k_{p},k_{p}-2(p+r)}\sigma _k^{k_{p},k_{p}-2(p+r)} ]$ .

On the other hand, considering the map $\delta ^{r}_{p+r,k+1-(p+r)}: \mathcal {E}^{r}_{p+r,k+1-(p+r)}\to \mathcal {E}^{r}_{p,k-p}$ , where $\mathcal {E}^{r}_{p+r,k+1-(p+r)}=0$ or

$$ \begin{align*}\mathcal{E}^{r}_{p+r,k+1-(p+r)}= \mathbb{Z}[\mu^{(k+1)_{p+r},{k+1_{p+r}-k_p}}\sigma_k^{(k+1)_{p+r},{k+1_{p+r}-k_p}}], \end{align*} $$

we have that

$$ \begin{align*}\mathrm{Im} \ \delta^{r}_{p+r,k+1-(p+r)}= \mathbb{Z}[\mu^{(k\!+\!1)_{p+r},(k\!+\!1)_{p+r}\!-\!k_p} \Delta^{(k\!+\!1)_{p+r}\!-\!k_p}_{k_{{p}},(k\!+\!1)_{p+r}} \sigma_k^{k_{p},(k\!+\!1)_{p+r}\!-\!k_p} ].\\[-40pt]\end{align*} $$

Corollary 4.7. The differential $d^{r}_{p}: \mathcal {E}^{r}_{p,k-p}\oplus \mathcal {E}^{r}_{p,k-1-p}\to \mathcal {E}^{r}_{p-r,k-1-(p-r)}\oplus \mathcal {E}^{r}_{p-r}, {}_{k-2-(p-r)} $ is induced by the matrix

(10) $$ \begin{align} \left(\begin{array}{cc} \Delta^r_{(k-2)_{p-r},(k-1)_p} & 0 \\ 0 & \Delta^r_{(k-1)_{p-r},k_p} \end{array} \right) \end{align} $$

in the following sense.

1. (1) If all modules are non-zero, then $d^{r}_p$ is the multiplication by the matrix in (10).

2. (2) If one or more modules is/are zero, then $d^{r}_p$ is the multiplication by the matrix obtained by replacing the entries corresponding to connections between the zero modules by zero entries in (10).

4.2 NMS-compatible filtrations

In this section, we introduce filtrations on an NMS chain complex that are consistent with the underlying cancellations and reductions occurring in the global dynamics through a spectral sequence approach and establish some of their properties.

Not all filtrations of a chain complex are NMS-compatible filtrations (see the next example).

Example 4.10. Consider the same chain complex endowed with the following filtration: $\mathcal {F}_0C= \{ h_{0}(\gamma _0^{2}), h_{1}(\gamma _0^{2}) \}$ , $\mathcal {F}_1C\setminus \mathcal {F}_{0}C= \{ h_{0}(\gamma _0^{1}) , h_{1}(\gamma _0^{1}) \}$ , $\mathcal {F}_2C\setminus \mathcal {F}_{1}C= \{ h_{1}(\gamma _1^{3}) , h_{2}(\gamma _1^{3}) \}$ , $\mathcal {F}_3C\setminus \mathcal {F}_{2}C= \{ h_{1}(\gamma _1^{5}) , h_{2}(\gamma _1^{5}) \}$ , $\mathcal {F}_4C\setminus \mathcal {F}_{3}C=\! \{ h_{1}(\gamma _1^{4}) , h_{2}(\gamma _1^{4}) \}$ and $\mathcal {F}_5C\setminus \mathcal {F}_{4}C=\! \{ h_{1}(\gamma _2^{6}) , h_{2}(\gamma _2^{6}) \}$ . This filtration is not NMS-compatible, since it does not satisfy the SF1 condition. More specifically, $\Delta ^{4}(\gamma _1^{3},\gamma _{0}^{2})_{22}$ is a primary pivot, but $\Delta ^{4}(\gamma _1^{3},\gamma _{0}^{2})_{11}$ is an entry above a primary pivot and hence it is neither a primary pivot nor a zero entry. Moreover, $\Delta ^{4}(\gamma _1^{5},\gamma _{0}^{1})_{22}$ is a primary pivot, but $\Delta ^{4}(\gamma _1^{5},\gamma _{0}^{1})_{11}$ is a change of basis pivot determined by a primary pivot $- E_{11}$ . See Figure 17.

Figure 17

SSSA for a non-NMS-compatible filtration: $\Delta ^2$ (left) and $\Delta ^4$ (right).

It is not known whether NMS-compatible filtrations always exist for a given NMS chain complex.

The next results show that NMS-compatible filtrations permit a characterization of non-zero differentials of the spectral sequence.

The last result characterizes the non-zero differentials $d^r_p$ induced by the submatrices $\Delta ^{r}(\gamma ^p_k,\gamma ^{p-r}_{k-1})$ corresponding to periodic orbits of consecutive indices. A non-zero differential induced by a submatrix $ \Delta ^{r}(\gamma ^p_2,\gamma ^{p-r}_{0})$ corresponding to a repelling periodic orbit and an attracting periodic orbit is given by $\pm E_{21}^{p}$ .

The next theorem is key in proving one of the main results of this paper, Theorem 5.1, since it is foundational in establishing dynamical results obtained in the next section.

Proof. Let $k=1$ and let $\Delta ^{r}(\gamma ^p_1,\gamma ^{p-r}_{0})$ be a matrix that induces a non-zero differential $d^r_p$ . Suppose that $\Delta ^{r}(\gamma ^p_1,\gamma ^{p-r}_{0})$ appears owing to a change of basis determined by $\Delta ^{r}(\gamma ^t_1,\gamma ^{i}_{0})= \pm E_{11}^p$ , where $t<p$ and $i>p-r$ . By Theorem 3.7, $\Delta ^{p-i}(\gamma ^p_1,\gamma ^{i}_{0})$ must be either $ \pm E_{11}$ , $\pm {I}$ or $\pm \widetilde {I}$ . Also, $\Delta ^{p-i}(\gamma ^t_1,\gamma ^{p-r}_{0})$ is either $ \pm E_{11}$ , $\pm {I}$ or $\pm \widetilde {I}$ ; in particular, $\Delta ^{p-i}(\gamma ^t_1,\gamma ^{p-r}_{0})_{11}=\pm 1$ . Therefore, by performing the corresponding change of basis, the column associated to $h_2(\gamma ^p_1)$ remains unaltered during this change of basis and one has the following possibilities.

1. (a) If $\Delta ^{r}(\gamma ^p_1,\gamma ^{p-r}_{0})_{22}$ is zero, then $\Delta ^{r}(\gamma ^p_1,\gamma ^{p-r}_{0})$ is either null or $\pm E_{11}$ .

2. (b) If $\Delta ^{r}(\gamma ^p_1,\gamma ^{p-r}_{0})_{22}$ is equal to $\pm 1$ , then $\Delta ^{p-i}(\gamma ^p_1,\gamma ^{p-r}_{0})_{11} =\pm 1$ , by Theorem 3.7. Thus, after the change of basis, $\Delta ^{r}(\gamma ^p_1,\gamma ^{p-r}_{0})_{11}$ is zero, which means that $\Delta ^{r}(\gamma ^p_1,\gamma ^{p-r}_{0})$ is $\pm E_{22}^p$ , which contradicts (SF1).

See Figure 18. Therefore, $\pm {I}^{pp}$ , $\pm \widetilde {I}^{pp}$ and $\pm E_{22}^p$ never appear owing to a change of basis determined by $\pm E_{11}^p$ .

Figure 18

Change of basis determined by $\pm E_{11}^p$ .

Let $k=2$ and let $\Delta ^{r}(\gamma ^p_2,\gamma ^{p-r}_{1})$ be a matrix that induces a non-zero differential $d^r_p$ . Suppose that $\Delta ^{r}(\gamma ^p_2,\gamma ^{p-r}_{1})$ appears owing to a change of basis determined by $\Delta ^{r}(\gamma ^t_2,\gamma ^{i}_{1})= \pm E_{22}^p$ , where $t<p$ and $i>p-r$ . By Theorem 3.8, $\Delta ^{p-i}(\gamma ^p_2,\gamma ^{i}_{1})$ must be either $ \pm E_{22}$ , $\pm {I}$ or $\pm \widetilde {I}$ and, analogously, $\Delta ^{p-i}(\gamma ^t_2,\gamma ^{p-r}_{1})$ is either $\pm E_{22}$ , $\pm {I}$ or $\pm \widetilde {I}$ ; in particular, $\Delta ^{p-i}(\gamma ^t_2,\gamma ^{p-r}_{1})_{22}=\pm 1$ . Therefore, by performing the corresponding change of basis, the column associated to $h_2(\gamma ^p_2)$ remains unaltered during this change of basis and one has the following possibilities.

1. (a) If $\Delta ^{r}(\gamma ^p_2,\gamma ^{p-r}_{1})_{11}$ is zero, then $\Delta ^{r}(\gamma ^p_2,\gamma ^{p-r}_{1})$ is either null or $\pm E_{22}$ .

2. (b) If $\Delta ^{r}(\gamma ^p_2,\gamma ^{p-r}_{1})_{11}$ is equal to $\pm 1$ , then $\Delta ^{p-i}(\gamma ^p_2,\gamma ^{p-r}_{1})_{22} =\pm 1$ , by Theorem 3.8. Thus after the change of basis, $\Delta ^{r}(\gamma ^p_2,\gamma ^{p-r}_{1})_{22}$ is zero, which means that $\Delta ^{r}(\gamma ^p_2,\gamma ^{p-r}_{1})$ is $\pm E_{11}^p$ , which contradicts (SF2).

See Figure 19. Therefore, $\pm {I}^{pp}$ , $\pm \widetilde {I}^{pp}$ and $\pm E_{11}^p$ never appear owing to a change of basis determined by $\pm E_{22}^p$ .

Note that the hypothesis of an NMS-compatible filtration is essential in Theorem 4.13 (see the next example).

Figure 19

Change of basis determined by $\pm E_{22}^p$ .

Example 4.14. Consider the following filtration for the NMS chain complex given in Example 3.9: $\mathcal {F}_0C= \{ h_{0}(\gamma _0^{1}), h_{1}(\gamma _0^{1}) \}$ , $\mathcal {F}_1C\setminus \mathcal {F}_{0}C= \{ h_{0}(\gamma _0^{2}) , h_{1}(\gamma _0^{2}) \}$ , $\mathcal {F}_2C\setminus \mathcal {F}_{1}C= \{ h_{1}(\gamma _1^{4}) , h_{2}(\gamma _1^{4}) \}$ , $\mathcal {F}_3C\setminus \mathcal {F}_{2}C= \{ h_{1}(\gamma _1^{3}) , h_{2}(\gamma _1^{3}) \}$ , $\mathcal {F}_4C\setminus \mathcal {F}_{3}C= \{ h_{1}(\gamma _1^{5}) , h_{2}(\gamma _1^{5}) \}$ and $\mathcal {F}_5C\setminus \mathcal {F}_{4}C= \{ h_{1}(\gamma _2^{6}) , h_{2}(\gamma _2^{6}) \}$ . This filtration is not NMS-compatible, since it does not satisfy the SF1 condition in Definition 4.8. More specifically, $\Delta ^{8}(\gamma _1^{5},\gamma _{0}^{1})_{22}$ is a primary pivot, but $\Delta ^{8}(\gamma _1^{5},\gamma _{0}^{1})_{11}=0$ such that $\Delta ^{6}(\gamma _1^{5},\gamma _{0}^{2})_{11} $ is a change of basis pivot determined by a primary pivot $- E_{11}^p$ . Consequently, $E_{22}^p$ appeared owing to a change of basis determined by $- E_{11}^p$ . See Figure 20.

Figure 20

Theorem 4.13 does not hold for non-NMS-compatible filtrations.

An additional requirement on an NMS-compatible filtration is reduction ordering.

For example, the filtration presented in Figure 20 does not satisfy the reduction ordering condition, since $\Delta ^{8}(\gamma _1^{4},\gamma _{0}^{1})= E_{22}^p$ and $\Delta ^{6}(\gamma _2^{6},\gamma _{1}^{4})= E_{11}^p$ .

5.1 Cancellations and reductions determined by a spectral sequence

In this section, one of the main results in this paper, Theorem 5.1, is proved. It establishes a correspondence between the dynamics of cancellations and reductions of periodic orbits in an NMS flow and the differentials of the spectral sequence of a filtered chain complex associated to the flow.

The hypothesis on the filtration of Theorem 5.1 will guarantee that, by following the unfolding of the associated spectral sequence, we can always dynamically perform corresponding cancellations and reductions in an NMS flow. First, one will require that $\mathcal {F}$ is an NMS-compatible filtration for $(C^{\mathrm {NMS}}(\widetilde {\varphi }),\Delta ^{\mathrm {NMS}})$ . Second, one must ensure that the spectral sequence determines, at each stage, at most one cancellation or reduction of a given periodic orbit. This is precisely what is ensured by requiring that the filtration satisfies the reduction ordering condition. Recall that $FM(\varphi )$ is the class of $h_2-h_1$ free morsifications as in Definition 3.4.