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This article is cited in 1 scientific paper (total in 1 paper) Classification of nonsingular four-dimensional flows with a untwisted saddle orbit V. D. Galkin, O. V. Pochinka, D. D. Shubin National Research University Higher School of Economics, Nizhny Novgorod, Russia
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    § 1. Introduction and statements of resultsIn this paper we consider so-called NMS-flows $f^t$, that is, nonsingular (without fixed points) Morse–Smale flows on closed orientable $n$-manifolds $M^n$, $n\geqslant 2$. The nonwandering set of such a flow consists of a finite number of periodic hyperbolic orbits. In the case when there are just few orbits the known invariants can greatly be simplified and, more importantly, the classification problem can be brought to an effective solution by indicating whether the invariants obtained are admissible. In [17] flows with two orbits on arbitrary closed $n$-manifolds were exhaustively classified. In [17] a complete topological classification was obtained for flows with three periodic orbits on orientable 3-manifolds. In [20] and [24] the question of classification was answered for three-dimensional Morse–Smale flows with a finite number of singular trajectories. The topological equivalence of nonsingular flows under assumptions of various generality was established, for instance, in [5] and [22]. It was shown in [21] that the only orientable 4-manifold that admits NMS-flows with a unique saddle periodic orbit which is, moreover, twisted (that is, its invariant manifolds are nonorientable) is $\mathbb S^3\times\mathbb S^1$. It was also proved there that such flows fall into eight equivalence classes. We note straight away that for a nontwisted orbit the number of equivalence classes of such orbits is infinite, as follows from [16], and there are flows with wildly embedded invariant manifolds of the saddle orbit among them. This paper is devoted to the topological equivalence of four-dimensional NMS-flows with a unique saddle periodic orbit, provided that this orbit is nontwisted. We turn to the statements of results. Let $M^4$ be a closed connected orientable 4-manifold, $f^t\colon M^4\to M^4$ be an NMS-flow and $\mathcal O$ be a periodic orbit of this flow. In a neighbourhood of the hyperbolic periodic orbit $\mathcal O$ the flow has a simple description (up to topological equivalence), namely, there exists a tubular neighbourhood $V_{\mathcal O}$ of $\mathcal O$ that is homeomorphic to $\mathbb D^3\times \mathbb S^1$ and such that the flow in it is topologically equivalent to a suspension over a linear diffeomorphism of $\mathbb R^3$ whose matrix has a positive determinant and real eigenvalues of modulus distinct from one (see Proposition 7 below). The number of eigenvalues with modulus greater than one is called the Morse index of $\mathcal O$; it is denoted by $i_{\mathcal O}$. If $i_{\mathcal O}=0$ ($i_{\mathcal O}=3$), then the periodic orbit under consideration is attracting (respectively, repelling), otherwise it is said to be saddle. A saddle orbit is twisted if just two eigenvalues are negative, one of which has modulus greater than one and the other has modulus less than one; otherwise the orbit is nontwisted. We put the minis sign before the Morse index $i_{\mathcal O}$ in the case when $\mathcal O$ is nontwisted. We let denote the number of windings along ${\mathcal O}$ made by the knot $c\subset V_{\mathcal O}$. Set
$$
\begin{equation*}
\Sigma_{\mathcal O}=\partial V_{\mathcal O}\cong\mathbb S^2\times\mathbb S^1,
\end{equation*}
\notag
$$
and let $e_{\mathcal O}\subset\Sigma_{\mathcal O}$ denote the generator of the fundamental group $\pi_1(\Sigma_{\mathcal O})$ such that Recall that a knot in $\mathbb S^2\times\mathbb S^1$ is standard if there exists a homeomorphism of $\mathbb S^2\times\mathbb S^1$ taking it to $\{s\}\times\mathbb S^1$. Consider the class $G^+_1(M^4)$ of NMS-flows $f^t\colon M^4\to M^4$ with a unique saddle orbit $S$ which is, moreover, untwisted. For the saddle orbit $S$ of a flow $f^t\in G^+_1(M^4)$ two cases are possible: (1) ${\dim W^{\mathrm{u}}_S=3}$; (2) $\dim W^{\mathrm{u}}_S=2$. Let $G^{+1}_1(M^4)$ and $G^{+2}_1(M^4)$ denote the classes of flows of types (1) and (2), respectively. In view of the different dimensions of unstable saddle manifolds it is clear that no flow in $G^{+1}_1(M^4)$ is equivalent to a flow in $G^{+2}_1(M^4)$. Moreover, $G^{+2}_1(M^4)=\{f^{-t}\colon f^t\in G^{+1}_1(M^4)\}$ and the flows $f^t$ and $f'^t$ are equivalent if and only if $f^{-t}$ and $f'^{-t}$ are. As a direct consequence, the solution of the classification problem in $G^+_1(M^4)$ reduces to its solution in the class $G=G^{+1}_1(M^4)$. In § 3.1 we establish the following fact. Because a flow is equivalent to a suspension in a neighbourhood of a periodic orbit, the set is homeomorphic to a 2-torus (Figure 1; in what follows we present the manifold as a spherical annulus $\mathbb S^2\times [0, 1]$ with bases identified by the rule $(x,0)\sim (x,1)$). In addition, if $N_{T}\subset\Sigma_S$ is a closed tubular neighbourhood of $T$, then the set $\Sigma_S\setminus \operatorname{int} N_{T}$ consist of a pair of solid tori $V^{-}$, $V^{+}$ that are tubular neighbourhoods of knots $L^\pm$ such that Throughout what follows the knot $L^\pm$ is oriented so that $\langle L^\pm\rangle_S=1$.Let $\mathcal R$ denote the set of repelling orbits of the flow $f^t$, and set
$$
\begin{equation*}
\Sigma_{\mathcal R}=\bigsqcup_{R\in\mathcal R}\Sigma_R.
\end{equation*}
\notag
$$
In § 3.3 we show that we can choose $N_{T}$ so that
$$
\begin{equation*}
N_{T}\subset \Sigma_A\quad\text{and} \quad (\Sigma_S\setminus \operatorname{int} N_{T})\subset \Sigma_{\mathcal R}.
\end{equation*}
\notag
$$
In Figure 2 we show some possible embeddings of $T$ in $\Sigma_A$. In § 3.4 we establish the following fact on the number of repelling orbits and the position of the knots $L^-$ and $L^+$ in the manifold $\Sigma_{\mathcal R}$ of the flow $f^t$ (Figure 3). Lemma 2. For flows in $G$ just two cases are actually possible: We assume without loss of generality that the knot $L^+$ is standard. We set $L=L^-$, and in the case when $\mathcal R$ consists of two orbits we set $R=R^-$ and let $B\subset\Sigma_R$ be a 3-ball bounded by a 2-sphere separating $L^-$ and $L^+$ in $\Sigma_{R}$. To each flow $f^t\in G$ we assign a pair of numbers $\rho_1\in\{-1,1\}$, $\rho_2\in\{-2,-1,1,2\}$ by the following rule (see Figure 3). If $f^t$ has two repelling orbits, then $\rho_1=\langle L^+\rangle_{R^+}$ and $\rho_2=2\langle e_{A}\rangle_{R^-}$. If $f^t$ has one repelling orbit, then $\rho_1=\langle L^+\rangle_{R}$ and $\rho_2=1$ ($\rho_2=- 1)$ in the case when the generator $e_A$ in $\Sigma_{R}$ points inward the ball $B$ (respectively, outward of $B$). We call the tuple the scheme of the flow $f^t\in G$ (Figure 4).The schemes $\mathcal S_{f^t}=(\Sigma_{R},L,\rho_1,\rho_2)$ and $\mathcal S_{f'^t}=(\Sigma_{R'},L',\rho'_1,\rho'_2)$ of two flows $f^t,f'^t\in G$ are said to be equivalent if there exists a homeomorphism $\varphi\colon \Sigma_{R}\to\Sigma_{R'}$ such that
The central result in our paper is the following theorem, which we prove in § 4. Theorem 1. Two flows $f^t, f'^t\in G$ are topologically equivalent if and only if their schemes $\mathcal S_{f^t}$ and $\mathcal S_{f'^t}$ are equivalent. To solve the problem of the realization of flows in the class under consideration we describe an abstract scheme. We call a tuple an abstract scheme if
As a direct consequence of Lemma 2, the scheme of any flow $f^t\in G$ is equivalent to an abstract scheme. We complete the classification of flows in the class under consideration by the following theorem, which we prove constructively in § 5. Theorem 2. For each abstract scheme $\mathcal S$ there exists a flow $f^t\in G$ whose scheme is equivalent to $\mathcal S$. A special case of flows under consideration is a suspension over a Morse–Smale 3-diffeomorphism with a unique saddle orbit. By [14] the supporting manifold of such a diffeomorphism is a sphere $\mathbb S^3$, and it follows from [3] and [4] that a complete invariant of topological conjugacy of these diffeomorphisms is also an embedding of a knot in $\mathbb S^2\times\mathbb S^1$. The following result is a direct consequence of this classification and Theorem 1. Theorem 3. A flow $f^t\in G$ with invariant $\mathcal S_{f^t}=(\Sigma_{R},L,\rho_1,\rho_2)$ is equivalent to a suspension over a Morse–Smale 3-diffeomorphism if and only if In particular, a flow $f^t\in G$ with invariants $\langle L\rangle_{R}=1$, $\rho_1=1$ and $\rho_2=2$ is equivalent to a suspension over a so-called Pixton diffeomorphism [15]. In this case $L$ is a Hopf knot (that is, it belongs to the homotopy class $\langle e_R\rangle_R$). It follows from [1] that there exist countably many inequivalent (ambiently nonhomeomorphic) Hopf knots (Figure 5). By [3] each Hopf knot can be realized by a Pixton diffeomorphism. It was shown in [16] that a suspension over a Pixton diffeomorphism realized for a knot not equivalent to a standard knot $L_0$, is a flow with wildly embedded invariant manifolds of the saddle orbit. Although invariant manifolds of the saddle orbit can be embedded wildly, the supporting manifold of a flow $f^t\in G$ that is a suspension over a 3-diffeomorphism is homeomorphic to $\mathbb S^3\times\mathbb S^1$. Surprisingly, in this paper we prove the following result. Theorem 4. The supporting manifold of each flow $f^t\!\in\! G$ is homeomorphic to ${\mathbb S^3\!\times\!\mathbb S^1}$. § 2. Embeddings in the manifold $ {\mathbb S}^2\times {\mathbb S}^1$Recall that by a $C^{r}$-embedding $(r\geqslant 0)$ of a manifold $X$ in a manifold $Y$ we mean a map $\lambda\colon X \to Y$ such that $\lambda \colon X\to \lambda(X)$ is a $C^r$-diffeomorphism. A $C^0$-embedding is also called a topological embedding. A topological embedding $\lambda\colon X \to Y$ of an $m$-manifold $X$ in an $n$–manifold $Y$ ($m\leqslant n$) is said to be locally flat at a point $\lambda(x)$, $x \in X$, if $\lambda(x)$ lies in a coordinate chart $(U,\psi)$ of $Y$ such that either $\psi(U \cap \lambda(X)) =\mathbb{R}^{m}$, where $\mathbb{R}^{m}\subset\mathbb{R}^{n}$ is the set of points with the last $n-m$ coordinates equal to zero, or $\psi(U \cap \lambda(X)) = R^{m}_{+}$, where $\mathbb{R}^{m}_{+} \subset\mathbb{R}^{m}$ is the set of points with nonnegative last coordinate. The embedding $\lambda$ is said to be locally flat and $X$ is said to be locally flatly embedded if $\lambda$ is locally flat at each point $\lambda(x)$, $x\in X$. Otherwise $\lambda$ is said to be wild, and $X$ is wildly embedded. A point $\lambda(x)$ at which $\lambda$ is not locally flat is called a point of wildness. Set $\Sigma=\mathbb S^2\times\mathbb S^1$. Proposition 1 ([3], Lemma 3.1). Let $\Lambda$ be a 2-sphere embedded in $\Sigma$ locally flatly. Then either $\Lambda$ bounds a 3-ball on $\Sigma$, or it is ambiently isotopic to the sphere $\mathbb S^2\times\{s_0\}$, $s_0\in\mathbb S^1$. Recall that a solid torus $V$ is a 3-manifold with boundary that is homeomorphic to $ \mathbb D^2\times\mathbb S^1$. A meridian in the solid torus $V$ is a knot $\mu\subset \partial V$ bounding a 2-disc $d\subset V$ such that $d\cap \partial V=\mu$. Proposition 2 ([18]). Let $V_1$ and $V_2$ be solid tori and $h\colon \partial V_1\to\partial V_2$ be a homeomorphism. Then $h$ extends to a homeomorphism $h\colon V_1\to V_2$ if and only if it takes a meridian in $V_1$ to a meridian in $V_2$. Proposition 3 ([2]). Let $V_1$ and $V_2$ be solid tori embedded locally flatly in a 3-manifold $M^3$ so that $M^3=V_1\cup V_2$ and $V_1\cap V_2=\partial V_1=\partial V_2$. Then $M^3\cong\Sigma$ if and only if a meridian in $V_1$ is incidentally a meridian in $V_2$. Proposition 4 ([2]). A locally flat knot $L\subset\Sigma$ is standard if and only if the complement in $\Sigma$ of a tubular neighbourhoods of $L$ is homeomorphic to a solid torus. A locally flat knot $L\subset\mathbb S^3$ is said to be trivial if there exists a homeomorphism $\mathbb S^3$ taking it to $\mathbb S^1\subset\mathbb S^3$. Otherwise the knot is said to be nontrivial. Proposition 5 (see [7], Theorem 3, and [6], Remark 6). Let $W$ be a set homeomorphic to the complement in $\mathbb S^3$ of a tubular neighbourhood of a nontrivial knot $L \subset \mathbb S^3$, and let $M^3 = W \cup V$, where $V$ is a solid torus such that $W \cap V = \partial W = \partial V$. Then $M^3$ us not homeomorphic to $\Sigma$. Let $T$ be a 2-torus locally flatly embedded in $\Sigma$ and $i_T\colon T\to\Sigma$ be the embedding. We say that $T$ is embedded homotopically trivially (nontrivially) if $i_{T*}(\pi_1(T))=0$ (respectively, if $i_{T*}(\pi_1(T))\neq 0$). Proposition 6 ([3], Theorem 4). Let $T$ be a 2-torus embedded in $\Sigma$ homotopically nontrivially. Then $T$ bounds a solid torus in $\Sigma$. To describe a 2-torus $T$ embedded in $\Sigma$ homotopically trivially note that, given such a torus $T$, there always exists a smoothly embedded 2-sphere $\Lambda\subset \Sigma$ that is homotopic to a layer $\mathbb S^2\times\{s_0\}$ and disjoint from $T$. Lemma 3. Let $T$ be a 2-torus homotopically trivially embedded in $\Sigma$. Then the following cases are possible: Proof. We represent $\mathbb S^3$ as a compactification of $\mathbb S^2\times\mathbb R$ by a north pole $N$ and a south pole $S$. Let $p\colon \mathbb S^2\times\mathbb R\to\Sigma$ be the projection defined by $p(s,r)=(s,r\pmod 1)$. Then the manifold $\Sigma$ is the orbit space of the action on $\mathbb S^2\times\mathbb R$ of the group of diffeomorphisms $\{g^n,\,n\in\mathbb Z\}$, where the diffeomorphism $g\colon\mathbb S^2\times \mathbb R\to \mathbb S^2\times\mathbb R$ is defined by
Let $\widetilde\Lambda\subset\mathbb S^3$ denote a connected component of $p^{-1}(\Lambda)$ and $\widetilde K\subset \mathbb S^3$ denote the spherical shell bounded by the 2-spheres $\widetilde \Lambda$ and $g(\widetilde \Lambda)$. Here the sphere $\widetilde \Lambda$ ($g(\widetilde \Lambda)$) bounds a 3-ball $B_N$ in $\mathbb S^3$ containing the pole $N$ (respectively, a 3-ball $B_S$ containing $S$); Figure 6. Let $\widetilde T\subset\widetilde K$ denote a connected component of $p^{-1}(T)$, which is a locally flatly embedded 2-torus because of the homotopic triviality of the torus $T$. Since $\mathbb S^3$ is an irreducible manifold (each 2-sphere embedded in it locally flatly bounds a 3-ball), the torus $\widetilde T$ bounds a solid torus $\widetilde V\subset\mathbb S^3$ (for instance, see [13]). Set $\widetilde W=\mathbb S^3\setminus \operatorname{int}\widetilde V$. As $\widetilde T$ is disjoint from $B_N$ and $B_S$, the following cases are possible for $\widetilde V$ and $\widetilde W$:
The following options for $T$ are direct consequences of the above: (1), (2) $T$ bounds a solid torus $V=p(\widetilde V)$ in $\Sigma$; (3), (4) $T$ does not partition $\Sigma$, and $\Sigma\setminus(\Lambda\cup T)$ has two connected components such that the closure of one of them, $p(\widetilde V\setminus \operatorname{int}B_N)$, is homeomorphic to a solid torus without a 3-ball, while the closure of the other, $p(g(\widetilde \Lambda)\setminus \operatorname{int}\widetilde V)$, is homeomorphic to a 3-ball without a solid torus; (5) $T$ partitions $\Sigma$ into two connected components one of which is homeomorphic to $p(\mathbb S^3\setminus \operatorname{int}\widetilde V)$. The proof is complete. § 3. Dynamics of flows in the class $G$3.1. The uniqueness of an attracting orbitIn this subsection we prove Lemma 1: the nonwandering set of a flow $f^t\in G$ contains a unique attracting orbit $A$. Proof of Lemma 1. The proof is based on the following representation for the supporting manifold $M^4$ of the NMS-flow $f^t$ with set of periodic orbits $\operatorname{Per}_{f^t}$ (for instance, see [19]):
$$
\begin{equation}
M^4 = \bigcup_{\mathcal O \in \operatorname{Per}_{f^t}} W^{\mathrm{u}}_{\mathcal O}=\bigcup_{\mathcal O \in \operatorname{Per}_{f^t}} W^{\mathrm{s}}_{\mathcal O},
\end{equation}
\tag{3.1}
$$
and on the asymptotic behaviour of the invariant manifolds
$$
\begin{equation}
\operatorname{cl}(W^{\mathrm{u}}_{\mathcal O}) \setminus W^{\mathrm{u}}_{\mathcal O} = \bigcup_{\widetilde{\mathcal O} \in \operatorname{Per}_{f^t}\colon W^{\mathrm{u}}_{\mathcal O}\cap W^{\mathrm{s}}_{\widetilde{\mathcal O}}\neq \varnothing} W^{\mathrm{u}}_{\widetilde{\mathcal O}}
\end{equation}
\tag{3.2}
$$
and
$$
\begin{equation}
\operatorname{cl}(W^{\mathrm{s}}_{\mathcal O}) \setminus W^{\mathrm{s}}_{\mathcal O} = \bigcup_{\widetilde{\mathcal O} \in \operatorname{Per}_{f^t}\colon W^{\mathrm{s}}_{\mathcal O}\cap W^{\mathrm{u}}_{\widetilde{\mathcal O}}\neq \varnothing} W^{\mathrm{s}}_{\widetilde{\mathcal O}}.
\end{equation}
\tag{3.3}
$$
Similarly to Theorem 1 in [9], the union of all source orbits and the stable manifolds of all saddle orbits of $f^t$ if a repeller.1 Since the stable manifolds have dimension 2, the repeller arising (which we denote by $R$) has topological dimension 2. By [10], Ch. 4, § 5, Corollary 1, the manifold $M^4\setminus R$ is connected, as also is $M^4\setminus \operatorname{int}U_R$, where $U_R$ is a trapping neighbourhood of $R$. On the other hand the set $U_A=M^4\setminus \operatorname{int}U_R$ is a trapping neighbourhood of the union $A$ of all attracting orbits of $f^t$, which is an attractor. Since $A=\bigcap_{t\geqslant 0} f^{t}(U_{A})$ and $U_A$ is connected, it follows that $A$ is connected, and therefore it consists of a single orbit.
The lemma is proved. 3.2. Canonical neighbourhoods of periodic orbitsRecall the definition of a suspension. Let $\varphi\colon M^3\to M^3$ be a diffeomorphism of a 3-manifold. Consider the diffeomorphism $g_{\varphi}\colon M^3\times \mathbb R \to M^3\times \mathbb R$ defined by
$$
\begin{equation*}
g_{\varphi}(x_1,x_2,x_3, x_4) = (\varphi(x_1,x_2, x_3),x_4-1).
\end{equation*}
\notag
$$
Then the group $\{g_{\varphi}^n\}\cong\mathbb Z$ acts freely and discontinuously on $M^3\times \mathbb R$, so the orbit space $\Pi_\varphi = M^3\times \mathbb R/ g_{\varphi}$ is a smooth 4-manifold and the natural projection $v_\varphi\colon M^3\times \mathbb R\to \Pi_\varphi$ is a covering. Furthermore, the flow $\xi^t\colon M^3\times \mathbb R\to M^3\times \mathbb R$ defined by induces the flow $[\varphi]^t= v_\varphi \xi^t v^{-1}_\varphi\colon\Pi_\varphi\to\Pi_\varphi$, called a suspension over the diffeomorphism $\varphi$. Consider the diffeomorphisms $a_0,a_{\pm 1},a_{\pm 2},a_3\colon \mathbb R^{3}\to \mathbb R^{3}$ defined by
$$
\begin{equation*}
\begin{gathered} \, a_3(x_1, x_2, x_3) = (2x_1, 2x_2, 2x_3), \qquad a_0 = a_3^{-1}, \\ a_{\pm 1}(x_1, x_2, x_3) = \biggl(\pm 2x_1, \pm\frac{x_2}{2}, \frac{x_3}{2}\biggr)\quad\text{and} \quad a_{\pm 2} = a_{\pm 1}^{-1}. \end{gathered}
\end{equation*}
\notag
$$
Set
$$
\begin{equation*}
\begin{aligned} \, \overline V_{0}&= \{ (x_1,x_2, x_3, x_4)\in \mathbb R^3 \mid 4^{x_4} x_1^2 + 4^{x_4}x^2_2 + 4^{x_4}x^2_3 \leqslant 1 \}, \\ \overline V_{\pm 1}&= \{ (x_1,x_2, x_3, x_4)\in \mathbb R^3 \mid 4^{-x_4} x_1^2 + 4^{x_4}x^2_2 + 4^{x_4}x^2_3 \leqslant 1 \}, \\ \overline V_{\pm 2}&= \{ (x_1,x_2, x_3, x_4)\in \mathbb R^3 \mid 4^{-x_4} x_1^2 + 4^{-x_4}x^2_2 + 4^{x_4}x^2_3 \leqslant 1 \} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\overline V_{3}= \{ (x_1,x_2, x_3, x_4)\in \mathbb R^3 \mid 4^{-x_4} x_1^2 + 4^{-x_4}x^2_2 + 4^{-x_4}x^2_3 \leqslant 1 \}.
\end{equation*}
\notag
$$
For $i\in\{0,\pm 1,\pm 2,3\}$ we set $v_i = v_{a_i}$, $\overline\Sigma_i=\partial{\overline V_i}$, $V_i=v_i(\overline V_i)$ and $\Sigma_{i} = v_i(\overline\Sigma_i)$. Also set $e_{i}=v_i(Ox_4)$. We let $\langle c\rangle_i$ denote the number of windings the knot $c\subset \Sigma_{i}$ makes along the generator $e_{i}$. The following result, due to Irwin [11], describes the behaviour of flows in a neighbourhood of a hyperbolic periodic orbit. Proposition 7. Given a hyperbolic periodic orbit $\mathcal O$ of a flow $f^t\colon M^4\to M^4$ on a closed orientable manifold $M^4$, there exist a tubular neighbourhood $V_{\mathcal O}$ of $\mathcal O$ and an integer $i_{\mathcal O}\in\{0,\pm 1,\pm 2,3\}$ such that the flow $f^t|_{V_{\mathcal O}}$ is topologically equivalent to $[a_{i_{\mathcal O}}]^t|_{V_{i_{\mathcal O}}}$ via a homeomorphism $H_{\mathcal O}$. We call the neighbourhood $V_\mathcal O=H_{\mathcal O}(V_{i_\mathcal O})$ in Proposition 7 a canonical neighbourhood of the periodic orbit $\mathcal O$. In the proof of topological equivalence we use the statement below, which follows from the proof of Theorem 4 and Lemma 4 in [17]; the reader can also find it in [20], Theorem 1.1. Proposition 8. A homeomorphism $h_i\colon \Sigma_i\to \Sigma_i$, where $i\in\{0,3\}$, extends to a homeomorphism $H_i\colon V_i\to V_i$ establishing the equivalence of the flow $[a_i]^t$ to itself if and only if the induced isomorphism $h_{i*}\colon \pi_1(\Sigma _i)\to \pi_1(\Sigma _i)$ is identical. 3.3. Partitioning a supporting manifold into canonical neighbourhoodsSet $\overline\Gamma = \{(x_1, x_2, x_3, x_4)\in \overline\Sigma_{2}\mid 4^{x_4}x_3^2 = 1/2\}$ and $\overline T = Ox_2x_3x_4\cap\overline\Sigma_{2}$. By construction the set $\overline\Sigma_{2}$ is homeomorphic to $\mathbb S^2\times\mathbb R$, and the set $\overline\Gamma$ consists of two surfaces, each homeomorphic to $\mathbb S^1\times\mathbb R$, which divide $\overline\Sigma_{2}$ into three connected components, one of which, $N_{\overline T}$, contains the cylinder $\overline T\cong\mathbb S^1\times\mathbb R$, while each of the other two is homeomorphic to $\mathbb D^2\times\mathbb R$. Then Consider the flow $f^t\in G$. By Lemma 1 $f^t$ has a unique attracting orbit $A$ with canonical neighbourhood $V_A=H_A(V_0)$ bounded by the manifold $\Sigma_A=\partial V_A\cong\mathbb S^2\times\mathbb S^1$. Let $S$ be a saddle orbit of the flow $f^t$. Then $V_S=H_S(V_{2})$ is a canonical neighbourhood of $S$. Set
$$
\begin{equation*}
T=H_S(\mathbb T), \qquad N_{T}=H_S(N_{\mathbb T}), \qquad V^-=H_S(\mathbb V^-), \qquad V^+=H_S(\mathbb V^+)
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\Sigma^{\mathrm{u}}_A=\biggl(\bigcup_{t>0,\,w\in N_{T}}f^{t}(w)\biggr)\cap \Sigma_A.
\end{equation*}
\notag
$$
Let $\lambda^\pm\subset\partial V^\pm$ denote a generator of the fundamental group of the solid torus $V^\pm$ and $\mu^\pm$ denote a meridian. Consider a continuous function $\tau_{A}\colon \Sigma_A\to\mathbb R^+$ such that $f^{\tau_{A}(a)}(a)\in N_{T}$ for $a\in \Sigma^{\mathrm{u}}_A$. Set
$$
\begin{equation*}
\widetilde V_A=V_A\cup\biggl(\bigcup_{a\in \Sigma_A}\biggl(\bigcup_{t\in[0,\tau_A(a)]}f^{-t}(a)\biggr)\biggr).
\end{equation*}
\notag
$$
It is immediate to verify that the flow $f^t|_{\widetilde V_A}$ is topologically equivalent to the suspension $[a_0]^t|_{V_0}$. So we assume throughout that $V_A=\widetilde V_A$. By (3.1) each connected component of the set contains a unique repelling orbit $R$, and the flow $f^t$ on its closure is equivalent to the suspension $[a_3]^t|_{V_3}$, so we denote this component by $V_R$. Let $\mathcal R$ denote the set of repelling orbits of the flow $f^t$. Then the supporting manifold $M^4$ is the union of canonical neighbourhoods with disjoint interiors:
$$
\begin{equation}
M^4 = V_A \cup V_S \cup\bigsqcup_{R\in\mathcal R}V_R.
\end{equation}
\tag{3.4}
$$
3.4. The number of repelling orbitsThis subsection is devoted to the proof of Lemma 2, stating that for flows $G$ precisely two cases are possible:
Proof of Lemma 2. It follows from (3.4) that
$$
\begin{equation}
\Sigma_A\setminus \operatorname{int}N_T=\Sigma_{\mathcal R}\setminus \operatorname{int}(V^+\sqcup V^-).
\end{equation}
\tag{3.5}
$$
Set $Y=\Sigma_A\setminus \operatorname{int}N_T$. Since $N_T$ is a tubular neighbourhood of the torus $T$, $\partial Y$ consists of two 2-tori, $T^+$ and $T^-$. In addition, by (3.5) where $h_{\pm}\colon \partial V^\pm\to T^\pm$ is a homeomorphism Let $\mu^\pm$ be a meridian in the solid torus $V^\pm$. Then by Proposition 6 and Lemma 3 the following cases are possible for $Y$:
Now we consider each case separately. (1) Let $Y^+$ and $Y^-$ denote connected components of $Y$, which are bounded by the tori $T^+$ and $T^-$, respectively. Then it follows from (3.6) that the set $\Sigma_{\mathcal R}$ also has two connected components. Hence $\mathcal R$ consists of precisely two orbits $R^-$ and $R^+$ such that $\Sigma_{R^\pm}=V^\pm\cup_{h_\pm}Y^\pm$. Since at least one of the sets $Y^+$ and $Y^-$ (for definiteness, $Y^+$) is homeomorphic to a solid torus, the manifold $\Sigma_{R^+}$ is obtained by gluing together two solid tori: $\Sigma_{R^+}=V^+\cup_{h_+}Y^+$. Since $\Sigma_{R^+}$ is homeomorphic to $\Sigma$, by Proposition 3 $h_+(\mu^+)$ is a meridian of the solid torus $Y^+$, and by Proposition 4 the knot $L^+$ is standard in $\Sigma_{R^+}$. Moreover, by construction $h_-(\mu^-)$ is a meridian of the solid torus $\Sigma_A\setminus Y^-$, so that by Proposition 2 the manifold $V^-\cup_{h_-}Y^-$ is homeomorphic to $\Sigma_A$ (Figure 7). Thus, case (1) is indeed possible, and we have proved the lemma in this case. (2) Since the set $Y$ is connected, it follows from (3.6) that the set $\Sigma_{\mathcal R}$ is too. Hence $\mathcal R$ consists of a unique orbit $R$ such that $\Sigma_{R}=V^+\cup_{h_+}Y\cup_{h_-}V^-$. In addition, the 2-sphere $\Lambda$ separates the solid tori $V^+$ and $V^-$ in $\Sigma_{R}$, so it also separates $L^+$ and $L^-$. On the other hand, by Proposition 1 it bounds a 3-ball $B\subset \Sigma_{R}$ containing one of these solid tori; for definiteness, let it be $V^-$ (Figure 8). Then $B\setminus \operatorname{int}V^-=\dot Y^-$ and $\Sigma_{R}\setminus \operatorname{int}(B\cup V^+)=\dot Y^+$. It follows from the second equality that $\Sigma_{R}\setminus \operatorname{int}V^+=\dot Y^+\cup B$. Since $\dot Y^+\cup B$ is a solid torus, the knot $L^+$ is standard in $\Sigma_{R}$. (3) Since $Y$ consists of two connected components one of which is homeomorphic to the complement in a 3-sphere $S^3$ to a tubular neighbourhood of a nontrivial knot, we can obtain from it a connected component of $\Sigma_{\mathcal R}$ by gluing a solid torus to its boundary. By Proposition 5 the result cannot be homeomorphic to $\Sigma$, which shows that case (3) cannot occur. Lemma 2 is proved. § 4. Necessary and sufficient conditions for the equivalence of flows in the class $G$In this section we prove Theorem 1: two flows $f^t, f'^t\in G$ are topologically equivalent if and only if their schemes $\mathcal S_{f^t}$ and $\mathcal S_{f'^t}$ are equivalent. Proof of Theorem 1. $\Rightarrow$. Let $f^t\colon M^4\!\to\! M^4\!\in\! G$ and $f'^t\colon M'^4\!\to\! M'^4\!\in\! G$ be two flows which are topologically equivalent by means of a homeomorphism $h\colon M^4\to M'^4$. Since $h$ takes periodic orbits of the flow $f^t$ to periodic orbits of $f'^t$ while preserving the direction of motion and $h$ preserves also the invariant manifolds of orbits, it follows that $\Omega_{f'^t}=\{\mathcal O'=h(\mathcal O)$, $\mathcal O\in\Omega_{f^t}\}$. Then we can assume without loss of generality that $V_{\mathcal O'}=h(V_{\mathcal O})$. Therefore, $\varphi= h|_{\Sigma_R}\colon \Sigma_R\to \Sigma_{R'}$ is the required homeomorphism establishing the equivalence of the schemes $\mathcal S_{f^t}$ and $\mathcal S_{f'^t}$, and ${\rho_i=\rho'_i}$, $i=1,2$.
$\Leftarrow$. Let $\varphi\colon \Sigma_R\to\Sigma_{R'}$ be a homeomorphism establishing the equivalence of the schemes $\mathcal S_{f^t}$ and $\mathcal S_{f'^t}$ of two flows $f^t\colon M^4\to M^4\in G$ and $f'^t\colon M'^4\to M'^4\in G$. We construct a homeomorphism $h\colon M^4\to M'^4$ establishing the equivalence of these flows in a few steps. By Proposition 7 there is a homeomorphism $H_S$ establishing the equivalence of $f^t|_{V_S}$ to the suspension $[a_{+1}]|_{V_{+1}}$. Set Then the homeomorphism $\varphi_S$ establishes the equivalence of the flows $f^t|_{V_S}$ and $f'^t|_{V_{S'}}$, which shows that $\varphi_S(T)={T'}$ and $\varphi_S(N_T)=N_{T'}$, and $\varphi_S$ takes the pair of knots $L^-$, $L^+$ to a pair of knots ${L'}^-$, $L'^+$ while preserving their orientations. On the other hand the homeomorphism $\varphi\colon \Sigma_{R}\to\Sigma_{R'}$ takes $L^-$ to $L'^-$ while preserving the orientation. We will assume without loss of generality that $\varphi(V^-)=V'^-$. Then the isomorphism induced by the homeomorphism $\varphi|_{T^-}\colon T^-\to T'^-$ has the following matrix in the generators $\lambda^-$, $\mu^-$, $\lambda'^-$ and $\mu'^-$:
$$
\begin{equation*}
\mathcal A=\begin{pmatrix} 1 & k \\ 0 & \delta \end{pmatrix}, \qquad k\in\mathbb Z, \quad \delta\in\{-1,1\}.
\end{equation*}
\notag
$$
Consider a homeomorphism $Q\colon\mathbb V_{1}\to\mathbb V_{1}$ defined by
$$
\begin{equation*}
Q = v_{a_{2}}\overline Q v_{a_{2}}^{-1}, \quad\text{where } \overline Q(x_1, x_2, x_3,x_4) = (\delta_1x_1,\widetilde Q(x_2, x_3),x_4)\colon V_{1}\to V_{1},
\end{equation*}
\notag
$$
$\delta_1\in\{-1,1\}$ and $\widetilde Q\colon Ox_2x_3\to Ox_2x_3$ is the linear diffeomorphism with matrix
$$
\begin{equation*}
\begin{pmatrix} 1 & k_2 \\ 0 & \delta_2 \end{pmatrix}, \qquad k_2\in\mathbb Z, \quad \delta_2\in\{-1,1\}.
\end{equation*}
\notag
$$
It is straightforward to verify that $Q$ establishes an equivalence of the flow $[a_{2}]^t$ to itself. Set where we choose $\delta_1$ so that $h_S(V^-)=V'^-$ and we choose $\delta_2$ and $k_2$ so that the isomorphism induced by the homeomorphism $h_S|_{T^-}\colon T^-\to T'^-$ has the matrix $\mathcal A$ in the generators $\lambda^-$, $\mu^-$ and $\lambda'^-$, $\mu'^-$.
Thus, the homeomorphisms $\varphi|_{T^-},h_S|_{T^-}\colon T^-\to {T'}^-$ are isotopic (for instance, see [18]), so that there exists a homeomorphism $h_-\colon \Sigma_{R}\to\Sigma_{R'}$ coinciding with $h_S$ on $V_-$ and with $\varphi$ outside a neighbourhood of the solid torus $V_-$. Now we consider separately two cases: (1) $|\rho_2|=2$; (2) $|\rho_2|=1$. In case (1) the flow $f^t$ ($f'^t$) has two repelling orbits $R^-$ and $R^+$ (respectively, $R'^-$ and $R'^+$), and a tubular neighbourhood $N_T$ ($N_{T'}$) of the torus $T$ (respectively, $T'$) divides $\Sigma_A$ (respectively, $\Sigma_{A'}$) into two connected components $Y^-$ and $Y^+$ (respectively, $Y'^-$ and $Y'^+$) bounded by $T^-$ and $T^+$ (respectively, bounded by $T'^-$ and $T'^+$). By Lemma 2 the knot $L^+$ ($L'^+$) is standard in $\Sigma_{R^+}$ (respectively, in $\Sigma_{R'^+}$), so that the set $Y^+$ (respectively, $Y'^+$) is homeomorphic to a solid torus. Furthermore, $\Sigma_{R^+}=V^+\cup Y^+$ (respectively, $\Sigma_{R'^+}=V'^+\cup Y'^+$). On the set $V^+$ we have the homeomorphism $h_{R^+}=h_S|_{V^+}\colon V^+\to V'^+$. Since the isomorphism induced by $h_{R^+}|_{T^+}\colon T^+\to T'^+$ has the matrix $\mathcal A$ in the generators $\lambda^+$, $\mu^+$ and $\lambda'^+$, $\mu'^+$, the restriction $h_{R^+}|_{T^+}$ takes a meridian in the solid torus $Y^+$ to a meridian in $Y'^+$. In this case (for instance, see [18]) $h_{R^+}$ extends to $Y^+$ as a homeomorphism $h_{R^+}\colon \Sigma_{R^+}\to\Sigma_{R'^+}$. Set $h_{R^-}=h_-$. Then a homeomorphism $h_A\colon \Sigma_A\to \Sigma_{A'}$ is defined which coincides with $h_S$ on $N_T$, with $h_{R^-}$ on $\Sigma_{R^-}\setminus\operatorname{int}V^-$ and with $h_{R^+}$ on $\Sigma_{R^+}\setminus\operatorname{int}V^+$. Since $\rho_i=\rho'_i$, $i=1,2$, by Proposition 8 the homeomorphism $h_{\mathcal O}\colon \Sigma_{\mathcal O}\to \Sigma_{\mathcal O'}$, $\mathcal O\in\{R^-,A,R^+\}$, extends to a homeomorphism $h_{\mathcal O}\colon V_{\mathcal O}\to V_{\mathcal O'}$ establishing the equivalence of the flows $f^t|_{V_{\mathcal O}}$ and $f'^t|_{V_{\mathcal O'}}$. Thus, the required homeomorphism $h\colon M^4\to M'^4$ coincides with $h_{\mathcal O}$, $\mathcal O\in\{S,A,R^-,R^+\}$, on $V_{\mathcal O}$. In case (2) the flow $f^t$ ($f'^t$) has one repelling orbit $R$ (respectively, $R'$). By Lemma 2 the knot $L^+$ ($L'^+$) is standard in $\Sigma_{R}$ (in $\Sigma_{R'}$), and the solid torus $V^-$ ($V'^-$) lies in the 3-ball $B\subset(\Sigma_{R}\setminus V^+)$ (respectively, in $B'\subset(\Sigma_{R'}\setminus V'^+)$). Since the homeomorphism $h_-\colon \Sigma_{R^-}\to\Sigma_{R'^-}$ takes the solid torus $V^-$ to the solid torus $V'^-$, we can assume without loss of generality that $B'=h_-(B)\subset(\Sigma_{R'}\setminus V'^+)$. Set $\widetilde L^+=h^{-1}_-(L'^+)$ and $\widetilde V^+=h^{-1}_-(V'^+)$. Since the knots $L^+$ and $\widetilde L^+$ are standard in $\Sigma_R$, the manifold $\Sigma_R$ can be partitioned into solid tori $W^+$ and $W^-$ so that $B\subset\operatorname{int} W^-$ and $V^+,\widetilde V^+\subset \operatorname{int} W^+$. Then the spaces $W^+\setminus\operatorname{int} V^+$ and $W^+\setminus\operatorname{int} \widetilde V^+$ are homeomorphic to $\mathbb T^2\times [0,1]$. Hence there exists a homeomorphism $\psi\colon \Sigma_R\to\Sigma_R$ that is identical on $W^-$ and satisfies $\psi(L^+)=\widetilde L^+$ and $\psi(V^+)=\widetilde V^+$. Set $\widetilde h_-=h_-\psi\colon \Sigma_R\to\Sigma_R$. Then $\widetilde h_-(V^+)=V'^+$. Set $\widetilde W^-=\Sigma_R\setminus\operatorname{int}V^+$ and $\widetilde W'^-=\Sigma_{R'}\setminus\operatorname{int}V'^+$. Since $B\subset\widetilde W^-$ is a 3-ball and $\widetilde W^-$ and $\widetilde W'^-$ are solid tori, there exists a homeomorphism $\widetilde\psi\colon \widetilde W^-\to\widetilde W^-$ identical on $B$ such that the isomorphism induced by $\widetilde h_{-}\widetilde\psi|_{T^+}\colon T^+\to T'^+$ has the matrix $\mathcal A$ in the generators $\lambda^+$, $\mu^+$ and $\lambda'^+$, $\mu'^+$. Then there is a homeomorphism $h_R\colon \Sigma_R\to\Sigma_{R'}$ equal to $h_S$ on $V^+$ and to $\widetilde h_{-}\widetilde\psi$ outside a neighbourhood of the solid torus $V^+$. Thus, there is a homeomorphism $h_A\colon \Sigma_A\to\Sigma_{A'}$ coinciding with $h_S$ on $N_T$ and with $h_{R}$ on $\Sigma_{R}\setminus\operatorname{int}(V^-\cup V^+)$. Since $\rho_i=\rho'_i$, $i=1,2$, by Proposition 8 the homeomorphism $h_{\mathcal O}\colon \Sigma_{\mathcal O}\to \Sigma_{\mathcal O'}$, $\mathcal O\in\{R,A\}$, extends to a homeomorphism $ h_{\mathcal O}\colon V_{\mathcal O}\to V_{\mathcal O'}$ establishing the equivalence of the flows $f^t|_{V_{\mathcal O}}$ and $f'^t|_{V_{\mathcal O'}}$. Thus, the required homeomorphism $h\colon {M^4\to M'^4}$ coincides with $h_{\mathcal O}$ on $V_{\mathcal O}$, $\mathcal O\in\{S,A,R\}$. Theorem 1 is proved. § 5. Recovery of flows in $G$ from an admissible abstract schemeRecall that by an abstract scheme we mean a tuple with the following properties:
In this section we prove Theorem 2: for each abstract scheme $\mathcal S$ there exists a flow $f^t\in G$ whose scheme is equivalent to $\mathcal S$. Proof of Theorem 2. Let $\mathcal S=(\Sigma,L,\rho_1,\rho_2)$ be an abstract scheme. We let $\langle L\rangle$ denote the homotopy type of the knot $L$, which is the number of windings made along the generator $e$ of $\Sigma$ (taking account of orientation). Now we describe how a scheme can be realized by a flow $f^t\in G$ with an equivalent scheme. We consider the cases (1) $\langle L\rangle\neq 0$, and (2) $\langle L\rangle=0$ separately.
In case (1) we begin by realizing an arbitrary scheme of the form $\mathcal S=(\Sigma,L,1,2)$, $\langle L\rangle> 0$, by a suspension. First we describe the construction of a diffeomorphism $f_L\colon \mathbb S^3\to\mathbb S^3$ for the given knot $L$. The required flow will be a suspension over this diffeomorphism. Next we show how the suspension can be modified to obtain a flow realizing an arbitrary scheme of the form (1). Since by [3] each Morse–Smale diffeomorphism with a unique saddle orbit is topologically conjugate to $f_L$, this will prove Theorem 3. Constructing a diffeomorphism $f_L\colon \mathbb S^3\to\mathbb S^3$. Let $\mathbf x=(x_1,x_2,x_3)\in\mathbb R^3$ and $\| \mathbf x \|=\sqrt{x_1 ^2 + x_2 ^ 2 + x_3 ^2}$, and let $\nu\colon \mathbb R^3\to\mathbb R^ 3$ be a diffeomorphism described by the formula Next we define a map $p\colon\mathbb R^3\setminus O\to\Sigma$ by
$$
\begin{equation*}
p(\mathbf x)=\biggl(\frac{x}{\|\mathbf x\|}, \log_2(\|\mathbf x\|)\ (\operatorname{mod} 1)\biggr).
\end{equation*}
\notag
$$
Let $L\subset\Sigma$ be a knot such that $\langle L\rangle=m\in\mathbb N$. Then the set $\overline L=p^{-1}(L)$ is a $\nu$-invariant union of $m$ arcs:
$$
\begin{equation*}
\overline L=\overline L_0\sqcup\dots\sqcup \nu^{m-1}(\overline L_0).
\end{equation*}
\notag
$$
Let $U(L)\subset\Sigma$ be a tubular neighbourhood of $L$. Then $U(\overline L)=p^{-1}(U(L))$ is a $\nu$-invariant neighbourhood of the arcs in $\overline L$, which has $m$ connected components $U(\overline L_0),\dots, \nu^{m-1}(U(\overline L_0))$, each of which is diffeomorphic to $\mathbb{D}^{2}\times\mathbb R$.
Let $C=\{(x_1,x_2,x_3)\in\mathbb R^3\colon x_2^2+x_{3}^2\leqslant 4\}$, and let $g\colon C\to C$ be a diffeomorphism defined by Let $g_m\colon C\times\mathbb Z_m\to C\times\mathbb Z_m$ be a diffeomorphism defined by
$$
\begin{equation*}
g_m(\mathbf x,j)=(g(\mathbf x),j+1 \ (\operatorname{mod} m)), \qquad \mathbf x\in C, \quad j\in\mathbb Z_m.
\end{equation*}
\notag
$$
Then there exists a diffeomorphism ${\zeta}\colon {U(\overline L)}\to C\times\mathbb Z_m$ that conjugates $\nu\vert_{{U(\overline L)}}$ to $g_m$. Let $\phi^t$ be a flow on $C$ defined by
$$
\begin{equation*}
\begin{cases} \dot{x}_1= \begin{cases} -\cos\biggl(\dfrac{\pi\|\mathbf x\|}{2}\biggr), & \|\mathbf x\| \leqslant 2, \\ 1, & \|\mathbf x\| > 2; \end{cases} \\ \dot{x}_2=\begin{cases} x_2,& \|\mathbf x\|<1, \\ {x_2}\biggl(1+\cos\biggl(\dfrac{3\pi\|\mathbf x\|}{2}\biggr)\biggr), & 1\leqslant\|\mathbf x\|\leqslant 2, \\ 0, & \|\mathbf x\| > 2; \end{cases} \\ \dot{x}_3=\begin{cases} x_3,& \|\mathbf x\|<1, \\ {x_3}\biggl(1+\cos\biggl(\dfrac{3\pi\|\mathbf x\|}{2}\biggr)\biggr), & 1\leqslant\|\mathbf x\|\leqslant 2, \\ 0, & \|\mathbf x\| > 2. \end{cases} \end{cases}
\end{equation*}
\notag
$$
By construction the diffeomorphism $\phi=\phi^1$ has two fixed points: the saddle point $P(-1,0,0)$ and the source $Q(1,0,0)$ (Figure 9); both are hyperbolic. One stable separatrix of $P$ coincides with the open interval $\bigl\{(x_1,x_2,x_3)\in \mathbb R^3$: $|x_1|<1,\,x_2=x_3=0\bigr\}$ in the basin of $Q$, and the other is the ray $\bigl\{(x_1,x_2,x_3)\in \mathbb R^3\colon x_1<-1,\,x_2=x_3=0\bigr\}$. Moreover, $\phi$ coincides with $g$ outside the ball $\{(x_1,x_2,x_3)\in C\colon x_1^2+x_2^2+x_3^2\leqslant 4\}$. Let $\phi_m\colon C\times\mathbb Z_m\to C\times\mathbb Z_m$ denote the diffeomorphism defined by
$$
\begin{equation*}
\phi_m(\mathbf x,j)=(\phi(\mathbf x),j+1\ (\operatorname{mod} m)), \qquad \mathbf x\in C, \quad j\in\mathbb Z_m.
\end{equation*}
\notag
$$
We define a diffeomorphism $\overline f_{L}\colon \mathbb R^3\to\mathbb R^3$ so that $\overline{f}_{L}$ coincides with $\nu$ outside $U(\overline L)$ and with ${\zeta}^{-1}\phi_m{\zeta}$ on $U(\overline L)$. Then $\overline f_{L}$ has two periodic orbits with period $m$ in $U(\overline L)$: the source orbit $\overline\alpha\sqcup\overline f_L(\overline\alpha)\sqcup\dots\sqcup\overline f^{m-1}_L(\overline\alpha)={\zeta}^{-1}(Q\times\mathbb Z_m)$ and the saddle orbit $\overline\sigma\sqcup\overline f_L(\overline\sigma)\sqcup\dots\sqcup\overline f^{m-1}_L(\overline\sigma)={\zeta}^{-1}(P\times\mathbb Z_m)$, both of which are hyperbolic (Figure 10). Next we project the dynamics onto a $3$-sphere. To do this we let $N(0,0,0,1)$ denote the north pole of the sphere $\mathbb S^3=\{s=(x_1,x_2,x_3,x_{4})\colon x_1^2+x_2^2+x_3^2+x_{4}^2=1\}$. For each point $s\in(\mathbb{S}^3\setminus N)$ there exists a unique line through $N$ and $s$ in $\mathbb R^{4}$; it intersects $\mathbb R^3$ at a unique point $\vartheta(s)$. The stereographic projection $\vartheta\colon \mathbb S^3\setminus N\to\mathbb R^3$ taking $s$ to $\vartheta(s)$ is a diffeomorphism. The corresponding formula is
$$
\begin{equation*}
\vartheta(x_1,x_2,x_3,x_{4})=\biggl(\frac{x_1}{1-x_{4}}, \frac{x_{2}}{1-x_{4}},\frac{x_3}{1-x_{4}}\biggr).
\end{equation*}
\notag
$$
By construction the diffeomorphism $\overline{f}_{L}$ coincides with $\nu$ in a neighbourhood of the point $O$ and in a neighbourhood of the point at infinity; hence it induces a Morse–Smale diffeomorphism on $\mathbb{S}^3$:
$$
\begin{equation*}
f_{{L}}(s)=\begin{cases} \vartheta^{-1}\overline f_{L}\vartheta(s),& s\neq N, \\ N,& s=N. \end{cases}
\end{equation*}
\notag
$$
The nonwandering set of $f_L$ consists of four orbits: the fixed sink $N$, the fixed source $\vartheta^{-1}(O)$, the source orbit $\mathcal O_\alpha=\alpha\sqcup f_L(\alpha)\sqcup\dots\sqcup f^{m-1}_L(\alpha)=\vartheta^{-1}({\zeta}^{-1}(Q\times\mathbb Z_m))$ with period $m$ and the saddle orbit $\mathcal O_\sigma=\sigma\sqcup f_L(\sigma)\sqcup\dots\sqcup f^{m-1}_L(\sigma)=\vartheta^{-1}({\zeta}^{-1}(P\times\mathbb Z_m))$ with period $m$. Constructing and modifying a suspension. Let $[f_L]^t\colon \mathbb S^3\times\mathbb S^1$ denote the suspension over the diffeomorphism $f_L$. Then the nonwandering set of the flow $f^t=[f_L]^t$ consists of four periodic orbits: an attracting orbit $A$, two repelling ones $R^-$ and $R^+$, and a saddle orbit $S$, which are suspensions over the orbits $N,\vartheta^{-1}(O),\mathcal O_\alpha$ and $\mathcal O_\sigma$, respectively. We can see directly from the construction that this flow belongs to $G$, its scheme is equivalent to $(\Sigma,L,1,2)$ and $\langle L\rangle=m$. To realize a flow with negative parameters in the scheme we modify the above flow as follows in neighbourhoods of periodic orbits. Let be an attracting or a repelling periodic orbit of $f^t$ and $V_{\mathcal O}$ be a canonical neighbourhood of $\mathcal O$ with boundary $\Sigma_{\mathcal O}$. Set
$$
\begin{equation*}
V^{t}_{\mathcal O} =f^t(V_{\mathcal O})\quad\text{and} \quad \Sigma^{t}_{\mathcal O} =f^t(\Sigma_{\mathcal O}), \quad t\in\mathbb R, \quad\text{and} \quad K_{\mathcal O}=\bigcup_{t\in[-1,1]}f^t(\Sigma_{\mathcal O}).
\end{equation*}
\notag
$$
Then $K_{\mathcal O}\cong\Sigma\times[-1,1]$. Let $x\in\Sigma$, $t\in[-1,1]$, let $\vec v_{\mathcal O}(x,t)$ be the vector field induced by $f^t$ on $K_{\mathcal O}$ and $\vec n_{{\mathcal O}}(x,t)$ be the vector field of unit outward normals to the hypersurfaces $f^t(\Sigma_{\mathcal O})\cong\Sigma\times\{t\}$. Consider the vector field $\vec w_{\mathcal O}(x,t)$ on $K_{\mathcal O}$ defined by
$$
\begin{equation*}
\vec w_{\mathcal O}(x,t)=(1-|t|)\vec n_{\mathcal O}(x,t)+|t|\vec v_{\mathcal O}(x,t).
\end{equation*}
\notag
$$
Let $\phi^t_{\mathcal O}$ denote the flow on $K_{\mathcal O}$ generated by the field $\vec w_{\mathcal O}$. Without loss of generality we assume that the neighbourhoods $V^{-1}_A, V^1_{R^-}$ and $V^1_{R^+}$ are pairwise disjoint, and we let $\phi^t$ denote the flow on $\mathbb S^3\times \mathbb S^1$ coinciding with $\phi^t_{\mathcal O}$ on the sets $K_{\mathcal O}$ and $\mathcal O\in\{A,R^+,R^-\}$ and with $f^t$ away from these sets. By construction $\phi^t$ belongs to $G$, its periodic orbits are $A,R^-,R^+$ and $S$, and its scheme is equivalent to $(\Sigma,L,1,2)$. Let $\mathcal O$ be an attracting or a repelling periodic orbit of $\phi^t$. Then $V_{\mathcal O}\cong\mathbb D^3\times\mathbb S^1$. Let $d\in\mathbb D^3$ and $e^{i\varphi}\in\mathbb S^1$. For $\delta\in\{-1,+1\}$ let $g_{\mathcal O,\delta}\colon V_{\mathcal O}\to V_{\mathcal O}$ be the diffeomorphism defined by
$$
\begin{equation*}
g_{\mathcal O,\delta}(d,e^{i\varphi}) = (d,e^{i\delta\varphi}).
\end{equation*}
\notag
$$
For the scheme $\mathcal S=(\Sigma,\delta_0L,\delta_1,2\delta_2)$, where $\delta_i\in\{-1,+1\}$, $i=0,1,2$, consider the flow $f^t_{\mathcal S}$ on $\mathbb S^3\times \mathbb S^1$ defined by
$$
\begin{equation*}
f^t_{\mathcal S}(x) = \begin{cases} g_{R^-,\delta_0}\phi^t g_{R^-,\delta_0} (x), & x\in V_{R^-}, \\ g_{R^+,\delta_1}\phi^t g_{R^+,\delta_1} (x), & x\in V_{R^+}, \\ g_{A,\delta_0\delta_2}\phi^t g_{A,\delta_0\delta_2} (x), & x\in V_{A}, \\ \phi^t(x) & \text{otherwise}. \end{cases}
\end{equation*}
\notag
$$
By construction $f^t_{\mathcal S}$ has the invariant $\mathcal S$, and its supporting manifold is $\mathbb S^3\times \mathbb S^1$. In case (2) we realize the flow by ‘attaching’ canonical neighbourhoods. Let $\mathcal S=(\Sigma,L,\rho_1,\rho_2)$ be a scheme in which $L\subset\Sigma$ is a knot such that $\langle L\rangle=0$. We carry out the further constructions separately in the following two cases: (2i) $|\rho_2|=1$; (2ii) $|\rho_2|=2$. In case (2i), as $\langle L\rangle=0$, there exists a 3-ball $B\subset\Sigma$ containing $L$ in its interior. Consider a standard knot $L^+\subset(\Sigma\setminus B)$ such that $\langle L^+\rangle=\rho_1$. Let ${U^-,U^+\subset\Sigma}$ be disjoint tubular neighbourhoods of the knots $L$ and $L^+$, respectively. Set $E^+_R= \operatorname{cl}(\Sigma\setminus (U^+\cup B))$ and $E^-_R=\operatorname{cl}(B\setminus U^-)$. Let $p_{\rho_2}\colon \mathbb S^2\times[0,1]\to \Sigma_0$ be a smooth map that is a diffeomorphism onto $\mathbb S^2\times(0,1)$ such that $p_{\rho_2}(s,0)=p_{\rho_2}(s,1)$ and $\langle p_{\rho_2}(\{s\}\times[0,1])\rangle_0=\rho_2$. In $\mathbb S^2\times[0,1]$ we consider a smooth submanifold $N\cong\mathbb T^2\times[-1,1]$ such that the set $\mathbb S^2\times[0,1]\setminus \operatorname{int} N$ has two connected components $E^-_A\cong E^-_R$ and $E^+_A\cong E^+_R$ which contain the spheres $\mathbb S^2\times\{0\}$ and $\mathbb S^2\times\{1\}$ on their respective boundaries. Recall that $\mathbb{T}\!=\!v_{2}(\overline T)$ is a 2-torus with a tubular neighbourhood $N_{\mathbb T}\!=\!v_{2}(\operatorname{cl}(N_{\overline T}))$, whose complement in $\Sigma_{2}\cong\mathbb S^2\times\mathbb S^1$ consists of two solid tori $\mathbb V^-$ and $\mathbb V^+$ (see § 3.3). Set $U=p_{\rho_2}(N)$, $U_A=\operatorname{cl}(\Sigma_0\setminus U)$ and $U_R=\operatorname{cl}(\Sigma\setminus (U^+\cup U^-))$. Then there exist diffeomorphisms $j_R\colon U^-\,{\sqcup}\, U^+\to \mathbb V^-\,{\sqcup}\,\mathbb V^+$, $j_A\colon U\to N_T$ and $j_{RA}\colon U_R\to U_A$ such that
$$
\begin{equation*}
\begin{gathered} \, \langle j_R(L)\rangle_2=\langle j_R(L^+)\rangle_2=1, \\ j_{RA}(E^-_R)=E^-_A, \qquad j_{RA}(E^+_R)=E^+_A\quad\text{and}\quad j_{RA}|_{\partial U_R}=j_Aj_R|_{\partial U_R}. \end{gathered}
\end{equation*}
\notag
$$
Assume without loss of generality that $\Sigma=\Sigma_3$ and $e=e_3$. We set $\widetilde M^3=\mathbb V_0 \sqcup\mathbb V_2\sqcup\mathbb V_3$ and introduce a minimal equivalence relation $\sim$ on the set $\widetilde M^3$ by setting
$$
\begin{equation*}
\begin{gathered} \, \widetilde x \sim j_R(\widetilde x) \quad\text{for } \widetilde x\in(U^-\sqcup U^+), \\ \widetilde x \sim j_A(\widetilde x) \quad\text{for } \widetilde x\in U \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\widetilde x \sim j_{RA}(\widetilde x) \quad\text{for } \widetilde x\in U_R.
\end{equation*}
\notag
$$
Let $M^3$ denote the set of equivalence classes modulo the above relation and $p\colon \widetilde M^3\to M^3$ be the natural projection. Consider the flow $f^t\colon M^3\to M^3$ defined by
$$
\begin{equation*}
f^t(x) = \begin{cases} p[a_{0}]^t(p|_{\mathbb V_0})^{-1}(x), &x\in p(\mathbb V_0), \\ p[a_{2}]^t(p|_{\mathbb V_2})^{-1}(x), &x\in p(\mathbb V_2), \\ p [a_{3}]^t(p|_{\mathbb V_3})^{-1}(x), &x\in p(\mathbb V_3). \end{cases}
\end{equation*}
\notag
$$
It is continuous. Using methods similar to the ones in part (1) we can smoothen it and obtain a required flow $f^t_{\mathcal S}$ with invariant $\mathcal S$. In case (2ii), since $\langle L\rangle=0$, we can assume without loss of generality that ${e\cap L=\varnothing}$. Consider a standard knot $L^+\subset(\Sigma_3\setminus B)$ such that $\langle L^+\rangle_3=\rho_1$. Let $U^-\subset\Sigma$ and $U^+\subset\Sigma_3$ be disjoint tubular neighbourhoods of the knots $L$ and $L^+$, respectively. Set $E_{R^+}=\operatorname{cl}(\Sigma_3\setminus U^+)$ and $E_{R^-}=\operatorname{cl}(\Sigma\setminus U^-)$. In $\Sigma_0$ we consider a smooth submanifold $U\cong\mathbb T^2\times[-1,1]$ such that the set $\Sigma_0\setminus \operatorname{int} U$ has two connected components $E^-_A\cong E_{R^-}$ and $E^+_A\cong E_{R^+}$ and there exists a diffeomorphism $j^-_{RA}(E_{R^-})=E^-_A$ such that $\langle j^-_{RA}(e)\rangle_0={\rho_2}/{2}$. Then there exist diffeomorphisms $j_R\colon U^-\sqcup U^+\to \mathbb V^-\sqcup\mathbb V^+$, $j_A\colon U\to N_T$ and $j^+_{RA}\colon E_{R^+}\to E^+_A$ such that
$$
\begin{equation*}
\langle j_R(L)\rangle_2=\langle j_R(L^+)\rangle_2=1\quad\text{and} \quad j^\pm_{RA}|_{\partial U^\pm}=j_A j_R|_{\partial U^\pm}.
\end{equation*}
\notag
$$
Assume without loss of generality that $\Sigma=\Sigma_3$ and $e=e_3$. We set $\widetilde M^3=\mathbb V_0 \sqcup\mathbb V_2\sqcup\mathbb V_3\sqcup\mathbb V_3$ and introduce the minimal equivalence relation $\sim$ on $\widetilde M^3$ such that
$$
\begin{equation*}
\begin{gathered} \, \widetilde x \sim j_R(\widetilde x) \quad\text{for } \widetilde x\in(U^-\sqcup U^+), \\ \widetilde x \sim j_A(\widetilde x) \quad\text{for } \widetilde x\in U \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\widetilde x \sim j^\pm_{AR}(\widetilde x) \quad\text{for } \widetilde x\in E_{R^\pm}.
\end{equation*}
\notag
$$
Let $M^3$ denote the set of equivalence classes modulo this relation and $p\colon \widetilde M^3\to M^3$ be the natural projection.Then we define a flow $f^t\colon M^3\to M^3$ similarly to case (2i). Theorem 2 is proved. § 6. Topology of the supporting manifold for flows in $G$In this section we prove Theorem 4. But first we establish the following fact, which is essentially used in the proof. Lemma 4. Let Then there exists a homeomorphism $H\colon P\to\widetilde P$ that is identical outside an arbitrary fixed neighbourhood $V_Q\subset P$ of $Q$ and such that $V_Q\cap\partial P=Q$. Proof. If $\partial Q=\varnothing$, then the required result follows from Lemma 3 in [8]. Otherwise for $t\in[0,1]$ consider the isotopy $z_t\colon [0,1]\to[0,1]\times[0,1]$ defined by
$$
\begin{equation*}
z_t(s)=\begin{cases} ((1-t)s,2ts), &s\in\biggl[0,\dfrac12\biggr], \\ (1+(t+1)(s-1),t),& s\in\biggl[\dfrac12,1\biggr]. \end{cases}
\end{equation*}
\notag
$$
We write the isotopy $z_t$ as Consider a ‘collar’ of $\partial Q$ in $Q$, that is, an embedding $\mu\colon \partial Q \times [-1,0]\to Q$ such that $\mu(q,0) = q$ for all $ q\in \partial Q$. Set $\check Q=\mu(\partial Q \times [-1,0])$ and $\widehat Q =\operatorname{cl}(Q\setminus\check Q)$. For $t\in[0,1]$ we define an isotopy $Z_t\colon Q\to Q\times[0,1]$ by We express $Z_t$ in the form Consider a ‘collar’ of $Q$ in $V_Q$, that is, an embedding $\nu\colon Q \times [-1,0]\to V_Q$ such that $\nu(q,0) = q$ for $ q\in Q$. Set $R^-=\nu(Q\times[-1,0])$. For $t\in[-1,0]$ set $Z^-_t(q)=(X_t(q),-Y_t(q))$ and $Z_t=\nu Z^-_t\colon Q\to R^-$. Then the required homeomorphism $H\colon P\to\widetilde P$ is defined by
The proof is complete. Now we are ready to prove Theorem 4: the supporting manifold of each flow $f^t\in G$ if homeomorphic to $\mathbb S^3\times\mathbb S^1$. Proof of Theorem 4. Let the flow $f^t$ belong to the class $G$. By Theorem 1 the equivalence class of the scheme $\mathcal S_{f^t}=(\Sigma_R,L,\rho_1,\rho_2)$ is a complete invariant of the flow. By Theorem 2 each flow $f^t\in G$ is equivalent to the flow recovered from the abstract scheme $\mathcal S=(\Sigma_R,L,\rho_1,\rho_2)$, which is equivalent to $\mathcal S_{f^t}$.
The further proof is carried out separately in the following two cases: (1) $\langle L\rangle\neq 0$; (2) $\langle L\rangle=0$. In case (1) the supporting manifold of the flow constructed from the knot $\langle L\rangle\neq 0$, is homeomorphic to $\mathbb S^3\times\mathbb S^1$, so Theorem 4 is known to hold in this case. In case (2) set $D^{\mathrm{s}}_S=W^{\mathrm{s}}_S\cap V_S$. By construction $D^{\mathrm{s}}_S$ is diffeomorphic to an annulus $\mathbb S^1\times[-1,1]$ bounded by the knots $L^-$ and $L^+$. Consider a tubular neighbourhood $V^{\mathrm{s}}_S\subset V_S$ of $D^{\mathrm{s}}_S$ which is diffeomorphic to $\mathbb D^2\times\mathbb S^1\times[-1,1]$, and consider a diffeomorphism $\nu\colon \mathbb D^2\times\mathbb S^1\times[-1,1]\to V^{\mathrm{s}}_S$ such that $\nu(\mathbb D^2\times\mathbb S^1\times\{-1\})=V^-$, $\nu(\mathbb D^2\times\mathbb S^1\times\{+1\})=V^+$ and the set $\nu(\mathbb S^1\times\mathbb S^1\times[-1,1])$ is transversal to trajectories of $f^t$. Set $W_R=V^{\mathrm{s}}_S\cup V_{\mathcal R}$ and $W_A=M^3\setminus\operatorname{int}W_R$. By construction $\partial W_A$ intersects all trajectories of $f^t$ lying in the set $W^{\mathrm{s}}_A\setminus A$, and therefore $W_A\cong V_A\cong\mathbb D^3\times\mathbb S^1$. We show below that $W_R\cong\mathbb D^3\times\mathbb S^1$. Then $M^3=W_A\cup W_R\cong\mathbb D^3\times\mathbb S^1$ by [12]. We prove that $W_R\cong\mathbb D^3\times\mathbb S^1$ separately in the following two cases: (2i) $|\rho_2|=1$; (2ii) $|\rho_2|=2$. In case (2ii) we have $V_{\mathcal R}=V_{R^-}\sqcup V_{R^+}$. Since $L^+$ is a standard knot in $V_{R^+}$, there exists a homeomorphism $\nu_+\colon \mathbb D^2\times\mathbb S^1\times[1,2]\to V_{R^+}$ such that $\nu_+(\mathbb D^2\times\mathbb S^1\times\{1\})=V^+$. Then (for instance, see [8], Lemma 3) $V^{\mathrm{s}}_S\cup V_{R^+}\cong \mathbb D^2\times\mathbb S^1\times[-1,2]$. Thus, $V^{\mathrm{s}}_S\cup V_{R^+}\cong V^-\times[-1,2]$, and therefore $W_R\cong V_{R^-}\cup V^-\times[-1,2]$, where $V_{R^-}\cap V^-\times[-1,2]=V^-$. By Lemma 4 In case (2i) we have $V_{\mathcal R}=V_{R}$, $L^+$ is a standard knot in $V_{R}$, and there exists a 3-ball $B\subset \Sigma_R$ such that $V^-\subset \operatorname{int}B$ and $B\cap V^+=\varnothing$. Consider a 4-ball $\widetilde B\subset V_R$ such that $\widetilde B\cap\Sigma_R=\partial\widetilde B\cap\Sigma_R=B$. Then $B_0=\operatorname{cl}(\partial\widetilde B\setminus B)\cong\mathbb D^3$. Set $\overline W_R=\operatorname{cl}(W_R\setminus B_0)$. By construction $\overline W_R$ is a connected 4-manifold with boundary and $W_R$ is obtained from it by identifying two disjoint copies of the ball $B_0$ on its boundary (by means of an orientation-reversing homeomorphism). Now the proof of the theorem reduces to showing that $\overline W_R\cong \mathbb D^4$. To do this we represent $\overline W_R$ as the union $\overline W_R=\widetilde V_R\cup\widetilde V^{\mathrm{s}}_S$, where $\widetilde V_R=\operatorname{cl}(V_R\setminus \widetilde B)$ and $\widetilde V^{\mathrm{s}}_S=V^{\mathrm{s}}_S\cup\widetilde B$. By construction $\widetilde V_R \cap \widetilde V^{\mathrm{s}}_S = V^-$. Similarly to (2ii), we show that $\widetilde V_R\cup V^{\mathrm{s}}_S \cong V^-\times[-1,2]$, and therefore $\overline W_R\cong \widetilde B\cup V^-\times[-1,2]$, where $\widetilde B\cap V^-\times[-1,2]=V^-$. By Lemma 4 Theorem 4 is proved.
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\Bibitem{GalPocShu24} |
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