Vector field on 3-sphere

This is not possible if the sources and sinks form a Morse decomposition. That means that there are no other invariant sets of the flow of the vector field. This happens if one can find a function $f:S^3\rightarrow\mathbb{R}$ that strictly decreases along the flow, except at the sources and sinks. For example if the flow is the gradient flow of a function (not necessarily Morse). The Morse-Conley relations state that, for a Morse decomposition of the three sphere

$$ \sum P_t(S_i)=P_t(S^3)+(1+t)Q_t $$

Where $P_t(S^3)=1+t^3$ is the Poincaré polynomial of the three sphere, $P_t(S_i)$ are the Poincaré polynomials of the Conley indices of the invariant sets, and $Q_t$ is a polynomial with non-negative coefficients. The Poincaré polynomial of a source is $t^3$ and of a sink it is $1$. Convince yourself that it is not possible to solve the equation, if there is more than one source or sink.

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However, it is possible if the sources and sinks do not form a Morse decomposition. I think I found an example of a vector field on $S^3$ with two sources, two sinks, and one periodic orbit. The idea is to start with the standard heightfunction, and change the sink into two sinks, a source and a periodic orbit, using a Hopf bifurcation.

Replace the a neighbourhood of the source of the standard height function by the vector field whose flow lines are like this:

The vector field has two sources, two sinks and a periodic orbit. Of course one can iterate this procedure around the sinks to obtain a vector field with $n$ sinks, $n$ sources and $n-1$ periodic orbits

The answer is that $\{N\} = 2\mathbb{N}$ such that both of the number of sinks and sources are $0.5 N$. Poincare-Hopf Theorem promises that $\{N\} \subset 2\mathbb{N}$ and the fact both of the number of sinks and sources are $0.5 N$.

Lemma 3.4 of this paper based on the main Theorem of this paper ensures that for any $K\in \mathbb{N}$, there exists a smooth flow whose singularities are composed of $K$ sinks and $K$ sources.

Actually, we can construct such a flow directly.

Claim: For any $K\in \mathbb{N}$, there exist a smooth flow $\phi_t$ on $S^3$ whose limit set is composed of $K$ sinks, $K$ sources and $2K$ saddle closed orbits.

Here, I only provide the main ideas.

'Proof:'

• Suppose $W$ is a genus $K$ handlebody with $K$ small holes. $\partial W =\Sigma \cup S_1 \cup \dots \cup S_K$ where $S_i$ is a 2-shpere and $\Sigma$ is a genus $K$ closed orientable surface.
• To prove the claim, one only need to construct a smooth flow $\psi_t$ on $W$ such that, (1) $\psi_t$ is transverse inward to $\Sigma$ and outward to $S_i$. (2) The limit set of $\psi_t$ is composed of $K$ saddle closed orbits.

(If this is true, one can glue $W$ and $W'$ along $\Sigma$ (in some sense Heegaard splitting), then glue some neighborhoods of sinks and sources to form $\phi_t$ on $S^3$ in the Claim.Here $W'$ is $W$ with converse orientation).

• For any $k\in \mathbb{N}$, there exists $V_k$ with smooth flow $\psi_t^k$ such that,

  (1) $\partial V_k = \Sigma_k^+ \cup T^+ \cup \Sigma_{k+1}^- \cup S^- $. $\Sigma_k^+$ and $\Sigma_{k+1}^-$ are genus $k$ and genus $k+1$ surfaces correspondingly. $T^+$ and $S^-$ are a torus and a shpere correspondingly.

  (2) The topology of $V_k$ is very well, see

  .

  (3)$\psi_t^k$ is transverse inward to $\Sigma_k^+$ and $T^+$ and outward to $\Sigma_{k+1}^-$ and $S^-$.

  (4)The maximal invariant set of $\psi_t^k$ is a saddle closed orbit $\gamma_i$.

• Then we can glue $(V_k, \psi_t^k)$ ($k\in \{1,\dots, (N-1)\}$) together by attaching $\Sigma_{k}^-$ to $\Sigma_{k}^+$ suitablely. We can obtain $(W,\psi_t)$ which we expected.