A universal extension of helicity to topological flows

Let $(Y^{3},\mu)$ be a closed smooth $3$-manifold equipped with a volume form $\mu$, and let $X$ be a smooth vector field on $Y$ that preserves $\mu$. That is, the flow $\varphi^{t}_{X}$ of $X$ satisfies $(\varphi^{t}_{X})^{*}\mu=\mu$ for all $t$. This condition is equivalent to the $2$-form

being closed. The vector field $X$ is said to be exact if $\omega$ is exact.

The helicity of an exact volume-preserving vector field $X$ is defined as

where $\alpha$ is any primitive $1$-form of $\omega$ and $Y$ is oriented via the volume form $\mu$. This integral turns out to be independent of the choice of primitive $1$-form $\alpha$.

A simple but fundamental property of helicity is that it is preserved under volume- and orientation-preserving diffeomorphisms; that is,

In fact, any other functional on the space of exact volume-preserving vector fields that is invariant under the action of $\operatorname{Diff}^{+}(Y,\mu)$ and satisfies certain natural regularity conditions must be a function of helicity [29, 16].

Two volume-preserving smooth vector fields $X_{1}$ and $X_{2}$ on $(Y,\mu)$ are said to be topologically conjugate if their flows are conjugate via a volume- and orientation-preserving homeomorphism, i.e. if there exists $f\in\operatorname{Homeo}^{+}(Y,\mu)$ such that

For examples of volume-preserving smooth vector fields that are topologically conjugate but not smoothly (or even $C^{1}$) conjugate, see [42, §10 & 11].

In his 1973 article [2], Arnold, having derived an ergodic interpretation of helicity, posed the following questions concerning the topological invariance of helicity.

In this paper we address the above questions for fixed-point-free volume-preserving flows, which is the setting considered by Ghys in [22, §5.4].

We provide a preliminary version of our results here, postponing the statement of our main result, Theorem 1.5, to Section 1.5, as it requires some preparatory material.

We briefly comment on the assumptions in the theorem. The notion of exactness for topological flows generalizes its smooth counterpart and is essential for defining helicity, even in the smooth setting; see Definitions 7.2 & 8.20. The condition that flow lines have zero measure is quite natural; it rules out certain pathological situations (see Example 8.12). The same assumption appears also in other works [20, 10]. The significance of the fixed-point-free condition is that it yields a $C^{0}$ Hamiltonian structure, enabling us to build on recent advances in $C^{0}$ symplectic topology and the understanding of the algebraic structure of groups of area-preserving homeomorphisms [13, 11, 14].

Now, fix a topological volume-preserving flow $\psi^{t}$ on $(Y,\mu)$ and suppose that we have a topological volume-preserving embedding

which intertwines the flow on $(0,1)\times\Sigma$ generated by the vector field $\partial_{t}$ and the flow $\psi^{t}$. Consider a $C^{0}$ Hamiltonian isotopy $\varphi^{t}\in\overline{\operatorname{Ham}}(\Sigma)$ and note that its suspension to $(0,1)\times\Sigma$ is volume preserving. We refer to the tuple $\mathcal{P}:=(\Sigma,\omega_{\Sigma},\alpha,\varphi^{t})$ as a plug. Given a plug $\mathcal{P}$, one can define a new volume-preserving flow $\psi^{t}\#\mathcal{P}$ on $Y$ by replacing the flow $\psi^{t}$ inside $\operatorname{im}(\alpha)$ with the suspension of $\varphi^{t}$. The compatibility condition between our helicity extension $\overline{\mathcal{H}}$ and the Calabi extension $\overline{\operatorname{Cal}}_{\Sigma}$ is given by

Our story begins in 1973, when Arnold [1] interpreted the helicity of a vector field as the average asymptotic linking number of flow lines, which motivated Questions 1.1 & 1.2; see Sections 2-4 of the English translation [2] and Problem 1973-23 in [3]. Two trajectories, beginning at two randomly chosen points in space, are followed for a long time and then closed into loops using a well-chosen system of short geodesic arcs.¹¹1The existence of the system of geodesic arcs is a subtle point which was proven rigorously in [54]. The linking number of these loops—averaged over time and all pairs of initial points—converges to the helicity as the time tends to infinity. Since linking numbers are preserved under homeomorphisms, at first glance this seems to imply topological invariance of helicity. However, as remarked by Ghys in [21], “… one should be cautious that a homeomorphism might entangle the small geodesic arcs that were used to close the trajectories.”

Since their formulation, Questions 1.1 and 1.2 have reappeared in various works [4, 20, 21, 51, 42], including Arnold and Khesin’s textbook [4, III, Problem 4.8], Ghys’ plenary ICM address [21, Section 1.4], and Tao’s blog [51]. The case of flows without rest points is emphasized again in [22, Sec. 5.4]. One reason for this ongoing interest is that real-world flows often lack smoothness. Therefore, a positive answer to these questions would highlight the significance of helicity as a meaningful invariant, even in low-regularity settings. As Tao notes [51], “This would be of interest in fluid equations, as it would suggest that helicity remains invariant even after the development of singularities in the flow.”

We should mention that Arnold’s influential article has prompted substantial further work in various directions; see, for example, [19, 9, 20, 10, 46, 28, 32, 42, 30]. Among these, Gambaudo & Ghys [20] and Müller & Spaeth [42] contain results towards Questions 1.1 & 1.2. Both cases in [20, 42] involve exact volume-preserving flows without fixed points, which are covered by Theorem 1.3. The Gambaudo–Ghys approach establishes the topological invariance of helicity for certain suspension flows by relating it to the Calabi invariant—a connection that is also central to our work.

The extension of helicity to low-regularity settings is the subject of ongoing research in fluid dynamics: In [23], Giri, Kwon and Novack extend helicity to a class of weak solutions of the Euler equations with regularity $H^{\frac{1}{2}-\varepsilon}$. Using convex integration, they further show that helicity need not be conserved for such solutions. This does not contradict our result, since the vector fields they construct are too irregular to generate a flow. It would be interesting to compare their extension of helicity with ours on the overlap of their respective domains.

Maximal non-degeneracy of $\omega$ means that $\ker\omega$ defines a line field on $Y$, called the characteristic line field, which integrates to a $1$-dimensional foliation on $Y$, called the characteristic foliation. The Hamiltonian structure $\omega$ naturally equips this foliation with a coorientation and a transverse measure locally diffeomorphic to the standard $2$-dimensional Lebesgue measure. Conversely, every such foliation uniquely determines a Hamiltonian structure.

The relevance of Hamiltonian structures to our discussion is as follows: Equip $Y$ with a volume form $\mu$. Then, a vector field $X$ is nowhere vanishing and volume-preserving if and only if $\omega\coloneqq\iota_{X}\mu$ is a Hamiltonian structure. In this case, the flow $\varphi_{X}^{t}$ is tangent to the characteristic foliation of $\omega$.

We say a Hamiltonian structure $\omega$ is exact if $\omega$ is an exact $2$-form. We define the helicity of an exact Hamiltonian structure on an oriented closed smooth $3$-manifold $Y$ to be

where $\alpha$ is any choice of primitive $1$-form of $\omega$. As already mentioned, this integral is independent of the choice of $\alpha$. By definition, a nowhere vanishing volume-preserving vector field $X$ is exact if and only if the induced Hamiltonian structure $\omega$ is exact. If this is the case, we have

This perspective allows us to reformulate Question 1.1 concerning the topological invariance of helicity in terms of Hamiltonian structures. Specifically, the question becomes whether two exact Hamiltonian structures related by pullback under an orientation-preserving homeomorphism necessarily have the same helicity. To make sense of pullbacks by homeomorphisms, we rely on the interpretation of Hamiltonian structures as cooriented measured foliations.

In the smooth setting, Gambaudo–Ghys [20] discovered a natural connection between helicity and the Calabi invariant. We construct our extensions of helicity such that this connection continues to hold in the $C^{0}$ setting.

Let $(\Sigma,\omega_{\Sigma})$ be an open surface with an area form, and let $\operatorname{Ham}(\Sigma)$ and $\overline{\operatorname{Ham}}(\Sigma)$ denote, respectively, the groups of compactly supported Hamiltonian diffeomorphisms and homeomorphisms of $(\Sigma,\omega_{\Sigma})$; see Section 2. We denote by $\operatorname{Cal}_{\Sigma}:\operatorname{Ham}(\Sigma)\rightarrow{\mathbb{R}}$ the Calabi homomorphism [6], which is defined as follows: for $\varphi=\varphi_{H}^{1}\in\operatorname{Ham}(\Sigma)$, generated by a compactly supported Hamiltonian $H\in C^{\infty}([0,1]\times\Sigma)$, we have³³3Different conventions exist for defining the Calabi invariant; we include the factor 2 for convenience.

The $2$-form $\omega_{\Sigma}$ induces a natural Hamiltonian structure on $(0,1)\times\Sigma$, which we denote by the same symbol $\omega_{\Sigma}$. Given an oriented $3$-manifold $Y$ equipped with a $C^{0}$ Hamiltonian structure $\Omega$, we define a plug to be a tuple

where $\alpha:((0,1)\times\Sigma),\omega_{\Sigma})\hookrightarrow(Y,\Omega)$ is an embedding of $C^{0}$ Hamiltonian structures and $\varphi^{t}\in\overline{\operatorname{Ham}}(\Sigma)$ is a $C^{0}$ Hamiltonian isotopy. Given a plug $\mathcal{P}$, we define an operation, called plug insertion, which creates a new $C^{0}$ Hamiltonian structure $\Omega\#\mathcal{P}$ on $Y$ by replacing the $C^{0}$ Hamiltonian structure $\Omega$ inside $\operatorname{im}(\alpha)$ with the pushforward of the $C^{0}$ Hamiltonian structure on $(0,1)\times\Sigma$ induced by $\omega_{\Sigma}$ via the embedding $\alpha\circ\Phi$, where $\Phi:(0,1)\times\Sigma\rightarrow(0,1)\times\Sigma$ is defined by $\Phi(t,p)\coloneqq(t,\varphi^{t}(p))$.⁴⁴4Plug insertion does not affect exactness of Hamiltonian structures, in smooth and $C^{0}$ settings. The effect of this operation on the characteristic foliations is as follows: Before plug insertion, the characteristic leaves of $\Omega$ inside $\operatorname{im}(\alpha)$ are of the form $\alpha((0,1)\times\{p\})$ for $p\in\Sigma$. After plug insertion, the characteristic leaves of $\Omega\#\mathcal{P}$ are of the form

for $p\in\Sigma$.

In the smooth setting, the following important identity, which can be deduced from [20], relates the helicities of $\Omega$ and $\Omega\#\mathcal{P}$:

The Calabi homomorphism was recently extended to the group of Hamiltonian homeomorphisms [14, 34]. In light of this, it is natural to require that identity (1.2) continues to hold for an extension of helicity to $C^{0}$ Hamiltonian structures. However, in the smooth setting, the Calabi homomorphism satisfies a certain naturality property—implicit in identity (1.2)—which was not known to hold in the $C^{0}$ category. Below, we establish this naturality in the $C^{0}$ setting via Theorem 1.4—a result of independent interest that plays a key role in our main theorem.

We are now in position to state our main result, which defines helicity $\overline{\mathcal{H}}(\Omega)$ for an arbitrary exact $C^{0}$ Hamiltonian structure $\Omega$ on an oriented closed topological $3$-manifold $Y$. The notion of exactness for $C^{0}$ Hamiltonian structures is defined as the vanishing of a certain cohomology class $\overline{\mathrm{Flux}}(\Omega)\in H^{2}(Y;{\mathbb{R}})$. In particular, if $Y$ is a rational homology three-sphere, then every $C^{0}$ Hamiltonian structure is exact. This is discussed in detail in Section 7.1.

Recall that $\Omega\#\mathcal{P}$ denotes the plug insertion operation introduced in Section 1.4. Moreover, $f^{*}\Omega$ stands for the pullback of $\Omega$ under a homeomorphism $f$.

As already mentioned, every choice of projection $\operatorname{pr}:\mathcal{R}\rightarrow{\mathbb{R}}$ gives rise to a real-valued extension of helicity.

We point out that it follows from the Extension and Invariance properties in Theorem 1.5 that two smooth Hamiltonian structures which are conjugated by an orientation-preserving homeomorphism have the same helicity. The proof of this makes essential use of the injectivity of the natural map ${\mathbb{R}}\cong\operatorname{Ham}^{\operatorname{ab}}({\mathbb{D}})\rightarrow\overline{\operatorname{Ham}}^{\operatorname{ab}}({\mathbb{D}})\cong\mathcal{R}$.

Let us elaborate further on the universality of our helicity extension $\overline{\mathcal{H}}$ and the naturality of the properties (Extension, Invariance, and Calabi) that characterize it in Theorem 1.5. The properties Extension and Invariance can be viewed as minimal requirements that any reasonable extension of helicity to $C^{0}$ Hamiltonian structures ought to satisfy. In light of identity (1.2), which describes the helicity change under plug insertion in the smooth setting, the Calabi property—though less obvious—is likewise a natural condition to impose.

To further motivate this property, observe that identity (1.2) has the following direct consequence: the helicity change $\mathcal{H}(\omega\#\mathcal{P})-\mathcal{H}(\omega)$ resulting from the insertion of a smooth plug $\mathcal{P}=(\Sigma,\omega_{\Sigma},\alpha,\varphi^{t})$ depends only on the time-one map $\varphi^{1}$, and is independent of both the ambient Hamiltonian structure $\omega$ and the embedding $\alpha$ of the plug. We refer to this fundamental property as plug homogeneity. Remarkably, imposing plug homogeneity—together with Extension and Invariance—already forces any helicity extension to be a function of our universal extension $\overline{\mathcal{H}}$.

This means that, once plug homogeneity is accepted as a fundamental property of helicity, one is essentially forced to arrive at our universal $\mathcal{R}$-valued helicity extension $\overline{\mathcal{H}}$. Moreover, insisting on plug homogeneity makes the problem of defining an ${\mathbb{R}}$-valued extension of helicity equivalent to specifying a projection $\operatorname{pr}:\mathcal{R}\rightarrow{\mathbb{R}}$, which, as explained in Subsection 1.4, cannot be done without making non-canonical choices. This indicates that $\mathcal{R}$ is indeed the natural target group for a helicity extension to exact $C^{0}$ Hamiltonian structures.

In Section 2, we review some preliminaries concerning area-preserving homeomorphisms, foliations, and transverse measures.

In Section 3, we prove a generalization of Theorem 1.4, which leads to the construction of our universal $\mathcal{R}$-valued extension of the Calabi homomorphism satisfying the naturality property stated in Section 1.4.

In Sections 4 and 5, we provide precise definitions of smooth and $C^{0}$ Hamiltonian structures, along with the plug insertion operation.

Section 6 presents two central results—Theorems 6.1 and 6.11—which describe the structure of $C^{0}$ Hamiltonian structures and their homeomorphisms. These theorems roughly state that such structures and maps can be smoothed up to the insertion of a carefully constructed plug.

In Section 7, we extend the definitions of flux and helicity to $C^{0}$ Hamiltonian structures, thereby completing the proof of Theorem 1.5. We also provide a proof of Proposition 1.6.

Finally, in Section 8, we carefully explain the relationship between Hamiltonian structures and volume-preserving flows, both in the smooth and in the $C^{0}$ setting. This is essential for deducing Theorem 1.3, which is formulated for topological volume-preserving flows, from our main result, Theorem 1.5, stated for $C^{0}$ Hamiltonian structures.

Let $(\Sigma,\omega)$ be a smooth $2$-manifold equipped with an area form. We assume that $\Sigma$ does not have boundary, but we allow it to be open.

Every compactly supported smooth Hamiltonian $H:[0,1]\times\Sigma\rightarrow{\mathbb{R}}$ induces a time-dependent Hamiltonian vector field $X_{H}$ generating a compactly supported Hamiltonian isotopy $(\varphi_{H}^{t})_{t\in[0,1]}$. We adopt the sign convention that $X_{H}$ is characterized by the identity $\iota_{X_{H_{t}}}\omega=dH_{t}$. The compactly supported Hamiltonian diffeomorphism group of $(\Sigma,\omega)$ is denoted by $\operatorname{Ham}(\Sigma,\omega)$.

Let $\operatorname{Homeo}_{c}(\Sigma,\omega)$ denote the group of all compactly supported homeomorphisms of $\Sigma$ preserving the measure induced by $\omega$. We topologize $\operatorname{Homeo}_{c}(\Sigma,\omega)$ as the direct limit of all the subgroups $\operatorname{Homeo}_{K}(\Sigma,\omega)$ of homeomorphisms supported inside some compact subset $K\subset\Sigma$, equipped with the compact-open topology. The group of Hamiltonian homeomorphisms $\overline{\operatorname{Ham}}(\Sigma,\omega)$ is defined to be the closure of $\operatorname{Ham}(\Sigma,\omega)$ inside $\operatorname{Homeo}_{c}(\Sigma,\omega)$. It is topologized as a subspace of $\operatorname{Homeo}_{c}(\Sigma,\omega)$.

Let $\operatorname{Homeo}_{0}(\Sigma,\omega)$ denote the identity component of $\operatorname{Homeo}_{c}(\Sigma_{,}\omega)$. Then $\overline{\operatorname{Ham}}(\Sigma,\omega)$ agrees with the kernel of the mass flow homomorphism [18, 48]

where $\Gamma\subset H^{1}(\Sigma;{\mathbb{R}})$ is a discrete subgroup whose precise definition will not be relevant for us.

Whenever there is no risk of confusion, we will abbreviate $\operatorname{Ham}(\Sigma)\coloneqq\operatorname{Ham}(\Sigma,\omega)$ and $\overline{\operatorname{Ham}}(\Sigma)\coloneqq\overline{\operatorname{Ham}}(\Sigma,\omega)$.

It is known that $\operatorname{Ham}(\Sigma)$ and $\overline{\operatorname{Ham}}(\Sigma)$, equipped with the $C^{0}$ topology, are both simply connected, unless $\Sigma$ is a sphere. If $\Sigma$ is a sphere, the fundamental group is $\mathbb{Z}/2\mathbb{Z}$. In the case of $\operatorname{Ham}(\Sigma)$, a proof of these facts is sketched in [44, Sec. 7.2] for closed $\Sigma$. The arguments therein can be adapted to the case of open $\Sigma$. As for $\overline{\operatorname{Ham}}(\Sigma)$, it can be shown that every loop in $\overline{\operatorname{Ham}}(\Sigma)$ is homotopic a loop in $\operatorname{Ham}(\Sigma)$ using the following two facts: first, every homeomorphism in $\overline{\operatorname{Ham}}(\Sigma)$ can be written as a $C^{0}$ limit of elements in $\operatorname{Ham}(\Sigma)$; and second, $\operatorname{Ham}(\Sigma)$ and $\overline{\operatorname{Ham}}(\Sigma)$ are both locally path connected; see, for example, [49, Cor. 2] or [50, Lem. 3.2].

The following two propositions will be used in the upcoming sections. Both are variants of known fragmentation and approximation results for area-preserving maps and can be established using classical techniques—specifically, fragmentation techniques and approximation of homeomorphisms by diffeomorphisms in the area-preserving setting; see, e.g., [33, 17]. We therefore omit the proofs.

Let $M^{n}$ be an oriented $C^{r}$ manifold for $r\in{\mathbb{Z}}_{\geq 0}\cup\{\infty\}$. For $k\geq 0$, let $B^{k}\subset{\mathbb{R}}^{k}$ denote the unit ball. We define a $d$-dimensional $C^{r}$ foliation $\mathcal{F}$ on $M$ to be a decomposition of $M$ into connected subsets $L\subset M$, called leaves, such that $M$ can be covered by $C^{r}$ coordinate charts

called foliation charts, with the property that, for every leaf $L$, the connected components of $\phi(L\cap U)$ are of the form $B^{d}\times\{*\}$. The collection of the foliation charts $\{(U,\phi)\}$ defining $\mathcal{F}$ forms a foliated atlas. Since the manifold $M$ is oriented, we may assume without loss of generality that the foliated atlas on $M$ is an oriented atlas of $M$, meaning that we require the transition maps between the foliation charts to be orientation preserving. Note that this does not imply that $\mathcal{F}$ is oriented.

We will consider foliations $\mathcal{F}$ which are equipped with two additional structures: a transverse measure $\Lambda$ and a transverse orientation, also referred to as a coorientation. A transverse measure $\Lambda$ is specified by a foliated atlas of $M$ consisting of foliation charts $\phi:U\rightarrow B^{d}\times B^{n-d}$ and a finite Borel measure $\lambda_{\phi}$ on $B^{n-d}$ for every chart $\phi$ such that the following is true: For $i\in\{1,2\}$, let $\phi_{i}:U_{i}\rightarrow B^{d}\times B^{n-d}$ be a chart and consider the transition map

Then, for every point $p_{0}=(x_{0},y_{0})\in\phi_{1}(U_{1}\cap U_{2})$, there exist an open neighborhood $V$ of $x_{0}$ in $B^{d}$, an open neighborhood $W$ of $y_{0}$ in $B^{n-d}$, and a topological embedding $\iota:W\hookrightarrow B^{n-d}$ such that the restriction of $\phi_{21}$ to $V\times W$ is of the form $\phi_{21}(x,y)=(*,\iota(y))$ and moreover

Requiring that the topological embeddings $\iota$ are orientation preserving determines a transverse orientation, or coorientation, of the foliation $\mathcal{F}$.

Given a foliation $\mathcal{F}$, a transversal is a compact topological submanifold with boundary $T^{n-d}\subset M$ that is transverse to the foliation $\mathcal{F}$. This means that every point in $T$ is contained in a foliation chart $\phi:U\rightarrow B^{d}\times B^{n-d}$ such that $\phi(T\cap U)$ is contained in a transverse slice of the form $\{*\}\times B^{n-d}$. A transverse measure $\Lambda$ induces a finite Borel measure on every transversal $T$ and maps between subsets of transversals obtained by sliding along leaves of $\mathcal{F}$ are measure preserving. In fact, one can equivalently define a transverse measure $\Lambda$ on $\mathcal{F}$ to be an assignment of a finite Borel measure to every transversal $T$ subject to the condition that sliding along leaves of $\mathcal{F}$ preserves measure. A transverse orientation admits a similar description: it is an assignment of an orientation to every transversal $T$ such that sliding along leaves of $\mathcal{F}$ preserves orientation.

Finally, note that since the ambient manifold $M$ is oriented, a coorientation of $\mathcal{F}$ induces a natural orientation on the foliation $\mathcal{F}$. Vice-versa, the orientation on $M$ together with an orientation of $\mathcal{F}$ determine a coorientation of $\mathcal{F}$.