Humilière Completion in Symplectic Topology

• Humilière completion is the metric completion of spaces of Lagrangian and Hamiltonian objects using spectral distances γ and c to adjoin limits of Cauchy sequences.
• It establishes a geometric framework where abstract completion points carry a γ-support that retains a trace of classical coisotropicity and intersection properties.
• The theory connects spectral invariants with dynamical applications, offering insights into invariant sets like generalized Birkhoff attractors in symplectic topology.

Searching arXiv for recent and foundational papers on the Humilière completion and related support/coisotropicity results. The Humilière completion is the metric completion of spaces of Lagrangian or Hamiltonian objects with respect to the symplectic spectral metric. In the formulation studied by Humilière and developed further by Viterbo and subsequent authors, one starts from smooth exact Lagrangians, exact graded Lagrangian branes, Hamiltonian diffeomorphisms, or Hamiltonian correspondences, equips these spaces with spectral distances such as $\gamma$ or $c$, and adjoins limits of Cauchy sequences. The resulting points are generally not genuine subsets, submanifolds, or maps, but they retain a geometric trace through the notion of $\gamma$-support, whose fundamental structural property is $\gamma$-coisotropicity (Viterbo, 2022, Bernardi et al., 10 Mar 2026).

1. Definition, scope, and ambient settings

In the general framework, the spectral distance $\gamma$ is considered on exact Lagrangians or, more precisely, exact graded Lagrangian branes, as well as on Hamiltonian diffeomorphisms and Hamiltonian correspondences. The corresponding completions are denoted

$\widehat{\LL}(M,\omega),\qquad \widehat{\mathcal L}(M,d\lambda),\qquad \widehat{\DHam}(M,\omega),$

and are explicitly described as completions first studied in Humilière’s work; the terminology Humilière completions is used for these spaces (Viterbo, 2022).

The ambient symplectic hypotheses are presented in two parallel forms. One is an aspherical setting, where $(M,\omega)$ is either closed or convex at infinity and satisfies

$[\omega]\pi_2(M)=0,\qquad c_1(TM)\pi_2(M)=0.$

The other is an exact setting $(M,d\lambda)$, convex at infinity. On the Lagrangian side, the case $T^*N$ with $c$0 closed is singled out as especially important, because exact Lagrangians, sheaf quantization, reduction, graph selectors, and pseudographs are available there (Viterbo, 2022).

A more specialized presentation appears in the cotangent-bundle setting. There the focus is on

$c$1

with $c$2 a closed smooth manifold, and on the class of closed, connected, exact Lagrangians Hamiltonian isotopic to the zero section $c$3. The underlying Lagrangians form $c$4, while the corresponding Lagrangian branes with primitive form $c$5 (Bernardi et al., 10 Mar 2026).

A basic subtlety of the completion is that its elements need not be compactly supported, because they may arise as limits of objects whose supports escape to infinity. More fundamentally, the completion is not a $c$6, Hausdorff, varifold, or current completion; it is induced by spectral min-max data. This distinguishes the Humilière completion from topological or measure-theoretic compactifications and explains why its geometry is encoded by support-detection via spectral invariants rather than by pointwise convergence (Viterbo, 2022, Bernardi et al., 10 Mar 2026).

This notion is unrelated to the categorical idempotent or Karoubi completion studied for $c$7-angulated categories in "Idempotent completion of $c$8-angulated categories" (Lin, 2017). The overlap is only terminological at the level of the word “completion.”

2. Spectral invariants and the metrics $c$9 and $\gamma$0

The construction is driven by spectral invariants. For exact graded Lagrangian branes $\gamma$1 and $\gamma$2, one considers

$\gamma$3

with

$\gamma$4

where $\gamma$5 is the fundamental class and $\gamma$6 the degree-zero unit. The spectral norm or distance is then

$\gamma$7

For branes the metric

$\gamma$8

is used before quotienting by additive constants in the primitive, and the descent relation is

$\gamma$9

This relation makes precise how $\gamma$0 is obtained from the brane-level metric $\gamma$1 (Viterbo, 2022).

In the cotangent-bundle formulation, the spectral invariants are defined using generating functions quadratic at infinity. If $\gamma$2 is Hamiltonian isotopic to the zero section, then Viterbo’s theorem gives a unique GFQI up to stabilization and fibered diffeomorphism. For a GFQI $\gamma$3 and $\gamma$4, one defines $\gamma$5 by min-max, and for a brane $\gamma$6 one sets

$\gamma$7

For two branes $\gamma$8 and $\gamma$9, with

$\gamma$0

the pairwise invariants are

$\gamma$1

The associated distances are

$\gamma$2

and

$\gamma$3

Theorem $\gamma$4 states that $\gamma$5 is a distance on $\gamma$6 and $\gamma$7 is a distance on $\gamma$8, and that $\gamma$9 acts by isometries on both metric spaces (Bernardi et al., 10 Mar 2026).

The Humilière completion is then simply the metric completion: $\widehat{\LL}(M,\omega),\qquad \widehat{\mathcal L}(M,d\lambda),\qquad \widehat{\DHam}(M,\omega),$0 The spaces being completed are not complete and, in the cotangent-bundle notes, are described as “not even Polish.” This indicates that the completion is not a formal redundancy but a genuinely larger symplectic category of generalized Lagrangian objects (Bernardi et al., 10 Mar 2026).

3. $\widehat{\LL}(M,\omega),\qquad \widehat{\mathcal L}(M,d\lambda),\qquad \widehat{\DHam}(M,\omega),$1-support as the geometric shadow of a completed object

A central problem is that an element of the completion is not literally a subset of the ambient manifold. The solution is the notion of $\widehat{\LL}(M,\omega),\qquad \widehat{\mathcal L}(M,d\lambda),\qquad \widehat{\DHam}(M,\omega),$2-support, defined operationally through local Hamiltonian perturbations. For

$\widehat{\LL}(M,\omega),\qquad \widehat{\mathcal L}(M,d\lambda),\qquad \widehat{\DHam}(M,\omega),$3

one sets

$\widehat{\LL}(M,\omega),\qquad \widehat{\mathcal L}(M,d\lambda),\qquad \widehat{\DHam}(M,\omega),$4

For a brane $\widehat{\LL}(M,\omega),\qquad \widehat{\mathcal L}(M,d\lambda),\qquad \widehat{\DHam}(M,\omega),$5, one similarly defines

$\widehat{\LL}(M,\omega),\qquad \widehat{\mathcal L}(M,d\lambda),\qquad \widehat{\DHam}(M,\omega),$6

and these notions agree after forgetting brane data: $\widehat{\LL}(M,\omega),\qquad \widehat{\mathcal L}(M,d\lambda),\qquad \widehat{\DHam}(M,\omega),$7 Thus support is intrinsic to the completed Lagrangian, not to a chosen brane enhancement (Viterbo, 2022).

In the cotangent-bundle notes the same definition is written in local-ball form: for

$\widehat{\LL}(M,\omega),\qquad \widehat{\mathcal L}(M,d\lambda),\qquad \widehat{\DHam}(M,\omega),$8

one has

$\widehat{\LL}(M,\omega),\qquad \widehat{\mathcal L}(M,d\lambda),\qquad \widehat{\DHam}(M,\omega),$9

iff for every $(M,\omega)$0 there exists $(M,\omega)$1 supported in $(M,\omega)$2 such that

$(M,\omega)$3

Since $(M,\omega)$4 is a metric, this is equivalent to $(M,\omega)$5 in the completed space. The support therefore detects the points near which localized Hamiltonian perturbations act nontrivially on the abstract completion point (Bernardi et al., 10 Mar 2026).

Several basic properties are established. The support is closed by definition. For honest smooth Lagrangians,

$(M,\omega)$6

Functoriality holds under symplectic maps: $(M,\omega)$7 and in particular under $(M,\omega)$8, hence under symplectic homeomorphisms. Products satisfy

$(M,\omega)$9

with equality

$[\omega]\pi_2(M)=0,\qquad c_1(TM)\pi_2(M)=0.$0

when $[\omega]\pi_2(M)=0,\qquad c_1(TM)\pi_2(M)=0.$1 is smooth. For correspondences,

$[\omega]\pi_2(M)=0,\qquad c_1(TM)\pi_2(M)=0.$2

Under reduction by a coisotropic $[\omega]\pi_2(M)=0,\qquad c_1(TM)\pi_2(M)=0.$3,

$[\omega]\pi_2(M)=0,\qquad c_1(TM)\pi_2(M)=0.$4

If $[\omega]\pi_2(M)=0,\qquad c_1(TM)\pi_2(M)=0.$5 in $[\omega]\pi_2(M)=0,\qquad c_1(TM)\pi_2(M)=0.$6, then

$[\omega]\pi_2(M)=0,\qquad c_1(TM)\pi_2(M)=0.$7

This lower-semicontinuity expresses that support cannot suddenly appear in a region where infinitely many approximants had no support (Viterbo, 2022).

A further localization statement sharpens the definition: if $[\omega]\pi_2(M)=0,\qquad c_1(TM)\pi_2(M)=0.$8, then

$[\omega]\pi_2(M)=0,\qquad c_1(TM)\pi_2(M)=0.$9

This identifies $(M,d\lambda)$0-support as the region where compactly supported Hamiltonian perturbations can be spectrally detected (Bernardi et al., 10 Mar 2026).

4. $(M,d\lambda)$1-coisotropicity and the main structural theorem

The support theory leads to the notion of a $(M,d\lambda)$2-coisotropic set. A subset $(M,d\lambda)$3 is said to be non-$(M,d\lambda)$4-coisotropic at $(M,d\lambda)$5 if for any ball $(M,d\lambda)$6 there exists a smaller ball $(M,d\lambda)$7 and a sequence of Hamiltonian maps $(M,d\lambda)$8 supported in $(M,d\lambda)$9 such that

$T^*N$0

and

$T^*N$1

Equivalently, $T^*N$2 is $T^*N$3-coisotropic at $T^*N$4 if there exists $T^*N$5 such that for every $T^*N$6 there is $T^*N$7 such that for all $T^*N$8,

$T^*N$9

In this sense, one cannot remove the set from arbitrarily small neighborhoods by Hamiltonians of arbitrarily small spectral norm (Viterbo, 2022).

The central theorem is that for any

$c$00

the set

$c$01

is $c$02-coisotropic. In the cotangent-bundle notes the same statement appears for

$c$03

This theorem gives the main geometric content of the Humilière completion: every completed Lagrangian carries a coisotropic shadow, even when the completion point itself is only an abstract metric limit (Viterbo, 2022, Bernardi et al., 10 Mar 2026).

For smooth submanifolds, $c$04-coisotropicity agrees with classical coisotropicity: $c$05 This identifies $c$06-coisotropicity as an extension of the standard notion to singular and merely closed subsets (Viterbo, 2022).

The comparison with other generalized isotropy notions is also explicit. If $c$07 is $c$08-coisotropic, then it is cone-coisotropic, and if it is cone-coisotropic, then it is Poisson coisotropic: $c$09 These implications are strict in general. Permanence properties include invariance under $c$10, locality in $c$11, stability under unions, and local heredity through $c$12-coisotropic submanifolds or Lagrangian germs (Viterbo, 2022).

5. Examples, large-support phenomena, and regularity

The simplest examples are smooth exact graphs. If $c$13 and

$c$14

then

$c$15

For two smooth functions $c$16,

$c$17

and

$c$18

These formulas show concretely how spectral convergence of graphs is controlled by oscillation and primitive data (Bernardi et al., 10 Mar 2026).

A genuine limit phenomenon appears for continuous functions. If $c$19 converges uniformly to $c$20, then the graphs $c$21 form a $c$22-Cauchy sequence and define

$c$23

Its support is no longer a smooth graph but the Vichery subdifferential: $c$24 In the broader exact setting the completed graph of a continuous differential satisfies

$c$25

This shows that a completion point can be represented by a continuous graph while its support becomes a nonsmooth generalized differential object (Viterbo, 2022, Bernardi et al., 10 Mar 2026).

The completion also contains elements with much larger support than a smooth Lagrangian. One result states that for any $c$26 there exists

$c$27

such that $c$28 contains

$c$29

The cotangent-bundle notes recall related “Peano Lagrangians” with

$c$30

These examples show that support can contain coisotropic blocks of dimension larger than $c$31, so the Humilière completion strictly exceeds the smooth Lagrangian category (Viterbo, 2022, Bernardi et al., 10 Mar 2026).

At the opposite extreme lie completed Lagrangians with “small” support. A completed Lagrangian is called regular if $c$32 is a smooth $c$33-dimensional manifold. Since the support is $c$34-coisotropic and smooth $c$35-coisotropic submanifolds are precisely classical coisotropic ones, such a support is automatically a smooth Lagrangian. In the exact cotangent setting, if $c$36 is regular and $c$37 is exact, then

$c$38

Similarly, if $c$39 and $c$40 is a compact exact Lagrangian submanifold of $c$41, then $c$42 is just that Lagrangian. This indicates that the completion does not create hidden structure above a smooth exact support (Viterbo, 2022, Bernardi et al., 10 Mar 2026).

Support is strong but not complete. There exist distinct

$c$43

with

$c$44

The cotangent-bundle notes likewise exhibit distinct limits with identical $c$45-support but positive spectral distance. Thus support captures the geometric footprint of a completion point without classifying the point uniquely (Viterbo, 2022, Bernardi et al., 10 Mar 2026).

6. Dynamical applications, Hamiltonian completions, and significance

One major application is to conformally exact symplectic dynamics. If $c$46 is conformally exact symplectic with conformal ratio $c$47, then

$c$48

for all $c$49. Hence

$c$50

is a strict contraction, so Banach’s fixed point theorem yields a unique fixed point

$c$51

Its geometric realization is the generalized Birkhoff attractor

$c$52

This set is invariant, closed, and $c$53-coisotropic. For dissipative maps of the annulus $c$54, the generalized notion recovers the classical Birkhoff attractor: $c$55 The Humilière completion is therefore not only a completion theorem but also a mechanism for producing canonical invariant sets in higher-dimensional dynamics (Bernardi et al., 10 Mar 2026).

Support also encodes non-displaceability and intersection remnants. If

$c$56

then

$c$57

Consequently, $c$58 is not displaceable, intersects every exact Lagrangian, and, in the cotangent setting, intersects every cotangent fiber. In the cotangent-bundle notes this is stated as: if $c$59 and $c$60, then

$c$61

and for every $c$62,

$c$63

These statements show that spectral completion preserves a strong residue of exact Lagrangian intersection theory (Viterbo, 2022, Bernardi et al., 10 Mar 2026).

The Hamiltonian-side completion has a parallel support philosophy. For $c$64, equality to the identity on an open set $c$65 is expressed through the support of the graph: $c$66 This leads to local quotient objects encoding flows on punctured domains and to extension theorems for singular Hamiltonians. If $c$67 is nowhere $c$68-coisotropic, then any element of $c$69 extends uniquely to an element of $c$70. In this formulation, $c$71-coisotropicity is the geometric obstruction to extension across singular sets (Viterbo, 2022).

Several open questions organize the current theory. The notes ask how large $c$72-support can be, what can be said when $c$73, and how many completion points can share the same support. They also raise connectivity questions for generalized Birkhoff attractors and emphasize that the completion of the Hamiltonian group is still not well understood. A striking flexibility statement nevertheless holds at the completed level: if $c$74 are two closed exact Lagrangians in $c$75, then there exists

$c$76

such that $c$77. This suggests that the Humilière completion enlarges symplectic topology enough to recover strong formal transitivity while still retaining rigid support-theoretic and coisotropic constraints (Bernardi et al., 10 Mar 2026).