We initiate the study of the fundamental group of natural completions of the group of Hamiltonian diffeomorphisms, namely its $C^0$-closure $\overline{\mathrm{Ham}}(M,\omega )$ in the set of all homeomorphisms, and its completion with respect to the spectral norm $\widehat{\mathrm{Ham}}(M,\omega )$. We prove that in some situations, namely complex projective spaces and rational Hirzebruch surfaces, certain Hamiltonian loops that were known to be non-trivial in $\pi _1(\mathrm{Ham}(M,\omega ))$ remain non-trivial in $\pi _1(\widehat{\mathrm{Ham}}(M,\omega ))$. This yields in particular cases, including $\mathbb{C}\mathrm{P}^2$ and the monotone $S^2\times S^2$, the injectivity of the map $\pi _1(\mathrm{Ham}(M,\omega ))\rightarrow \pi _1(\widehat{\mathrm{Ham}}(M,\omega ))$ induced by the inclusion. The same results hold for the Hofer completion of $\mathrm{Ham}(M,\omega )$. Moreover, whenever the spectral norm is known to be $C^0$-continuous, they also hold for $\overline{\mathrm{Ham}}(M,\omega )$.
Our method relies on computations of the valuation of Seidel elements and hence of the spectral norm on $\pi _1(\mathrm{Ham}(M,\omega ))$. Some of these computations were known before, but we also present new ones which might be of independent interest. For example, we show that the spectral pseudo-norm is degenerate when $(M,\omega )$ is any non-monotone $S^{2}\times S^{2}$. At the contrary, it is a genuine norm when $M$ is the 1-point blow-up of $\mathbb{C}\mathrm{P}^{2}$; it is unbounded for small sizes of the blow-up and become bounded starting at the monotone one.
Nous initions l’étude du groupe fondamental de certaines complétions naturelles du groupe des difféomorphismes hamiltoniens. Plus précisément, nous considérons son adhérence $\overline{\mathrm{Ham}}(M,\omega )$ dans le groupe des homéomorphismes pour la topologie $C^0$ et sa complétion pour la norme spectrale $\widehat{\mathrm{Ham}}(M,\omega )$. Nous montrons que dans certaines situations, comme les espaces projectifs complexes ou les surfaces de Hirzebruch rationnelles, certains lacets hamiltoniens connus pour être non-triviaux dans $\pi _1(\mathrm{Ham}(M,\omega ))$ le restent dans $\pi _1(\widehat{\mathrm{Ham}}(M,\omega ))$. Dans des cas particuliers, incluant $\mathbb{C}\mathrm{P}^2$ et le produit $S^2\times S^2$ monotone, ceci implique l’injectivité de l’application $\pi _1(\mathrm{Ham}(M,\omega ))\rightarrow \pi _1(\widehat{\mathrm{Ham}}(M,\omega ))$ induite par l’inclusion. Les mêmes résultats s’appliquent à la complétion de $\mathrm{Ham}(M,\omega )$ pour la norme de Hofer. De plus, ils restent valables pour $\overline{\mathrm{Ham}}(M,\omega )$ dans toute situation où la norme spectrale est $C^0$-continue.
Notre méthode repose sur le calcul des valuations des éléments de Seidel, et donc de la pseudo-norme spectrale sur $\pi _1(\mathrm{Ham}(M,\omega ))$. Certains de ces calculs étaient déjà connus, mais nous en présentons aussi de nouveaux, qui nous semblent intéressants en eux-mêmes. Par exemple, nous montrons que la pseudo-norme spectrale sur $\pi _1(\mathrm{Ham}(M,\omega ))$ est dégénérée lorsque $M$ est n’importe quel produit $S^{2}\times S^{2}$ non-monotone. À l’inverse, c’est une norme lorsque $M$ est l’éclaté en un point de $\mathbb{C}\mathrm{P}^{2}$ ; celle-ci est de plus non-bornée pour les éclatements de petite taille et devient bornée à partir de l’éclatement correspondant au cas monotone.
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Keywords: Hamiltonian diffeomorphisms, spectral norm, Seidel homomorphism
Vincent Humilière 1 ; Alexandre Jannaud 2 ; Rémi Leclercq 3
CC-BY 4.0
@article{AFST_2025_6_34_5_1295_0,
author = {Vincent Humili\`ere and Alexandre Jannaud and R\'emi Leclercq},
title = {Essential loops in completions of {Hamiltonian} groups},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {1295--1323},
year = {2025},
publisher = {Universit\'e de Toulouse, Toulouse},
volume = {Ser. 6, 34},
number = {5},
doi = {10.5802/afst.1833},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1833/}
}
TY - JOUR AU - Vincent Humilière AU - Alexandre Jannaud AU - Rémi Leclercq TI - Essential loops in completions of Hamiltonian groups JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2025 SP - 1295 EP - 1323 VL - 34 IS - 5 PB - Université de Toulouse, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1833/ DO - 10.5802/afst.1833 LA - en ID - AFST_2025_6_34_5_1295_0 ER -
%0 Journal Article %A Vincent Humilière %A Alexandre Jannaud %A Rémi Leclercq %T Essential loops in completions of Hamiltonian groups %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2025 %P 1295-1323 %V 34 %N 5 %I Université de Toulouse, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1833/ %R 10.5802/afst.1833 %G en %F AFST_2025_6_34_5_1295_0
Vincent Humilière; Alexandre Jannaud; Rémi Leclercq. Essential loops in completions of Hamiltonian groups. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 34 (2025) no. 5, pp. 1295-1323. doi: 10.5802/afst.1833
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