Essential loops in completions of Hamiltonian groups
We initiate the study of the fundamental group of natural completions of the group of Hamiltonian diffeomorphisms, namely its $C^0$-closure $\overline{\mathrm{Ham}}(M,\omega)$ 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 $\pi_1\big(\mathrm{Ham}(M,\omega)\big)$ remain non-trivial in $\pi_1\big(\widehat{\mathrm{Ham}}(M,\omega)\big)$. This yields in particular cases, including $\mathbb C\mathrm P^2$ and the monotone $S^2\times S^2$, the injectivity of the map $\pi_1\big(\mathrm{Ham}(M,\omega)\big)\to\pi_1\big(\widehat{\mathrm{Ham}}(M,\omega)\big)$ 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 $\pi_1\big(\mathrm{Ham}(M,\omega)\big)$. 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.