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 $\overline{\mathrm{H}\mathrm{a}\mathrm{m}}(M,\omega )$ and its completion with respect to the spectral norm $\widehat{\mathrm{H}\mathrm{a}\mathrm{m}}(M,\omega )$. 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 ${\pi }_1(\mathrm{H}\mathrm{a}\mathrm{m}(M,\omega ))$ remain non-trivial in ${\pi }_1(\widehat{\mathrm{H}\mathrm{a}\mathrm{m}}(M,\omega ))$. This yields in particular cases, including $\mathbb{C}{\mathrm{P}}^2$ and the monotone $S^2\times S^2$, the injectivity of the map ${\pi }_1(\mathrm{H}\mathrm{a}\mathrm{m}(M,\omega ))\to {\pi }_1(\widehat{\mathrm{H}\mathrm{a}\mathrm{m}}(M,\omega ))$ induced by the inclusion. The same results hold for the Hofer completion of $\mathrm{H}\mathrm{a}\mathrm{m}(M,\omega )$. Moreover, whenever the spectral norm is known to be $C^0$-continuous, they also hold for $\overline{\mathrm{H}\mathrm{a}\mathrm{m}}(M,\omega )$. Our method relies on computations of the valuation of Seidel elements and hence of the spectral norm on ${\pi }_1(\mathrm{H}\mathrm{a}\mathrm{m}(M,\omega ))$. 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.