  We initiate the study of the fundamental group of the group of Hamiltonian
homeomorphisms denoted by $\overline{\mathrm{Ham}}(M,\omega)$, i.e. the
$C^0$-closure of the group of Hamiltonian diffeomorphisms
$\mathrm{Ham}(M,\omega)$ in $\mathrm{Homeo}(M)$. 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
$\pi_1\overline{\mathrm{Ham}}(M,\omega)$. This yields in some cases, including
$\mathbb C\mathrm P^2$ and the monotone $S^2\times S^2$, the injectivity of the
map $\pi_1\mathrm{Ham}(M,\omega)\to\pi_1\overline{\mathrm{Ham}}(M,\omega)$
induced by the inclusion.
  Our method relies on results from $C^0$ symplectic topology and 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.
