  We show that the topology of the Fermi sea of a $D$-dimensional Fermi gas is
reflected in the multipartite entanglement characterizing $D+1$ regions that
meet at a point. For odd $D$ we introduce the multipartite mutual information,
and show that it exhibits a $\log^D L$ divergence as a function of system size
$L$ with a universal coefficient that is proportional to the Euler
characteristic $\chi_F$ of the Fermi sea. This provides a generalization, for a
Fermi gas, of the well-known result for $D=1$ that expresses the $\log L$
divergence of the bipartite entanglement entropy in terms of the central charge
$c$ characterizing a conformal field theory. For even $D$ we introduce a
charge-weighted entanglement entropy that is manifestly odd under a
particle-hole transformation. We show that the corresponding charge-weighted
mutual information exhibits a similar $\log^D L$ divergence proportional to
$\chi_F$. Our analysis relates the universal behavior of the multipartite
mutual information in the absence of interactions to the $D+1$'th order
equal-time density correlation function, which we show exhibits a universal
behavior in the long wavelength limit proportional to $\chi_F$. Our analytic
results are based on the replica method. In addition we perform a numerical
study of the charge-weighted mutual information for $D=2$ that confirms several
aspects of the analytic theory. Finally, we consider the effect of interactions
perturbatively within the replica theory. We show that for $D=3$ the $\log^3 L$
divergence of the topological mutual information is not perturbed by weak
short-ranged interactions, though for $D=2$ the charge-weighted mutual
information is perturbed. Thus, for $D=3$ the multipartite mutual information
provides a robust classification that distinguishes distinct topological Fermi
liquid phases.
