  Fermi sea in a metal can host exotic quantum topology, which determines its
conductance quantization and is characterized by Euler characteristic $\chi_F$.
Unlike gapped band topology described by the global feature of wave function,
this topology of gapless system is associated with the geometry of Fermi sea,
and thus probing and identifying $\chi_F$ are inherently difficult in
higher-dimensional systems. Here, we propose a dimensional reduction theory for
Fermi sea topology in $d$-dimensional metallic systems, showing that $\chi_F$
can be determined by the feature of so-called reduced critical points on Fermi
surfaces, with theoretical simplicity and observational intuitiveness. We also
reveal a nontrivial correspondence between the Fermi sea topology and the
gapped band topology by using an ingenious mapping, of which $\chi_F$ exactly
equals to the topological invariant of gapped topological phases. This provides
a potential way to capture $\chi_F$ through the topological superconductors.
Our work opens an avenue to characterize and detect the Fermi sea topology
using low-dimensional momentum information.
