  We present a fully many-body formulation of topological invariants for
various topological phases of fermions protected by antiunitary symmetry, which
does not refer to single particle wave functions. For example, we construct the
many-body $\mathbb{Z}_2$ topological invariant for time-reversal symmetric
topological insulators in two spatial dimensions, which is a many-body
counterpart of the Kane-Mele $\mathbb{Z}_2$ invariant written in terms of
single-particle Bloch wave functions. We show that an important ingredient for
the construction of the many-body topological invariants is a fermionic partial
transpose which is basically the standard partial transpose equipped with a
sign structure to account for anti-commuting property of fermion operators. We
also report some basic results on various kinds of pin structures -- a key
concept behind our strategy for constructing many-body topological invariants
-- such as the obstructions, isomorphism classes, and Dirac quantization
conditions.
