  We study the homotopy classification of symmetry representations to describe
the bulk topological invariants protected by the combined operation of a
two-fold rotation $C_{2z}$ and time-reversal $T$ symmetries. We define
topological invariants as obstructions to having smooth Bloch wave functions
compatible with a momentum-independent symmetry representation. When the Bloch
wave functions are required to be smooth, the information on the band topology
is contained in the symmetry representation. This implies that the
$d$-dimensional homotopy class of the unitary matrix representation of the
symmetry operator corresponds to the $d$-dimensional topological invariants.
Here, we prove that the second Stiefel-Whitney number, a two-dimensional
topological invariant protected by $C_{2z}T$, is the homotopy invariant that
characterizes the second homotopy class of the matrix representation of
$C_{2z}T$. As an application of our result, we show that the three-dimensional
bulk topological invariant for the $C_{2z}T$-protected topological crystalline
insulator proposed by C. Fang and L. Fu in Phys. Rev. B 91, 161105(R) (2015),
which we call the 3D strong Stiefel Whitney insulator, is identical to the
quantized magnetoelectric polarizability. The bulk-boundary correspondence
associated with the quantized magnetoelectric polarizability shows that the 3D
strong Stiefel-Whitney insulator has chiral hinges states as well as 2D
massless surface Dirac fermions. This shows that the 3D strong Stiefel Whitney
insulator has the characteristics of both the first order and the second order
topological insulators, simultaneously, which is consistent with the recent
classification of higher-order topological insulators protected by an order-two
symmetry.
