  In a tight-binding lattice model with $n$ orbitals (single-particle states)
per site, Wannier functions are $n$-component vector functions of position that
fall off rapidly away from some location, and such that a set of them in some
sense span all states in a given energy band or set of bands;
compactly-supported Wannier functions are such functions that vanish outside a
bounded region. They arise not only in band theory, but also in connection with
tensor-network states for non-interacting fermion systems, and for flat-band
Hamiltonians with strictly short-range hopping matrix elements. In earlier
work, it was proved that for general complex band structures (vector bundles)
or general complex Hamiltonians---that is, class A in the ten-fold
classification of Hamiltonians and band structures---a set of
compactly-supported Wannier functions can span the vector bundle only if the
bundle is topologically trivial, in any dimension $d$ of space, even when use
of an overcomplete set of such functions is permitted. This implied that, for a
free-fermion tensor network state with a non-trivial bundle in class A, any
strictly short-range parent Hamiltonian must be gapless. Here, this result is
extended to all ten symmetry classes of band structures without additional
crystallographic symmetries, with the result that in general the non-trivial
bundles that can arise from compactly-supported Wannier-type functions are
those that may possess, in each of $d$ directions, the non-trivial winding that
can occur in the same symmetry class in one dimension, but nothing else. The
results are obtained from a very natural usage of algebraic $K$-theory, based
on a ring of polynomials in $e^{\pm ik_x}$, $e^{\pm ik_y}$, . . . , which occur
as entries in the Fourier-transformed Wannier functions.
