  In this work, we examine the topological phases that can arise in triangular
lattices with disconnected elementary band representations. We show that,
although these phases may be "fragile" with respect to the addition of extra
bands, their topological properties are manifest in certain nontrivial
holonomies (Wilson loops) in the space of nontrivial bands. We introduce an
eigenvalue index for fragile topology, and we show how a nontrivial value of
this index manifests as the winding of a hexagonal Wilson loop; this remains
true even in the absence of time-reversal or sixfold rotational symmetry.
Additionally, when time-reversal and twofold rotational symmetry are present,
we show directly that there is a protected nontrivial winding in more
conventional Wilson loops. Crucially, we emphasize that these Wilson loops
cannot change without closing a gap to the nontrivial bands. By studying the
entanglement spectrum for the fragile bands, we comment on the relationship
between fragile topology and the "obstructed atomic limit" of B. Bradlyn et
al., Nature 547, 298--305 (2017). We conclude with some perspectives on
topological matter beyond the K-theory classification.
