  Topological phases, such as Chern insulators, are defined in terms of
additive indices that are stable against the addition of trivial degrees of
freedom. Such topology presents an obstruction to any Wannier representation,
namely, the representation of the electronic states in terms of symmetric,
exponentially localized Wannier functions. Here, we address the converse
question: Do obstructions to Wannier representation imply stable band topology?
We answer this in the negative, pointing out that some bands can also display a
distinct type of "fragile topology." Bands with fragile topology do not admit
any Wannier representation by themselves, but such a representation becomes
possible once certain additional trivial degrees of freedom are supplied. We
construct a physical model of fragile topology on the honeycomb lattice that
also helps resolve a recent puzzle in band theory. This model provides a
counterexample to the assumption that splitting of an "elementary band
representation" introduced in [Nature 547, 298--305 (2017)] leads to bands that
are individually topological. Instead, half of the split bands of our model
realizes a trivial band with exponentially localized symmetric Wannier
functions, whereas the second half possess fragile topology. Our work
highlights an important and previously overlooked connection between band
structure and Wannier functions, and is expected to have far reaching
consequences given the central role played by Wannier functions in the modeling
of real materials.
