Proof of necessity of Band Inversion in Topological Insulators Is there any rigorous proof/argument for the necessity of Band Inversion in topological insulators? From a mathematical point of view, I am trying to find an answer in terms of the topology of the Bloch Bundle. Is there any relevant work?
 A: As you specifically mention $\mathbb{Z}_2$ invariants in your comment, I will start my answer with the Fu-Kane $\mathbb{Z}_2$ invariant. In their original paper, they prove that the $\mathbb{Z}_2$ index can be expressed purely in terms of inversion eigenvalues at high-symmetry points.
If you maintain symmetry and dimensionality in your system, then the only way to change the inversion eigenvalue at a high-symmetry point is by performing a band inversion. Note that the inversion need to happen with two bands which have opposite inversion eigenvalues, otherwise the transition does not change topology, so not just any band inversion will do. I will refer to such band inversions as non-trivial.
In this setting, this is a sufficient condition to characterize topology. We have a $\mathbb{Z}_2$ indicator which is indicated by symmetry eigenvalues and is stable against symmetry preserving perturbations. So in this case, band eigenvalues can capture the topology.
This is not always the case however. Far from all topology can be identified purely based on symmetry eigenvalues. Generally, however, gap closings signal interesting topological features because the Bloch bundle description assumes that a gap exists everywhere, so that we can talk about the topology of the valence and conduction subspace separately. More mathematically, the manifold of states you are describing (usually a Grassmannian of some sort) changes if the gap closes. If found the first few sections of this paper helpful in clarifying the mathematical structure. This is why physicists focus so much on band closures.
In general, proving whether or not a system has topological features is a difficult problem, and depends to some extent on your definition of topological (see e.g. fragile topology or the very recently introduced subdimensional topology). Not all topology can be determined based purely off symmetry eigenvalues, so you can have nontrivial topology even if you do not have a nontrivial band inversion.

So how do you determine topology more generally?
The standard way of computing at least stable topological invariants (stable here refers to invariant under addition of trivial bands) is K-theory. Roughly speaking, in K-theory you characterize the topology of the Bloch bundle, but allow arbitrary subtraction and addition of trivial bands.
In general computing K-theory results is very hard. There are some results available for restricted symmetry settings. Perhaps most famously, there is the tenfold way classification, which computes the stable indicators for systems with time-reversal and particle-hole symmetry.
There are some extensions of this, by e.g. including order-two symmetries (which includes inversion), but a complete classification for all crystalline symmetries is still missing.
The crystalline phases which can be determined purely from symmetry eigenvalues have been tabulated: these are the so-called symmetry indicator/topological quantum chemistry methods (here, here, here and here). A good review is here.
So to summarize: a nontrivial band inversion can identify a topological phase transition, but far from all topological phases can be identified by band inversions.

A final comment: This state of affairs is mirrored in the mathematical field of topology. There are a myriad of different topological invariants for vector bundles, and if you have two vector bundles with different invariants, then they have different topology. But showing that two vector bundles have the same topology is much harder to show in general, as you would have to check all possible invariants.
