I have been learning topological insulators recently, and I become more and more curious about the link between topological insulators and mathematical theory these days. I know topological insulators have something to do with fiber bundles and K-theory. I have a relatively good background of undergraduate topology and recently I have read some introduction about fiber bundles and K-theory. What is missing in my mind is the link between math and physics. That is, what exactly do we regard as fiber bundles and classify them as trivial and non-trivial? Can someone kindly give me the answer or some references?