  We give a self-contained and enriched review about topology properties in the
rapidly growing field of topological states of matter (TSM). This review is
mainly focus on the beautiful interplay of topology mathematics and condensed
matter physics that issuing TSM. Fiber bundle theory is a powerful concept to
describe the non-trivial topology properties underlying the physical system. So
we briefly present some motivation of fiber bundle theory and following that
several effective topological methods have been introduced to judge whether a
fiber bundle is trivial or not. Next, we give some topological invariants that
characterizes the non-trivial TSM in the non-interacting systems in all
dimensions, which is called topological band theory. Following that, we review
and generalize the topological response using topological field theory called
Chern-Simons effective theory.
  Finally, the classification of free-fermion systems have been studied by loop
space and K-theory.
