## ABSTRACT

Given a fibre F, a structure group G and a base space M, we may construct many fibre bundles over M, depending on the choice of the transition functions. Natural questions we may ask ourselves are how many bundles there are over M with given F and G, and how much they differ from a trivial bundle M × F. For example, we observed in section 10.5 that an SU(2) bundle over S
^{4} is classified by the homotopy group
π
3
(
SU
(
2
)
)
≅
ℤ
https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315275826/bc862b2f-4261-4389-a878-bcfe0d59a882/content/eq3482.tif"/>
. The number n ∈ ℤ tells us how the transition functions twist the local pieces of the bundle when glued together. We have also observed that this homotopy group is evaluated by integrating F
^{2} ∈ h
^{4}(S
^{4}) over S
^{4}, see theorem 10.7.