Knowlpedia

Principal bundle transition function

The group-valued cocycle on overlaps that relates two equivariant trivializations of a principal bundle.

Fix a π:P→M\pi:P\to M and an {Ui}i∈I\{U_i\}_{i\in I} together with

ψi:π−1(Ui)→Ui×G.\psi_i:\pi^{-1}(U_i)\to U_i\times G.

On an overlap Uij=Ui∩UjU_{ij}=U_i\cap U_j, the principal bundle transition function is the (necessarily unique)

gij:Uij→Gg_{ij}:U_{ij}\to G

characterized by either of the equivalent conditions:

  • (Trivialization comparison) For all x∈Uijx\in U_{ij} and h∈Gh\in G,
    ψi∘ψj−1(x,h)=(x, gij(x) h).\psi_i\circ\psi_j^{-1}(x,h)=(x,\,g_{ij}(x)\,h).
  • (Local section comparison) If si(x)=ψi−1(x,e)s_i(x)=\psi_i^{-1}(x,e) and sj(x)=ψj−1(x,e)s_j(x)=\psi_j^{-1}(x,e), then
    sj(x)=si(x)⋅gij(x)(x∈Uij).s_j(x)=s_i(x)\cdot g_{ij}(x)\qquad (x\in U_{ij}).

The functions gijg_{ij} satisfy the cocycle identities on overlaps:

  1. gii(x)=eg_{ii}(x)=e on UiU_i,
  2. gij(x)=gji(x)−1g_{ij}(x)=g_{ji}(x)^{-1} on UijU_{ij},
  3. gij(x) gjk(x)=gik(x)g_{ij}(x)\,g_{jk}(x)=g_{ik}(x) on triple overlaps Ui∩Uj∩UkU_i\cap U_j\cap U_k.

These identities are the compatibility conditions ensuring that the local products Ui×GU_i\times G glue to a global principal bundle.

Examples
  1. Trivial bundle. If P=M×GP=M\times G with the standard trivializations on each UiU_i, then all transition functions are gij≡eg_{ij}\equiv e.
  2. Möbius band as an O(1)O(1)-bundle. Over S1S^1 covered by two arcs, the nontrivial has transition function g12≡−1∈O(1)g_{12}\equiv -1\in O(1) on the overlap.
  3. Hopf bundle. For S3→S2S^3\to S^2 with the usual two-chart cover of S2S^2, the transition function on the overlap (homotopic to the equator) is a nontrivial map S1→S1S^1\to S^1 representing the generator of π1(S1)\pi_1(S^1).