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A section forces the structure group to retract from a principal bundle

Terminology

Let $H$ be a group with a topology making multiplication $H \times H \to H$ and inversion $H \to H$ continuous, and write $e$ for its identity. A continuous right $H$ -action on $E$ is a continuous map $E \times H \to E$ , $(z, h) \mapsto z h$ , satisfying $z e = z$ and $(z h_{1}) h_{2} = z (h_{1} h_{2})$ .

A locally trivial right principal $H$ -bundle is a surjective continuous map $p : E \to B$ of topological spaces with a continuous right $H$ -action satisfying $p (z h) = p (z)$ , such that the action is free ( $z h = z$ implies $h = e$ ) and transitive (for each $z, z^{'} \in p^{- 1} (b)$ there is $h \in H$ with $z h = z^{'}$ ) on each fiber, and $B$ has an open cover $(U_{i})$ with $H$ -equivariant homeomorphisms $p^{- 1} (U_{i}) ≅ U_{i} \times H$ over $U_{i}$ . Here $H$ acts on the second factor by right multiplication. A space $E$ is weakly contractible when it is path connected and $π_{j} (E, z_{0}) = 0$ for every $z_{0} \in E$ and $j \geq 1$ .

Statement

Let $p : E \to B$ be a nonempty locally trivial principal topological $H$ -bundle with right $H$ -action, and suppose it has a continuous section. For any $b_{0} \in B$ , the fiber $H$ is a retract of $E$ . If $E$ is weakly contractible, then $H$ is path connected and $π_{j} (H, e) = 0$ for every $j \geq 1$ .

The retraction is expressed by the commutative diagram $\begin{array}{ccc} H & \overset{i}{\to} & E \\ ↓ {id}_{H} & ↓ r \\ H & = & H \end{array}$ with TikZ source

Statement references

May, A Concise Course in Algebraic Topology, Chapter 7

Hatcher, Algebraic Topology, Chapter 4

Facts & Assumptions

Given: The locally trivial principal bundle, its right action, and continuous section $s : B \to E$ from the Statement. For the final conditional conclusion additionally assume that $E$ is weakly contractible.

[F1]

A locally trivial fiber bundle has local product charts over an open cover. The free and transitive right $H$ -action and the $H$ -equivariance of the charts are the additional principal-bundle hypotheses stated above. Locally trivial fiber bundle

[F2]

Based homotopy groups are functorial and invariant under based homotopy. Higher homotopy groups are functorial and based homotopy invariant

[A1]

In the final implication, assume the weak-contractibility condition defined in Terminology.

Proof

1.1F1givenconstruct

Define $Φ : B \times H \to E$ by $Φ (b, h) = s (b) h$ . Every $e \in E$ lies over a unique $b = p (e)$ and the principal action is free and transitive on $p^{- 1} (b)$ , so there is a unique $h$ with $e = s (b) h$ . Thus $Φ$ is bijective. In each local principal chart, $s$ is a continuous local section and the group coordinate $h$ is continuous; these charts cover $B$ , so the inverse $e \mapsto (p (e), h (e))$ is continuous. The forward map is continuous by the action. Hence $Φ$ is a homeomorphism.

2.1A1step 1.1given

Fix $b_{0} \in B$ . Under $Φ$ , the inclusion $i : H \to E$ , $h \mapsto s (b_{0}) h$ , and the map $r : E \to H$ , $r (e) = {pr}_{H} (Φ^{- 1} (e))$ , satisfy $r \circ i = {id}_{H}$ . Thus $H$ is a retract. If $E$ is weakly contractible, it is path connected by [A1], so its continuous image $r (E) = H$ is path connected.

3.1A1F2step 2.1algebra∎

Choose basepoint $e_{0} = s (b_{0})$ . The maps $i$ and $r$ are based at $e_{H}$ and $e_{0}$ , respectively, and [F2] gives $r_{*} i_{*} = (r \circ i)_{*} = {id}_{π_{j} (H, e_{H})}$ . But $π_{j} (E, e_{0}) = 0$ by [A1], so $i_{*}$ has zero target; the identity on $π_{j} (H, e_{H})$ factors through zero and is zero. Therefore $π_{j} (H, e_{H}) = 0$ for each $j \geq 1$ .

A section forces the structure group to retract from a principal bundle

Terminology

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources