associated bundle




Given a right principal GG-bundle π:PX\pi: P\to X and a left GG-action on some FF, all in a sufficiently strong category CC (such as Top), one can form the quotient object P× GF=(P×F)/P \times_G F = (P\times F)/{\sim}, where P×FP \times F is a product and \sim is the smallest congruence such that (using generalized elements) (pg,f)(p,gf)(p g,f)\sim (p,g f); there is a canonical projection P× GFXP\times_G F\to X where the class of (p,f)(p,f) is mapped to π(p)X\pi(p)\in X, hence making P× GFXP\times_G F\to X into a fibre bundle with typical fiber FF, and the transition functions belonging to the action of GG on FF. We say that P× GFXP\times_G F\to X is the associated bundle to PXP\to X with fiber FF.

In geometric homotopy theory

In the context of higher topos theory there is an elegant and powerful definition and construction of associated bundles. We discuss here some basics and how this recovers the traditional definition. For more see at associated infinity-bundle and at geometry of physics -- representations and associated bundles.

At geometry of physics -- principal bundles in the section Smooth principal bundles via smooth groupoids is discussed how smooth principal bundles for a Lie group GG over a smooth manifold XX are equivalently the homotopy fibers of morphisms of smooth groupoids (smooth stacks) of the form

XBG. X \stackrel{}{\longrightarrow} \mathbf{B}G \,.

Now given an action ρ\rho of GG on some smooth manifold VV, and regardiing this action via its action groupoid projection p ρ:V//GBGp_\rho \colon V//G \to \mathbf{B}G as discussed above, then we may consider these two morphisms into BG\mathbf{B}G jointly

V//G p ρ X g BG \array{ && V//G \\ && \downarrow^{\mathrlap{p_\rho}} \\ X &\stackrel{g}{\longrightarrow}& \mathbf{B}G }

and so it is natural to construct their homotopy fiber product.

We now discuss that this is equivalently the associated bundle which is associated to the principal bundle PXP \to X via the action ρ\rho.


For GG a smooth group (e.g. a Lie group), XX a smooth manifold, PXP \to X a smooth GG-principal bundle over XX and ρ\rho a smooth action of GG on some smooth manifold VV, then the associated VV-fiber bundle P× GVXP \times_G V\to X is equivalently (regarded as a smooth groupoid) the homotopy pullback of the action groupoid-projection p ρ:V//GBGp_\rho \colon V//G \to \mathbf{B}G along a morphism g:XBGg \colon X\to\mathbf{B}G which modulates PP

P× GV V//G p ρ X g BG. \array{ P\times_G V &\longrightarrow& V//G \\ \downarrow && \downarrow^{\mathrlap{p_\rho}} \\ X &\stackrel{g}{\longrightarrow}& \mathbf{B}G } \,.

By the discussion at geometry of physics -- principal bundles in the section Smooth principal bundles via smooth groupoids, the morphism gg of smooth groupoids is presented by a morphism of pre-smooth groupoids after choosing an open cover {U iX}\{U_i \to X\} over wich PP trivialize and choosing a trivialization, by the zig-zag

C({U i}) g (BG) lwe X \array{ C(\{U_i\})_\bullet &\stackrel{g_\bullet}{\longrightarrow}& (\mathbf{B}G)_\bullet \\ \downarrow^{\mathrlap{\simeq_{lwe}}} \\ X }

where the top morphism is the Cech cocycle of the given local trivialization regarded as a morphism out of the Cech groupoid of the given cover.

Moreover, by this proposition the morphism (p ρ) (p_\rho)_\bullet is a global fibration of pre-smooth groupoids, hence, by the discussion at geometry of physics -- smooth homotopy types, the homotopy pullback in question is equivalently computed as the ordinary pullback of pre-smooth groupoids of (p ρ) (p_\rho)_\bullet along this g g_\bullet

C({U i}) ×(BG) (V//G) (V//G) (p ρ) C({U i}) g (BG) lwe X. \array{ C(\{U_i\})_\bullet \underset{(\mathbf{B}G)_\bullet}{\times} (V//G)_\bullet &\longrightarrow& (V//G)_\bullet \\ \downarrow && \downarrow^{\mathrlap{(p_\rho)_\bullet}} \\ C(\{U_i\})_\bullet &\stackrel{g_\bullet}{\longrightarrow}& (\mathbf{B}G)_\bullet \\ \downarrow^{\mathrlap{\simeq_{lwe}}} \\ X } \,.

This pullback is computed componentwise. Hence it is the pre-smooth groupoid whose morphisms are pairs consisting of a morphism (x,i)(x,j)(x,i)\to (x,j) in the Cech groupoid as well as a morphism sgρ(s)(g)s \stackrel{g}{\to} \rho(s)(g) in the action groupoid, such that the group label gg of the latter equals the cocycle g ij(x)g_{i j}(x) of the cocycle on the former. Schematically:

C({U i}) ×(BG) (V//G) ={((x,i),s)g ij(x)((x,j),ρ(s)(g))}. C(\{U_i\})_\bullet \underset{(\mathbf{B}G)_\bullet}{\times} (V//G)_\bullet = \left\{ ((x,i),s) \stackrel{g_{i j}(x)}{\longrightarrow} ((x,j),\rho(s)(g)) \right\} \,.

This means that the pullback groupoid has at most one morphism between every ordered pair of objects. Accordingly this groupoid is equivalence of groupoids equivalent to the quotient of its space of objects by the equivalence relation induced by its morphisms:

(iU i×V)/ . \cdots \simeq \left( \underset{i}{\coprod} U_i \times V \right)/_\sim \,.

This is a traditional description of the associated bundle in question.



  • Norman Steenrod, The topology of fibre bundles, Princeton Mathematical Series 14, 1951. viii+224 pp. MR39258; reprinted 1994

  • Dale Husemöller, Fibre bundles, McGraw-Hill 1966 (300 p.); Springer Graduate Texts in Math. 20, 2nd ed. 1975 (327 p.), 3rd. ed. 1994 (353 p.)

Revised on August 9, 2016 09:27:43 by Dexter Chua (