# nLab Milnor construction

### Context

#### Bundles

bundles

fiber bundles in physics

## Context

• dependent type theory

• ## Classes of bundles

• covering space

• numerable bundle

• ## Presentations

• bundle gerbe

• groupal model for universal principal ∞-bundles

• microbundle

• ## Examples

• trivial vector bundle

• tautological line bundle

• basic line bundle on the 2-sphere?
• Hopf fibration

• canonical line bundle

• ## Constructions

• clutching construction

• inner product on vector bundles

• dual vector bundle

• projective bundle

• # Contents

## Idea

Given a (Hausdorff) topological group $G$, the Milnor construction of the universal principal bundle for $G$ (also known as Milnor’s join construction) constructs the join of infinitely many copies of $G$, i.e., the colimit of joins

$(E G)_{Milnor} := colim \; G \ast G \ast \ldots \ast G,$

and canonically equips it with a continuous and free right action of $G$ that yields the structure of a CW-complex such that the action of $G$ permutes the cells. Consequently, the natural projection $(E G)_{Milnor} \to (E G)_{Milnor}/G$ is a model for the universal bundle $E G \to B G$ of locally trivial principal $G$-bundles over paracompact Hausdorff spaces?, or equivalently, of numerable bundle $G$-principal bundles over all Hausdorff topological spaces.

There is a generalisation of Milnor’s construction that works for topological groupoids, and reduces to the above case when the groupoid is $\mathbf{B}G$, the delooping of the group $G$.

## References

Last revised on May 26, 2017 at 05:38:27. See the history of this page for a list of all contributions to it.