REFERENCE SHEET

Coverings & van Kampen

Lift loops upstairs, or assemble a fundamental group from overlapping pieces.

Before this The fundamental group, Homomorphisms & quotients

π₁(U∪V) ≅ π₁(U) *π₁(U∩V) π₁(V)

Definitions

Covering map

A continuous map p:E→X such that every x has an open neighborhood U whose preimage is a disjoint union of open sets each mapped homeomorphically onto U.

Path lift

A path upstairs mapping to a given path downstairs, with a specified starting point in its fiber.

Amalgamated free product

A group formed from two groups by identifying images of a common group. In the display, the maps come from inclusions of the overlap.

Key theorems

Path lifting

For a covering map, a path and a starting point above its initial point determine a unique lift.

Injectivity on π₁

A covering map induces an injective homomorphism on fundamental groups at corresponding basepoints.

Seifert–van Kampen

If U,V are open and path connected, U∩V is nonempty and path connected, and the basepoint lies in the intersection, the displayed amalgamated-product description holds for X=U∪V.

Notable applications

Circle covers

The map z↦zⁿ from S¹ to S¹ realizes the subgroup nℤ of ℤ, for n≥1.

Computing presentations

Decompose spaces into pieces and translate overlap identifications into generators and relations.

Work it out

Try before revealing

01 / Lift

For p(t)=e²πⁱᵗ from ℝ to S¹, lift γ(s)=e⁴πⁱˢ starting at 0.

Hint

Find a real-valued function with the correct exponential.

Solution

The unique lift is γ̃(s)=2s. It ends at 2, so the lift is not a loop.

02 / Compute

What homomorphism on fundamental groups is induced by z↦z³ on S¹?