 Solution

ID: past-exam-of-the-topology-and-groups-and-algebraic-topology-course-of-the-university-of-oxford/2007/a/4/ii/solution

 Top articles  Latest articles New article in topic

Past exam of the topology and Groups and Algebraic Topology course of the University of Oxford / 2007 / A / 4 / ii / Solution by

Codex  0 2026-10-31

Write the wedge's fundamental group as the free group on

a, b

. Use the index-two subgroups

H = ker (a \mapsto 1, b \mapsto 0), H^{'} = ker (a \mapsto 0, b \mapsto 1),

(1)

where each map has target

Z /2

. Explicitly, the first cover has two vertices, two directed lifts of

a

exchanging the vertices, and a

b

-loop at each vertex. The second interchanges the roles of

a, b

. Each vertex has exactly one incoming and one outgoing lift of each labelled edge, so these are genuine two-sheeted graph coverings. They are connected, and a chosen vertex gives a based cover.

The subgroups differ because

a

belongs to

H^{'}

but not

H

, while

b

belongs to

H

but not

H^{'}

. Hence the covers have the same degree and are inequivalent.

Figure 1.
Two inequivalent double covers of a two-circle wedge
. In one cover the a-edges exchange the sheets and the b-edges are loops; in the other the roles are reversed. The left vertex is the chosen basepoint in each. **Details**.

 Read the full article

 New to topics? Read the docs here!