Planar coverings of surfaces
These
doodles were made while thinking about the proof of a theorem of Maskit
(see [1]) which
classifies the regular planar coverings of a surface of finite type (that is a
surface whose fundamental group is finitely generated). Let S be a surface with
fundamental group p. Recall that any covering space of S is uniquely
described as the quotient of the universal cover of S by some subgroup H < p. We shall let SH denote the covering
space associated to the subgroup H. The covering is said to be regular when H
is a normal subgroup. In this case the group G= p/H
acts on SH with quotient space S. (The circles in the pictures
represent hypothetical translates of an essential simple closed curve in SH
under the action of G, in the case where SH is a planar surface).
The statement of the theorem is as follows:
Theorem:
(Maskit) Let SH à
S be a regular covering of surfaces where S is of
finite type. Then SH is a planar surface if and only if there exists
a (necessarily finite) collection of mutually disjoint simple closed
orientation preserving curves C1 ,.., Cn
in S and integers p1 ,..,
pn such that H is normally generated (as a subgroup
of p)
by the conjugacy classes [Ci]
pi
for i=1,..,n.
Just one
reason that this theorem is interesting is that it can be used in order to give
a proof of the Loop Theorem for 3-manifolds with boundary (this is because each
boundary component of the universal cover of a 3-manifold M with boundary is a
planar regular covering of a component of the boundary of M – the Loop
Theorem states that if B is a component of the boundary of M and the inclusion
of B in M induces a map on fundamental groups which fails to be injective then
there is an essential simple closed curve in B which is null-homotopic in M. Maskit’s
Theorem implies that there exists an essential closed curve some multiple of
which is null-homotopic in M. A further Lemma, due to
Whitehead and later Papakyriakopoulos,
is needed to complete the proof of the Loop Theorem). This theorem of Maskit was used in much the same way in my recent paper [2]
which gave a version of the Loop Theorem in the context of Poincaré
duality pairs.
Here is a sketch of the proof
of Maskit’s Theorem. Suppose that SH
à S is a regular covering of
surfaces with covering automorphism group G, and suppose
that S is of finite type. The theorem can be restated as follows
:
SH
is a planar surface if and only if there exists a family F of mutually disjoint
simple closed curves in SH which is G-invariant and which generates the
fundamental group of SH.
It is known
that a surface S is planar if and only if no pair of simple closed curves in S can
have intersection number 1. It follows that the above condition is sufficient
for SH to be planar (since the family of curves F will necessarily generate
the homology of SH). On the other hand, if we suppose that SH
is planar we can use the fact that no two simple closed curves intersect exactly
once in order to construct such a family of curves F.
Let C be
any essential simple closed curve in the planar surface SH, and
denote by G(C) the union of all translates of C by elements of G. Up to a small
perturbation of C we may suppose that it intersects each of its translates
transversely and at most one at a time. Define the complexity of C to be
the number of times that a curve close and parallel to C will intersect G(C). This is just the total number of times
that C crosses another of its translates. More
generally we define the length of any simple path or loop in G(C) to be
the number of times the path crosses right through an intersection point (without
turning left or right). Note that when C has complexity 0 the G-translates of C
are mutually disjoint simple closed curves. We
claim that, given any essential simple closed curve C we can find an essential
simple closed C0 which has complexity 0 and which lies in the
neighbourhood of G(C). This allows us to complete the proof by an inductive
process – suppose we have already found a G-invariant family F of mutually
disjoint simple closed curves, then either F already generates the fundamental group
or there is a further essential simple closed curve C which is disjoint from and
not parallel to any element of F. In the latter case we find a C0 with
zero complexity which is also disjoint from (and not parallel to) F and we
simply add G(C0) to F. The process terminates because S was supposed
to be of finite type.
To prove
the claim, we suppose that C has non-zero complexity and consider a translate gC which intersects C nontrivially.
By planarity, the two curves
intersect at least twice and cut each other up into a union of segments, as illustrated
below. Amongst all of these segments, choose A to be one of minimal length (in
the sense, relative to G(C), defined above). We may as well suppose that A
belongs to gC so that the endpoints
p and q of A decompose C into a union of paths B1 and B2. Define simple loops
C1, and C2, by taking the union of A with B1, and B2 respectively. Then each of
C1 and C2 has complexity strictly less than C, and at least one of them is
essential. The claim follows by a straightforward induction.
Since C1
lies in G(C) its complexity is at most its length in G(C) which is at most len(B1) + len(A) £
len(B1) + len(B2) £
len(C) – 2.
[1] B. Maskit,
A theorem on planar covering surfaces with applications to 3-manifolds, Ann.
of Math. 81
(1965), 341—355