The Birman–Hilden property of covering spaces of nonorientable surfaces
UDC 517.5 Let $p: \widetilde{N} \rightarrow N$ be a finite covering space of nonorientable surfaces, where $\chi(\widetilde{N}) < 0$. We search whether or not $p$ has the Birman–Hilden property.  
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| author | Atalan, F. Medetogullari, E. Atalan, F. Medetogullari, E. Atalan, F. Medetogullari, E. |
| author_facet | Atalan, F. Medetogullari, E. Atalan, F. Medetogullari, E. Atalan, F. Medetogullari, E. |
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| description | UDC 517.5
Let $p: \widetilde{N} \rightarrow N$ be a finite covering space of nonorientable surfaces, where $\chi(\widetilde{N}) < 0$. We search whether or not $p$ has the Birman–Hilden property.
  |
| doi_str_mv | 10.37863/umzh.v72i3.6044 |
| first_indexed | 2026-03-24T03:25:46Z |
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UDC 517.5
F. Atalan (Dep. Math., Atilim Univ., Ankara, Turkey),
E. Medetogullari (Dep. Math., TED Univ., Ankara, Turkey)
THE BIRMAN – HILDEN PROPERTY OF COVERING SPACES
OF NONORIENTABLE SURFACES
ВЛАСТИВIСТЬ БIРМАНА – ХIЛЬДЕНА ДЛЯ ПРОСТОРIВ НАКРИТТЯ
НЕОРIЄНТОВНИХ ПОВЕРХОНЬ
Let p : \widetilde N \rightarrow N be a finite covering space of nonorientable surfaces, where \chi ( \widetilde N) < 0. We search whether or not p has
the Birman – Hilden property.
Нехай p : \widetilde N \rightarrow N — скiнченний простiр накриття неорiєнтовних поверхонь, де \chi ( \widetilde N) < 0. Ми з’ясуємо, чи має p
властивiсть Бiрмана – Хiльдена.
1. Introduction and statements of the results. Let \Sigma (resp., N ) denote the connected, orientable
(resp., nonorientable) surface of genus g with n marked points. Let \mathrm{M}\mathrm{o}\mathrm{d}(\Sigma ) (resp., \mathrm{M}\mathrm{o}\mathrm{d}(N))
denote the mapping class group of \Sigma (resp., N ), which is the group of isotopy classes of orientation-
preserving (resp., all) diffeomorphisms of \Sigma (resp., N ), where diffeomorphisms and isotopies fix the
set of marked points. In this paper, we assume that the set of marked points is the set of branched
points. Hence, the mapping classes must fix the set of branch points.
Our aim is to get a better understanding of the algebraic structure of the mapping class groups.
In this paper, we will take into consideration mainly nonorientable surfaces and their mapping class
groups, which are far less understood compared to their orientable counter parts.
In this work, we will consider the Birman – Hilden property of branched covers of surfaces, both
orientable and nonorientable. Let S denote a surface (orientable or nonorientable). Fix a covering
(possibly branched) p : \~S \rightarrow S. Let \mathrm{L}\mathrm{M}\mathrm{o}\mathrm{d}(S) be the finite index subgroup of \mathrm{M}\mathrm{o}\mathrm{d}(S) consisting of
mapping classes of S, which lift to diffeomorphisms of \~S. Let \mathrm{S}\mathrm{M}\mathrm{o}\mathrm{d}( \~S) be the subgroup of \mathrm{M}\mathrm{o}\mathrm{d}( \~S)
consisting of fiber preserving (or symmetric) mapping classes of \~S. If the surjective homomorphism
\Phi : \mathrm{L}\mathrm{M}\mathrm{o}\mathrm{d}(S) \rightarrow \mathrm{S}\mathrm{M}\mathrm{o}\mathrm{d}( \~S)/\mathrm{D}\mathrm{e}\mathrm{c}\mathrm{k}(p) is an isomorphism, then we say that the covering space p :
\~S \rightarrow S has the Birman – Hilden property. This property plays an important role in understanding the
algebraic structure of mapping class groups.
It is important to note that, in Birman – Hilden theory branch points in a surface S and their
preimages in \~S are treated differently. Any liftable diffeomorphism should leave the set of branch
points invariant and therefore, we regard the branch points in S as marked points. However, on \~S the
preimages of the branch points are considered as ordinary points, not as marked points. Therefore,
the diffeomorphisms of \~S do not need to leave invariant the preimage of the set of branch points.
It is known that all unbranched coverings and regular coverings are fully ramified so that they
satisfy the Birman – Hilden property [1 – 3, 6]. However, there are irregular branched covering spaces
which have the Birman – Hilden property and there are coverings that do not have the Birman – Hilden
property (see [8]).
Winarski gave one necessary condition and one sufficient condition for a covering space of
orientable surfaces to have the Birman – Hilden property in [8]. Later, Margalit and Winarski present
c\bigcirc F. ATALAN, E. MEDETOGULLARI, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3 307
308 F. ATALAN, E. MEDETOGULLARI
a new article related to the Birman – Hilden property [7]. In this paper, our goal is to investigate
whether or not similar results for nonorientable surfaces hold. The organization of the paper is as
follows. In Section 2, we will give some notations and definitions, secondly, we will state some
results of Winarski. In Section 3, we extend these results to nonorientable surfaces. In particular, we
prove the following theorem in this section.
Theorem 1.1. Let p : \widetilde N \rightarrow N be a finite covering space of nonorientable surfaces, where
\chi ( \widetilde N) < 0. If p is fully ramified, then it has the Birman – Hilden property.
In Section 4, we will introduce a blowing up process to obtain covering spaces of nonorientable
surfaces from those of orientable surfaces. By using this process, we prove the following theorem.
Theorem 1.2. If a covering space does not have the Birman – Hilden property then the covering
space that is obtained by blowing up the given covering spaces does not have the Birman – Hilden
property either.
In the same section, using Theorem 1.2 repeatedly, we construct a covering space of nonorientable
surfaces, which does not have the Birman – Hilden property. We also prove in Theorem 4.1 that the
blowing up process preserves the weak curve lifting property.
Lastly, the final section contains some immediate consequences of these results.
2. Preliminaries. Throughout this paper, let S denote a surface (orientable or nonorientable).
2.1. A branched covering space over a surface. Let S be a surface. A space \~S is said to be a
branched covering space over S if there is a map p : \~S \rightarrow S such that other than the inverse image
of some finite set B in S, p is a covering map, where for all s \in B there is an open neighborhood U
such that U \cap B = \{ s\} . Moreover, p - 1(U) is the disjoint finite union some open subsets Vi so that
the restriction map of p on Vi\setminus \{ \~si\} , p| : Vi\setminus \{ \~si\} \rightarrow U\setminus \{ s\} is a degree ri covering map and each Vi
contains exactly one point \~si \in p - 1(s), and at least one ri > 1.
2.2. Ramification number. The number ri is called a ramification number. A preimage \~si is
said to be ramified, if ri > 1, otherwise it is said to be unramified.
2.3. Fully ramified. A covering space is said to be fully ramified, if all points in p - 1(B) are
ramified.
2.4. Simple cover. A degree n branched covering space is said to be a simple cover if every
branch point has n - 1 preimages.
2.5. Essential simple closed curves. Let a be an unoriented simple closed curve on any surface
(marked surface or not) S. According to whether a regular neighbourhood of a is an annulus or a
Möbius band, we call a two-sided or one-sided, respectively. An essential simple closed curve on an
orientable surface \Sigma is a curve which is not isotopic to a single point on the surface \Sigma (see [8]). We
note that all simple closed curves on \Sigma are two-sided. On the other hand, on a nonorientable surface
N, an essential simple closed curve is a two-sided curve, which is not isotopic to a single point or
which is not the boundary of a Möbius band. For example, on Fig. 1, the simple closed curve c on
the surface N is essential but the preimages c1 and c2 on the cover \widetilde N are inessential, because each
ci bounds a Möbius band. The other preimage c3 on \widetilde N is also inessential, since it bounds a disc. As
it is indicated in the introduction, on the cover \~S the preimages of the branch points are considered
as ordinary points, not as marked points.
2.6. The weak curve lifting property. A covering space of surfaces is said to have the weak
curve lifting property if the preimage in \widetilde \Sigma (resp., \widetilde N ) of every essential simple closed curve on \Sigma
(resp., N ) has at least one essential connected component (see [8]).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
THE BIRMAN – HILDEN PROPERTY OF COVERING SPACES OF NONORIENTABLE SURFACES 309
. . . . . . . .
..
. . . . . . . .
.
.
. . . .
.
.
c
1
c
2
c
3
N
N
c
*
*
*
**
*
Fig. 1
2.7. The orientation double cover of \bfitN . Let \pi : \Sigma \rightarrow N be the orientation double cover of the
nonorientable surface N with the deck transformation \tau : \Sigma \rightarrow \Sigma . Then \pi \ast (\Pi 1(\Sigma , x)) is a normal
subgroup of index 2 in \Pi 1(N, \pi (x)) and the deck transformation group is \langle \tau | \tau 2 = identity\rangle \sim = \BbbZ 2.
If F \prime is a diffeomorphism of \Sigma such that \tau \circ F \prime = F \prime \circ \tau , then F \prime induces a diffeomorphism F on
N. On the other hand, any diffeomorphism F of the nonorientable surface N has a unique lift to an
orientation preserving diffeomorphism F \prime : \Sigma \rightarrow \Sigma .
Indeed, any isotopy of the nonorientable surface N lifts to the surface \Sigma . Moreover, if the Euler
characteristic of the surface is negative (hence it carries a complete hyperbolic metric), then any
isotopy of orientation preserving diffeomorphisms of \Sigma , whose end maps commute with the deck
transformation \tau , can be deformed into one which commutes with \tau , and hence descends to an
isotopy of diffeomorphisms of N (cf. Lemma 5 in [9]).
2.8. Pull back map. Let \pi : E \rightarrow B be a fiber bundle, which is a continuous surjective locally
trivial map so that for each point b \in B, \pi - 1(p) = F. Obviously covering spaces are fiber bundles.
If f : B\prime \rightarrow B is a continuous map, the set
f\ast E =
\bigl\{
(b\prime , e) \in B\prime \times E | f(b\prime ) = \pi (e)
\bigr\}
which gives the following commutative diagram is called pull back bundle:
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
310 F. ATALAN, E. MEDETOGULLARI
f\ast E
\pi \ast
\rightarrow B\prime
f\ast \downarrow \downarrow f
E
\pi \rightarrow B ,
where \pi \ast : f\ast E \rightarrow B\prime and f\ast : f\ast E \rightarrow E are projection maps, first one projects onto first component
and the second one projects onto second component.
2.9. One necessary condition and one sufficient condition for orientable surfaces. Winarski
proved the following two results.
Theorem 2.1 [8]. Let p : \widetilde \Sigma \rightarrow \Sigma be a finite covering space of orientable surfaces such that
\chi (\widetilde \Sigma ) < 0. If p is fully ramified, then p has the Birman – Hilden property.
Proposition 2.1 [8]. Let p : \widetilde \Sigma \rightarrow \Sigma be a finite covering space of orientable surfaces such that
\chi (\widetilde \Sigma ) < 0. If p : \widetilde \Sigma \rightarrow \Sigma has the Birman – Hilden property, then it has the weak curve lifting property.
3. One necessary condition and one sufficient condition for nonorientable surfaces. 3.1. The
necessary part. The following proposition and its proof are analogous to the orientable case (see
Proposition 2.1 [8]) and, therefore, we only state the proposition and omit its proof.
Proposition 3.1. Let p : \widetilde N \rightarrow N be a finite covering space of nonorientable surfaces such that
\chi ( \widetilde N) < 0. If p : \widetilde N \rightarrow N has the Birman – Hilden property, then it has the weak curve lifting
property.
3.2. The sufficient part . 3.2.1. Proof of Theorem 1.1. The proof has two steps:
Step 1. Let \pi : \Sigma \rightarrow N be the orientation double cover and let \~\pi : \widetilde \Sigma \rightarrow \widetilde N be the pull back of \pi :
\Sigma \rightarrow N by p: \widetilde \Sigma \~p\rightarrow \Sigma
\~\pi \downarrow \downarrow \pi \widetilde N p\rightarrow N ,
where p\ast (\Sigma ) = \widetilde \Sigma =
\bigl\{
(x, y) \in \widetilde N \times \Sigma | p(x) = \pi (y)
\bigr\}
. Since \pi : \Sigma \rightarrow N is double covering, \~\pi :\widetilde \Sigma \rightarrow \widetilde N is double covering where \~\pi (x, y) = x.
Now, if \tau : \Sigma \rightarrow \Sigma is the deck transformation, then \~\tau : \widetilde \Sigma \rightarrow \widetilde \Sigma given by \widetilde \tau (x, y) = (x, \tau (y)) is
the deck transformation of \~\pi : \widetilde \Sigma \rightarrow \widetilde N. Indeed,
\widetilde \tau 2(x, y) = \widetilde \tau (\widetilde \tau (x, y)) = \widetilde \tau (x, \tau (y)) = (x, \tau 2(y)) = (x, y).
Thus, \widetilde \tau 2 is the identity.
The map \~p : \widetilde \Sigma \rightarrow \Sigma given by \~p(x, y) = y is also fully ramified. Since \Sigma is orientable, \widetilde \Sigma is
orientable. Indeed, since \~\pi : \widetilde \Sigma \rightarrow \widetilde N is a double covering and \widetilde \Sigma is orientable, we deduce that \~\pi :\widetilde \Sigma \rightarrow \widetilde N is the orientation double cover. Now, since \~p : \widetilde \Sigma \rightarrow \Sigma given by \~p(x, y) = y is fully
ramified, \~p has the Birman – Hilden property by Theorem 2.1.
Step 2. We will show that p : \widetilde N \rightarrow N has the Birman – Hilden property. Suppose not. Then
there is a diffeomorphism f : N \rightarrow N such that f is not isotopic to the identity, and its lift \~f :\widetilde N \rightarrow \widetilde N is isotopic to a deck transformation.
Let g : \Sigma \rightarrow \Sigma and \~g : \widetilde \Sigma \rightarrow \widetilde \Sigma be the unique orientation preserving lifts f and \~f to the orientation
double covers, respectively. Since f is not isotopic to the identity, g is not isotopic to the identity. On
the other hand, the unique orientation preserving lift \~g of \~f is isotopic to a deck transformation. So,
\~p : \widetilde \Sigma \rightarrow \Sigma does not have the Birman – Hilden property. By Winarski’s result, this is a contradiction,
since \~p is fully ramified.
Theorem 1.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
THE BIRMAN – HILDEN PROPERTY OF COVERING SPACES OF NONORIENTABLE SURFACES 311
4. Blow up process and the Birman – Hilden property. In this section, we will introduce
blow up of coverings and prove Theorem 1.2 which enables us to construct coverings not having the
Birman – Hilden property. Furthermore, we prove Theorem 4.1 and we give Theorem 4.2.
4.1. Blow up process. Let p : \widetilde \Sigma \rightarrow \Sigma be a d-fold covering space of orientable surfaces. We
delete one disc D \subset \Sigma and denote its lifts (discs) as D1, D2, . . . , Dd in \widetilde \Sigma and identify the boundary
points of each deleted disc via antipodal map. We denote the resulting covering space q : \widetilde N \rightarrow N.
This is said to be a blow up process.
4.2. Proof of Theorem 1.2. Let p : \widetilde \Sigma \rightarrow \Sigma and q : \widetilde N \rightarrow N be the coverings given as in
Subsection 4.1. We will show if p : \widetilde \Sigma \rightarrow \Sigma does not have the Birman – Hilden property, q : \widetilde N \rightarrow N
does not have the Birman – Hilden property.
Let f : \Sigma \rightarrow \Sigma be a diffeomorphism not isotopic to the identity but it lifts to some \~f : \widetilde \Sigma \rightarrow \widetilde \Sigma
which is isotopic to a deck transformation. By isotopy we may assume that f is identity on some disc
on which we blow up \Sigma to get N. Then f induces some diffeomorphism g such that the restriction
of g on N \setminus \BbbR P 2 is equal to f. Moreover, the following diagram commutes:
N
g\rightarrow N
B\downarrow \downarrow B
\Sigma
f\rightarrow \Sigma ,
where B : N \rightarrow \Sigma is the blow up projection. Obviously, g lifts to \~g, which is also a diffeomorphism
isotopic to a deck transformation of \widetilde N \rightarrow N. Therefore, \widetilde N \rightarrow N does not have the Birman – Hilden
property.
Theorem 1.2 is proved.
Theorem 4.1. Let p : \widetilde S \rightarrow S be a finite covering space of surfaces such that \chi (\widetilde S) < 0. If p :\widetilde S \rightarrow S has the weak curve lifting property, then so does its blow up.
Proof. Suppose not. Then, there is an inessential (two-sided) simple closed curve \gamma on the
surface S, but \gamma is an essential (two-sided) simple closed curve on S\sharp \BbbR P 2 and all components of
the preimage are inessential on \widetilde S\sharp n\BbbR P 2, where n is the degree of p : \widetilde S \rightarrow S. Now, there are two
cases to consider:
Case 1. The curve \gamma bounds a punctured disc, say U, on the surface S. Since \gamma is an essential
simple closed curve on S\sharp \BbbR P 2, the point we blow up lies inside the punctured disc bounded by \gamma .
Any component of the preimage of U is still a punctured disc. Each component is a connected cover
of U and since the puncture is a branch point, one of the components is at least a two fold cover.
After applying the blow up process to the covering space, U becomes a punctured Möbius band and
each connected component of the preimage of this Möbius band is a nonorientable surface with one
boundary component. Moreover, at least one of them has genus at least two. Let \~\gamma be the boundary
of one such connected component of genus at least two on the surface \widetilde S. The curve \~\gamma , which is of
course a component of the preimage of \gamma , is an essential curve. To see this note that the surface \widetilde S
has Euler characteristic at most - 1. After the blow up process its genus is increased by at least two.
Hence, the blown up nonorientable surface has genus at least five. Therefore, the boundary curve \~\gamma
is essential on \widetilde S\sharp n\BbbR P 2, which is a contradiction.
Case 2. The curve \gamma bounds a Möbius band, call again U, on the surface S. Now, any component\widetilde U of the preimage of U is either a Möbius band or a cylinder, depending on whether the degree
of the covering \widetilde U \rightarrow U is odd or even. Hence, after the blow up process each component of the
preimage of U is a nonorientable surface of genus at least two with one or two boundary components
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
312 F. ATALAN, E. MEDETOGULLARI
on the surface \widetilde S\sharp n\BbbR P 2. Finally, an Euler characteristic argument as in the above case finishes the
proof.
Theorem 4.1 is proved.
4.3. An application of Proposition 3.1. We deal with the threefold simple branched cover of
surfaces p : \widetilde \Sigma \rightarrow \Sigma , where \widetilde \Sigma is a closed surface of genus g and \Sigma is a sphere with 2g + 4 branch
points. It is known that this covering space does not have the Birman – Hilden property [4, 5]. Also,
the example with g = 3 in Fig. 3 in [8] does not have the Birman – Hilden property since it leaks the
weak curve lifting property.
Now, we blow up this covering space and obtain the following covering space as in Fig. 1. As
it is indicated in the introduction, the preimages of the branch points which are shown in Fig. 1, are
not thought as marked points in \~N. The preimage of the essential simple closed curve c in Fig. 1
is the union of c1, c2 and c3 in \~N and every ci is inessential (see Subsection 2.5). Therefore, this
covering space does not have the weak curve lifting property and so, it does not have the Birman –
Hilden property by Proposition 3.1.
The two figures in the following subsection aim to present that proving a covering space does
not have the Birman – Hilden property by using weak curve lifting property may not be easy. On the
other hand, Theorem 1.2 will be easier to use instead.
4.4. An application of Theorem 1.2. We consider again the threefold simple branched cover of
surfaces p : \widetilde \Sigma \rightarrow \Sigma , where \widetilde \Sigma is a closed surface of genus g and \Sigma is a sphere with 2g + 4 branch
points and we take g = 2. Then by blowing up this one twice gives a covering space as in Figs. 2
and 3.
In this situation, the preimage of the essential simple closed curve c in Fig. 2 is the union of c1,
c2 and c3 in \~N and at least one of the components of the preimage of c is essential (c1 and c2 are
essential, because each ci bounds a Klein Bottle). Therefore, we could not say immediately that this
covering space does not have the weak curve lifting property, so we could not use Proposition 3.1.
We need to check whether or not this covering space has the weak curve lifting property for other
essential curves.
Winarski described an algorithm to check whether or not a covering space has the weak curve
lifting property (Proposition 2 and Theorem 4 in [8]). It follows from the proofs of these results that
they are valid for nonorientable surfaces also (for the sake of completeness we include a sketch of
her proof):
Theorem 4.2. Let p : \widetilde S \rightarrow S be a finite covering space of surfaces such that \chi (\widetilde S) < 0. There
is an algorithm to check whether or not p has the weak curve lifting property.
Sketch of Proof. First we assume that the covering space is unbranched of degree n. Fix base
points x0 \in S and \~x0 \in p - 1(x0) \subset \~S. Clearly, a diffeomorphism of S lifts to \~S if and only if it
preserves the image subgroup H0 = p\ast (\pi 1( \~S, \~x0)). Let \mathrm{L}\mathrm{M}\mathrm{o}\mathrm{d}(S, x0) denote the subgroup of ele-
ments that have liftable representatives. There is an action of \mathrm{M}\mathrm{o}\mathrm{d}(S, x0) on the set of all subgroups
\pi 1
\bigl(
\~S, \~x0
\bigr)
of index n. Denote the orbit of H0 by \scrH , which is clearly finite. Moreover, from the
lifting criteria for maps to the covering spaces we see that the stabilizer of H0 is \mathrm{L}\mathrm{M}\mathrm{o}\mathrm{d}(S, x0).
Denote the homomorphism P : \mathrm{M}\mathrm{o}\mathrm{d}(S, x0) \rightarrow \Sigma \scrH , which describes the restriction of the action to
the orbit \scrH .
Now using the finiteness of the group \Sigma \scrH , we choose a finite set F in \mathrm{M}\mathrm{o}\mathrm{d}(S, x0) so that for
each H \in \scrH , there is some f \in F with f \cdot H = H0. Finally, using the forgetfull homomorphism
\mathrm{M}\mathrm{o}\mathrm{d}(S, x0) \rightarrow \mathrm{M}\mathrm{o}\mathrm{d}(S) one obtains a set of coset representatives for the subgroup \mathrm{L}\mathrm{M}\mathrm{o}\mathrm{d}(S) in
\mathrm{M}\mathrm{o}\mathrm{d}(S). Next we use this algorithm to finish the proof of the theorem for (un)branched coverings.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
THE BIRMAN – HILDEN PROPERTY OF COVERING SPACES OF NONORIENTABLE SURFACES 313
. . . . . .
. . . .
.
.
. .
.
.
1
2
3
N
N
. .
c
c
c
c
*
* *
**
*
Fig. 2
To do this let S\circ denote the surface S, where the branched points are removed and let \~S\circ = p - 1(S\circ ).
Let \scrG (S) denote the set of isotopy classes of two-sided simple closed curves in S and let \scrM (S)
denote the set of isotopy classes of two-sided multicurves in S. The embedding S\circ \rightarrow S clearly
takes simple closed curves to simple closed curves. So there is sequence of maps
\Psi : \scrG (S) \rightarrow \scrG (S\circ ) \rightarrow \scrM (S\circ ) \rightarrow \scrM (S),
where the first one is a bijection, the second one is given by taking preimage and the last one simply
fills in the preimages of the branch points back.
Now if two elements c, c\prime \in \scrG (S) are so that f(c) = c\prime for some f \in \mathrm{L}\mathrm{M}\mathrm{o}\mathrm{d}(S), then \Psi (c)
and \Psi (c\prime ) have the same number of essential and inessential components. Hence, checking the weak
curve lifting property amounts to compute \Psi (c) for a set of representatives of \scrG (S)/\mathrm{L}\mathrm{M}\mathrm{o}\mathrm{d}(S).
Finally, the proof concludes using the fact that by the classification of surfaces the set \scrG (S)/\mathrm{M}\mathrm{o}\mathrm{d}(S)
is finite.
Theorem 4.2 is proved.
One should check the weak curve lifting property for essential (two-sided) simple closed curves
corresponding to each topological type. Therefore, in order to clarify the construction of this covering
space and identify topologically different curves on the surfaces, we need to distinguish dots from
stars although they are equivalent topologically.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
314 F. ATALAN, E. MEDETOGULLARI
. . . . . .
. . . .
.
.
. .
.
.
N
N
. .
d
d
3
d
2
* *
* *
**
d
1
Fig. 3
Now, let us consider another essential simple closed curve d in Fig. 3. Its preimage is the union
of d1, d2 and d3 in \~N and one of the components of the preimage of d is essential (d2 is essential,
because it bounds a nonorientable surface of genus 3).
It turns out that there are 29 topological types one of which is nonseparating and all others are
separating for this covering space. One can determine all components of the preimage of every
essential curve and one can check whether or not at least one of the components of the preimage of
each essential curve is essential. However, this may not always be easy. In this situation, Theorem 1.2
will be easier to use as a criterion to say that this covering space does not have the Birman – Hilden
property.
5. Final remarks. Let p : \widetilde \Sigma \rightarrow \Sigma denote a covering (possibly branched) of surfaces. Let z \in \Sigma
and \{ \~z1, \~z2, . . . , \~zn\} = p - 1(z), where z is not a branch point. Let N be the blow up of \Sigma at z and\widetilde N be the blow up of \widetilde \Sigma at all \~zi’s. Then we get a covering pB : \widetilde N \rightarrow N. In this case, we have the
following observations:
1. If p : \widetilde \Sigma \rightarrow \Sigma is a regular covering, then so is pB : \widetilde N \rightarrow N. Thus, both coverings have the
Birman – Hilden property.
2. If p : \widetilde \Sigma \rightarrow \Sigma is a fully ramified, then so is pB : \widetilde N \rightarrow N. Hence, both coverings have the
Birman – Hilden property, by Theorems 2.1 and 1.1.
3. p : \widetilde \Sigma \rightarrow \Sigma is a simple cover(see Subsection 2.4)
\bigl(
\chi (\widetilde \Sigma ) < 0
\bigr)
if and only if pB : \widetilde N \rightarrow N
is a simple cover
\bigl(
\chi ( \widetilde N) < 0
\bigr)
. If p : \widetilde \Sigma \rightarrow \Sigma is a simple cover
\bigl(
\chi (\widetilde \Sigma ) < 0
\bigr)
, p does not have the
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
THE BIRMAN – HILDEN PROPERTY OF COVERING SPACES OF NONORIENTABLE SURFACES 315
Birman – Hilden property by Theorem 2 in [8]. If pB : \widetilde N \rightarrow N is a simple cover
\bigl(
\chi ( \widetilde N) < 0
\bigr)
, then
pB does not have the Birman – Hilden property. Because, if pB : \widetilde N \rightarrow N is a simple cover, then p :\widetilde \Sigma \rightarrow \Sigma is a simple cover. By Winarski’s result, p does not have the Birman – Hilden property. Since
p does not have the Birman – Hilden property, pB does not have the Birman – Hilden property by
Theorem 1.2.
References
1. J. Aramayona, C. J. Leininger, J. Souto, Injections of mapping class groups, Geom. Topol., 13, № 5, 2523 – 2541
(2009).
2. J. S. Birman, H. M. Hilden, Lifting and projecting homeomorphisms, Arch. Math. (Basel), 23, 428 – 434 (1972).
3. J. S. Birman, H. M. Hilden, On isotopies of homeomorphisms of Riemann surfaces, Ann. Math., 2, № 97, 424 – 439
(1973).
4. J. S. Birman, B. Wajnryb, 3-fold branched coverings and the mapping class group of a surface, Geometry and
Topology (College Park, Md., 1983/84), Lect. Notes Math., 1167, 24 – 46 (1985).
5. T. Fuller, On fiber-preserving isotopies of surface homeomorphisms, Proc. Amer. Math. Soc., 129, № 4, 1247 – 1254
(2001).
6. C. Maclachlan, W. J. Harvey, On mapping class groups and Teichmüller spaces, Proc. London Math. Soc. (3), 30,
№ 4, 496 – 512 (1975).
7. D. Margalit, R. R. Winarski, The Birman – Hilden Theory, arXiv:1703.03448 (2017).
8. R. R. Winarski, Symmetry, isotopy, and irregular covers, Geom. Dedicata, 177, 213 – 227 (2015).
9. W. Yingqing, Canonical reducing curves of surface homeomorphism, Acta Math. Sinica (N.S.), 3, № 4, 305 – 313
(1987).
Received 22.01.17
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
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| id | umjimathkievua-article-6044 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:46Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/f6/eb932627d3f4e85e1ca3f4ead31e69f6.pdf |
| spelling | umjimathkievua-article-60442022-03-26T11:01:30Z The Birman–Hilden property of covering spaces of nonorientable surfaces The Birman–Hilden property of covering spaces of nonorientable surfaces The Birman–Hilden property of covering spaces of nonorientable surfaces Atalan, F. Medetogullari, E. Atalan, F. Medetogullari, E. Atalan, F. Medetogullari, E. UDC 517.5 Let $p: \widetilde{N} \rightarrow N$ be a finite covering space of nonorientable surfaces, where $\chi(\widetilde{N}) &lt; 0$. We search whether or not $p$ has the Birman–Hilden property. &nbsp; УДК 517.5 Нехай $p: \widetilde{N} \rightarrow N$ — скiнченний простiр накриття неорiєнтовних поверхонь, де $\chi(\widetilde{N}) &lt; 0$. Ми з’ясуємо, чи&nbsp; має $p$ властивiсть Бiрмана–Хiльдена. Institute of Mathematics, NAS of Ukraine 2020-03-28 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6044 10.37863/umzh.v72i3.6044 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 3 (2020); 307-315 Український математичний журнал; Том 72 № 3 (2020); 307-315 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6044/8684 |
| spellingShingle | Atalan, F. Medetogullari, E. Atalan, F. Medetogullari, E. Atalan, F. Medetogullari, E. The Birman–Hilden property of covering spaces of nonorientable surfaces |
| title | The Birman–Hilden property of covering spaces of nonorientable surfaces |
| title_alt | The Birman–Hilden property of covering spaces of nonorientable surfaces The Birman–Hilden property of covering spaces of nonorientable surfaces |
| title_full | The Birman–Hilden property of covering spaces of nonorientable surfaces |
| title_fullStr | The Birman–Hilden property of covering spaces of nonorientable surfaces |
| title_full_unstemmed | The Birman–Hilden property of covering spaces of nonorientable surfaces |
| title_short | The Birman–Hilden property of covering spaces of nonorientable surfaces |
| title_sort | birman–hilden property of covering spaces of nonorientable surfaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6044 |
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