FIXED POINTS ON NON TRIVIAL SURFACE BUNDLES
D. L. GONÇALVES, A. K. M. LIBARDI, D. PENTEADO AND J. P. VIEIRA
Abstract. The main purpose of this work is to study fixed points of fibrepreserving maps over S 1 of a surface bundles S → E → S 1 , where S is a closed
surface of negative Euler characteristic. When the bundle is trivial this has
already been solved. For non trivial bundles we ????????
Introduction
Given a fibration p : E → B and f : E → E a fibre-preserving map over B, the
question if f can be deformed over B (by a fibrewise homotopy) to a fixed point
free map has been considered for several years by many authors. Among others, see
for example [?], [?], [?], [?], [?], [?], [?], [?] and [?]. More recently also the fibrewise
coincidence case has been considered in [?], [?], [?], [?] and [?], which certainly has
intersection with the fixed point case.
For the case where the fibre is a surface see more details in [?] and [?] and
references therein.
In the present work we study the case on a non trivial surface bundles S → E →
1
S where S is a closed surface with Euler characteristic negative.
The main result of this paper is: ??????
This paper is organized into ??? sections. In section 1
1. Fibre preserving maps of S-bundles over S 1
Let f : E → E be a fibre-preserving map over B, i.e., p ◦ f = p where p : E → B
is a fibre bundle with fibre a surface denoted by S (also called an S-bundle over B).
0
When is f deformable over B to a fixed point free map f by a fibrewise homotopy
over B? We remark that in order to have a positive answer a necessary condition
is that the map f restricted to a fibre is deformable to a fixed point free map.
Now we review some known facts of the literature and show few general results
about fibre preserving maps of S-bundles over S 1 which are useful for our purpose.
Date: October 14, 2015.
2010 Mathematics Subject Classification. Primary 55M20, Secondary 55R10.
Key words and phrases. fixed point; fibre bundle; fibrewise maps and homotopy; surface;
braids.
1
2
D. L. GONÇALVES, A. K. M. LIBARDI, D. PENTEADO AND J. P. VIEIRA
Given h : S → S a homeomorphism let us consider the bundles Eh → S 1 where
Eh = S × [0, 1]/ ∼ and ∼ is the relation (x, 0) ∼ (h(x), 1). From now on we will
denote Eh by E, for short.
Lemma 1.1. Given g : S → S, then there is a fibre preserving map f : E → E
such that restricted to the fibre is g if and only if the two maps g ◦ h and h ◦ g are
homotopic.
Proof. The hypothesis implies that g and h ◦ g ◦ h−1 are homotopic and let H
be a homotopy. Consider the map G : S × [0, 1] → S × [0, 1] defined by G(x, t) =
(H(x, t), t). This map induces on the fibre bundle a desirable map because if (x, 0) ∼
(h(x), 1) then (g(x), 0) ∼ (h ◦ g ◦ h−1 (h(x)), 1) = (h ◦ g(x), 1). The converse follows
easily.
Now let us consider the induced homomorphism on the fundamental groups
g# : π1 (S) → π1 (S).
Proposition 1.2. Let S be a closed surface with χ(S) < 0, and g : S → S be a
map. Then:
a) If g# is the trivial homomorphism, then the map g is homotopic to the constant
map. The map g extends to a fibre preserving map and the set of fibrewise homotopy
classes of extensions is into one-to-one correspondence with the set of conjugacy
classes of elements of π1 (S).
b) If the image of the homomorphism g# is a free group of rank one, generated by an
element v, then if the map g extends to a fibre-preserving map the set of fibrewise
homotopy classes of extensions is into one-to-one correspondence with this set of
elements of the centralizer of Im(g# ), which in turn is isomorphic to Z generated
by an element w which is a maximal root of the generator v.
c) If the image of the homomorphism g# is either π1 (S) or a free group of rank
greater than one, then there is at most one homotopy class of extension f : E → E.
Proof. For the part a) for each element α ∈ π1 (S) consider the map f : E →
E given as composite of the projection p : E → S 1 and a map ` : S 1 → E
which represents (α, 1) ∈ π1 (E) = π1 (S) o Z. Since p ◦ ` = IdS 1 , the composite
` ◦ p is a fibre-preserving map. Now given any other map f 0 which extends g, the
induced homomorphism by f 0 on the fundamental group coincide with the induced
homomorphism of one of the maps constructed above. So we claim that the two
maps are indeed fibrewise homotopic. We use the homotopy lifting property for
the homotopy H 0 : E × I × I → S 1 where H 0 in E × I × 0 is the given homotopy
composite with p and in E ×I ×1 is the projection into E followed by p. The number
of equivalent extensions are classified by the homomorphisms up to conjugacy.
FIXED POINTS ON NON TRIVIAL SURFACE BUNDLES
3
For the case b) let us construct an extension as above but now we have the
restriction that the element α above must commute with v.
If (α, u) ∈ π1 (E) = π1 (S)oπ1 (S 1 ) where u is a generator of π1 (S 1 ) then utu−1 =
h# (t), ∀t ∈ π1 (S). Therefore f# (u)f# (t)f# (u)−1 = f# ◦ h# (t) = g# ◦ h# (t). So
(α, u)g# (t)(α, u)−1 = f# ◦h# (t) = g# ◦h# (t) and therefore (α, 1)(1, u)g# (t)(1, u−1 )(α−1 , 1) =
g# ◦ h# (t).
So by lemma ??, θ−1 (α, 1)h# ◦ g# (t)(α, 1)−1 θ = h# ◦ g# (t) where θ ∈ π1 (S).
Then θ−1 (α, 1) commutes with h# ◦ g# (t).
−1
−1 −1
−1
So h−1
(α, 1))h−1
(α, 1) or equivalently
# (θ
# ◦ h# ◦ g# (t) = h# ◦ h# ◦ g# (t)h# θ
−1
−1
h−1
(α, 1)) ◦ g# (t) = g# (t)h−1
(α, 1).
# (θ
# θ
−1
(α, 1)) commutes with g# (t).
Therefore h−1
# (θ
−1
−1
But h−1
(α, 1)) = wr because h−1
(α, 1)) ∈ C(g# (π1 (S))) =< w >,
# (θ
# (θ
where wr = v which implies that θ−1 (α, 1) = h# (wr ) and then (α, 1) = θh#(wr )) ∀r ∈
Z.
The rest of the proof is as in case a).
The case c) is similar and we use the fact that the centralizer of the image of the
homomorphism g# is trivial. This force (α, 1) = θ and the result follows.
The cases we are interested are b) and c) since in case a) the maps g has always
a fixed point. For a given non trivial bundles E and a map g : S → S which can
be deformed to fixed point free, we will be interested to know if there is a fibrewise
map f which is fixed point free and the restriction to S is homotopic to g. Further
in case such map exist, how many homotopy class of such maps there exist. We
will treat these question on the next two sections.
2. Non trivial surface bundles and fibrewise maps
Now we consider surface bundles over S 1 , S → E → S 1 where E =
S×I
(x,0)∼(h(x),1)
for h : S → S a homeomorphism where E is not homeomorphic to the product
S × S 1 . A major question is: Given a homotopy class α ∈ [S, S] which has minimal
number of fixed point µ, i.e., there is g 0 ∈ α which has µ fixed points and any other
map on the homotopy class has at least µ fixed points, we would like to classify
how many homotopy classes of fibre maps f : E → E there exist which restrict to
the fibre is the class of α and over each fiber the map has exactly µ fixed points. In
this note we study this question for the case µ = 0. Our main result of this section
is to show that there is at most one class of fibre maps. In principle may happen
that the number of homotopy classes [f ] is zero and this question will be studied
in the next section.
4
D. L. GONÇALVES, A. K. M. LIBARDI, D. PENTEADO AND J. P. VIEIRA
We begin with few generalities and notation which helps to formulate more clear
the problem. Let [E, E]B denote the set of homotopy class of the fibre maps of E
over B. Then there is a well defined map ψ : [E, E]B → [S, S] which associate to a
homotopy class of a fibre-preserving map [f ] the homotopy class of the restriction
g|S : S → S (chosen fibre over a base point of B).
Definition 2.1. A map g is is called of type I if rank(Im(g# )) 6= 0, 1. In case
rank(Im(g# )) = 1 we say that f is of type II.
We remark that [S, S] = [s, s]I t [S, S]II where [S, S]I and [S, S]II denote the
set of homotopy class of the fibre maps of S ober B of type I and II, respectively.
Using Lemma ?? one can show that the map above is certainly not surjective. In
case the hypothesis of the Lemma ?? is satisfied and [g] ∈ [S, S] satisfies µ[g] = 0,
we would like to compute the number of classes [f ] ∈ ψ −1 ([g]) such that f can be
deformed fibrewise to a fixed point free map.
Call W ⊂ [S, S] the image of ψ and W0 ⊂ W the elements of W which have
minimal number of fixed points zero. From now on let us consider a class α ∈ W0 .
Lemma 2.2. Given φ : π1 (S) → π1 (S × S − ∆) such that the diagram
φ
π1 (S)
id
/ π1 (S × S − ∆)
&
p2 | #
π1 (S)
commutes, i.e. p2 |# ◦ φ = id, and if θ ∈ π1 (S × S − ∆) commutes with Imφ then
θ = 1.
Proof. This follows from the proof of Theorem 2.3 in [?].
Proposition 2.3. Suppose that f, f1 : E → E are fixed point free maps over S 1
and f |S = f1 |S. Then f is fibrewise homotopic to f1 over S 1 (which we shall denote
by f ∼S 1 f1 ).
Proof. Suppose that f, f1 : E → E are fixed point free maps over S 1 and f |S =
f1 |S. Then there exist φ˜1 , φ̃ : E → E ×S 1 E − ∆ that commute the diagram
S − {x0 }
ι
φ˜1
/ E ×S 1 E − ∆
5
;
φ̃
E
(f,1)
(f1 ,1)
/ 1 E ×S 1 E
p2
/E
FIXED POINTS ON NON TRIVIAL SURFACE BUNDLES
5
namely, φ̃(x) = (f (x), x) and φ˜1 (x) = (f1 (x), x)).
We have that
π1 (E) = c, b¯1 , . . . , b¯n ; cb¯i c−1 = h# (b¯i ), other relations from S ,
where c is the generator from π1 (S 1 ) and b¯i are generators of π1 (S).
Since p2 # (φ̃# (c)) = p2 # (φ˜1 # (c)) = c then φ˜1 # (c) = φ̃# (c)θ where θ belongs to
the image of π1 (S − {x0 }) into π1 (E ×S 1 E − ∆) by the homomorphism ι# .
From the relation cb¯i c−1 = h# (b¯i ) we have φ̃# (c)φ̃# (b¯i )φ̃# (c)−1 = φ̃# (h# (b¯i ))
and φ˜1 (c)φ˜1 (b¯i )φ˜1 (c)−1 = φ˜1 (h# (b¯i )).
#
#
#
#
Since f |S = f1 |S follows φ̃# (b¯i ) = φ˜1 # (b¯i ) and therefore φ̃# (h# (b¯i )) = φ˜1 # (h# (b¯i )).
Then we have φ˜1 (c)−1 φ̃# (c)φ̃# (b¯i )φ̃# (c)−1 φ˜1 (c) = φ̃# (b¯i ).
#
#
So θ = φ̃# (c) φ˜1 # (c) ∈ π1 (E ×S 1 E − ∆) satisfies θ−1 φ̃# (b¯i )θ = φ̃# (b¯i ).
Moreover from p2 # (θ) = (p2 # φ̃# (c))−1 p2 # φ˜1 # (c) = c−1 c = 1 follows that q# ◦
−1
p2 # (θ) = q# (1) = 1 and therefore θ ∈ Ker(q# ◦ p2 # ) = Imj# .
Then θ = j# (θ2 ) for a θ2 ∈ π1 (S × S − ∆), where j# is the induced by the
inclusion j : S × S − ∆ → E ×S 1 E − ∆.
We have that j# (θ2−1 )φ̃# (b¯i )j# (θ2 ) = φ̃# (b¯i ). But j# φ̃# (b¯i ) = φ̃# (b¯i ). So
j# (θ−1 φ̃# (b¯i )θ2 ) = j# (φ̃# (b¯i )) and since j# is an injective map we have that
2
θ2−1 φ̃# (b¯i )θ2 = φ̃# (b¯i ).
Therefore θ2 commutes with imφ˜# and so θ2 = 1 by Lemma ??, which implies
θ = 1.
So φ̃# = φ˜1 # and therefore φ̃ ∼S 1 φ˜1 and then f ∼S 1 f1 .
Proposition 2.4. Let f1 , f2 : E → E be fibrewise fixed point free maps such
0
that f1 |S is homotopic to f2 |S with f1 |S of type I. Then there exists f1 : E → E
0
fibrewise homotopic to f1 : E → E such that f1 is a fixed point free map over S 1
0
and f1 |S = f2 |S .
Proof. Given a homotopy H : S ×I → S such that H(x, 0) = f1 |S (x) and H(x, 1) =
f2 |S (x) we consider L : E × 0 ∪ S × I → E defined by L(x, 0) = f1 (x), ∀x ∈ E and
L(x, t) = ι ◦ H(x, t), ∀(x, t) ∈ S × I, where ι : S ,→ E is the inclusion.
We have p ◦ L(x, 0) = p(x), ∀x ∈ E and p ◦ L(x, t) = p|S ◦ H(x, t) = p ◦
L(x, 0), ∀(x, t) ∈ S × I.
Therefore, by the fibrewise homotopy extension property, see [?], L extends to a
map L̃ : E × I → E such that p ◦ L̃(x, 0) = p ◦ L(x, 0) = p ◦ f1 (x) = p(x).
0
0
0
Let f1 = L̃ |E×1 . Therefore f1 is a fixed point free map and f1 |S = f2 |S .
Corollary 2.5. Suppose that f, f1 : E → E are fixed point free maps over B
and f |S is homotopic to f1 |S(which we shall denote by f |S ∼ f1 |S ). Then f is
fibrewise homotopic to f1 over B (which we shall denote by f ∼B f1 ).
6
D. L. GONÇALVES, A. K. M. LIBARDI, D. PENTEADO AND J. P. VIEIRA
Proof. Suppose that f, f1 : E → E are fixed point free maps over B and f |S ∼
0
f1 |S . Then by Proposition ?? there exists f : E → E fibrewise homotopic to
0
0
f : E → E over B such that f is a fixed point free map over S 1 and f |S = f1 |S .
0
0
Therefore by the Proposition ?? f ∼B f1 . Since f ∼B f the result follows.
Theorem 2.6. Given α ∈ [S, S] and g ∈ α, if g can be deformed to fixed point free
then there is at most one [f ] ∈ [E, E] such that [f |S ] = α and f is fixed point free
over S 1 .
Proof. We already observed that g can not be homotopic to the constant map. By
Proposition 1.2 it suffices to consider the case where g is of type I. This case follows
from the results above.
3. Existence of fibre maps fixed point free on non trivial surface
bundles
Let f : E → E be a fibre-preserving map.
Definition A fibre-preserving map is called of type I if rank(Im(g# ) 6= 0, 1
where g is the restriction of f to a fibre S. In case rank(Im(g# ) = 1 we say that
f is of type II.
In this final section we make some consideration about the problem of the image
of ψ where ψ : [E, E]B → [S, S] was defined in the previous section. Recall that E
is the mapping torus of a homeomorphism h.
We begin by give an algebraic characterization of the elements on the image of
ψ.
Lemma 3.1. Let us consider α = [g] ∈ [S, S]. Then α ∈ Im(ψ) if and only if
h ◦ g is homotopic to g ◦ h]. Further this is equivalent to say that the induced
homomorphisms by h ◦ g and g ◦ h on the fundamental group are conjugated.
Proof. If α ∈ Imψ, there exists f : E → E over S 1 , such that ψ([f ]) = [f |S ] = α =
[g]. Let K : S × [0, 1] → S be the homotopy between f |S and g, that is, K0 = f |S
and K1 = g.
From the Homotopy Lifting Property on the diagram
f ∪K
E × 0 ∪ S × [0, 1]
ι
L
E × [0, 1]
p̄
/8 E
p
/ S1
where p̄(x, t) = p(x) for all x ∈ E and f ∪ K is defined by f ∪ K(x, 0) = f (x) and
if x ∈ S then f ∪ K(x, t) = K(x, t), there exists L comutting it.
FIXED POINTS ON NON TRIVIAL SURFACE BUNDLES
7
It follows that L1 : E → E over S 1 , L1 (x) = L(x, 1) satisfies: L1 |S = g.
From Lemma??, we have that h ◦ f |S is homotopic to f |S ◦ h. The conversely
follows from the Lemma ??.
Further, if (α, u) ∈ π1 (E) = π1 (S)oπ1 (S 1 ) where u is a generator of π1 (S 1 ) then
utu−1 = h# (t), ∀t ∈ π1 (S). Therefore g# ◦h# (t) = f# ◦h# (t) = f# (u)f# (t)f# (u)−1 =
(α, u)g# (t)(α, u)−1 = (α, 1)(1, u)g# (t)(1, u−1 )(α−1 , 1), ∀t ∈ π1 (S).
But h# (g# (t)) = ug# (t)u−1 . So g# ◦ h# (t) = (α, 1)h# ◦ g# (t)(α−1 , 1).
Using the result above should not be difficult to show an element which is not
in the image of ψ. But in some cases do not exist, for example if [h] = [IdS ]. But
perhaps this is the only case.
Lemma 3.2. If g ◦ h = h ◦ g, then there is an extension of g which is the induced
of the product map g × id : S × [0, 1] → S × [0, 1] given by ĝ < x, t >→< g(x), t >.
In particular if g is fixed point free then the extension ĝ is also fixed point free.
Proof. It suffices to verify that ĝ < x, 0 >= ĝ < h(x), 1 >. In fact we have
ĝ < x, 0 >=< g(x), 0 > and ĝ < h(x), 1 >=< g ◦ h(x), 1 >. How g ◦ h = h ◦ g then
ĝ < h(x), 1 >=< h ◦ g(x), 1 >=< g(x), 0 >. So the result follows.
3.1. Fibre maps for maps g of type I.
This is the case where rank(Im(g# ) 6= 0, 1. The type of result that we expected,
based in the result of the trivial case, is as follows:
Given g there is at most one homotopy class of fibre map among all extensions.
Further, if there is an extension, then the class of this extension is fixed point free.
We will show this using a refinement of Lemma ?? above and a result by ???
GK ??? which say that the subspace of the fixed point free maps in the homotopy
class of a map g where rank(im(g# )) 6= 0, 1 is path connected.
Lemma 3.3. If g ◦ h is homotopic to h ◦ g, then there is an extension of g given
by ĝ < x, t >→< Ht (x), t > where H is a homotopy between g and h ◦ g ◦ h−1 .
In particular if g is fixed point free and of type I then the extension ĝ is also fixed
point free.
Proof. It suffices to verify that ĝ < x, 0 >= ĝ < h(x), 1 >. In fact we have
ĝ < x, 0 >=< H0 (x), 0 >=< g(x), 0 > and ĝ < h(x), 1 >=< H1 (h(x)), 1 >=<
h ◦ g ◦ h−1 (h(x)), 1 >=< h ◦ g(x), 1 >=< g(x), 0 >. So the result follows.
3.2. Fibre maps for maps g of type II. This is the case where rank(Im(g# ) =
1. The type of result that we expected, based in the result of the trivial case, is as
follows:
8
D. L. GONÇALVES, A. K. M. LIBARDI, D. PENTEADO AND J. P. VIEIRA
Given g there is at most one homotopy class of fibre map among all extensions,
such that this map is fixed point free. In the case of the trivial bundles, we just
replace ” there at most” by ”there is exactly one”.
Here is an specific example that we must make a test.
First of all the formulation of the problem is already more elaborate. More precisely, let us consider the map ψ : [E, E]B → [S, S] which associate to a homotopy
class of a fibre preserve map [f ] the homotopy class of the restriction f |S : S → S.
Then one would like to know first which homotopy class [g] ∈ [S, S] which contains
a fixed point free map are in the image of ψ. Second, for a class [g] in the image
how many classes [f ] ∈ [E, E] we would like to compute the pre-image of [g], i.e.
ψ −1 [g]. For example in the case that we solved, we have that the [g] is in the image
for all maps g which are fixed point free and the pre-image contains exactly one
element.
O LEMA ABAIXO É O MESMO QUE O LEMA ACIMA
Lemma 3.4. If g ◦ h is homotopic to h ◦ g, then there is an extension of g given
by ĝ < t, x >→< t, Ht (x) > where H is a homotopy between g and h ◦ g ◦ h−1 . In
particular if g is fixed point free then the extension ĝ is also fixed point free.
Proof. It suffices to verify that ĝ < 0, x >= ĝ < 1, h(x) >. In fact we have
ĝ < 0, x >=< 0, H0 (x) >=< 0, g(x) > and ĝ < 1, h(x) >=< 1, H1 (h(x)) >=<
1, h ◦ g ◦ h−1 (h(x)) >=< 1, h ◦ g(x) >=< 0, g(x) >. So the result follows.
3.3. Final comments. Dear Dan
Here is Daciberg. I hope you are well. I have a quick question and would like to
know your opinion. Just if you have on top of your head. Let S be a surface with
negative Euler characteristic and h a homeomorphism. Question: If g is another
homeomorphism such that h ◦ g and g ◦ h are isotopic, can one find h0 isotopic to
h, g 0 isotopic to g, such that h0 ◦ g 0 = g 0 ◦ h0 (on the nose)?
Question 2 (in case the above is true. If g : S → S is continuous self map of S
such that h ◦ g and g ◦ h are homotopic, can one find h0 isotopic to h, g 0 homotopic
to g, such that h0 ◦ g 0 = g 0 ◦ h0 (on the nose)?
Thanks in advance for any of your comments
Best regards,
Daciberg
Message from Dan Margalit
Abelian subgroups of the mapping class group can be lifted to Diff,
so that implies the first answer to your question is yes. This basically
FIXED POINTS ON NON TRIVIAL SURFACE BUNDLES
9
follows from the classification of abelian subgroups by Birman-LubotzkyMcCarthy. Luis Paris told me that he and Christian Bonnati wrote up
the details, but as far as I know this has not appeared.
I have not thought about the second question. It sounds too good to
be true, because the answer to the first question requires a very deep
theorem. Dan
We consider here two subcases, subcase 1 if g is homotopic to a homeomorphism
and subcase 2 if Im(g# ) is free and rank(Im(g# ) > 1.
Case g is a homeomorphism
This subcase is similar simplest one. The main result is:
Proposition 3.5. If g is homemorphism which is fixed point free then g is extends
to a fibre map if and only if there is g 0 ∈ [g], g 0 a homeomorphism such that
g 0 ◦ h0 = h0 ◦ g 0 . Therefore there is a unique extension of g 0 where one model for
this extension is the map which is constant and equal to g 0 as t vary. Problem- I
do not know if we can assume to be fixed point free.
Proof. The main idea of the above is to use the classification of abelian subgroups
by Birman-Lubotzky-McCarthy which says that Abelian subgroups of the mapping
class group can be lifted to Diff.
Remark: For the moment we do not know if given g which is extendable if this
extension is fixed point free. All we know is that the extension is unique.
Case Im(g# ) is free and rank(Im(g# ) > 1.
This case is similar to the previous one and worse. I do not even the results
about the mapping class and Diff. So the statement is similar and slight week. The
only commun part is that the extension is unique and we have for the moment no
means to measure if the extension can be done by a fixed point free or not.
We observe that the cases above does not appear when the buldle is trivial, since
there we automatically have an obvious extension which is fixed point free.
But perhaps specific calculations on braids can provide some results.
Acknowledgment The first author would like to thank Dan Margaret for many
helpful and fruitful discussions about mapping class group close related with the
subject of the manuscript.
XXXXXXXXXXXXXXXXXXXXXXXXXX
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(Daciberg Lima Gonçalves) Dept. de Matemática - IME - USP, Caixa Postal 66.281 CEP 05314-970, São Paulo - SP, Brazil
E-mail address: [email protected]
(Alice Kimie Miwa Libardi) Departamento de Matemática, I.G.C.E - Unesp - Univ Estadual Paulista, Caixa Postal 178, Rio Claro 13500-230, Brazil
E-mail address: [email protected]
FIXED POINTS ON NON TRIVIAL SURFACE BUNDLES
11
(Dirceu Penteado) Departamento de Matemática, Universidade Federal de São Carlos,
Rodovia Washington Luiz, Km 235, São Carlos 13565-905, Brazil
E-mail address: [email protected]
(João Peres Vieira) Departamento de Matemática, I.G.C.E - Unesp - Univ Estadual
Paulista, Caixa Postal 178, Rio Claro 13500-230, Brazil
E-mail address: [email protected]