Fixed points on trivial surface bundles
The main purpose of this work is to study fixed points of fiber preserving maps over $S^1$ on the trivial surface bundles $S^1\times S_2$, where $S_2$ is the closed orientable surface of genus 2. We classify all such maps that can be deformed fiberwise to a fixed point free map.
Gespeichert in:
| Datum: | 2013 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Інститут математики НАН України
2013
|
| Online Zugang: | https://trim.imath.kiev.ua/index.php/trim/article/view/309 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Transactions of Institute of Mathematics of NAS of Ukraine |
| Завантажити файл: |  |
Institution
Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552812956614656 |
|---|---|
| author | Gonçalves, D. L. Libardi, A. K. M. Penteado, D. Vieira, J. P. Gonçalves, D. L. Libardi, A. K. M. Penteado, D. Vieira, J. P. |
| author_facet | Gonçalves, D. L. Libardi, A. K. M. Penteado, D. Vieira, J. P. Gonçalves, D. L. Libardi, A. K. M. Penteado, D. Vieira, J. P. |
| author_institution_txt_mv | [
{
"author": "D. L. Gonçalves",
"institution": "Dept. de Matemática - IME - USP"
},
{
"author": "A. K. M. Libardi",
"institution": "Departamento de Matemática, I.G.C.E - Unesp - Univ Estadual Paulista"
},
{
"author": "D. Penteado",
"institution": "Departamento de Matemática, Universidade Federal de São Carlos"
},
{
"author": "J. P. Vieira",
"institution": "Departamento de Matemática, I.G.C.E - Unesp - Univ Estadual Paulista"
}
] |
| author_sort | Gonçalves, D. L. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2018-02-10T20:56:26Z |
| description | The main purpose of this work is to study fixed points of fiber preserving maps over $S^1$ on the trivial surface bundles $S^1\times S_2$, where $S_2$ is the closed orientable surface of genus 2. We classify all such maps that can be deformed fiberwise to a fixed point free map. |
| first_indexed | 2026-08-04T01:05:23Z |
| format | Article |
| fulltext |
Збiрник праць Iн-ту математики НАН України 2013, том 6, N 6, 67–85
D. L. Gonçalves, A. K. M. Libardi, D.
Penteado, João Peres Vieira
( Dept. de Matemática - IME - USP, Caixa Postal 66.281 - CEP
05314-970, São Paulo - SP, Brazil, Departamento de Matemática,
I.G.C.E - Unesp - Univ Estadual Paulista, Caixa Postal 178, Rio Claro
13500-230, Brazil, Departamento de Matemática, Universidade Federal
de São Carlos, Rodovia Washington Luiz, Km 235, São Carlos
13565-905, Brazil Departamento de Matemática, I.G.C.E - Unesp -
Univ Estadual Paulista, Caixa Postal 178, Rio Claro 13500-230, Brazil )
Fixed Points on Trivial Surface
Bundles
dlgoncal@ime.usp.br, alicekml@rc.unesp.br, dirceu@dm.ufscar.br,
jpvieira@rc.unesp.br
The main purpose of this work is to study fixed points of fiber- preserv-
ing maps over S1 on the trivial surface bundles S1 × S2, where S2 is the
closed orientable surface of genus 2. We classify all such maps that can be
deformed fiberwise to a fixed point free map.
Introduction
Given a fibration E → B and f : E → E a fiber-preserving map over
B, the question if f can be deformed over B (by a fiberwise homotopy) to
c© D. L. Gonçalves, A. K. M. Libardi, D. Penteado, J. P. Vieira, 2013
68 Fixed Points on Trivial Surface Bundles
a fixed point free map has been considered for several years by many au-
thors. Among others, see for example [Dol74], [FH81], [Gon87], [Pen97],
[GPV04], [GPV09I] and [GPV09II]. More recently also the fiberwise co-
incidence case has been considered in [Kos11], [GK09], [GPV10], [SV12],
[Vie12] and [GKLN], which certainly has intersection with the fixed point
case.
In [FH81], Fadell, E. and Husseini, S. showed that the fiberwise fixed
point problem can be stated in terms of obstructions (including higher
ones) if the fibration satisfies certain hypothesis. This is the case if the
base space, the total space and the fiber F are manifolds, and the dimen-
sion of F is greater or equal to 3. The project to study fixed point of
fiberwise maps for surface bundles has been considered mainly in the case
where the base is S1 and it can be divided into several cases as follows.
If the fiber F is the projective real space RP 2 we never obtain a fixed
point free fiberwise map, because RP 2 has the fixed point property. This
case leads to a natural question about the minimal size of the fixed point
set, namely, when is possible to have the fixed point set connected. Close
related, if not equivalent, is the problem of classify maps which can be
deformed to a map with exactly one fixed point in each fiber.
The case of fiber S2, despite the fact that the approach of [FH81] can
be used, using different techniques, it was studied in [Kos11], [GPV10]
and [GKLN].
We note that if the fiber is a closed surface S distinct of S2 and RP 2,
the approach of [FH81] can not be used. A project to study surface
bundles for closed surface distinct of S2 and RP 2 has started looking
the case where the base is S1. The case where S is the torus has been
solved by other methods in [GPV04] (see also [Kos11]). For S the Klein
bottle the results were obtained in [GPV09II] (see also [SV12]) by similar
methods as the case of the torus.
In the present work we start the case of a surface bundle over S1
D. L. Gonçalves, A. K. M. Libardi, D. Penteado, João Peres Vieira 69
where the fiber is S2 and S2 is the closed orientable surface of genus 2.
More precisely we study fiberwise maps of the trivial bundle S1 × S2.
Let us consider the fibration S1×S2 → S1 and h : S1×S2 −→ S1×S2
a fiber-preserving map over S1, where h(x, y) = (x, f(x, y)), ∀(x, y) ∈
S1 × S2 and f is a map from S1 × S2 into S2.
The main result of this paper is:
Theorem 4.3 A fiberwise map h can be deformed over S1 to a fixed
point free map if and only if h is fiberwise homotopic to id × g where
g : S2 → S2 is a fixed point free map homotopic to f restricted to 1×S2.
This paper is organized into 4 sections. In section 1 we review an
approach to study fixed point of fiberwise maps and we adapt it for the
case to be analyzed. In section 2 we make the main calculations where we
compute the fundamental group of several spaces and homomorphisms
to study a certain algebraic diagram. The main result of this section is
Theorem 3.5. In section 3 we proof the main result of this work, which is
Theorem 4.3. In section 4 we give a very brief view of the continuation
of the study of the problems for the majority of the cases, which are still
to be analyzed.
2 Preliminaries
Let h : E → E be a fiber-preserving map over B, i.e., p ◦ h = p where
p : E → B is a fiber bundle with fiber a surface denoted by S. When is h
deformable over B to a fixed point free map h
′
by a fiberwise homotopy
over B ? We remark that in order to have a positive answer a necessary
condition is that the map h restricted to a fiber is deformable to a fixed
point free map.
Now we review an approach which was used in [GPV04] and
[GPV09II]. Assuming the necessary condition, h is deformable over B to
a fixed point free map h
′
by a fiberwise homotopy over B if and only if
70 Fixed Points on Trivial Surface Bundles
there exists a lifting ψ such that the following diagram is commutative,
up to homotopy:
F
��
E(E ×B E −∆)
e1
��
E
ψ
88
(h,1)
// E ×B E
(2.1)
Here E ×B E is the pullback of p by p, ∆ is the diagonal in E ×B E
and the inclusion E ×B E − ∆ ↪→ E ×B E is changed by the fibration
e1 : E(E ×B E −∆)→ E ×B E with fiber F , where πi(F) ' πi+1(E ×B
E,E ×B E − ∆). Also E(E ×B E − ∆) is the pullback of the fibration
e0 : (E ×B E)[0,1] → E ×B E by the inclusion E ×B E −∆ → E ×B E.
The fibration e0 : (E ×B E)[0,1] → E ×B E is the evaluation at 0 and
e1 : E(E ×B E −∆)→ E ×B E is the evaluation at 1.
Let us observe that if E,B and S are closed manifolds then πi+1(E×B
E,E ×B E −∆) ' πi+1(S, S − y0) (see [FH81]).
When E = B × S is the trivial bundle and h : B × S → B × S
is a fiber-preserving map over B, the map h can be write in the form
h(x, y) = (x, f(x, y)) for some f : B × S → S. Then the diagram 2.1 can
be modified and becomes equivalent to the following diagram:
D. L. Gonçalves, A. K. M. Libardi, D. Penteado, João Peres Vieira 71
F
��
E(B × (S × S −∆))
e1
��
B × S
ψ
66
(1,f,1)
// B × S × S
(2.2)
3 Trivial S−bundles over S1 with χ(S) < 0
Let S be a surface with χ(S) < 0 and let us consider the fibration
S1 × S → S1 and h : S1 × S −→ S1 × S a fiber-preserving map over S1,
where h(x, y) = (x, f(x, y)), ∀(x, y) ∈ S1×S and f is a map from S1×S
into S. We also consider x0 and y0 base points of S1 and S, respectively,
and f : (S1 × S, (x0, y0)) −→ (S, f(x0, y0)), with f(x0, y0) 6= y0. From
the map f we obtain the maps g = f |{x0}×S and l = f |S1×{y0}. Recall
that we are assuming the necessary condition: the map g is deformable
to a fixed point free map.
Using the approach developed in [GPV04] and [GPV09II] we will
study in our case the existence of an algebraic lifting ψ to the diagram
72 Fixed Points on Trivial Surface Bundles
1
��
π1(F)
��
' π2(S, S − x0)
π1(E(S1 × (S × S −∆)))
q#=1π1(S1)×j#
��
' π1(S1 × (S × S −∆))
π1(S1 × S)
ψ
55
(1,f,1)#
// π1(S1 × S × S)
��
1
(3.1)
where π1(F) ' π1(S × S −4) is the pure braid group of S on 2-strings.
The existence of the lifting mentioned above is equivalent to find lift-
ings θ and φ described in diagrams 3.2 and 3.3 below where θ and φ
satisfy certain conditions. Since we are assuming the necessary con-
dition then the lifting φ exists. So, we have the following two dia-
grams, where i1#, i2# and j# are induced homomorphisms on funda-
mental groups by the injective maps i1 : S1 → S1 × S, i2 : S → S1 × S
and j : S × S −∆ → S × S, respectively, and q2# and pi# are induced
homomorphisms by the projection maps q2 : S1 × S × S → S × S and
pi : S × S → S, respectively.
π1(S × S −∆)
j#
''
pi|# // π1(S)
π1(S1)
θ
33
i1#
// π1(S1 × S)
(1,f,1)#
// π1(S1 × S × S)
q2#
// π1(S × S)
pi#
OO (3.2)
D. L. Gonçalves, A. K. M. Libardi, D. Penteado, João Peres Vieira 73
π1(S × S −∆)
j#
''
pi|# // π1(S)
π1(S)
φ
33
i2#
// π1(S1 × S)
(1,f,1)#
// π1(S1 × S × S)
q2#
// π1(S × S)
pi#
OO (3.3)
We remark that in these diagrams we are omitting base points.
The following theorem provides some conditions that the liftings θ
and φ must satisfy which are equivalent to a positive solution of the fixed
point problem for the trivial bundle.
Theorem 3.1. There exists ψ on the diagram 3.1 if and only if there
exist θ and φ in the diagrams 3.2 and 3.3, respectively, such that Imθ
commutes with Imφ.
Proof. Let us suppose that there exists a lifting ψ in the diagram (2.1).
Define φ = q2|#◦ψ◦i2# and θ = q2|#◦ψ◦i1#, where i1 : S1 → S1×S and
i2 : S → S1 × S denote the inclusion maps and q2| : S1 × (S × S −∆)→
(S × S − ∆) denotes the projection on the second factor. Therefore
θ and φ are lifting for the diagrams 3.2 and 3.3, respectively, because
q# ◦ ψ = (1, f, 1)# and q2# ◦ q# = j# ◦ q2|#.
Now, for all x ∈ Imθ and for all y ∈ Imφ we have
xy = q2|# ◦ ψ ◦ i1#([b])q2|# ◦ ψ ◦ i2#([s])
= q2|# ◦ ψ(i1#([b])i2#([s]))
= q2|# ◦ ψ(([b], 1)(1, [s]))
= q2|# ◦ ψ((1, [s])([b], 1))
= q2|# ◦ ψ(i2#([s])i1#([b]))
= q2|# ◦ ψ ◦ i2#([s])q2|# ◦ ψ ◦ i1#([b])
= yx
Conversely, suppose that θ and φ exist and we define ψ by ψ([b], [s]) =
([b], θ([b]) φ([s])) where b : S1 → S1 and s : S1 → S denote loops
74 Fixed Points on Trivial Surface Bundles
based at x0 and at y0, respectively. Since Imθ commutes with Imφ,
we have that ψ is a homomorphism and denoting by s0 : S1 → S and
b0 : S1 → S1 the constant maps at y0 and at x0, respectively, it follows
that (q# ◦ ψ)([b], [s]) = (1, f, 1)#([b], [s]),∀([b], [s]) ∈ π1(S1 × S).
From now on we specialize for the case where the fiber S is the surface
S2. So we consider the trivial bundle S1 × S2.
If φ is a lifting of the diagram 3.3 to discuss the existence of the
lifting θ we will denote by 1 a generator of π1(S1) ≡ Z and by θ(1) = ω ∈
π1(S2×S2−∆). A presentation of π1(S2×S2−∆) is given in [FH82]. We
will use the following notation: let ai = ρ1,i ∈ π1(S2×S2−∆), i = 1, 2, 3, 4
and by bi = ρ2,i ∈ π1(S2 × S2 −∆).
So π1(S2 × S2 −∆) has the following presentation:
(I) [a1, a
−1
2 ][a3, a
−1
4 ] =: B1 = B−1
2 := [b1, b
−1
2 ][b3, b
−1
4 ] (which defines
the elements B1 and B−1
2 ).
(II) blajb−1
l = aj where 1 ≤ j, l ≤ 4, and j < l(resp. j < l − 1) if l is
odd (resp. l is even).
(III) bkakb−1
k = ak[a−1
k , B1] and b−1
k akbk = ak[B−1
1 , ak] for all 1 ≤ k ≤ 4.
(IV) bkak+1b
−1
k = B1ak+1[a−1
k , B1] and b−1
k ak+1bk =
B−1
1 [B1, ak]ak+1[B−1
1 , ak], for all k odd, 1 ≤ k ≤ 4.
(V) bk+1akb
−1
k+1 = akB
−1
1 , and b−1
k+1akbk+1 = akB1[B−1
1 , ak+1], for all
k odd, 1 ≤ k ≤ 4.
(VI) blajb−1
l = [B1, a
−1
l ]aj [a
−1
l , B1] and b−1
l ajbl = [al, B
−1
1 ]aj [B
−1
1 , al]
for all 1 ≤ l < j ≤ 4 and (j, l) 6= (2t, 2t− 1) for all t ∈ {1, 2}.
We also observe that from the fibration p2 |: S2 × S2 −∆ −→ S2 we
get the following exact sequence:
1 // π1(S2 − y0) // π1(S2 × S2 −∆)
p2|# // π1(S2) // 1 .
(3.4)
D. L. Gonçalves, A. K. M. Libardi, D. Penteado, João Peres Vieira 75
The group π1(S2 − y0) is free and from the sequence above it is
identified with the subgroup of π1(S2 × S2 − ∆) freely generated by
a1, a2, a3, a4. Also the image of the set of elements b1, b2, b3, b4 projects
to a set of generators of π1(S2) giving a presentation of π1(S2) =<
b̄1, b̄2, b̄3, b̄4 | [b̄1, b̄
−1
2 ][b̄3, b̄
−1
4 ] >. More details see [FH82].
Given a group G the central series of G is defined recursively by
G1 = G,Gn+1 = [G,Gn], n = 1, 2, . . . .
For any group G we have that Gm is a normal subgroup of Gn for all
n ≤ m. In case G is free group of finite rank r then it is well known that
Gn/Gn+1 is a free abelian of rank
Nn =
1
n
∑
d|n
µ(d)r
n
d
(see [[MKS76],Theorem 5.11, p.330]). Here µ(d) denotes the Moebius
Function defined for all positive integers by µ(1) = 1, µ(p) = −1 if p is
a prime number, µ(pk) = 0 for k > 1, and µ(b · c) = µ(b) · µ(c) if b and c
are coprime integers.
For any group G denote the commutator [[a, b], c] by (a, b, c). If a, b, c
are elements of a group G and k,m, n are positive integers such that
a ∈ Gk, b ∈ Gm, c ∈ Gn then (a, b, c)·(b, c, a)·(c, a, b) ≡ 1 mod Gk+m+n+1
(see [[MKS76], Theorem 5.3, p.293]).
For the next Lemma, let G = G1 = π1(S2− y0) which is a free group,
and it is identified with a subgroup of π1(S2 × S2 − ∆) using the short
exact sequence 3.4.
Lemma 3.2. If v ∈ G2 = [G1, G1], then [bj , v] ∈ G3, for j = 1, 2, 3, 4.
Proof. We will prove the statement for b3. The other cases are simi-
lar. If v ∈ G2 = [G1, G1], then v is a finite product of [ai, aj ] and
of its inverses. If v1, v2 ∈ G2 then [b3, v1v2] = b3v1v2b
−1
3 v−1
2 v−1
1 =
b3v1b
−1
3 v−1
1 v1b3v2b
−1
3 v−1
2 v−1
1 = [b3, v1]v1[b3, v2]v−1
1 . We know that the
76 Fixed Points on Trivial Surface Bundles
conjugation by v1 preserves the central series and then if we prove that
[b3, [ai, aj ]] = 1 mod G3 the result follows.
Now in G1/G3, we have that
[b3, [ai, aj ]] = b3[ai, aj ]b
−1
3 [ai, aj ]
−1 = b3aiaja
−1
i a−1
j b−1
3 [ai, aj ]
−1.
In the case where i 6= 4 and j 6= 4 and recalling that in G1/G3, b3
commutes with ai and aj (i.e, the action is trivial) we have the desired
result.
If i = 4 and j 6= 4 we also have that b3 commutes with aj because
j 6= 4 and the action in a4 results in B1a4. Therefore in G1/G3 the action
of b3 in [a4, aj ] is b3a4aja
−1
4 a−1
j b−1
3 = B1a4aja
−1
4 B−1
1 a−1
j and so
b3[a4, aj ]b
−1
3 [a4, aj ]
−1 = B1a4aja
−1
4 B−1
1 a4a
−1
j a−1
4 = [B1 , a4aja
−1
4 ]
= [a4aja
−1
4 , B1]−1 ∈ G3.
The case where i 6= 4 and j = 4 is analogue.
Let C(θ(1)) be the centralizer of θ(1) in π1(S2 × S2 −∆).
Proposition 3.3. Let θ be a lifting such that (p2 |#)(C(θ(1)) =
π1(S2, y0). Then θ(1) ∈ G and there exist u1, u2, u3, u4 elements of G
such that ujbj ∈ C(θ(1)), j = 1, 2, 3, 4. If θ(1) = xv, with v ∈ G2, x ∈ G,
then we have that [u−1
j , x−1][x−1, bj ] = 0 in G2/G3.
Proof. From the hypothesis (p2 |#)(C(θ(1))) = π1(S2, y0) follows that
p2#(θ(1)) is in the centralizer of π1(S). This implies that this element
is trivial and then θ(1) ∈ G. Also, given b̄j from the hypothesis follows
that there exists xj which is in the centralizer of θ(1) which projects to
b̄j . Therefore xj = ujbj for some uj ∈ G.
Since ujbj ∈ C(θ(1)), j = 1, 2, 3, 4 we have that
xvujbj = ujbjxv
bjxb
−1
j [bj , v] = u−1
j xuj [u
−1
j , v]
u−1
j x−1ujbjxb
−1
j = [u−1
j , v][v, bj ]
[u−1
j , x−1][x−1, bj ] = [u−1
j , v][v, bj ]
D. L. Gonçalves, A. K. M. Libardi, D. Penteado, João Peres Vieira 77
where v ∈ G2, uj ∈ G1. Then in G2/G3 it follows from the Lemma 3.2
that
[u−1
j , x−1][x−1, bj ] = 0. (3.5)
The group G2/G3 is a Z- free module and let us consider the basis
{[a1, a2], [a1, a3], [a1, a4], [a2, a3], [a2, a4], [a3, a4]}
which we refer as the canonical basis of G2/G3.
Lemma 3.4. In G2/G3 we have:
a) [aiaj , x] = [ajai, x], where ai, aj are generators of G and x ∈ G.
b) If B = [a1, a
−1
2 ][a3, a
−1
4 ] ∈ G, then B = −[a1, a2]−[a3, a4] and its coor-
dinate in relation to the canonical basis is given by (−1, 0, 0, 0, 0,−1).
c) The element [axii , a
xj
j ] is given in the following form:
[axii , a
xj
j ] =
0 se i = j
xixj [ai, aj ] se i < j
−xixj [aj , ai] se i > j
Proof. Since [aiaj , x] = [ai, [aj , x]][aj , x][ai, x] and [ajai, x] =
[aj , [ai, x]][ai, x][aj , x], the result of item a) follows by observing that in
G2/G3 we have that [ai, [aj , x]] = 0 = [aj , [ai, x]] which is commutative.
The items b) and c) are easy.
Theorem 3.5. If there exists θ and (p2 |#)(C(θ(1)) = π1(S2, y0) then
θ(1) ∈ G2 = [G1, G1].
Proof. It follows from the exact sequence 1 → [G1, G1] → G1 →
G1/[G1, G1] → 0 that θ(1) ∈ G1 is of the form θ(1) = xv with
78 Fixed Points on Trivial Surface Bundles
x = aa1a
b
2a
c
3a
d
4 and v ∈ G2 = [G1, G1]. We are going to prove that
the exponents a = b = c = d = 0.
It follows from Proposition 3.3 that in G2/G3 we have
[u−1
j , x−1][x−1, bj ] = 0 with uj ∈ G1.
In fact this is a system with j equations and four variables x:(a, b, c, d)
and for each j four variables (ej , fj , gj , hj) corresponding to u−1
j =
a
ej
1 a
fj
2 a
gj
3 a
hj
4 .
Writing [u−1, x−1] in the canonical basis of G2/G3 and observing that
the exponents of u−1 = ae1a
f
2a
g
3a
h
4 must to appear with sub-index j (We
are omitting such sub-index) we obtain:
[u−1, x−1] = (af − be)[a1, a2] + (ag − ce)[a1, a3] + (ah− de)[a1, a4] +
(bg − cf)[a2, a3] + (bh− df)[a2, a4] + (ch− dg)[a3, a4]
Calculating [x−1, bj ] ∈ G2 with x = aa1a
b
2a
c
3a
d
4 we obtain:
x−1b1xb
−1
1 = x−1aa1(B1a2)bac3a
d
4
= a−d4 a−c3 a−b2 (B1a2)bac3a
d
4
= a−d4 a−c3 a−b2 (ab2a
−b
2 B1a
b
2 . . . a
−2
2 B1a
2
2a
−1
2 B1a2)ac3a
d
4
x−1b2xb
−1
2 = x−1(a1B
−1
1 )aab2a
c
3a
d
4
= a−d4 a−c3 a−b2 a−a1 (a1B
−1
1 )aab2a
c
3a
d
4
= a−d4 a−c3 a−b2 a−a1 (a1B
−1
1 a−1
1 a2
1B
−1
1 a−2
1 . . . aa1B
−1
1 a−a1 )×
×aa1ab2ac3ad4
x−1b3xb
−1
3 = x−1aa1a
b
2a
c
3(B1a4)d
= a−d4 (B1a4)d
= a−d4 (ad4a
−d
4 B1a
d
4 . . . a
−2
4 B1a
2
4a
−1
4 B1a4)
x−1b4xb
−1
4 = x−1aa1a
b
2(a3B
−1
1 )cad4
= a−d4 a−c3 (a3B
−1
1 )cad4
= a−d4 a−c3 (a3B
−1
1 a−1
3 a2
3B
−1
1 a−2
3 . . . ac3B
−1
1 a−c3 ac3)ad4
Therefore in relation to the canonical basis of
G2/G3 and by using Lemma 3.4 b) we obtain
(−b, 0, 0, 0, 0,−b), (a, 0, 0, 0, 0, a), (−d, 0, 0, 0, 0,−d), (c, 0, 0, 0, 0, c) as
the coordinates of [x−1, bj ] respectively for j = 1, 2, 3, 4.
D. L. Gonçalves, A. K. M. Libardi, D. Penteado, João Peres Vieira 79
So, the system to be solved is
−be +af = b,−a, d,−c respectively for b1, b2, b3, b4
−ce +ag = 0
−de +ah = 0
−cf +bg = 0
−df +bh = 0
−dg +ch = b,−a, d,−c respectively for b1, b2, b3, b4
(3.6)
understanding that in the letters (e, f, g, h) must to appear sub-index
j, but not in the letters (a, b, c, d).
a-) d 6= 0 in the system (3.6)
a1) If b = 0 we have that L5 implies f = 0, making L1 without
solution.
a2) If b 6= 0, in the system L3 → dL1 − bL3 produces an incompati-
bility : new L3 and L5 .
b-) b 6= 0 in the system (3.6)
b1) If d = 0 we have that L6 implies c 6= 0 and h 6= 0. Then L3
implies a = 0 and from L5 we conclude that b = 0, making L1
without solution.
b2) If d 6= 0, in the system L4 → bL6 + dL4 produces an incompati-
bility: new L4 and L5.
c-) c 6= 0 in the system (3.6)
c1) If a = 0 then L1 implies that b 6= 0 and e 6= 0 and from L3
we obtain d = 0 and from L5 we have that h = 0. Therefore L6 is
impossible.
c2) If a 6= 0 the system is impossible. In the system we make L4 →
cL1 − aL4 and obtain an incompatibility: new L4 and L2.
80 Fixed Points on Trivial Surface Bundles
d-) a 6= 0 in the system (3.6)
d1) If c = 0 the system is impossible. It follows from the fact that
L6 implies d 6= 0 and g 6= 0. Also L4 implies b = 0 and L5 implies
f = 0 making L1 impossible.
d2) If c 6= 0 the system is impossible, because in the system we make
L3 → aL6− cL3 which produces an incompatibility: new L3 and L2.
From the considerations above we conclude that a = b = c = d = 0
and therefore x = 1 and θ(1) ∈ G2.
4 Main Result
Let h : S1 × S2 → S1 × S2 given by h(x, y) = (x, f(x, y)). Let us
consider l : (S1, x0) → (S2, f(x0, y0)) and g : (S2, y0) → (S2, f(x0, y0))
given by l(x) = f(x, y0) and g(y) = f(x0, y), respectively. Without loss
of generality we are assuming that g is a fixed point free map.
To prove our main result we need the following
Lemma 4.1. Let t : S2 → S2 be a continuous map and t# : π1(S2) →
π1(S2) the induced homomorphism of the map t. Suppose that t#(bi) =
αni , where bi, i = 1, 2, 3, 4 is a generator of π1(S2). If the map t can be
deformed to a fixed point free map then
∑4
1 ni|α|i = 1 where |α|i denotes
the sum of the exponents of bi in the word α.
Proof. Let ι : S1 → S2 be a map which represents the element α ∈
π1(S2). We can define t′ : S2 → S1 such that ι ◦ t′ = t. By the com-
mutativity property for fixed point we know that the Nielsen number of
t is the same as the Nielsen number of t′ ◦ ι, which is a self map of the
circle. So if t is deformable to a fixed point free map then we have that
the Nielsen number of t′ ◦ ι is trivial which is equivalent to say that the
D. L. Gonçalves, A. K. M. Libardi, D. Penteado, João Peres Vieira 81
Lefschetz number of t′ ◦ ι is 0, which is the same to say
∑4
1 ni|α|i = 1.
So the result follows.
The main result will follows from the Proposition below.
Proposition 4.2. The fiberwise map h is deformable to a fixed point
free map over S1 if and only if l#(1) = e, where l# : π1(S1;x0) →
π1(S2; f(x0, y0)) .
Proof. Let h be a fiberwise map where h(x, y) = (x, f(x, y)). To prove
that h can be deformed fiberwise to a fixed point free map it is suffice to
show that f is homotopic to the map f
′
(x, y) = f(x0, y).
Because S1×S2 and S2 are K(π, 1) the two maps are homotopic if the
induced homomorphisms on the fundamental group are equal. Because
π1(S1×S2) = π1(S1)×π1(S2) to show that the two homomorphisms are
the same it suffices to show that these homomorphisms coincide when
restricted to each of the two subgroups π1(S1), π1(S2). By hypothesis
l#(1) = e follows that they coincide on π1(S1). By the definition of f
′
also follows that they coincide on π1(S2), and this concludes the proof of
one implication.
Reciprocally, let h be a map deformable to a fixed point free map over
S1. Then by Theorem 3.1 exist φ and θ such that the image of θ commutes
with the image of φ. From the diagrams 3.2 and 3.3, p1|# ◦ θ = l# and
p1|# ◦ φ = g#. It is known that g#(π1(S2)) is a subgroup of π1(S2)
isomorphic to one of the following groups:
1. {e}.
2. a free group of rank 2 (see [LS89]and [Zie62]).
3. π1(S)
4. Z = 〈β〉
82 Fixed Points on Trivial Surface Bundles
The first item does not occur, otherwise g is homotopic to the constant
map so it can not be deformed to a fixed point free map.
In the second and third cases, from above l#(1) commutes with all
elements of g#(π1(S)) but the centralizer of these two subgroups is trivial.
Therefore l#(1) = e.
For the last case we have that g#(π1(S)) = Z = 〈β〉 = 〈αk〉 where
α 6= 0, α has no roots and αk = β. Since l#(1) commutes with the
elements of g#(π1(S)) then l#(1) = αr. If r = 0 the proof follows. So
suppose that r 6= 0.
Writing g#(bi) = αni we have by the lemma 4.1 that if g is homotopic
to a fixed point free map then
∑4
1 ni|α|i 6= 0.
We have that p1|# ◦θ(1) = l#(1). We also have that im φ ⊂ C(θ(1)),
and p2|#(Im(φ))) = π1(S). Therefore p2|#(C(θ(1))) = π1(S) and by
theorem 3.5 follows that θ(1) ∈ G2 = [G1, G1].
So, αr = l#(1) ∈
[
p1|#(G1), p1|#(G1)
]
.
Therefore α ∈
[
p1|#(G1), p1|#(G1)
]
and then |α|i = 0 and by using
the above result we conclude that g is not homotopic to a fixed point
free map, which contradicts the initial condition on g. So the result
follows.
In fact the proof above shows that if θ(1) = e then f does not depend
of x, i.e. h is the unique fiberwise map homotopic to id × g where g :
S2 → S2 is a fixed point free map homotopic to f restricted to 1×S. So
we state the main result.
Theorem 4.3. A fiberwise map h can be deformed over S1 to a fixed
point free map if and only if h is fiberwise homotopic to id × g where
g : S2 → S2 is a fixed point free map homotopic to f restricted to 1×S2.
D. L. Gonçalves, A. K. M. Libardi, D. Penteado, João Peres Vieira 83
5 Other surface bundles
Here let us make few comments about the fixed point question studied
in the previous sections in the case we have a more general surface bundle.
Let S → E → B be a surface bundle over a space B where S is a
closed surface of negative Euler characteristic. We can consider three
subfamilies of the family of these bundles, namely: I) let S be an arbitrary
closed surface(orientable or nonorientable) of arbitrary genus g > 1, and
E = S1 × S; II) let E = B × S be a bundle for B any connected CW
complex; III) let E be a S−bundle over S1.
The subcases I) and II) we expect that the answer of the problem
should be similar to the answer of the case studied here where S = S2.
The subcase III) is more subtle. First of all the formulation of the
problem is already more elaborate. More precisely, let us consider the
map φ : [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 φ. 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. φ−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.
The study and full calculation of the questions above are in progress
and should appear somewhere.
Acknowledgments
The authors were partly supported by the Projeto Temático - Topolo-
gia Algébrica, Geométrica e Diferencial - FAPESP (08/57607-6)
84 Fixed Points on Trivial Surface Bundles
References
[Dol74] A. Dold; The fixed point index of fibre- preserving maps, Inven-
tiones Math., 25 (1974), 281–297.
[FH81] E. Fadell and S. Husseini; A fixed point theory for fiber-preserving
maps, Lecture Notes in Mathematics, vol. 886, Springer Verlag,
(1981), 49–72.
[FH82] E. Fadell and S. Husseini; The Nielsen number on surfaces, in:
Topological methods in non linear functional analysis, Toronto, Ont.,
1982, in : Contemporary Mathematics, vol. 21, Amer. Math. Soc.,
Providence, RI, 1983, 59–98.
[Gon87] D. L. Gonçalves; Fixed Point of S1- Fibrations, Pacific J. Math-
ematics 129(1987), 297–306.
[GK09] D.L. Gonçalves and U. Koschorke; Nielsen coincidence theory
of fibre-preserving maps and Dold’s fixed point index, Topological
Methods in Nonlinear Analysis, Journ. of Juliusz Schauder Center
33 (2009), 85–103
[GKLN] D. L. Gonçalves, U. Koschorke, A. K. M. Libardi and O.M. Neto;
Coincidences of fiberwise maps between sphere bundles over the cir-
cle, to appear in PEMS.
[GPV04] D. L. Gonçalves, D. Penteado and J. P. Vieira; Fixed Points
on Torus Fiber Bundles over the Circle, Fundamenta Mathematicae,
vol.183(1)(2004), 1–38.
[GPV09I] D. L. Gonçalves, D. Penteado and J. P. Vieira; Abelian-
ized Obstruction for fixed points of fiber-preserving maps of surface
bundles, Methods Nonlinear Anal. vol.33(2)(2009), 293–305.
D. L. Gonçalves, A. K. M. Libardi, D. Penteado, João Peres Vieira 85
[GPV09II] D. L. Gonçalves, D. Penteado and J. P. Vieira; Fixed points
on Klein bottle fiber bundles over the circle, Fundamenta Mathemat-
icae, vol.203(3)(2009), 263–292.
[GPV10] D. L. Gonçalves et al; Coincidence Points of fiber maps on Sn-
bundles, Topology and its Applications, vol.157(2010), 1760–1769.
[Kos11] U. Koschorke; Fixed points and coincidences in torus bundles, J.
of Topology and Analysis, 3, nr.2 (2011), 177–212.
[LS89] R. C. Lyndon and P. E. Schupp; Combinatorial Group Theory,
Ergebnisse der Mathematik und ihrer Grenzgebiete 89, Springer Ver-
lag, (1970).
[MKS76] W. Magnus, A. Karrass and D. Solitar; Combinatorial Group
Theory - Presentations of Groups in Terms of Generators and Re-
lations, Dover Publications,Inc., New York (1976).
[Pen97] D. Penteado; Fixed points on surface fiber bundles, Matemática
Contemporânea, Proceedings of the 10th Brazilian Topology Meet-
ing 13(1997), 251-267.
[SV12] W.L. Silva and J.P. Vieira; Coincidences of self-maps on Klein
Bottle fiber bundles over the circle, JP Journal of Geometry and
Topology, vol.12(1)(2012), 55–97.
[Vie12] J.P. Vieira; Coincidence of maps on Torus Fiber Bundles over
the Circle, Preprint, Unesp - Univ. Estadual Paulista, Rio Claro -
São Paulo, Brazil (2012).
[Zie62] H. Zieschang;Über Worte Sa11 Sa22 ...S
ap
p mit p freien Erzeugenden,
Math. Ann. 147(1962), 143-153.
|
| id | oai:trim.imath.kiev.ua:article-309 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:05:23Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/6c/3f6a61c2738b62640f4e0757ee43bb6c.pdf |
| spelling | oai:trim.imath.kiev.ua:article-3092018-02-10T20:56:26Z Fixed points on trivial surface bundles Нерухомі точки на тривіальних розшаруваннях з шарами поверхнями Gonçalves, D. L. Libardi, A. K. M. Penteado, D. Vieira, J. P. Gonçalves, D. L. Libardi, A. K. M. Penteado, D. Vieira, J. P. The main purpose of this work is to study fixed points of fiber preserving maps over $S^1$ on the trivial surface bundles $S^1\times S_2$, where $S_2$ is the closed orientable surface of genus 2. We classify all such maps that can be deformed fiberwise to a fixed point free map. Основною метою роботи є вивчення нерухомих точок відображень, що зберігають шари над $S^1$ на тривіальних розшаруваннях, шари яких є поверхнями $S^1\times S_2$, де $S_2$ - замкнена орієнтована поверхня роду 2. Ми класифікуємо всі такі відображення, що можна пошарово деформувати до відображення з нерухомою точкою. Інститут математики НАН України 2013-06-26 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/309 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 10 No. 6 (2013): Brasilian-Polish Topology Workshop; 67-85 Сборник Трудов Института математики НАН Украины; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 67-85 Збірник Праць Інституту математики НАН України; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 67-85 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/309/288 Авторське право (c) 2013 Праці Інституту математики НАН України |
| spellingShingle | Gonçalves, D. L. Libardi, A. K. M. Penteado, D. Vieira, J. P. Gonçalves, D. L. Libardi, A. K. M. Penteado, D. Vieira, J. P. Fixed points on trivial surface bundles |
| title | Fixed points on trivial surface bundles |
| title_alt | Нерухомі точки на тривіальних розшаруваннях з шарами поверхнями |
| title_full | Fixed points on trivial surface bundles |
| title_fullStr | Fixed points on trivial surface bundles |
| title_full_unstemmed | Fixed points on trivial surface bundles |
| title_short | Fixed points on trivial surface bundles |
| title_sort | fixed points on trivial surface bundles |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/309 |
| work_keys_str_mv | AT goncalvesdl fixedpointsontrivialsurfacebundles AT libardiakm fixedpointsontrivialsurfacebundles AT penteadod fixedpointsontrivialsurfacebundles AT vieirajp fixedpointsontrivialsurfacebundles AT goncalvesdl fixedpointsontrivialsurfacebundles AT libardiakm fixedpointsontrivialsurfacebundles AT penteadod fixedpointsontrivialsurfacebundles AT vieirajp fixedpointsontrivialsurfacebundles AT goncalvesdl neruhomítočkinatrivíalʹnihrozšaruvannâhzšaramipoverhnâmi AT libardiakm neruhomítočkinatrivíalʹnihrozšaruvannâhzšaramipoverhnâmi AT penteadod neruhomítočkinatrivíalʹnihrozšaruvannâhzšaramipoverhnâmi AT vieirajp neruhomítočkinatrivíalʹnihrozšaruvannâhzšaramipoverhnâmi AT goncalvesdl neruhomítočkinatrivíalʹnihrozšaruvannâhzšaramipoverhnâmi AT libardiakm neruhomítočkinatrivíalʹnihrozšaruvannâhzšaramipoverhnâmi AT penteadod neruhomítočkinatrivíalʹnihrozšaruvannâhzšaramipoverhnâmi AT vieirajp neruhomítočkinatrivíalʹnihrozšaruvannâhzšaramipoverhnâmi |