The Borsuk-Ulam Theorem for Double Coverings of Seifert Manifolds
We study a Borsuk-Ulam type theorem for pairs $(M,\tau)$ with $\tau$ a fixed point free involution of $M$, and such that both $M$ and $N\colon = M/\tau$ are Seifert  manifolds. In this note our point of view will be to start with a Seifert manifold $N$. Any non-trivial element $\xi\in H...
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Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552801121337344 |
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| author | Bauval, A. Gonçalves, D. L. Hayat, C. Zvengrowski, P. Bauval, A. Gonçalves, D. L. Hayat, C. Zvengrowski, P. |
| author_facet | Bauval, A. Gonçalves, D. L. Hayat, C. Zvengrowski, P. Bauval, A. Gonçalves, D. L. Hayat, C. Zvengrowski, P. |
| author_institution_txt_mv | [
{
"author": "A. Bauval",
"institution": "Institut de Mathématiques de Toulouse Equipe Emile Picard,"
},
{
"author": "D. L. Gonçalves",
"institution": "Departamento de Matemática - IME-USP"
},
{
"author": "C. Hayat",
"institution": "Toulouse Mathematics Institute"
},
{
"author": "P. Zvengrowski",
"institution": "Department of Mathematics and Statistics University of Calgary"
}
] |
| author_sort | Bauval, A. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2018-02-10T20:56:26Z |
| description | We study a Borsuk-Ulam type theorem for pairs $(M,\tau)$ with $\tau$ a fixed point free involution of $M$, and such that both $M$ and $N\colon = M/\tau$ are Seifert  manifolds. In this note our point of view will be to start with a Seifert manifold $N$. Any non-trivial element $\xi\in H^1(N;Z_2)$ then gives rise to a pair $(M_\xi, \tau_\xi) = (M , \tau) $ with $M$ (necessarily) also a Seifert  manifold, and a double covering  $p \colon M \twoheadrightarrow N$,  with $\tau$ being the fixed point free involution on $M$ associated to this double covering as the non-trivial deck transformation. We then seek the largest value of $n$, called the $\mathbb{Z}_2$-index of $(M,\tau)$, such that the Borsuk-Ulam property holds for maps into $\mathbb{R}^n$, i.e. such that for every continuous map $f\colon M\to \mathbb{R}^n$, there is an $x\in M$ such that $f(x)=f(\tau(x))$. In case $M$ is a $3$-manifold (such as a Seifert manifold), the$\mathbb{Z}_2$-index can take only the values $1,2,3$. |
| first_indexed | 2026-08-04T01:05:12Z |
| format | Article |
| fulltext |
Збiрник праць Iн-ту математики НАН України 2013, том 6, N 6, 165–189
A. Bauval, D. L. Gonçalves, C. Hayat and P.
Zvengrowski
The Borsuk-Ulam Theorem for
Double Coverings of Seifert Manifolds
We study a Borsuk-Ulam type theorem for pairs (M, τ) with τ a fixed point
free involution of M , and such that both M and N : = M/τ are Seifert
manifolds. In this note our point of view will be to start with a Seifert
manifold N . Any non-trivial element ξ ∈ H1(N ;Z2) then gives rise to a
pair (Mξ, τξ) = (M, τ) with M (necessarily) also a Seifert manifold, and a
double covering p : M � N , with τ being the fixed point free involution on
M associated to this double covering as the non-trivial deck transformation.
We then seek the largest value of n, called the Z2-index of (M, τ), such that
the Borsuk-Ulam property holds for maps into Rn, i.e. such that for every
continuous map f : M → Rn, there is an x ∈M such that f(x) = f(τ(x)).
In case M is a 3-manifold (such as a Seifert manifold), the Z2-index can
take only the values 1, 2, 3.
1 Introduction
The study of involutions on manifolds has been of great interest and
importance within topology, as illustrated by the books of J. Matoušek
[12] and S. L. de Medrano [11] (and in particular, for involutions on
Seifert manifolds, cf. the book of Montesinos [13]). The most famous
theorem in the subject is undoubtedly the classical Borsuk-Ulam theorem,
which applies to the antipodal involution of a sphere. This theorem
together with various generalizations and applications continues to be of
great interest. For example, a generalization of the Borsuk-Ulam theorem
that applies to a fixed point free involution on any manifold has recently
been studied by Gonçalves, Hayat, and Zvengrowski [7]. The case of
manifolds of dimension 2 and the corresponding Borsuk-Ulam theorem
c© A. Bauval, D. L. Gonçalves, C. Hayat and P. Zvengrowski, 2013
166 Borsuk-Ulam Theorem
has also been recently studied by Gonçalves and Guaschi [6]. The above
mentioned book of Matoušek gives an extensive set of references related to
the Borsuk-Ulam theorem; in addition to these further interesting aspects
and generalizations of the classical Borsuk- Ulam theorem appear (among
others) in work by K. D. Joshi [10], J. Jaworowski [9], A. Dold [5], and
more recently in work of P. L. Q. Pergher, D. de Mattos, E. L. dos Santos
[16], P. L. Q. Pergher, H. K. Singh, T. B. Singh [17], as well as survey
papers among which we mention H. Steinlein [22], and I. Nagasaki [14].
In this paper we attempt to initiate this study for the Seifert man-
ifolds, a large and important class of 3-manifolds introduced by Seifert
[19] in 1933. This is possible, using the aforementioned paper [7] and
the knowledge of the Z2-cohomology rings of these manifolds, cf. [2], [3],
[4] for the orientable case and more recently [1] for all Seifert manifolds.
We will suppose throughout that all manifolds under consideration are
closed and connected.
Given a (closed, connected) m-manifold N , any non-trivial element
ξ ∈ H1(N ;Z2) gives rise to an epimorphism φ : π1N � Z2 and a pair
(Mξ, τξ) = (M, τ), where p : M � N is a double covering,M is a (closed,
connected) m-manifold, and τ is the fixed point free involution on M
associated to this double covering as the non-trivial deck transformation.
This correspondence is via the sequence of isomorphisms
hom(π1(N),Z2) ≈ hom((π1(N))ab,Z2) ≈ hom(H1(N),Z2) ≈ H1(N ;Z2).
(1)
Definition 1.1. (i) We say that the Borsuk-Ulam property BU(n) holds
for (M, τ) if for every continuous map f : M → Rn, there is an x ∈ M
such that f(x) = f(τ(x)).
(ii) The Z2-index indZ2(M, τ) is then defined as the largest n ≤ ∞
such that BU(n) holds.
>From [7] it is known that indZ2(M, τ) ≥ 1 always holds, and
indZ2(M, τ) = 1 if and only if ξ ∈ Im(ρ : H1(N ;Z)→ H1(N ;Z2), where
ρ is the coefficient homomorphism induced by the surjection Z � Z2.
Furthermore, it is shown there that indZ2
(M, τ) ≤ m = dim(M) and
indZ2
(M, τ) = m if and only if ξm 6= 0 ∈ Hm(N ;Z2). It follows that the
inequality 1 ≤ indZ2(M, τ) ≤ m is always satisfied. In particular, for
m = 3, the Z2-index can only equal 1, 2, or 3. These facts are formally
stated in Section 2 as Theorem 2.1.
In the present work, we suppose that N is a Seifert manifold (of
dimension m = 3), presented in the usual way by its Seifert invariants
A. Bauval, D. L. Gonçalves, C. Hayat, and P. Zvengrowski 167
(cf. [15], [19]). The presentation of π1(N), associated to these invariants,
is the standard presentation found in [15], and allows one to list the (non-
trivial) homomorphisms φ : π1N � Z2. We classify the φ’s for which the
Z2-index equals 1, equals 2, or equals 3. The main results are expressed
in terms of the Seifert invariants of N and the homomorphism φ.
This work contains five sections. In Section 2, we recall some basic
facts about Seifert manifolds. In Section 3 we consider the situation of
maps into R2; the main results are Proposition 3.4 and Theorem 3.5.
The former gives necessary and sufficient conditions for indZ2
(M, τ) = 1,
and the latter (which is essentially the negation of the former) for
indZ2(M, τ) ≥ 2. In Section 4 we consider the situation of maps into
R3; the main result is Theorem 4.3 which gives necessary and sufficient
conditions for indZ2
(M, τ) = 3. In Section 5 we make some general com-
ments about the relation between the Z2-index = 2 and the Z2-index =
3 cases. In this section we also study several specific examples that effec-
tively illustrate the techniques, for a variety of Seifert manifolds, and also
show that the distinction between the various cases can be surprisingly
delicate.
Another (and probably more natural) approach to these questions is
to start with the manifold M and fixed point free involution τ , then
construct N as the orbit space M/τ . For Seifert manifolds M this can
lead to cases that are not covered in the present paper, indeed cases
where N is not a Seifert manifold in the classical sense, depending on the
geometry (in the sense of Thurston) ofM . The authors hope to complete
the study, from this point of view, in subsequent research, with [7] being
the first step in this direction and the present note the second step. We
also note that the condition ξm 6= 0 mentioned above becomes ξ3 6= 0
for a 3-manifold, and for orientable 3-manifolds this condition also arises
in the study of general relativity (where one says such 3-manfolds have
“type 1"), cf. [21]. The condition ξ3 6= 0 is equivalent to the existence
of a degree 1 (or odd degree) map of the 3-manifold onto RP 3.
2 Introductory Remarks and Notation for 3-
manifolds
Let N be a 3-manifold. In Section 1 the isomorphism (1) between
H1(N ;Z2) and hom(π1(N),Z2) was introduced. Under this correspon-
dence, the image in H1(N ;Z2) of a homomorphism φ : π1(N) → Z2 will
168 Borsuk-Ulam Theorem
be denoted by ξφ = ξ. Any non-zero element ξ ∈ H1(N ;Z2) corresponds
to an epimorphism φ : π1N � Z2 which induces a short exact sequence:
1→ Kerφ� π1N � Z2 → 0.
>From the theory of covering spaces, we know that there exists a
connected 3-manifold M = Mφ such that Kerφ = π1(M) is a normal,
index 2, subgroup of π1(N), and M � N is the regular double covering
of N corresponding to Kerφ. We also know that the non-trivial deck
transformation is a fixed point free involution τφ = τ on M such that the
quotient M/τ is homeomorphic to N . We will use this correspondence
freely whenever necessary.
From [7] Theorems (3.1) and (3.2) we have:
Theorem 2.1. Let N be a 3-manifold and φ : π1(N) � Z2 an epimor-
phism. Let (M, τ) and ξ ∈ H1(N ;Z2) be determined as above.
(i) One has indZ2
(M, τ) = 1 if and only if the homomorphism
φ : π1(N) � Z2 factors through the projection Z � Z2 (equivalently
ξ ∈ Im(ρ : H1(N ;Z)→ H1(N ;Z2))), otherwise indZ2
(M, τ) ∈ {2, 3},
(ii) One has indZ2
(M, τ) = 3 if and only if ξ3 6= 0.
We now focus on the situation where N is any Seifert manifold (ori-
entable or not), as introduced in [19]. We shall answer the following
question: given a presentation of N in terms of Seifert invariants, for
which φ is indZ2(M, τ) = 1, 2, or 3 ?
Following the notation of Orlik [15], from now on, N will be a Seifert
manifold described by a list of Seifert invariants
{e; (∈, g); (a1, b1), . . . , (an, bn)}
(note that Orlik uses b for the Euler number e). We do not need them to
be “normalized” as in [15] and [19]: we only assume that e is an integer,
the type ∈ will described below, g is the genus of the base surface (the
orbit space obtained by identifying each S1 fibre of N to a point), and
for each k, the integers ak, bk are coprime with ak 6= 0 (in case bk = 0
then ak = ±1).
As in [15], p.74 (and elsewhere), it is convenient to add an additional
(non-exceptional) fibre a0 = 1, b0 = e. We shall then use the following
A. Bauval, D. L. Gonçalves, C. Hayat, and P. Zvengrowski 169
presentation of the fundamental group of N :
π1(N) =
〈s0, . . . , sn
v1, . . . , vg′
h
∣∣∣∣∣∣
[sk, h] and sakk h
bk , 0 ≤ k ≤ n
vjhv
−1
j h−εj , 1 ≤ j ≤ g′
s0 . . . snV
〉
, (2)
where the generators and g′, V are described below. Also note that if
e = 0 then the relation sa00 h
b0 reduces to s0 = 1, so in this case s0 is
usually omitted.
• The type ∈ of N equals:
o1 if both the base surface and the total space are orientable
(which forces all εj ’s to equal 1);
o2 if the base surface is orientable and the total space is non-
orientable, hence g ≥ 1 (which forces all εj ’s to equal −1);
n1 if both the base surface and the total space are non-orientable
(hence g ≥ 1) and moreover, all εj ’s equal 1;
n2 if the base surface is non-orientable (hence g ≥ 1) and the
total space is orientable (which forces all εj ’s to equal −1);
n3 if both the base surface and the total space are non-orientable
and moreover, all εj ’s equal −1 except ε1 = 1, and g ≥ 2;
n4 if both the base surface and the total space are non-orientable
and moreover, all εj ’s equal −1 except ε1 = ε2 = 1, and g ≥ 3.
We note that these six types, in Seifert’s original notation, are
respectively denoted Oo, No, Nn, On, NnI, NnII, where the
first (capital) letter refers to the orientability or non-orientability
of the total space N , while the second (lower case) letter refers to
the same for the base surface.
• The orientability of the base surface and its genus g determine the
number g′ of the generators vj ’s and the word V in the last relator
of π1(N) as follows:
– when the base surface is orientable, g′ = 2g and V =
[v1, v2] . . . [v2g−1, v2g];
– when the base surface is non-orientable, g′ = g and V =
v21 . . . v
2
g .
170 Borsuk-Ulam Theorem
• The generator h corresponds to the generic regular fibre.
• The generators sk for 0 ≤ k ≤ n correspond to (possibly) excep-
tional fibres.
Throughout this paper, we shall use the following notations (the last
one Sφ depends on φ, all the previous ones only on N).
Notation 2.2. Let N be a Seifert manifold described by a list of Seifert
invariants
{e; (∈, g); (a1, b1), . . . , (an, bn)}.
• Denoting by a the least common multiple of the ak’s,
c =
n∑
k=0
bk(a/ak).
• The number of even ak’s will be denoted by d.
• We distinguish three cases:
– Case 1, d = 0 and c is even;
– Case 2, d = 0 and c is odd;
– Case 3, d > 0.
• In Case 3, the indices k are reordered by decreasing 2-valuation
ν2(ak). Hence the set of even ak’s will be {a0, . . . , ad−1} and the
set of k’s for which ak has maximal 2-valuation, denoted by SN ,
will be {0, . . . , J − 1} for some 0 < J ≤ d. Note that after this
reordering, in Case 3, a0 6= 1.
• Sφ will denote the set of k’s for which φ(sk) = 1.
Note that these cases are not related to the type ∈, each of the three
cases can occur with any of the six types. The next lemma will be useful
in Section 3.
Lemma 2.3. In Case 3 (d > 0), c has the same parity as J . Further-
more, one also has Sφ ⊆ {0, . . . , d− 1}, |Sφ| is even, and φ(h) = 0.
A. Bauval, D. L. Gonçalves, C. Hayat, and P. Zvengrowski 171
Proof. With the above notational conventions, a/ak is odd if and only
if k < J , and for such k’s, bk is also odd since it is coprime to ak.
Hence, modulo 2, c =
∑
bk(a/ak) ≡
∑
0≤k<J 1 = J . The fact that
Sφ ⊆ {0, . . . , d − 1} follows directly from the definition of d and the
reordering convention in Case 3. If we take any k ∈ {0, . . . , d − 1} we
have ak even and bk odd, hence 0 = φ(sakk h
bk) = akφ(sk) + bkφ(h) =
φ(h). Finally, note that φ(V ) = 0 in both the case of orientable or non-
orientable base surface, since φ is a homomorphism and Im(φ) ⊆ Z2.
Then 0 = φ(s0 · · · snV ) = φ(s0) + . . . φ(sn) implies |Sφ| is even.
We close this section with an abelianized version of (2), which gives
a presentation of H1(N) = H1(N ;Z). This will also be useful for the
work in Section 3.
H1(N) =
〈s0, . . . , sn
v1, . . . , vg′
h
∣∣∣∣∣∣
aksk + bkh, 0 ≤ k ≤ n
(1− εj)h, 1 ≤ j ≤ g′
s0 + . . .+ sn + V
〉
, (3)
where V = 0 for types o1 and o2, and V = 2(v1 + . . .+ vg) for the four
remaining types.
3 Study of indZ2
(M, τ) ≥ 2
As before, let φ : π1(N) � Z2 be an epimorphism and ξ ∈ H1(N ;Z2)
the corresponding cohomology class as given in (1). By Theorem 2.1,
the set of ξ’s for which indZ2
(M, τ) = 1 is the image of the coefficient
homomorphism ρ : H1(N ;Z) → H1(N ;Z2), so our initial goal in this
section is to compute Im(ρ) (a less direct method, leading to the same
results, would be to compute the kernel of the Bockstein homomorphism
H1(N ;Z2) → H2(N ;Z)). This is done in Propositions 3.1, 3.3, and 3.4.
Then, in 3.5, we determine when ξ 6∈ Im(ρ), and this is equivalent to
indZ2
(M, τ) ≥ 2.
>From the presentation (3) of H1(N) we shall compute H1(N ;Z2)
(Proposition 3.1, which will be repeated later as a small part of The-
orem 4.1), and similarly compute H1(N ;Z) (Proposition 3.3). We use
the fact that H1(N ;Z2) naturally identifies to the subspace of cocycles
contained in C1(N,Z2) := hom(C1(N),Z2), where C1(N) is the free
abelian group with generators vj , sk, h. Furthermore, using the isomor-
phism (1), we see that the 1-cocycles are simply the 1-cochains that
172 Borsuk-Ulam Theorem
vanish on the abelianized relations for π1(N), as given in (3). We denote
by v̂j (1 ≤ j ≤ g′), ŝk (0 ≤ k ≤ n), ĥ, the elements of the dual basis
of C1(N,Z2) corresponding respectively to vj , sk, and h. In Proposition
3.3, the same notations and identifications will be used, replacing Z2
by Z, recalling also that H1(X;Z) is a free abelian group for any finite
CW -complex X.
Proposition 3.1. Let α = ĥ+
∑n
k=0 bkŝk and αk = ŝk + ŝ0, 1 ≤ k ≤ n.
A basis of the Z2-vector space H1(N ;Z2) ⊆ C1(N,Z2) is (with Notation
2.2):
– Case 1 : {v̂1, . . . , v̂g′ , α},
– Case 2 : {v̂1, . . . , v̂g′},
– Case 3 : {v̂1, . . . , v̂g′ , α1, . . . , αd−1}.
Proof. Consider an arbitrary element
u = xĥ+
n∑
k=0
zkŝk +
g′∑
j=1
yj v̂j ∈ C1(N ;Z2)
(with zk, yj , x ∈ Z2). Due to the presentation (2) of π1(N), u ∈
H1(N ;Z2) if and only if the following n + 2 equations, coming from
the relations in (3), are satisfied:
akzk + bkx = 0, k = 0, . . . , n, and z0 + . . .+ zn = 0.
When d = 0 all ak and a are odd, so c =
∑
bk and this system is
thus equivalent to:
zk = bkx (k = 0, . . . , n) and cx = 0.
The elements of H1(N ;Z2) are therefore the u’s of the form:
u = x
(
ĥ+
n∑
k=0
bkŝk
)
+
g′∑
j=1
yj v̂j ,
with no restriction on x in Case 1 (d = 0 and c even), but with x = 0 in
Case 2 (d = 0 and c odd).
A. Bauval, D. L. Gonçalves, C. Hayat, and P. Zvengrowski 173
When d > 0, the system is equivalent to:
x = 0, zk = bkx (k = d, . . . , n) and z0 + . . .+ zn = 0,
which simplifies to:
x = zd = . . . = zn = 0 and z0 = z1 + . . .+ zd−1.
So, in Case 3, the elements of H1(N ;Z2) are the u’s of the form:
∑
1≤k≤d−1
zk(ŝk + ŝ0) +
g′∑
j=1
yj v̂j ,
which completes the proof.
Remark 3.2. In future use of this proposition and the following ones
it will be important to note that if the cohomology class u ∈ H1(N ;Z2)
corresponds to the epimorphism φ : π1(N) � Z2 via the isomorphism (1),
then zk = φ(sk), yj = φ(vj), and x = φ(h). In this case we also write
u = ξφ = ξ, as in Section 1.
Proposition 3.3. The abelian group H1(N ;Z) is free and generated by
the following elements of C1(N,Z):
• if ∈= o2: {v̂1, . . . , v̂g′},
• if ∈= n2, n3, n4: {v̂2 − v̂1, . . . , v̂g′ − v̂1},
• if ∈= o1:
– if c = 0: {v̂1, . . . , v̂g′ , aĥ−
∑n
k=0 bk(a/ak)ŝk},
– if c 6= 0: {v̂1, . . . , v̂g′},
• if ∈= n1:
– if c is even: {(c/2)v̂1+aĥ−
∑n
k=0 bk(a/ak)ŝk, v̂2−v̂1, . . . , v̂g′−
v̂1},
– if c is odd: {cv̂1 + 2aĥ− 2
∑n
k=0 bk(a/ak)ŝk, v̂2− v̂1, . . . , v̂g′ −
v̂1}.
174 Borsuk-Ulam Theorem
Proof. It was noted earlier in this section that H1(N ;Z) is free. As in
the proof of Proposition 3.1, consider an arbitrary element
u = xĥ+
n∑
k=0
zkŝk +
g′∑
j=1
yj v̂j ∈ C1(N ;Z),
now with zk, yj , x ∈ Z. We obtain that u ∈ H1(N ;Z) if and only if the
following equations are satisfied:
akzk + bkx = 0, k = 0, . . . , n;
(1− εj)x = 0, j = 1, . . . , g′;
n∑
k=0
zk = 0 if ∈= o1, o2;
n∑
j=0
zk + 2
g′∑
j=1
yj = 0 if ∈= n1, n2, n3, n4.
Let us first treat the four easiest cases. As soon as some εj equals −1 (i.e.
∈= o2, n2, n3, n4), the equation involving such a εj implies x = 0, which,
by the first n+ 1 equations, forces all zk’s to be also zero. The remaining
last equation thus reduces to 0 = 0 if ∈= o2 and to y1 = −
∑
j>1 yj
if ∈= n2, n3, n4. This already enables us to assert that the elements of
H1(N ;Z) are the u’s of the form:
• if ∈= o2:
∑
yj v̂j
• if ∈= n2, n3, n4:
∑
j>1 yj(v̂j − v̂1).
In the two remaining cases ∈= o1, n1 (where the conditions (1− εj)x = 0
are vacuous since εj = 1), first note that the first n+ 1 equations imply
that each ak divides x, hence so does a (their l.c.m.). Letting x = ax′,
these equations may be rewritten:
zk = −bk(a/ak)x′, k = 0, . . . , n.
The remaining last equation thus becomes:
cx′ = 0 if ∈= o1; cx′ = 2
∑
yj if ∈= n1.
When ∈= o1, this last equation forces x′ (hence also the zk’s) to be 0
if and only if c 6= 0. Hence the elements of H1(N ;Z) are the u’s of the
form:
A. Bauval, D. L. Gonçalves, C. Hayat, and P. Zvengrowski 175
• if ∈= o1 and c 6= 0:
∑
yj v̂j
• if ∈= o1 and c = 0: x′
(
aĥ−
∑n
k=0 bk(a/ak)ŝk
)
+
∑
yj v̂j .
In the last remaining case (∈= n1), the last equation forces x′ to be even
whenever c is odd, which naturally leads us to consider two subcases:
• if c is even, this equation amounts to y1 = (c/2)x′ −
∑
j>1 yj ;
• if c is odd, letting x′ = 2x′′ allows rewriting the equation as y1 =
cx′′ −
∑
j>1 yj .
Hence the elements of H1(N ;Z) are the u’s of the form:
• if ∈= n1 and c is even:
x′
(
aĥ−
n∑
k=0
bk(a/ak)ŝk + (c/2)v̂1
)
+
∑
j>1
yj(v̂j − v̂1)
• if ∈= n1 and c is odd:
x′′
(
2aĥ− 2
n∑
k=0
bk(a/ak)ŝk + cv̂1
)
+
∑
j>1
yj(v̂j − v̂1),
which completes the proof.
>From Theorem 2.1 and Propositions 3.1 and 3.3 we deduce:
Proposition 3.4. With the notations of Proposition 3.1 and Notation
2.2, the subspace Im(ρ) ⊆ H1(N ;Z2) has basis:
• if ∈= o2: {v̂1, . . . , v̂g′},
• if ∈= n2, n3, n4: {v̂2 + v̂1, . . . , v̂g′ + v̂1},
• if ∈= o1: if c 6= 0 then {v̂1, . . . , v̂g′},
– if c = 0 and d = 0 then {v̂1, . . . , v̂g′ , α},
– if c = 0 and d > 0 then {v̂1, . . . , v̂g′ ,
∑
1≤k≤J−1 αk},
• if ∈= n1:
176 Borsuk-Ulam Theorem
– if c is odd: {v̂1, . . . , v̂g′},
– when c is even and d = 0: {v̂2 + v̂1, . . . , v̂g′ + v̂1, (c/2)v̂1 +α},
– if c is even and d > 0: {v̂2 + v̂1, . . . , v̂g′ + v̂1, (c/2)v̂1 +∑
1≤k≤J−1 αk}.
Proof. Most of this statement follows immediately from Propositions 3.1
and 3.3; we shall address the only non-obvious parts which are the two
possibilities (∈= o1, c = 0) and (∈= n1, c even). Note that in these two
cases, Case 2 (d = 0, c odd) does not occur.
If ∈= o1 and c = 0, we must compute the image in H1(N ;Z2) of
u := aĥ −
∑n
k=0 bk(a/ak)ŝk ∈ H1(N ;Z), in terms of the generators of
H1(N ;Z2).
• If d = 0 (Case 1, a, ak are all odd): ρ(u) = ĥ+
∑n
k=0 bkŝk = α.
• If d > 0 (Case 3, a/ak is odd only for 0 ≤ k ≤ J − 1, a is even, and
b0, . . . , bd−1 are odd ):
ρ(u) = 0 · ĥ+
n∑
k=0
bk(a/ak)ŝk =
J−1∑
k=0
bkŝk =
J−1∑
k=0
ŝk.
By Lemma 2.3 J is even, so we may rewrite this sum as
∑J−1
k=1 (ŝk+
ŝ0) =
∑J−1
k=0 αk, as desired.
• If ∈= n1 and c is even, the proofs in the two cases (Case 1 and Case
3) are identical to the corresponding previous two cases for ∈= o1,
except that (c/2)v̂1 is added to u and hence also to ρ(u).
We now take into account Remark 3.2, Notation 2.2, and Proposition
3.4 (in its negated form), to prove the main theorem of this section.
Theorem 3.5. One has indZ2
(Mφ, τφ) ∈ {2, 3} in exactly the following
cases:
• Either ∈= o1 and c 6= 0, or ∈= n1 and c is odd, or ∈= o2, and in
addition {φ(h), φ(s0), . . . , φ(sn)} 6= {0},
• Either ∈= n2 or ∈= n3 or ∈= n4, and in addition
{φ(
∑
vj), φ(h), φ(s0),
. . . , φ(sn} 6= {0},
A. Bauval, D. L. Gonçalves, C. Hayat, and P. Zvengrowski 177
• ∈= o1, c = 0, d > 0 and Sφ 6= ∅, SN
• ∈= n1, c is even and:
– if d = 0:
∑g′
j=1 φ(vj) 6= (c/2)φ(h)
– if d > 0: either Sφ 6= ∅, SN , or Sφ = ∅ and
∑g′
j=1 φ(vj) 6= 0,
or Sφ = SN and
∑g′
j=1 φ(vj) 6= (c/2).
Proof. Writing as usual ξ = ξφ ∈ H1(N ;Z2), the condition given in
Theorem 2.1(1) tells us that indZ2
(Mφ, τφ) ∈ {2, 3} if and only if ξ 6∈
Im(ρ). Now Proposition 3.4 identifies Im(ρ), so in each case we simply
have to negate the conditions given in Proposition 3.4.
• When either ∈= o1 and c 6= 0, or ∈= n1 and c is odd, or ∈= o2,
Im(ρ) = 〈v̂1, . . . , v̂g′〉. Therefore ξ = xĥ+
∑n
k=0 zkŝk+
∑g′
j=1 yj v̂j 6∈
Im(ρ) if and only if some zk or x is non-zero, which is identical to
the given condition (see Remark 3.2).
• When either ∈= n2 or ∈= n3 or ∈= n4, Im(ρ) = 〈v̂2 + v̂1, . . . , v̂g′ +
v̂1〉. Therefore ξ ∈ Im(ρ) if and only if ξ =
∑g′
j=2 yj(v̂j + v̂1), or
equivalently ξ =
∑g′
j=1 yj v̂j with
∑g′
j=1 yj = 0. So ξ 6∈ Im(ρ) if
and only if some xk 6= 0 or x 6= 0 or
∑g′
j=1 yj 6= 0, which is identical
to the given condition.
• When ∈= o1, c = 0, d = 0, we see from Propositions 3.4 and 3.1
(Case 1) that ρ is surjective. So ξ ∈ Im(ρ), i.e. indZ2
(M,φ) = 1,
and hence this case does not appear on the list in Theorem 3.5.
• When ∈= o1, c = 0, d > 0, we have Case 3 so Lemma 2.3 applies,
and we shall use it several times here. In particular we will use
0 = c ≡ J (mod 2) and x = φ(h) = 0 without further mention.
Here Im(ρ) = 〈v̂1, . . . , v̂g′ ,
∑J−1
k=1 αk〉, hence ξ ∈ Im(ρ) if and only
if, for some yj , z ∈ Z2,
ξ =
g′∑
j=1
yj v̂j + z
J−1∑
k=1
(ŝk + ŝ0) =
g′∑
j=1
v̂j +
J−1∑
k=0
zŝk.
Since, as already noted, x = 0, we deduce ξ =
∑g′
j=1 yj v̂j +∑J−1
k=0 zŝk 6∈ Im(ρ) if and only if either φ(ŝk) = zk 6= 0 for
178 Borsuk-Ulam Theorem
some k ≥ J , or zJ = . . . = zn = 0 and {φ(ŝ0), . . . , ϕ(ŝJ−1)} =
{z0, . . . , zJ−1} = {0, 1} (i.e. ϕ(ŝ0), . . . , φ(ŝJ−1) are not all equal).
These conditions are easily seen, recalling Notation 2.2, to be equiv-
alent to Sφ 6= ∅, SN , as stated.
• When ∈= n1, c even, and d = 0, we have Case 1 so zk = bkx, 0 ≤
k ≤ n, as seen in the proof of Proposition 3.1. Here Im(ρ) =
〈v̂2 + v̂1, . . . , v̂g′ + v̂1, (c/2)v̂1 + α〉. Noting that α(h) = x and
v̂1(h) = 0, this gives that ξ ∈ Im(ρ) if and only if
ξ =
g′∑
j=2
yj(v̂j + v̂1) + x[(c/2)v̂1 + α] =
g′∑
j=1
yj v̂j + xα
=
g′∑
j=1
yj v̂j + xĥ+
n∑
k=0
zkŝk,
where y1 = x(c/2) +y2 + . . . yg′ , or equivalently
∑g′
j=1 yj = x(c/2).
It follows that ξ 6∈ Im(ρ) if and only if
∑g′
j=1 yj 6= (c/2)x, and this
is the same as the stated condition.
• When ∈= n1, c even, and d > 0 we again have Case 3 so as in the
previous Case 3, J is even and x = 0. Now
Im(ρ) = 〈v̂2 + v̂1, . . . , v̂g′ + v̂1, (c/2)v̂1 +
J−1∑
k=1
ŝk + ŝ0〉.
Then ξ ∈ Im(ρ) if and only if ξ =
∑g′
j=2 yj(v̂j + v̂1) + tc1v̂1 +
t
∑n
k=0 ŝk =
∑g′
j=1 yj v̂j + t
∑n
k=0 ŝk, where y1 = y2 + . . . + yg′ +
t(c/2), or equivalently
∑g′
j=1 yj = t(c/2). It follows that ξ 6∈ Im(ρ)
if and only if either Sφ 6⊆ SN , or Sφ ⊆ SN and
∑g′
j=1 φ(vj) 6=
(c/2)φ(sk) for at least one k, 1 ≤ k ≤ J − 1. Again, these
conditions are easily seen to be equivalent to the stated conditions
in this case.
A. Bauval, D. L. Gonçalves, C. Hayat, and P. Zvengrowski 179
4 Study of indZ2
(M, τ) = 3
According to 2.1(ii), one has indZ2(M, τ) = 3 if and only if ξ3 6= 0.
We therefore begin this section by stating known results (cf. [1], [2], [3],
[4]) for the Z2-cohomology ring of a Seifert manifold N . For H1(N ;Z2)
this necessarily overlaps with some of the computations done in Section
3, and the notations used in Section 3 are consistent with those in the
references (where we now will write v̂j = θj). For types ∈= o1, o2 what
we now call θ1, θ2, θ3, θ4, . . . correspond respectively to θ1, θ
′
1, θ2, θ
′
2, . . .
in [4], while the notation is identical for the remaining four types. As
far as the cup products it suffices to list just the non-zero products in
positive dimensions, on the generators, also taking account that xy = yx
in H∗( ; Z2).
Theorem 4.1. Let N be any Seifert manifold described by a list of Seifert
invariants
{e; (∈, g); (a1, b1), . . . , (an, bn)},
the type ∈ being o1, o2, n1, n2, n3, n4.
Using Notation 2.2, the cohomology groups H∗(N ;Z2) are: H0 =
Z2{1} (the unit for the cup-product), H3 = Z2{γ}, and (with 1 ≤ j ≤
g′ = 2g for the types o1 and o2, and 1 ≤ j ≤ g′ = g for the other types):
– Case 1 (if d = 0 and c is even): H1 = Z2{θ1, . . . θg′ , α = ĥ +∑n
k=0 bkŝk}, H2 = Z2{ϕ1, . . . , ϕg′ , β}.
– Case 2 (if d = 0 and c is odd): H1 = Z2{θ1, . . . , θg′}, H2 =
Z2{ϕ1, . . . , ϕg′},
– Case 3 (if d > 0): H1 = Z2{θ1, . . . θg′ , α1, . . . αd−1} ,
H2 = Z2{ϕ1, . . . , ϕg′ , β1, . . . , βd−1}.
The non-trivial cup-products, on the generators of H1⊗H1 and H1⊗
H2, are:
– In all three Cases, for the types o1 and o2, θ2i−1ϕ2i = θ2iϕ2i−1 = γ,
while for the other types θjϕj = γ.
– in Case 1, for the types o1 and o2, θ2i−1θ2i = β, while for the
other types θ2j = β.
180 Borsuk-Ulam Theorem
– in Case 1, θjα = ϕj, αβ = γ, αϕj = γ when εj = −1 (as specified
in Section 2 for each of the types), and
α2 = (c/2)β +
∑
εj=−1
ϕj .
– in Case 3 (i.e. d > 0), αkβk = γ, k > 0, and, for k, l > 0,
αkα` =
a0
2
β0 + δk,`
ak
2
βk,
where β0 denotes
∑
1≤k≤d−1 βk.
>From this theorem, we deduce:
Proposition 4.2. With the same notations, let ξ = ξφ ∈ H1(N ;Z2).
• In Case 1,
ξ3 =
φ(h)(c/2) · γ when ∈= o1,
φ(h)((c/2) +
∑
φ(vi)) · γ when ∈= o2, n1,
φ(h)((c/2) + g) · γ when ∈= n2,
φ(h)((c/2) + φ(v1) + g − 1) · γ when ∈= n3,
φ(h)((c/2) + φ(v1) + φ(v2) + g) · γ when ∈= n4.
• In Case 2, ξ3 = 0.
• In Case 3, ξ3 = (
∑
φ(sk)(ak/2)) · γ.
Proof.
Case 1. Let ξ = x · α+
∑
yj · θj with x = φ(h) and yj = φ(vj), then
ξ2 = x · α2 +
∑
yj · θ2j = x((c/2) · β +
∑
εj=−1
ϕj) + y · β,
with y = 0 when ∈= o1, o2, y =
∑
yj when ∈= n1, n2, n3, n4, and∑
εj=−1 ϕj = 0 for types o1, n1. For the various types, this now
gives :
– when ∈= o1, ξ3 = (xα+
∑2g
j=1 yjθj) ∪ x(c/2)β = x(c/2) · γ.
A. Bauval, D. L. Gonçalves, C. Hayat, and P. Zvengrowski 181
– when ∈= o2,
ξ3 = (xα+
2g∑
j=1
yjθj) ∪ x((c/2)β +
2g∑
j=1
ϕj)
= x((c/2) + 2g +
2g∑
j=1
yj) · γ = x((c/2) +
2g∑
j=1
yj) · γ.
– when ∈= n1, n2, n3, n4,
ξ3 = (xα+
g∑
j=1
yjθj) ∪
(x(c/2) + y)β + x
∑
εj=−1
ϕj
= x
(c/2) + y + #{j | εj = −1}+
∑
εj=−1
yj
· γ
= x
(c/2) +
∑
εj=1
yj + #{j | εj = −1}
· γ.
Case 2. ξ =
∑
yjθj , hence ξ2 =
∑
yjθ
2
j = 0 and ξ3 = 0.
Case 3. Letting zk = φ(sk), recall from the proof of Proposition 3.1 that
zk = 0 for k ≥ d, z0 =
∑
k>0 zk, and ξ =
∑
1≤k≤d−1 zkαk +
∑
yjθj ,
hence
ξ2 =
∑
1≤k≤d−1
zkα
2
k +
∑
yjθ
2
j
=
∑
1≤k≤d−1
zk(
a0
2
β0 +
ak
2
βk) + 0
=
a0
2
(
∑
1≤k≤d−1
zk)β0 +
∑
1≤k≤d−1
zk
ak
2
βk
=
a0
2
z0β0 +
∑
1≤k≤d−1
zk
ak
2
βk
=
∑
0≤k≤d−1
zk
ak
2
βk
182 Borsuk-Ulam Theorem
and
ξ3 = (
∑
1≤k≤d−1
zkαk +
∑
yjθj) ∪
∑
0≤k≤d−1
zk
ak
2
βk
= z0
a0
2
(
∑
1≤k≤d−1
zkαk) ∪ β0 +
∑
1≤k≤d−1
zk
ak
2
· γ
= z0
a0
2
(
∑
1≤k≤d−1
zk)γ +
∑
1≤k≤d−1
zk
ak
2
· γ
= z0
a0
2
γ +
∑
1≤k≤d−1
zk
ak
2
· γ
=
∑
0≤k≤d−1
zk
ak
2
· γ.
Using Proposition 4.2, we conclude:
Theorem 4.3. One has indZ2
(Mφ, τφ) = 3 if and only if
• either N satisfies Case 3 (i.e. d > 0) and
∑
φ(sk)=1 ak is not a
multiple of 4,
• or N satisfies Case 1 (i.e. d = 0 and c is even), and φ(h) = 1, and
the following element of Z2 is nonzero:
– when ∈= o1: c/2
– when ∈= o2, n1: (c/2) +
∑
φ(vj)
– when ∈= n2: (c/2) + g
– when ∈= n3: (c/2) + φ(v1) + g − 1
– when ∈= n4: (c/2) + φ(v1) + φ(v2) + g.
5 Remarks and examples
In this section we give a brief discussion of the class ξ2 and several
examples. The first few examples tend to involve relatively simple Seifert
manifolds for which the full machinery of the previous sections is not
strictly needed. The final two examples are more involved and the full
A. Bauval, D. L. Gonçalves, C. Hayat, and P. Zvengrowski 183
machinery will be necessary. These examples cover three of the six possi-
ble Seifert manifold types, namely ∈= o1, n1, n3, as well as various Euler
numbers e and genus g′.
Proposition 5.1. (a) If indZ2
(M, τ) = 1, then ξ2 = 0.
(b) If indZ2
(M, τ) = 3, then ξ2 6= 0.
Proof. (a) Consider the Bockstein homomorphisms B : H1(N ;Z2) →
H2(N ;Z) and β = Sq1 : H1(N ;Z2) → H2(N ;Z2), and recall that
under the coefficient homomorphism ρ′ : H2(N ;Z) → H2(N ;Z2) one
has ρ′ ◦ B = β. From Theorem 2.1(i) we know indZ2
(M, τ) = 1 if and
only if ξ ∈ Im(ρ). Since Im(ρ) = Ker(B), the condition is equivalent to
B(ξ) = 0. And this implies 0 = Sq1(ξ) = ξ2.
(b) This is immediate from Theorem 2.1(ii).
Based on 5.1 (a), it is interesting to have examples where ξ2 = 0 and
where the indZ2
(M, τ) could equal 1 or equal 2. In fact such examples
are already considered in [7], Section 5, and we will recall them here.
Example 5.2.
(a) Let N = L(4, 1), and ξ ∈ H1(N ;Z2) ≈ Z2 be the generator.
Then indZ2
(M, τ) = 2 and ξ2 = 0.
(b) Let N = S1 × V , V being is any closed surface, and ξ = π∗(u),
where π : N � S1 is the projection and u generates H1(S1;Z2). Then
indZ2
(M, τ) = 1 and ξ2 = 0.
(c) As a special case of (b) let N = S1×RP 2, thenH1(N ;Z2) has the
generator u as in (b), and the additional generator x corresponding to the
(pull-back) of the generator of H1(RP 2;Z2). Now, in addition to ξ = u as
in (b), we have two further possible choices ξ = v or ξ = u+ v. For each
of these latter two choices we have indZ2
(M, τ) = 2 since ξ2 = v2 6= 0
and ξ3 = 0.
Of course, the conclusions in Example 5.2 as well as the following
Example 5.3 also follow easily from our main theorems. As an illustration,
in 5.2(a) we have L(4, 1) = {4; (o1, 0)} (cf. [15] 5.4(i)). Here a0 = 1, b0 =
4, whence d = 0, c = 4, and this implies we are in Case 1. By Theorem
4.1 the only non-zero element in H1(N ;Z2) is α, hence ξ = α. Again by
Theorem 4.1 we have α2 = (c/2)β = 0. Now applying Theorem 3.5 (first
case) and Theorem 4.3 (second case), we obtain that indZ2
(M, τ) = 2.
184 Borsuk-Ulam Theorem
We also remark that in 5.2(b) and 5.2(c) one has e = 0, and the type is
o1 if V is orientable, n1 if V is non-orientable.
Our next example illustrates to some extent the delicacy of the
Borsuk-Ulam situation. The example shows that one can have two dou-
ble covers of a Seifert manifold N by the same Seifert manifold M but
with different Z2-indices for (M, τ). Indeed the example already arises
at the level of surface topology.
Example 5.3.
Let N = RP 2#RP 2#RP 2, one has π1(N) =
〈v1, v2, v3|v21v22v23〉, H1(N ;Z2)
≈ Z3
2 with generators θ1, θ2, θ3 and H2(N ;Z2) ≈ Z2 with generator
β. Furthermore θ2i = β whereas θiθj = 0, i 6= j (cf. [8] Section 3.2).
The characteristic class ξ1 = θ2 + θ3 corresponds to the homomorphism
φ1 : π1(N) � Z2 given by φ1(v1) = 0, φ1(v2) = φ1(v3) = 1. Similarly
the characteristic class ξ2 = θ1 corresponds to φ2 : π1(N) � Z2 with
φ2(v1) = 1, φ2(v2) = φ2(v3) = 0. Using Proposition 4.2 of [7] we
obtain at once that indZ2 is 1 for ξ1 and 2 for ξ2 (this corresponds to
ξ21 = 0, ξ22 6= 0). The surface M that is the double cover of N must
have Euler characteristic χ(M) = 2χ(N) = −2. Since it is not hard to
see that in both cases M is non-orientable, it follows that in both cases
M = RP 2#RP 2#RP 2#RP 2.
By simply taking the product of M and N with S1, we obtain similar
examples with Seifert manifolds (where we take φi(h) = 0). Indeed,
writing N1 = N × S1, we have that N1 has ∈= n1, g′ = 3, and no
exceptional fibres whence d = c = 0. From 3.5, final case, we see that
φ1(v1)+φ1(v2)+φ1(v3) = 0 implies the Z2-index for φ1 equals 1, whereas
φ2(v1) + φ2(v2) + φ2(v3) = 1 implies the Z2 index for φ2 is 2 or 3. Since
ζ3 = 0 for any ζ ∈ H1(N1;Z2), the Z2-index of φ2 must be 2.
It should be noted, as was already done in Seifert’s original paper
[19], that the same 3-manifold (even S3) can often be fibred in different
ways, i.e. the Seifert “invariants" are not always true invariants in the
sense that they may not be unique. However the cohomology ring with
any coefficients, and fundamental group, are of course true invariants,
and the determination of the Z2-index is based upon these. We conclude
with a couple of deeper examples for which the techniques of Sections 3
and 4 must be utilized to answer the Borsuk-Ulam question.
Example 5.4.
A. Bauval, D. L. Gonçalves, C. Hayat, and P. Zvengrowski 185
Let N be the Seifert manifold given by the following Seifert invariants:
N = {0, (n3, 2); (9, 4), (5, 2), (7, 2)}.
Then, a presentation of π1(N) is:
π1(N) =
〈s1, s2, s3
v1, v2
h
∣∣∣∣∣∣
[sk, h] (k = 1, 2, 3)
[v1, h], v2hv
−1
2 h
s91h
4, s52h
2, s73h
2, s1s2s3v
2
1v
2
2
〉
.
Note that d = 0 (since 9, 5, 7 are odd) and c is even (since 4, 2, 2
are even), hence we are in Case 1 of Notation 2.2. The following table
shows the values of all possible non-zero φ’s on the generators of π1(N),
as well as the corresponding cohomology class ξ ∈ H1(N ;Z2) under the
isomorphism (1). Also recall that here, by Theorem 4.1 (or Proposition
3.1), H1(N ;Z2) has generators α, θ1, θ2 with α = ĥ+4ŝ1 +2ŝ2 +2ŝ3 = ĥ,
and finally that ∈= n3 implies all φ(sj) = 0. The final column in the
table gives the Z2-index, in each case, of (Mi, τi) := (Mξi , τξi). The
proofs for the data in the table are given in Proposition 5.5 below.
φi s1 s2 s3 h v1 v2 ξi indZ2(Mi, τi)
φ1 0 0 0 1 0 0 α 3
φ2 0 0 0 1 1 0 α+ θ1 2
φ3 0 0 0 1 0 1 α+ θ2 3
φ4 0 0 0 1 1 1 α+ θ1 + θ2 2
φ5 0 0 0 0 1 0 θ1 2
φ6 0 0 0 0 0 1 θ2 2
φ7 0 0 0 0 1 1 θ1 + θ2 1
Proposition 5.5. • For ξ = ξ7 one has indZ2
(Mi, τi) = 1.
• For ξ = ξ2, ξ4, ξ5, ξ6 one has indZ2(Mi, τi) = 2.
• For ξ = ξ1, ξ3 one has indZ2(Mi, τi) = 3.
Proof. By Theorem 3.5, indZ2(Mi, τi) = 1 if and only if φ(h) = φ(v1 +
v2) = 0, i.e. φ = φ7. Moreover, N is in Case 1, ∈= n3, c is a multiple
of 4 and g = 2 hence, by Theorem 4.3, indZ2
(Mi, τi) = 3 if and only if
φ(h) = φ(v1) + 1 = 1, i.e. φ = φ1, φ3.
186 Borsuk-Ulam Theorem
Our concluding example has (in contrast to the previous examples)
non-zero Euler number, arbitrary genus g ≥ 0, and a relatively large
number (seven) of singular fibres.
Example 5.6.
Let Ng, g ≥ 0, be the Seifert manifold given by the Seifert invariants
{−2; (o1, g); (16, 5), (16, 1), (16, 1), (16, 1), (2, 1), (3, 2), (3, 1)}.
With the conventions given in Notation 2.2, a presentation of π1(N) is
(note that according to these conventions the singular fibres are reordered
so that s0 corresponds to (16, 5), s1 to (16, 1) . . ., s6 to (3, 1), and s7 to
(1, e) = (1,−2) :
π1(N) =
〈s0, s1, s2, s3, s4, s5, s6, s7
v1, v2, . . . , v2g−1, v2g
h
∣∣∣∣∣∣
[sk, h] and sakk h
bk , 0 ≤ k ≤ 7
[vj , h], 1 ≤ j ≤ 2g
s0 · · · s7[v1, v2] · · · [v2g−1, v2g]
〉
.
One easily checks that here a = 48, c = 0, d = 5, J = 4, whence
SN = {0, 1, 2, 3} and we are in Case 3 of Notation 2.2. As usual, φ denotes
any surjective homomorphism φ : π1(Ng) � Z2 and τ the corresponding
involution of the double cover M arising from φ. It is also readily seen
that φ(h) = φ(s5) = φ(s6) = φ(s7) and φ(s0) + φ(s1) + φ(s2) + φ(s3) +
φ(s4) = 0 are necessary conditions for φ to be a homomorphism.
Proposition 5.7. • indZ2
(M, τ) = 1 iff either Sφ = ∅ (in which
case φ(sk) = 0, 0 ≤ k ≤ 7, g ≥ 1, and φ(vj) = 1 for at least one
j), or Sφ = SN (in which case φ(s0) = φ(s1) = φ(s2) = φ(s3) = 1).
• indZ2
(M, τ) = 3 iff φ(s4) = 1 (whence also φ(s0)+φ(s1)+φ(s2)+
φ(s3) = 1).
• In all remaining cases indZ2
(M, τ) = 2.
Proof. By Theorem 3.5 we have indZ2(M, τ) > 1 if and only if d > 0 and
Sφ 6= ∅, SN . Since here d = 5, the negation of the previous sentence gives
the first statement of the proposition.
By Theorem 4.3 we have indZ2
(M, τ) = 3 if and only if d > 0 (which
is the case) and
∑
{ak : k ∈ Sφ} is not divisible by 4. We have already
observed that Sφ ⊆ {0, 1, 2, 3, 4} and furthermore a0 = a1 = a2 = a3 =
16, hence
∑
{ak : k ∈ Sφ} ≡ 2 ·φ(s4) (mod 4), and this gives the second
statement of the proposition. The third and final statement follows by
default.
A. Bauval, D. L. Gonçalves, C. Hayat, and P. Zvengrowski 187
References
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homology ring of a class of Seifert manifolds, Topology and its
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188 Borsuk-Ulam Theorem
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A. Bauval, D. L. Gonçalves, C. Hayat, and P. Zvengrowski 189
Institut de Mathématiques de Toulouse
Equipe Emile Picard, UMR 5580
Université Toulouse III
118 Route de Narbonne, 31400 Toulouse - France
e-mail: bauval@math.univ-toulouse.fr
Departamento de Matemática - IME-USP
Caixa Postal 66281 - Ag. Cidade de São Paulo
CEP: 05314-970 - São Paulo - SP - Brasil
e-mail: dlgoncal@ime.usp.br
Institut de Mathématiques de Toulouse
Equipe Emile Picard, UMR 5580
Université Toulouse III
118 Route de Narbonne, 31400 Toulouse - France
e-mail: hayat@math.univ-toulouse.fr
Department of Mathematics and Statistics
University of Calgary
Calgary, Alberta, Canada T2N 1N4
e-mail: zvengrow@ucalgary.ca
|
| id | oai:trim.imath.kiev.ua:article-299 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:05:12Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/83/1b1a30cb558222a75cb130e1f7e3a483.pdf |
| spelling | oai:trim.imath.kiev.ua:article-2992018-02-10T20:56:26Z The Borsuk-Ulam Theorem for Double Coverings of Seifert Manifolds Теорема Борсука-Улама для дволистих накриттів многовидів Зейферта Bauval, A. Gonçalves, D. L. Hayat, C. Zvengrowski, P. Bauval, A. Gonçalves, D. L. Hayat, C. Zvengrowski, P. We study a Borsuk-Ulam type theorem for pairs $(M,\tau)$ with $\tau$ a fixed point free involution of $M$, and such that both $M$ and $N\colon = M/\tau$ are Seifert&nbsp; manifolds. In this note our point of view will be to start with a Seifert manifold $N$. Any non-trivial element $\xi\in H^1(N;Z_2)$ then gives rise to a pair $(M_\xi, \tau_\xi) = (M , \tau) $ with $M$ (necessarily) also a Seifert&nbsp; manifold, and a double covering&nbsp; $p \colon M \twoheadrightarrow N$,&nbsp; with $\tau$ being the fixed point free involution on $M$ associated to this double covering as the non-trivial deck transformation. We then seek the largest value of $n$, called the $\mathbb{Z}_2$-index of $(M,\tau)$, such that the Borsuk-Ulam property holds for maps into $\mathbb{R}^n$, i.e. such that for every continuous map $f\colon M\to \mathbb{R}^n$, there is an $x\in M$ such that $f(x)=f(\tau(x))$. In case $M$ is a $3$-manifold (such as a Seifert manifold), the$\mathbb{Z}_2$-index can take only the values $1,2,3$. Ми вивчаємо теорему типу Борсука-Улама для пар $(M,\tau)$, де $\tau$ - інволюція без нерухомих точок на $M$,&nbsp; та&nbsp; $M$ і $N\colon = M/\tau$ є многовидами Зейферта. Відповідно до цього ми розпочинаємо з многовиду Зейферта $N$. Будь-який нетривіальний елемент&nbsp; $\xi\in H^1(N;Z_2)$ дає пару t\ $(M_\xi, \tau_\xi) = (M , \tau) $, де $M$ (обов'язково) також многовид Зейферта, і дволисте накриття $p \colon M \twoheadrightarrow N$, де $\tau$ буде інволюцією без нерухомих точок на $M$ асоційована з цим дволистим накриттям як нетривіальним накриваючим перетворенням. Тоді ми шукаємо найбільше значення $n$, що називається $\mathbb{Z}_2$-індексом для $(M,\tau)$, таким, що має місце властивість Борсука-Улама для відображень на $\mathbb{R}^n$,&nbsp; тобто для довільного неперервного відображення $f\colon M\to \mathbb{R}^n$ існує $x\in M$ таке, що $f(x)=f(\tau(x))$. У випадку, коли $M$ є $3$-многовидом (наприклад многовидом Зейферта), $\mathbb{Z}_2$-індекс може досягати лише значень&nbsp; $1,2,3$. Інститут математики НАН України 2013-06-26 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/299 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 10 No. 6 (2013): Brasilian-Polish Topology Workshop; 165-189 Сборник Трудов Института математики НАН Украины; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 165-189 Збірник Праць Інституту математики НАН України; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 165-189 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/299/285 Авторське право (c) 2013 Праці Інституту математики НАН України |
| spellingShingle | Bauval, A. Gonçalves, D. L. Hayat, C. Zvengrowski, P. Bauval, A. Gonçalves, D. L. Hayat, C. Zvengrowski, P. The Borsuk-Ulam Theorem for Double Coverings of Seifert Manifolds |
| title | The Borsuk-Ulam Theorem for Double Coverings of Seifert Manifolds |
| title_alt | Теорема Борсука-Улама для дволистих накриттів многовидів Зейферта |
| title_full | The Borsuk-Ulam Theorem for Double Coverings of Seifert Manifolds |
| title_fullStr | The Borsuk-Ulam Theorem for Double Coverings of Seifert Manifolds |
| title_full_unstemmed | The Borsuk-Ulam Theorem for Double Coverings of Seifert Manifolds |
| title_short | The Borsuk-Ulam Theorem for Double Coverings of Seifert Manifolds |
| title_sort | borsuk-ulam theorem for double coverings of seifert manifolds |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/299 |
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