The Lefschetz Theorem for multivalued maps
A subcollection of multivalued maps called s-maps is introduced. Then to a self s-map $f$ of a finite connected CW-complex an integer $\mathcal{L}_{s}(f)$ is associated and an analog of the Lefschetz Fixed Point Theorem is proved.
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| author | Kiszkiel, J. M. Kiszkiel, J. M. |
| author_facet | Kiszkiel, J. M. Kiszkiel, J. M. |
| author_institution_txt_mv | [
{
"author": "J. M. Kiszkiel",
"institution": "Faculty of Mathematics and Computer Science, Nicolaus Copernicus University"
}
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| author_sort | Kiszkiel, J. M. |
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| description | A subcollection of multivalued maps called s-maps is introduced. Then to a self s-map $f$ of a finite connected CW-complex an integer $\mathcal{L}_{s}(f)$ is associated and an analog of the Lefschetz Fixed Point Theorem is proved. |
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Збiрник праць Iн-ту математики НАН України 2013, том 6, N 6, 118–129
J.M. Kiszkiel
(Faculty of Mathematics and Computer Science, Nicolaus Copernicus
University, Chopina 12/18, 87-100 Toruń, Poland)
The Lefschetz Theorem for
multivalued maps
A subcollection of multivalued maps called s-maps is introduced. Then
to a self s-map f of a finite connected CW-complex an integer Ls(f) is
associated and an analog of the Lefschetz Fixed Point Theorem is proved.
1 Introduction
The Lefschetz Fixed Point Theorem states that if X is a sufficiently
nice space and f : X → X is a singlevalued continuous map, then it is
possible to associate to f an integer L(f) such that if L(f) 6= 0, then f
has a fixed point. The number L(f) is called the Lefschetz number of f
and is a well known and very useful homotopical invariant.
In the literature one can find many tries of generalizations of the
Lefschetz number to the case of multivalued maps. The paper [2] presents
the Lefschetz number defined for maps for which the image of any two
different points has the same finite number of elements. More general
approches regarding to acyclic and admissible maps are presented in [3].
Authors of [6] consider multivalued maps with an additional algebraic
structure called the weight of a map. They construct for such maps the
Lefschetz number using the Darbo homology functor, but that number
essentially depends on the weight as well.
The main goal of this paper is to introduce a subcollection of multival-
ued maps, called s-maps for which it is possibile to define the Lefschetz
number in a new way. In Section 2 we recall some basic information about
c© J.M. Kiszkiel, 2013
J.M. Kiszkiel 119
the ordinary Lefschetz number and the Lefschetz set of admissible maps
presented in [3]. Then in Section 3 we show a categorical construction
which leads to an extension of some definitions to larger categories. We
apply such a construction to define a subcollection of multivalued maps
called s-maps and the Lefschetz number of them as well. Moreover, we
prove an analog of the Lefschetz Fixed Point Theorem for s-maps (Theo-
rem 3.9). Next in Section 4 we compare our approach with the Lefschetz
set of admissible maps. We present some examples of s-maps which are
not admissible. At the end of the section we use the categorical construc-
tion again to define s-admissible maps which generalize both admissible
and s-maps. Next we formulate the Lefschetz Fixed Point Theorem for
such maps (Theorem 4.9).
ACKNOWLEDGMENTS. I would like to thank Marek Golasiński for
many halpful conversations and useful suggestions.
2 Preliminaries
First we recall some basic information about multivalued maps. More
details one can find in [3].
Let X, Y be two topological spaces and assume that for every point
x ∈ X a nonempty compact subset ϕ(x) ⊆ Y is given. In this case, we
say that ϕ is a multivalued map from X to Y and we write ϕ : X ( Y .
Let ϕ : X ( Y be a multivalued map and A ⊆ X, then the image of
A under ϕ is the set
ϕ(A) =
⋃
x∈A
ϕ(x).
Let ϕ : X ( Y be a multivalued map and B ⊆ Y , then the large
preimage of B under ϕ is the set
ϕ−1(B) = {x ∈ X | ϕ(x) ∩B 6= ∅}.
If ϕ : X ( Y and ψ : Y ( Z are two multivalued maps, then for any
C ⊆ Z we have
(ψ ◦ ϕ)−1(C) = ϕ−1(ψ−1(C)).
A multivalued map ϕ : X ( Y is called upper semicontinuous (u.s.c.),
provided for every closed B ⊆ Y the set ϕ−1(B) is closed inX. If ϕ : X (
120 The Lefschetz Theorem for multivalued maps
Y and ψ : Y ( Z are u.s.c. maps, then the composition ψ ◦ ϕ : X ( Z
is also u.s.c..
Remark 2.1. Let f : X → Y be a singlevalued continuous map onto Y .
Then its inverse can be considered as a multivalued map f inv : Y ( X
defined by f inv(y) = f−1(y) for y ∈ Y . If f is closed, then f inv is u.s.c..
Later we write f−1 instead of f inv.
Now we recall the Lefschetz Fixed Point Theorem for singlevalued
maps. For details check [1], [4] and [5]. Denote by C the collection
of all finite connected CW-complexes. Let X ∈ C and f : X → X be
a singlevalued continuous map. Recall that we have a well defined in-
teger L(f) =
∑
k∈Z
(−1)ktr(fk), called the Lefschetz number of f , where
fk : Hk(X,Q) → Hk(X,Q) are maps induced by f on the rational ho-
mology groups and tr(fk) denotes the trace of the homomorphism fk.
Remark 2.2. When we work in C there is no difference which homol-
ogy functor we use, but if X is an arbitrary topological space, then by
Hk(X,Q) we mean the k-th Čech rational homology group of X.
Notice that the Lefschetz number has the following very useful prop-
erties, which are a simple consequences of analogous properties of the
trace:
Proposition 2.3. Let X, Y ∈ C, f : X → Y and g : Y → X be two
singlevalued continuous maps, then L(fg) = L(gf).
Corollary 2.4. Let X, Y ∈ C, g : X → Y be a homeomorphism and
f : X → X be a singlevalued continuous map, then L(f) = L(gfg−1).
There is also a property which connects the Lefschetz number of a
map with the Lefschetz number of its prime iteration:
Theorem 2.5 (The mod p Theorem [4]). Let X ∈ C, f : X → X be a
contiuous map and p be a prime. Then L(fp) ≡ L(f) mod p.
J.M. Kiszkiel 121
The most famouse and important application of the Lefschetz number
is:
Theorem 2.6 (Lefschetz Fixed Point Theorem [5]). Let X ∈ C and
f : X → X be a singlevalued continuous map. If L(f) 6= 0, then f has a
fixed point.
To recall the construction of the Lefschetz number of multivalued
admissible maps, we need some definitions. All results presented below
are stated in [3].
A space X is called acyclic, when:
(i) Hk(X,Q) = 0 for all k ≥ 1;
(ii) H0(X,Q) = Q.
A singlevalued continuous map f : X → Y is called proper, provided
for every compact K ⊆ Y the set f−1(K) is compact.
A singlevalued continuous map f : X → Y is called a Vietoris map,
provided the following conditions hold:
(i) f : X → Y is proper;
(ii) the set f−1(y) is acyclic for all y ∈ f(X).
Vietoris maps have the following important property:
Theorem 2.7 (Vietoris [3]). If X, Y ∈ C and f : X → Y is a Vietoris
map, then the induced homomorphism fk : Hk(X,Q) → Hk(Y,Q) is an
isomorphism for all k ≥ 0.
A multivalued map ϕ : X ( Y is called admissible, provided there
exist a space Z and two continuous maps p : Z → X and q : Z → Y such
that:
(i) p is a Vietoris map;
(ii) q(p−1(x)) ⊆ ϕ(x) for all x ∈ X.
We write (p, q) ⊆ ϕ when maps p and q are as above.
The Lefschetz set of an admissible map ϕ : X ( X is defined by:
La(ϕ) =
{∑
k∈Z
(−1)ktr(qkp−1k ) | (p, q) ⊆ ϕ
}
.
122 The Lefschetz Theorem for multivalued maps
Remark 2.8. Let X,Y ∈ C and ϕ : X ( Y be an admissible map. It is
easy to see, that if (p, q) ⊆ ϕ, then qp−1 : X ( Y is u.s.c. and the set
qp−1(x) is connected for all x ∈ X.
The most important application of the Lefschetz set is:
Theorem 2.9 (Lefschetz Fixed Point Theorem for admissible maps [3]).
Let X ∈ C and ϕ : X ( X be an admissible map. If La(ϕ) 6= {0}, then
ϕ has a fixed point.
Remark 2.10. In [3] a collection of spaces for which it is possible to
consider admissible maps is larger than C. Moreover, in [7] there is con-
sidered a broader class of maps for which it is possible to define the
Lefschetz set.
3 The Lefschetz number of s-maps
In this section we introduce a subcollection of multivalued maps called
s-maps and investigate their properties. First, we show a very usefull
categorical construction which helps us in defining s-maps.
Definition 3.1. Let D be a category and C its subcategory, not necessary
full. Define a category (D, C) as follows:
(i) object of (D, C) are quadruples (X,A, r, s), where X ∈ D, A ∈ C,
r : X → A and s : A→ X are morphisms in D such that rs = idA;
(ii) Mor(D,C)((X,A, r, s), (Y,B, t, q)) = {(ϕ, f) ∈ D × C | ϕ = qfr, f ∈
MorC(A,B)};
(iii) a composition law in (D, C) is induced from the composition laws in
D and C;
(iv) id(X,A,r,s) = (sr, idA).
Observe that the composition of morphisms in the category (D, C)
is well defined because if (ϕ, f) ∈ Mor(D,C)((X,A, r, s), (Y,B, t, q)) and
(ψ, g) ∈ Mor(D,C)((Y,B, t, q), (Z,C, l,m)), then ψϕ = lgtqfr = lgfr,
because tq = idY , so (ψϕ, gf) ∈ Mor(D,C)((X,A, r, s), (Z,C, l,m)).
Notice that if ϕ = qfr, then f = tϕs, so f is uniquely determinded
by ϕ (when suitable r, s, t and q are choosen).
J.M. Kiszkiel 123
Now let D be a category of topological spaces and multivalued u.s.c.
maps and C its subcategory of finite connected CW-complexes and sin-
glevalued continuous maps. Consider the category (D, C) for such D and
C.
Definition 3.2. Let X ∈ D and ϕ : X ( X be a multivalued u.s.c. map.
The map ϕ is called an s-map if there exist:
(i) A ∈ C;
(ii) a singlevalued continuous map fϕ : A→ A;
(iii) a singlevalued continuous surjection r : X → A;
(iv) a multivalued u.s.c. map s : A( X;
such that:
(a) (X,A, r, s) ∈ (D, C);
(b) (ϕ, fϕ) ∈ Mor(D,C)((X,A, r, s), (X,A, r, s)).
If ϕ : X ( X is an s-map and fϕ is a map like in the above definition,
then we say that the morphism (ϕ, fϕ) represents ϕ in (D, C).
If X ∈ C and ϕ : X → X is a singlevalued continuous map, then
clearly ϕ is an s-map, because it is enough to take A = X, fϕ = ϕ
and r = s = idX . This factorization is called standart. Of course for a
singlevalued continuous map there can exsist factorizations different to
the standart one.
Example 3.3. A map ϕ : [0, 2]→ [0, 2] given by ϕ(x) = 0 for all x ∈ [0, 2]
is singlevalued, so we have the standard factorization. On the other
hand, we can choose a different morphism in (D, C) which represet ϕ, for
example r : [0, 2]→ [0, 1] is given by
r(x) =
{
x for x ∈ [0, 1];
1 for x ∈ (1, 2];
s : [0, 1] ( [0, 2] is the inverse of r, so s(x) = r−1(x) for all x ∈ [0, 1] and
fϕ : [0, 1]→ [0, 1] is defined by fϕ(x) = 0 for all x ∈ [0, 1].
For selfmorphisms in category C we have a well defined Lefschetz
number. We can extend this definition to the category (D, C) by taking
L(ϕ, fϕ) = L(fϕ). Our goal is to show that if an s-map ϕ : X ( X is
represented by two different morphisms (ϕ, fϕ) and (ϕ, gϕ), then L(fϕ) =
L(gϕ). To show that we need first to prove some lemmas:
124 The Lefschetz Theorem for multivalued maps
Lemma 3.4. If an s-map ϕ : X → X is represented by pairs (ϕ, fϕ) and
(ϕ, gϕ) are such that ϕ = sfϕr and ϕ = tgϕq for suitable s, r and q, t,
then:
(i) fϕr = rtgϕq;
(ii) qsfϕ = gϕqs;
(iii) gϕqsrt = gϕ;
(iv) rtgϕ is a singlevalued continuous map;
(v) qsfϕ is a singlevalued continuous map;
(vi) gϕqs is a singlevalued continuous map.
Proof. (i), (ii) and (iii) are easy consequences of equalities sfϕr = tgϕq,
qt = id and rs = id. Let now prove (iv). Using (i) we have that rtgϕq is
a singlevalued continuous map. Moreover, q is a singlevalued continuous
surjection, so (iv) follows. Property (v) is analogous to (iv) and (vi) is a
consequence of (ii) and (v).
Lemma 3.5. If an s-map ϕ : X → X is represented in (D, C) by pairs
(ϕ, fϕ) and (ϕ, gϕ), then L(fnϕ ) = L(gnϕ) for n ≥ 2.
Proof. Let ϕ = sfϕr and ϕ = tgϕq, then using Proposition 2.3 and
Lemma 3.4 we obtain:
L(fnϕ ) = L((rtgϕqs)n) = L(rtgnϕqs) = L(rtgn−1ϕ gϕqs) =
L(gϕqsrtgn−1ϕ ) = L(gnϕ).
As an easy consequence of Theorem 2.5 and Lemma 3.5 we get:
Proposition 3.6. If an s-map ϕ : X → X is represented in (D, C) by
pairs (ϕ, fϕ) and (ϕ, gϕ), then L(fϕ) = L(gϕ).
Remark 3.7. Example 3.3 shows that we cannot prove the above propo-
sition directly as Lemma 3.5 because the composition sr : X ( X does
not have to be singlevalued.
Now we are in a position to define the Lefschetz number of s-maps:
J.M. Kiszkiel 125
Definition 3.8. Let ϕ : X ( X be an s-map. The Lefschetz number
of ϕ is a number Ls(ϕ) = L(fϕ), where (ϕ, fϕ) represents ϕ in (D, C).
According to Proposition 3.6 this number is well defined.
If f : X → X is a singlevalued continuous map then Ls(f) = L(f).
To show this it is enough to take the standart factorization.
Now we prove an analog of the Lefschetz Fixed Point Theorem for
s-maps:
Theorem 3.9 (Lefschetz Fixed Point Theorem). Let ϕ : X ( X be an
s-map and Ls(ϕ) 6= 0, then ϕ has a fixed point.
Proof. The map ϕ is an s-map. Let suitable A, fϕ, r and s be choosen.
Acoording to the definition Ls(ϕ) = L(fϕ), where ϕ = sfϕr. The map
fϕ : A → A is singlevalued continuous and A ∈ C, so the ordinary Lef-
schetz Fixed Point Theorem implies that fϕ has a fixed point a = r(x)
for some x ∈ X. Let z ∈ s(r(x)), then ϕ(z) = ϕ(x), because r(z) = r(x).
Therefore, we have z ∈ sr(x) = sfϕr(x) = ϕ(x) = ϕ(z), so z is a fixed
point of ϕ.
The easiest way to show that ϕ : X ( X is an s-map, it is to find an
equivalence relation R on X such that A = X/R and r : X → A is the
canonical projection. If ϕ : X ( X, then
R = {(x, y) ∈ X ×X | ϕ(x) = ϕ(y)
and
ϕ−1(x) = ϕ−1(y) 6= ∅} ∪ {(x, x) | x ∈ X}
is called a canonical relation for ϕ. If we use the canonical relation, then
we write XR istead of A.
Now we present some examples of s-maps and find their Lefschetz
numbers:
Example 3.10. Let Sn be the n-sphere and ϕ : Sn → Sn be such that
ϕ(x) = Sn for all x ∈ Sn. Let R be the canonical relation for ϕ. Then
XR = {∗}, where {∗} denotes the one point space. We have ϕ = sfϕr,
where r : Sn → {∗} is given by r(x) = {∗} for every x ∈ Sn, s : {∗}( Sn
is defined by s(∗) = Sn and fϕ = id{∗}. Therefore ϕ is an s-map and
Ls(ϕ) = L(fϕ) = L(id{∗}) = 1.
126 The Lefschetz Theorem for multivalued maps
Example 3.11. Let X ∈ C and f : S1 → S1 be a continuous singlevalued
map. Define ϕ : S1 ×X ( S1 ×X by ϕ(b, x) = {f(b)} ×X. Let R be
the canonical relation for ϕ. Then XR = S1, r : S1 × X → S1 is given
by r(b, x) = b for all (b, x) ∈ S1 × X, s : S1 ( S1 × X is given by
s(b) = {b} ×X and fϕ = f . We have Ls(ϕ) = L(f).
4 Comparision of the Lefschetz numbers
In this section we compare the Lefschetz number of s-maps with the
Lefschetz set of admissible maps. One may expect that there is some co-
incidence between those two conceptions, but as we see in some examples
they give different results. This leads to a definition of s-admissible maps
which generalizes both admissible and s-maps. We start this section with
analysing some examples.
Our first example shows a situation when La(ϕ) 6= {Ls(ϕ)}:
Example 4.1. Let ϕ : Sn ( Sn be such that ϕ(x) = Sn for all x ∈ Sn.
We have shown in Example 3.10, that this is an s-map and Ls(ϕ) = 1.
On the other hand, the map ϕ is admissible and La(ϕ) = Z, because
(idSn , f) ⊂ ϕ for all singlevalued continuous maps f : S1 → S1.
In the previous example we have {Ls(ϕ)} ⊆ La(ϕ), but that is not
true in general.
Example 4.2. Let ϕ : [0, 2] ( [0, 2] be given by:
ϕ(x) =
{
x for x ∈ (0, 2);
{0, 2} for x ∈ {0, 2}.
Then ϕ is both admissible and an s-map. We have
La(ϕ) =
{∑
k∈Z
(−1)ktr(qkp−1k ) | (p, q) ⊆ ϕ
}
= {L(id[0,2])} = {1},
because we cannot choose p and q such that qp−1 6= id[0,2] (see Remark
2.8). On the other hand, we have Ls(ϕ) = L(fϕ) = L(idS1) = 0, because
XR is homeomorphic to S1 and using Proposition 2.4 we can replace in
our calculations the map fϕ by idS1 .
J.M. Kiszkiel 127
In next two examples we show s-maps which are not admissible. As
a consequence for those maps only the Lefschetz number of s-maps is
possible to define.
Example 4.3. Let ϕ : [0, 2] ( [0, 2] be given by:
ϕ(x) =
x+ 1 for x ∈ [0, 1);
{0, 2} for x = 1;
x− 1 for x ∈ (1, 2].
This map is not admissible (see Remark 2.8). On the other hand, ϕ is
an s-map and we have Ls(ϕ) = L(fϕ) = L(idS1) = 0, because XR is
homeomorphic to S1 and using Proposition 2.4 we can think that fϕ is
a rotation by an angle π which is homotopic to the identity map on S1.
Example 4.4. Let ϕ : [0, 2]→ [0, 2] be given by:
ϕ(x) =
−x+ 1 for x ∈ [0, 1);
{0, 2} for x = 1;
−x+ 3 for x ∈ (1, 2].
Then ϕ is not admissible, but ϕ is an s-map and we have Ls(ϕ) = L(fϕ) =
2, because we can think that fϕ : S1 → S1 and has a degree equal −1.
Remark 4.5. After studing two previous examples one can easy see that
for any integer n it is possible to find a multivalued map ϕ which is not
admissible, but is an s-map and Ls(ϕ) = n. Namely, it is enough to
take a multivalued map ϕ : [0, 2] ( [0, 2] which is not admissible and a
suitable map fϕ : S1 → S1 has a degree 1− n.
Remark 4.6. The maps from the last two examples can be considered
as the multivalued weighted maps (check [6] for a definition), but only
trivial weight is possible for those maps.
Now we formulate a definition which generalizes both admissible and
s-maps. Let D be a category of spaces and multivalued maps and C be
its subcategory of finite connected CW-complexes and admissible maps.
128 The Lefschetz Theorem for multivalued maps
Definition 4.7. Let X ∈ D and ϕ : X ( X be a multivalued map. The
map ϕ is called an s-admissible map if there exist:
(i) A ∈ C;
(ii) an admissible map βϕ : A( A;
(iii) a singlevalued continuous surjection r : X → A;
(iv) a multivalued u.s.c. map s : A( X;
such that:
(a) sβϕr(x) ⊆ ϕ(x) for all x ∈ X;
(b) (X,A, r, s) ∈ (D, C).
Let ϕ : X ( X be s-admissible. Denote by (D, C)ϕ the set of all maps
βϕ which are like in the above definition.
Definition 4.8. The Lefschetz set of s-admissible map is a set:
Ls(ϕ) =
⋃
βϕ∈(D,C)ϕ
La(βϕ).
Theorem 4.9 (Lefschetz Fixed Point Theorem). Let X ∈ C and ϕ : X (
X be an s-admissible map. If Ls(ϕ) 6= {0}, then ϕ has a fixed point.
Proof. We choose suitable s, r and βϕ such that La(βϕ) 6= {0}. Than we
use Theorem 2.9 and following the proof of Theorem 3.9 we obtain that
sβϕr has a fixed point, which is also a fixed point of ϕ.
Remark 4.10. The easiest way to show that ϕ : X ( X is s-admissible
is to find an admissible map ψ : X ( X such that ψ(x) ⊆ ϕ(x) for all
x ∈ X or an s-map η : X ( X such that η(x) ⊆ ϕ(x) for all x ∈ X.
Now we present an example of an s-admissible map which is neither
admissible nor an s-map:
Example 4.11. Let ϕ : [0, 3] ( [0, 3] be given by:
ϕ(x) =
[−x+ 1,−x+ 2] for x ∈ [0, 1);
[0,−x+ 2] ∪ [−x+ 4, 3] for x ∈ [1, 2];
[−x+ 4,−x+ 5] for x ∈ (2, 3].
The map ϕ is not an s-map. Moreover, ϕ is not admissible, because the
graph of ϕ has two connected components and neither of them is a graph
J.M. Kiszkiel 129
of a multivalued map from [0, 3] to [0, 3]. On the other hand we have an
s-map η : [0, 3] ( [0, 3] given by:
η(x) =
−x+ 1 for x ∈ [0, 1);
{0, 3} for x = 1;
−x+ 4 for x ∈ (1, 3]
such that η(x) ⊆ ϕ(x) for all x ∈ X. Consequently ϕ is s-admissible.
We have Ls(η) = 2, so 2 ∈ Ls(ϕ). Moreover, it can be shown that
Ls(ϕ) = {2}.
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Fixed Point Theory Appl. 2 (2007), no. 1, 53-60.
[3] L.Górniewicz, Topological fixed point theory of multivalued mappings.
Second edition. Topological Fixed Point Theory and Its Applica-
tions, 4. Springer, Dordrecht, 2006.
[4] A.Granas, J.Dugundji, Fixed point theory. Springer Monographs in
Mathematics. Springer-Verlag, New York, 2003.
[5] J. Jezierski, W. Marzantowicz, Homotopy methods in topological
fixed and periodic points theory. Topological Fixed Point Theory and
Its Applications, 3. Springer, Dordrecht, 2006.
[6] J.Pejsachowicz, R.Skiba, Fixed point theory of multivalued weighted
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2011, Art. ID 216146, 11 p.
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| resource_txt_mv | trimimathkievua/f1/52acb3ae5882713e2a6fc4961893d8f1.pdf |
| spelling | oai:trim.imath.kiev.ua:article-2812018-02-10T20:56:26Z The Lefschetz Theorem for multivalued maps Теорема Лефшеца для багатозначних відображень Kiszkiel, J. M. Kiszkiel, J. M. A subcollection of multivalued maps called s-maps is introduced. Then to a self s-map $f$ of a finite connected CW-complex an integer $\mathcal{L}_{s}(f)$ is associated and an analog of the Lefschetz Fixed Point Theorem is proved. Введено сукупність багатозначних відображень, що називаються s-відображеннями. Тоді s-відображенню $ f $ скінченного звязного CW-комплексу ставиться у відповідність ціле число $ \mathcal{L}_ {s} (f) $ і доведено аналог теореми Лефшеца про нерухомі точки. Інститут математики НАН України 2013-06-26 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/281 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 10 No. 6 (2013): Brasilian-Polish Topology Workshop; 118-129 Сборник Трудов Института математики НАН Украины; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 118-129 Збірник Праць Інституту математики НАН України; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 118-129 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/281/283 Авторське право (c) 2013 Праці Інституту математики НАН України |
| spellingShingle | Kiszkiel, J. M. Kiszkiel, J. M. The Lefschetz Theorem for multivalued maps |
| title | The Lefschetz Theorem for multivalued maps |
| title_alt | Теорема Лефшеца для багатозначних відображень |
| title_full | The Lefschetz Theorem for multivalued maps |
| title_fullStr | The Lefschetz Theorem for multivalued maps |
| title_full_unstemmed | The Lefschetz Theorem for multivalued maps |
| title_short | The Lefschetz Theorem for multivalued maps |
| title_sort | lefschetz theorem for multivalued maps |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/281 |
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