Some results in quasitopological homotopy groups
UDC 515.4 We show that the $n$th quasitopological homotopy group of a topological space is isomorphic to $(n-1)$th quasitopological homotopy group of its loop space and by this fact we obtain some results about quasitopological homotopy groups. Finally, using the long exact sequence of a based pair...
Збережено в:
| Дата: | 2020 |
|---|---|
| Автори: | , , , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2020
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/564 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507047862206464 |
|---|---|
| author | Nasri, T. Mirebrahimi , H. Torabi , H. T. H. H. Nasri, T. Mirebrahimi , H. Torabi , H. |
| author_facet | Nasri, T. Mirebrahimi , H. Torabi , H. T. H. H. Nasri, T. Mirebrahimi , H. Torabi , H. |
| author_sort | Nasri, T. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:49:28Z |
| description | UDC 515.4
We show that the $n$th quasitopological homotopy group of a topological space is isomorphic to $(n-1)$th quasitopological homotopy group of its loop space and by this fact we obtain some results about quasitopological homotopy groups. Finally, using the long exact sequence of a based pair and a fibration in qTop introduced by Brazas in 2013, we obtain some results in this field. |
| doi_str_mv | 10.37863/umzh.v72i12.564 |
| first_indexed | 2026-03-24T02:03:06Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i12.564
UDC 515.14
T. Nasri (Dep. Pure Math., Faculty Basic Sci., Univ. Bojnord, Iran),
H. Mirebrahimi, H. Torabi (Dep. Pure Math., Center Excellence in Analysis on Algebraic Structures,
Ferdowsi Univ. Mashhad, Iran)
SOME RESULTS IN QUASITOPOLOGICAL HOMOTOPY GROUPS
ДЕЯКI РЕЗУЛЬТАТИ З КВАЗIТОПОЛОГIЧНИХ ГОМОТОПIЧНИХ ГРУП
We show that the nth quasitopological homotopy group of a topological space is isomorphic to (n - 1)th quasitopological
homotopy group of its loop space and by this fact we obtain some results about quasitopological homotopy groups. Finally,
using the long exact sequence of a based pair and a fibration in qTop introduced by Brazas in 2013, we obtain some results
in this field.
Доведено, що n-та квазiтопологiчна гомотопiчна група топологiчного простору є iзоморфною (n - 1)-й квазiтополо-
гiчнiй гомотопiчнiй групi його простору петель, та отримано деякi результати, що вiдносяться до квазiтопологiчних
гомотопiчних груп. Насамкiнець за допомогою довгої точної послiдовностi для базової пари та розшарування у
qTop, що запропонував Бразас у 2013 р., отримано деякi результати у цiй областi.
1. Introduction. Endowed with the quotient topology induced by the natural surjective map q :
\Omega n(X,x) \rightarrow \pi n(X,x), where \Omega n(X,x) is the nth loop space of (X,x) with the compact-open
topology, the familiar homotopy group \pi n(X,x) becomes a quasitopological group which is called
the quasitopological nth homotopy group of the pointed space (X,x), denoted by \pi qtop
n (X,x) (see
[3 – 5, 10]).
It was claimed by Biss [3] that \pi qtop
1 (X,x) is a topological group. However, Calcut and Mc-
Carthy [7] and Fabel [8] showed that there is a gap in the proof of [3] (Proposition 3.1). The misstep
in the proof is repeated by Ghane et al. [10] to prove that \pi qtop
n (X,x) is a topological group [10]
(Theorem 2.1) (see also [7]).
Calcut and McCarthy [7] showed that \pi qtop
1 (X,x) is a homogeneous space and more precisely,
Brazas [5] mentioned that \pi qtop
1 (X,x) is a quasitopological group in the sense of [1].
Calcut and McCarthy [7] proved that for a path connected and locally path connected space
X, \pi qtop
1 (X) is a discrete topological group if and only if X is semilocally 1-connected (see also
[5]). Pakdaman et al. [12] showed that for a locally (n - 1)-connected space X, \pi qtop
n (X,x) is
discrete if and only if X is semilocally n-connected at x (see also [10]). Also, they proved that the
quasitopological fundamental group of every small loop space is an indiscrete topological group. We
recall that a loop in X at x is called small if it is homotopic to a loop in every neighborhood U of
x. Also the topological space X with non trivial fundamental group is called a small loop space if
every loop of X is small.
In this paper, we obtain some results about quasitopological homotopy groups. One of the main
results of Section 2 is Theorem 2.1.
By this fact we can show that some properties of a space can be transferred to its loop space.
Also, we obtain several results in quasitopological homotopy groups. Moreover, we show that for a
fibration p : E - \rightarrow X with fiber F, the induced map f\ast : \pi qtop
n (B, b0) - \rightarrow \pi qtop
n - 1(F,
\~b0) is continuous.
Brazas in his thesis [6] exhibited two long exact sequences of based pair (X,A) and fibration p :
E - \rightarrow X in qTop. In Section 3, we use these sequences and obtain some results in this filed.
c\bigcirc T. NASRI, H. MIREBRAHIMI, H. TORABI, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12 1663
1664 T. NASRI, H. MIREBRAHIMI, H. TORABI
2. Quasitopological homotopy groups. It is well-known that for a pointed topological space
(X,x), for all n \geq 1 and 1 \leq k \leq n - 1, \pi n(X,x) \sim = \pi n - k(\Omega
k(X,x), ex). In this section, we
extend this result for quasitopological homotopy groups and we obtain some results about them. The
following theorem is one of the main results of this paper.
Theorem 2.1. Let (X,x) be a pointed topological space. Then, for all n \geq 1 and 1 \leq k \leq
\leq n - 1,
\pi qtop
n (X,x) \sim = \pi qtop
n - k(\Omega
k(X,x), ex),
where ex is the constant k-loop in X at x.
Proof. Consider the following commutative diagram:
\Omega n(X,x)
\phi - - - - \rightarrow \Omega n - k(\Omega k(X,x), ex) \downarrow q q
\downarrow
\pi qtop
n (X,x)
\phi \ast - - - - \rightarrow \pi qtop
n - k(\Omega
k(X,x), ex),
(1)
where \phi : \Omega n(X,x) - \rightarrow \Omega n - k(\Omega k(X,x), ex) given by \phi (f) = f \sharp is a homeomorphism with inverse
g \mapsto - \rightarrow g\flat in the sense of [13]. Since the map q is a quotient map, the homomorphism \phi \ast is an
isomorphism between quasitopological homotopy groups.
The following result is a consequence of Theorem 2.1.
Corollary 2.1. Let X be a locally (n - 1)-connected. Then X is semilocally n-connected at x
if and only if \Omega n - 1(X,x) is semilocally simply connected at ex, where ex is the constant loop in X
at x.
Proof. Since X is a locally (n - 1)-connected, by [12] (Theorem 6.7), X is semilocally n-
connected at x if and only if \pi qtop
n (X,x) is discrete. By Theorem 2.1, \pi qtop
n (X,x) \sim =
\sim = \pi qtop
1 (\Omega n - 1(X,x), ex). Also \pi qtop
1 (\Omega n - 1(X,x), ex) is discrete if and only if \Omega n - 1(X,x) is
semilocally simply connected at ex by [12] (Theorem 6.7).
Note that the above result has been shown by Hidekazu Wada [17] (Remark) and Authors [11]
(Lemma 3.1) with another methods.
Corollary 2.2. Let (X,x) = \mathrm{l}\mathrm{i}\mathrm{m}
\leftarrow
(Xi, xi) be the inverse limit of an inverse system \{ (Xi, xi), \varphi ij\} I .
Then, for all n \geq 1 and 1 \leq k \leq n - 1,
\pi qtop
n (X,x) \sim = \pi qtop
n - k(\mathrm{l}\mathrm{i}\mathrm{m}\leftarrow
\Omega k(Xi, xi), ex).
Virk [16] introduced the SG (small generated) subgroup of fundamental group \pi 1(X,x), denoted
by \pi sg
1 (X,x), as the subgroup generated by the following elements
[\alpha \ast \beta \ast \alpha - 1],
where \alpha is a path in X with initial point x and \beta is a small loop in X at \alpha (1). Recall that
a space X is said to be small generated if \pi 1(X,x) = \pi sg
1 (X,x), also a space X is said to
be semilocally small generated if for every x \in X there exists an open neighborhood U of x
such that i\ast \pi 1(U, x) \leq \pi sg
1 (X,x). Torabi et al. [15] proved that if X is small generated space,
then \pi qtop
1 (X,x) is an indiscrete topological group and the quasitopological fundamental group of a
semilocally small generated space is a topological group. By Theorem 2.1, we obtain several results
in quasitopological homotopy groups as follows:
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
SOME RESULTS IN QUASITOPOLOGICAL HOMOTOPY GROUPS 1665
Corollary 2.3. Let X be a topological space such that \Omega n - 1(X,x) is small generated. Then
\pi qtop
n (X,x) is an indiscrete topological group.
Proof. Since \Omega n - 1(X,x) is a small generated space, then \pi qtop
1 (\Omega n - 1(X,x), ex) is an indiscrete
topological group, by [15] (Remark 2.11). Therefore \pi qtop
n (X,x) \sim = \pi qtop
1 (\Omega n - 1(X,x), ex) implies
that \pi qtop
n (X,x) is an indiscrete topological group.
Corollary 2.4. Let X be a topological space such that \Omega n - 1(X,x) is a semilocally small ge-
nerated space. Then \pi qtop
n (X,x) is a topological group.
Proof. Since \Omega n - 1(X,x) is semilocally small generated, then \pi qtop
1 (\Omega n - 1(X,x), ex) is a topo-
logical group, by [15] (Theorem 4.1). Therefore \pi qtop
n (X,x) \sim = \pi qtop
1 (\Omega n - 1(X,x), ex) implies that
\pi qtop
n (X,x) is a topological group.
Fabel [8] proved that \pi qtop
1 (HE, x) is not topological group. By considering the proof of this
result it seems that if \pi 1(X,x) is an abelian group, then \pi qtop
1 (X,x) is a topological group. He [9]
also showed that for each n \geq 2 there exists a compact, path connected, metric space X such that
\pi qtop
n (X,x) is not a topological group. In the following example we show that there is a metric space
Y with Abelian fundamental group such that \pi qtop
1 (Y, y) is not a topological group.
Example 2.1. Let n \geq 2, X be the compact, path connected, metric space introduced in [9]
such that \pi qtop
n (X,x) is not a topological group. By Theorem 2.1 \pi qtop
1 (\Omega n - 1(X,x), ex) is not a
topological group. Since for every n \geq 2, \pi n(X,x) is an Abelian group, hence there is a metric
space Y = \Omega n - 1(X,x) with Abelian fundamental group such that \pi qtop
1 (Y, y) is not a topological
group.
In [4] (Proposition 3.25), it is proved that the quasitopological fundamental groups of shape
injective spaces are Hausdorff. By Theorem 2.1 we have the following result.
Corollary 2.5. Let X be a topological space such that \Omega n - 1(X,x) is shape injective space.
Then \pi qtop
n (X,x) is Hausdorff.
Proposition 2.1 [15]. For a pointed topological space (X,x), if \{ [ex]\} is closed (or equiva-
lently the topology of \pi qtop
1 (X,x) is T0), then X is homotopically Hausdorff.
We generalized the above proposition as follows:
Proposition 2.2. For a pointed topological space (X,x), if \{ [ex]\} is closed (or equivalently the
topology of \pi qtop
n (X,x) is T0), then X is n-homotopically Hausdorff.
Proof. By Theorem 2.1 since \pi qtop
n (X,x) is T0, hence \pi qtop
1 (\Omega n - 1(X,x), ex) is T0. There-
fore by previous proposition \Omega n - 1(X,x) is homotopically Hausdorff which implies that X is n-
homotopically Hausdorff by [11] (Lemma 3.5).
Corollary 2.6. Let X be a topological space such that \Omega n - 1(X,x) is shape injective space.
Then X is n-homotopically Hausdorff.
Proof. It follows from Corollary 2.5 and Proposition 2.2.
Let (B, b0) be a pointed space and p : E - \rightarrow B be a fibration with fiber F. Consider its
mapping fiber, Mp = \{ (e, \omega ) \in E \times BI : \omega (0) = b0 and \omega (1) = p(e)\} . If \~b0 \in p - 1(b0), then
the injection map k : \Omega (B, b0) - \rightarrow Mp given by k(\omega ) = ( \~b0, \omega ) induces a homomorphism f\ast :
\pi n(B, b0) - \rightarrow \pi n - 1(F, \~b0) [13].
Theorem 2.2. Let (B, b0) be a pointed space and p : E - \rightarrow B be a fibration. If \~b0 \in p - 1(b0),
then f\ast : \pi qtop
n (B, b0) - \rightarrow \pi qtop
n - 1(F,
\~b0) is continuous for all n \geq 1.
Proof. We consider the following commutative diagram:
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1666 T. NASRI, H. MIREBRAHIMI, H. TORABI
\Omega n - 1(\Omega (B, b0), eb0)
k\sharp - - - - \rightarrow \Omega n - 1(Mp, \ast ) \downarrow q q
\downarrow
\pi qtop
n - 1(\Omega (B, b0), eb0)
k\ast - - - - \rightarrow \pi qtop
n - 1(Mp, \ast ),
(2)
where q is the quotient map and k\sharp is the induced map of k : \Omega (B, b0) - \rightarrow Mp by the functor
\Omega n - 1. Since k\sharp is continuous and q is a quotient map, k\ast : \pi qtop
n - 1(\Omega (B, b0), eb0) - \rightarrow \pi qtop
n - 1(F,
\~b0)
is continuous. By Theorem 2.1, \pi qtop
n - 1(\Omega (B, b0), eb0) is isomorphic to \pi qtop
n (B, b0). Therefore, f\ast :
\pi qtop
n (B, b0) - \rightarrow \pi qtop
n - 1(F,
\~b0) is continuous.
3. Long exact sequence of \bfitpi \bfitq \bfitt \bfito \bfitp
\bfitn (\bfitX ). Brazas [6] (Theorem 2.49) proved that for every
based pair (X, A) with inclusion i : A - \rightarrow X, there is a long exact sequence in the category of
quasitopological groups as follows:
... - \rightarrow \pi qtop
n+1(A) \rightarrow \pi qtop
n+1(X) \rightarrow \pi qtop
n+1(X,A) \rightarrow \pi qtop
n (A) - \rightarrow ...
\cdot \cdot \cdot - \rightarrow \pi qtop
1 (X) - \rightarrow \pi qtop
1 (X,A) - \rightarrow \pi qtop
0 (A) - \rightarrow \pi qtop
0 (X).
He [6] (Proposition 2.20) also showed that for every fibration p : E - \rightarrow B of path connected spaces
with fiber F, there is a long exact sequence in the category of quasitopological groups as follows:
\cdot \cdot \cdot - \rightarrow \pi qtop
n (E) \rightarrow \pi qtop
n (B) \rightarrow \pi qtop
n - 1(F ) \rightarrow \pi qtop
n - 1(E) - \rightarrow \cdot \cdot \cdot
\cdot \cdot \cdot - \rightarrow \pi qtop
1 (B) - \rightarrow \pi qtop
0 (F ) - \rightarrow \pi qtop
0 (E) - \rightarrow \pi qtop
0 (B). (3)
In follow, we obtain some results and examples by these exact sequences.
Example 3.1. Consider the pointed pair (HA,HE), where HA is the harmonic archipelago and
HE is the hawaiian earring. Then by [6] (Theorem 2.49), there is a long exact sequence in qTop:
\cdot \cdot \cdot - \rightarrow \pi qtop
n+1(HE) \rightarrow \pi qtop
n+1(HA) \rightarrow \pi qtop
n+1(HA,HE) \rightarrow \pi qtop
n (HE) - \rightarrow ...
\cdot \cdot \cdot - \rightarrow \pi qtop
1 (HA) - \rightarrow \pi qtop
1 (HA,HE) - \rightarrow \pi qtop
0 (HE) - \rightarrow \pi qtop
0 (HA).
Recall that a short exact sequence E : 0 - \rightarrow H
i\rightarrow X
\pi \rightarrow G - \rightarrow 0 of topological Abelian
groups will be called an extension of topological groups if both i and \pi are continuous and open
homomorphisms when considered as maps onto their images. Also, the extension E is called split if
and only if it is equivalent to the trivial extension E0 : 0 - \rightarrow H
iH\rightarrow H \times G
\pi G\rightarrow G - \rightarrow 0 [2].
Theorem 3.1 ([2], Theorem 1.2). Let E : 0 - \rightarrow H
i\rightarrow X
\pi \rightarrow G - \rightarrow 0 be an extension of
topological Abelian groups. The following are equivalent:
(1) E splits.
(2) There exists a right inverse for \pi .
(3) There exists a left inverse for i.
The above results hold for quasitopological groups, too.
Proposition 3.1. If r : X - \rightarrow A is a retraction, then there are isomorphisms in quasitopological
groups, for all n \geq 2,
\pi qtop
n (X) \sim = \pi qtop
n (A)\times \pi qtop
n (X,A).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
SOME RESULTS IN QUASITOPOLOGICAL HOMOTOPY GROUPS 1667
Proof. Consider the pointed pair (X,A). By [6] (Theorem 2.49), there is a long exact sequence
\cdot \cdot \cdot - \rightarrow \pi qtop
n+1(X) - \rightarrow \pi qtop
n+1(X,A) - \rightarrow \pi qtop
n (A)
i\ast \rightarrow \pi qtop
n (X) - \rightarrow \pi qtop
n (X,A) - \rightarrow . . . .
Since r is a retraction and i\ast is an injection, there is a short exact sequence
0 - \rightarrow \pi qtop
n (A)
i\ast \rightarrow \pi qtop
n (X) \rightarrow \pi qtop
n (X,A) - \rightarrow 0.
Moreover, this sequence is an extension. Indeed, the map i\ast and \pi \ast are continuous and open
homomorphisms when considered as maps onto their images. Therefore,
\pi qtop
n (X) \sim = \pi qtop
n (A)\times \pi qtop
n (X,A).
Proposition 3.2. Let B \subseteq A \subseteq X be pointed spaces. Then there is a long exact sequence of the
triple (X,A,B) in qTop:
\cdot \cdot \cdot - \rightarrow \pi qtop
n+1(X,A) - \rightarrow \pi qtop
n (A,B) - \rightarrow \pi qtop
n (X,B) - \rightarrow \pi qtop
n (X,A) - \rightarrow \pi qtop
n - 1(A,B) - \rightarrow . . . .
Proof. Consider the following commutative diagram and chase a long diagram as follows:
The following results are immediate consequences of sequence (3).
Corollary 3.1. If p : E - \rightarrow B is a fibration with E contractible, then f\ast : \pi qtop
n (B, b0) - \rightarrow
- \rightarrow \pi qtop
n - 1(F,
\~b0) is an isomorphism in quasitopological groups for all n \geq 2 and f\ast : \pi qtop
1 (B, b0) - \rightarrow
- \rightarrow \pi qtop
0 (F ) is an isomorphism in Set.
Corollary 3.2. Let (X,x) be a pointed topological space. Then \pi qtop
n (X,x) \sim = \pi qtop
n - 1(\Omega (X,x), ex)
in quasitopological groups for all n \geq 2, where ex is the constant loop in X at x and \pi qtop
1 (X,x) \sim =
\sim = \pi qtop
0 (\Omega (X,x)) in Set.
Proof. By [14] (Proposition 4.3), the map p : PX - \rightarrow X is a fibration with fiber \Omega (X,x), where
PX = (X,x)(I,0). By [6] (Proposition 2.20), the sequence
\cdot \cdot \cdot - \rightarrow \pi qtop
n (PX, ex) - \rightarrow \pi qtop
n (X,x) - \rightarrow \pi qtop
n - 1(\Omega (X,x), ex) - \rightarrow \pi qtop
n - 1(PX, ex) - \rightarrow \cdot \cdot \cdot
\cdot \cdot \cdot - \rightarrow \pi qtop
1 (X,x) - \rightarrow \pi qtop
0 (\Omega (X,x)) - \rightarrow \pi qtop
0 (PX) - \rightarrow \pi qtop
0 (X)
is exact in qTop. By [14] (Proposition 4.4), (PX, ex) is contractible and therefore the result holds
by Corollary 3.1.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1668 T. NASRI, H. MIREBRAHIMI, H. TORABI
Corollary 3.3. If p : \~X - \rightarrow X is a covering projection, then for all n \geq 2, \pi qtop
n ( \~X) \sim = \pi qtop
n (X)
in quasitopological groups and \pi qtop
1 ( \~X) can be embedded in \pi qtop
1 (X).
Proof. This result follows by sequence (3) and this fact that the fiber F of the covering projection
p is discrete and therefore \pi qtop
n (F ) is trivial for all n \geq 1.
References
1. A. Arhangelskii, M. Tkachenko, Topological groups and related structures, Atlantis Stud. Math. (2008).
2. H. J. Belloa, M. J. Chascoa, X. Domnguezb, M. Tkachenko, Splittings and cross-sections in topological groups, J.
Math. Anal. and Appl., 435, № 2, 1607 – 1622 (2016).
3. D. Biss, The topological fundamental group and generalized covering spaces, Topology and Appl., 124, № 3,
355 – 371 (2002).
4. J. Brazas, The fundamental group as topological group, Topology and Appl., 160, № 1, 170 – 188 (2013).
5. J. Brazas, The topological fundamental group and free topological groups, Topology and Appl., 158, № 6, 779 – 802
(2011).
6. J. Brazas, Homotopy mapping spaces, Ph. D. Dissertation, Univ. New Hampshire (2011).
7. J. S. Calcut, J. D. McCarthy, Discreteness and homogeneity of the topological fundamental group, Topology Proc.,
34, 339 – 349 (2009).
8. P. Fabel, Multiplication is discontinuous in the Hawaiian earring group (with the quotient topology), Bull. Pol. Acad.
Sci. Math., 59, № 1, 77 – 83 (2011).
9. P. Fabel, Compactly generated quasi-topological homotopy groups with discontinuous multiplication, Topology Proc.,
40, 303 – 309 (2012).
10. H. Ghane, Z. Hamed, B. Mashayekhy, H. Mirebrahimi, Topological homotopy groups, Bull. Belg. Math. Soc. Simon
Stevin, 15, № 3, 455 – 464 (2008).
11. T. Nasri, B. Mashayekhy, H. Mirebrahimi, On quasitopological homotopy groups of inverse limit spaces, Topology
Proc., 46, 145 – 157 (2015).
12. A. Pakdaman, H. Torabi, B. Mashayekhy, On H -groups and their applications to topological fundamental groups,
preprint, arXiv:1009.5176v1.
13. J. J. Rotman, An introduction to algebraic topology, Grad. Textx in Math., 119, Springer-Verlag (1988).
14. R. M. Switzer, Algebraic topology homotopy and homology, Springer-Verlag (1975).
15. H. Torabi, A. Pakdaman, B. Mashayekhy, Topological fundamental groups and small generated coverings, Math.
Slovaca, 65, № 5, 1 – 12 (2011).
16. Z. Virk, Small loop spaces, Topology and Appl., 157, № 2, 451 – 455 (2010).
17. H. Wada, Local connectivity of mapping space, Duke Math. J., 22, № 3, 419 – 425 (1955).
Received 14.09.17
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
|
| id | umjimathkievua-article-564 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:03:06Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/5a/3dfb5283d365c46d80e206b6835b895a.pdf |
| spelling | umjimathkievua-article-5642025-03-31T08:49:28Z Some results in quasitopological homotopy groups Some results in quasitopological homotopy groups Nasri, T. Mirebrahimi , H. Torabi , H. T. H. H. Nasri, T. Mirebrahimi , H. Torabi , H. Homotopy group Quasitopological group Fibration Homotopy group Quasitopological group Fibration UDC 515.4 We show that the $n$th quasitopological homotopy group of a topological space is isomorphic to $(n-1)$th quasitopological homotopy group of its loop space and by this fact we obtain some results about quasitopological homotopy groups. Finally, using the long exact sequence of a based pair and a fibration in qTop introduced by Brazas in 2013, we obtain some results in this field. УДК 515.4 Деякi результати з квазiтопологiчних гомотопiчних груп Доведено, що $n$-та квазітопологічна гомотопічна група топологічного простору є ізоморфною $(n-1)$-й квазітопологічній гомотопічній групі його простору петель, та отримано деякі результати, що відносяться до квазітопологічних гомотопічних груп. Насамкінець за допомогою довгої точної послідовності для базової пари та розшарування у qTop, що запропонував Бразас у 2013 р., отримано деякі результати у цій області. Institute of Mathematics, NAS of Ukraine 2020-12-24 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/564 10.37863/umzh.v72i12.564 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 12 (2020); 1663-1668 Український математичний журнал; Том 72 № 12 (2020); 1663-1668 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/564/8874 |
| spellingShingle | Nasri, T. Mirebrahimi , H. Torabi , H. T. H. H. Nasri, T. Mirebrahimi , H. Torabi , H. Some results in quasitopological homotopy groups |
| title | Some results in quasitopological homotopy groups |
| title_alt | Some results in quasitopological homotopy groups |
| title_full | Some results in quasitopological homotopy groups |
| title_fullStr | Some results in quasitopological homotopy groups |
| title_full_unstemmed | Some results in quasitopological homotopy groups |
| title_short | Some results in quasitopological homotopy groups |
| title_sort | some results in quasitopological homotopy groups |
| topic_facet | Homotopy group Quasitopological group Fibration Homotopy group Quasitopological group Fibration |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/564 |
| work_keys_str_mv | AT nasrit someresultsinquasitopologicalhomotopygroups AT mirebrahimih someresultsinquasitopologicalhomotopygroups AT torabih someresultsinquasitopologicalhomotopygroups AT t someresultsinquasitopologicalhomotopygroups AT h someresultsinquasitopologicalhomotopygroups AT h someresultsinquasitopologicalhomotopygroups AT nasrit someresultsinquasitopologicalhomotopygroups AT mirebrahimih someresultsinquasitopologicalhomotopygroups AT torabih someresultsinquasitopologicalhomotopygroups |