Comparison of Topologies: Perfect Native View Test

Comparison of Topologies on Homotopy Groups with

Subgroup Topology Viewpoint

Naghme Shahami

, Behrooz Mashayekhy

1,∗

Department of Pure Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of

Mashhad, P.O.Box 1159, Mashhad 91775, Iran.

Abstract

By introducing various topologies on the homotopy groups of a topological space,

some researchers make these well known notions in algebraic topology more useful

and powerful. In this paper, ﬁrst we recall and review some known topologies on

homotopy groups. Then by reviewing some famous subgroups of homotopy groups

and using the concept of subgroup topology, we intend to compare these topologies

in order to present some results on topologized homotopy groups.

Keywords: Topologized homotopy groups, subgroup topology, compact-open

topology, Spanier topology, whisker topology, higher Spanier group.

2020 MSC: 55Q05, 55Q07, 55Q52.

1. Introduction and Motivation

During last two decades some researchers have shown that there are various useful,

interesting and functorial topologies on the fundamental group π

(X, x

) and the

homotopy groups π

(X, x

) of a topological space X which make them powerful

tools for studying some topological properties of spaces (see [1, 2, 5, 6, 7, 9, 10, 11,

12, 13, 21, 23]). Recently, the authors have studied and reviewed most of known

topologies on the fundamental group π

(X, x

) and summed up all comparison of

various topologies on the fundamental group in a diagram (see [21]). In this paper,

we are going to study and review some topologies on the homotopy groups π

(X, x

)

and then using the concept of subgroup topology with respect to some neighbourhood

∗

Corresponding author

Email addresses: na.shahami@mail.um.ac.ir (Naghme Shahami ), bmashf@um.ac.ir

(Behrooz Mashayekhy)

arXiv:2602.20992v1 [math.AT] 24 Feb 2026

families of famous subgroups of the homotopy groups π

(X, x

), we intend to compare

topologized homotopy groups with various topologies.

The paper is organized as follows. Inspired by some famous subgroups of the

fundamental group π

(X, x

), some researchers have introduced and deﬁned similar

subgroups for the homotopy group π

(X, x

) (see [5, 6]). In Section 2, we review

these subgroups including the n-small subgroup, π

(X, x

), the n-small generated

subgroup, π

(X, x

), the n-Spanier subgroup, π

(X, x

), the n-thick Spanier sub-

group, π

tsp

(X, x

), the n-weak thick Spanier subgroup, π

wtsp

(X, x

). Also, we are

inspired by [22] to deﬁne the n-path Spanier subgroup, eπ

(X, x

), and compare this

subgroup to the others as follows (in order to avoid confusion, from now on, we omit

the subscript n for the subgroups of the homotopy group π

(X, x

)):

(X, x

) ≤ π

(X, x

) ≤ eπ

(X, x

) ≤ π

(X, x

) ≤ π

tsp

(X, x

) ≤ π

wtsp

(X, x

In Section 3, ﬁrst we review various topologies that have been deﬁned on ho-

motopy groups. These topologies include the whisker topology [4], π

(X, x

the compact-open quotient topology [13, 14], π

qtop

(X, x

), the tau topology [7],

(X, x

), the lim topology [14], π

lim

(X, x

), the Spanier topology [6], π

Span

(X, x

the thick Spanier topology [6], π

tSpan

(X, x

), the weak thick Spanier topology [6],

wtSpan

(X, x

), the shape topology [9], π

(X, x

), and the pseudometric topology

[9], π

met

(X, x

). Then by reviewing the concept of subgroup topology of a neigh-

bourhood family introduced in [8] especially a type of subgroup topology with re-

spect a subgroup introduced in [21] and considering some famous subgroups of the

homotopy groups mentioned in Section 2, we introduce some topologies on the ho-

motopy groups π

(X, x

) such as the path Spanier topology, π

pSpan

(X, x

), the n-

small subgroup topology, π

(X,x

)

(X, x

), the n-small generated subgroup topol-

ogy, π

(X,x

)

(X, x

), the n-path Spanier subgroup topology, π

eπ

(X,x

)

(X, x

), the n-

Spanier subgroup topology, π

(X,x

)

(X, x

), the n-thick Spanier subgroup topology,

tsp

(X,x

)

(X, x

), the n-weak thick Spanier subgroup topology, π

wtsp

(X,x

)

(X, x

In Section 4, using some properties of the subgroup topology presented in [8, 21]

and some results of other researchers, we are going to compare the above mentioned

topologies on homotopy groups with each other as much as possible. Among some

results, we prove that

(X, x

) ≼ π

wtSpan

(X, x

) ≼ π

tSpan

(X, x

) ≼ π

Span

(X, x

) ≼ π

pSpan

(X, x

and

(X, x

) ≼ π

qtop

(X, x

) ≼ π

lim

(X, x

) = π

(X, x

) ≼ π

(X,x

)

(X, x

where A ≼ B mean that B is ﬁner than A. Also we show that

wtsp

(X,x

)

(X, x

) ≼ π

tsp

(X,x

)

(X, x

) ≼ π

(X,x

)

(X, x

) ≼ π

eπ

(X,x

)

(X, x

)

≼ π

(X,x

)

(X, x

) ≼ π

(X,x

)

(X, x

Moreover, we investigate some conditions under which some of the above inequalities

become equality. As an example, we deﬁne the concept n-semilocally H-connected

at x

for a subgroup H ≤ π

(X, x

) and we prove that a topological space X is

n-semilocally H-connected at x

if and only if H is an open subgroup of π

(X, x

)

which is a generalization of [1, Theorem 4.2]. We show that the equality π

(X, x

) =

(X,x

)

(X, x

) holds if and only if X is n-semilocally H-connected at x

. Finally, we

sum up all results of this section in a diagram in order to compare various topologies

on homotopy groups.

2. Some Subgroups of Homotopy Groups

In order to review some subgroups on the homotopy groups similar to famous

subgroups of the fundamental group, we recall some notions and notations from

[15]. A loop at x

in X is a map from I = [0, 1] into X that carries the boundary

∂I = {0, 1} onto x

. The fundamental group π

(X, x

) consists of all homotopy

classes, relative to ∂I, of such loops and admits a natural, and very useful, group

structure. This concept is generalized as follows. For each positive integer n, I

[0, 1]

= {(s

, ..., s

) ∈ R

: 0 ≤ s

≤ 1, i = 1, ..., n} is called the n-cube in R

The boundary ∂I

of I

consists of all (s

, ..., s

) ∈ I

for which s

= 0 or s

= 1

for at least one value of i. If X is a topological space and x

∈ X, then an n-loop

at x

is a continuous map α : I

→ X such that α(∂I

) = x

. The collection of

all n-loops at x

in X is partitioned into equivalence classes by homotopy relative

to ∂I

. The equivalence class containing an n-loop α is denoted by [α]. If α is an

n-loop at x

in X, deﬁne α

−1

: I

→ X by α

−1

, s

, ..., s

) = α(1 − s

, s

, ..., s

) for

all (s

, s

, ..., s

) ∈ I

(see [15, p. 136]). Let X be a topological space, x

∈ X and

n ≥ 2 a positive integer. Let π

(X, x

) be the set of all homotopy classes, relative to

∂I

, of n-loops at x

in X. For [α], [β] ∈ π

(X, x

), deﬁne [α] + [β] = [α + β]. Then,

with this operation, π

(X, x

) is an abelian group in which the identity element is

] and the inverse of any [α] is given by [α

−1

] (see [15, Theorem 2.5.3]). The group

(X, x

) is called the n-th homotopy group of X at x

We know that if there exists a path σ : I → X from x

to x

, then π

(X, x

) and

(X, x

) are isomorphism. A similar result exists for homotopy groups. Let n ≥ 2,

deﬁne σ

: π

(X, x

) → π

(X, x

) by σ

[α] = [F

], where F

(s) = F (s, 1) for any

s ∈ I

and F : I

× I → X is a homotopy with the following properties:

F (s, 0) = α(s); s ∈ I

F (s, t) = σ

−1

(t); s ∈ ∂I

, t ∈ I.

Then by [15, Theorem 2.5.6] σ

is a well-deﬁned isomorphism that depends only on

the path homotopy class of σ, i.e., if σ

′

≃ σ rel {0, 1}, then σ

′

= σ

In [5, 6], Bahredar et al. deﬁned some subgroups of homotopy groups similar to

some famous subgroups of the fundamental groups as follows.

1. The n-small subgroup: An n-loop α : (I

, ∂I

) → (X, x

) is said to be

small if for every open neighborhood U of x

, there exists an n-loop f : I

→ X

with the base point x

such that f(I

) ⊆ U and [α] = [f] [5, Deﬁnition 4.5]. A small

n-loop subgroup at x

, denoted by π

(X, x

), is a subgroup of π

(X, x

) consisting of

homotopy classes of all small n-loops at x

(see [5]).

2. The n-small generated subgroup: The set π

(X, x

) = {σ

([α])|[α] ∈

(X, x

), σ is a path with the end point x

} is a subgroup of π

(X, x

) which is

called the n-small generated subgroup (see [5]).

3. The n-Spanier subgroup: Let U be an open cover of a pointed space (X, x

Then π

(U, x

) is a subgroup of π

(X, x

) which is spanned by all homotopy classes

of the form

j=1

], where for every 1 ≤ j ≤ n, σ

(0) = x

and the n-loop v

lies

in one of the neighborhoods U

∈ U. This group is called the n-Spanier subgroup with

respect to U (see [5, Deﬁnition 3.1]). The intersection of the n-Spanier subgroups

relative to open covers of X is called the n-Spanier subgroup and it is denoted by

(X, x

) (see [5]).

4. The n-thick Spanier subgroup: Let U be an open cover of a pointed space

(X, x

). The n-thick Spanier subgroup with respect to U is the subgroup of π

(X, x

)

which is generated by elements of the form σ

⊛ f

], where f

⊛ f

(z) = f

(z)

if z ∈ S

and f

⊛ f

(z) = f

(z) if z ∈ S

−

, f

: (D

, s

) → (X, x

) are pointed

continuous maps such that for any z ∈ ∂D

, f

(z) = f

(z), and Imf

⊆ U

for some

∈ U (i = 1, 2) and it is denoted by π

tSp

(U, x

). The intersection of the n-thick

Spanier subgroups relative to open covers of X is called the n-thick Spanier subgroup

which is denoted by π

tsp

(X, x

) (see [6]).

5. The n-weak thick Spanier subgroup: Let U be an open cover of a pointed

space (X, x

). The n-weak thick Spanier subgroup with respect to U is the subgroup

of π

(X, x

) which is generated by elements of the form σ

[α], where α(I

) ⊆ U

∪U

for some U

, U

∈ U and it is denoted by π

wtsp

(U, x

). The n-weak thick Spanier

subgroup of X is the intersection of the n-weak thick Spanier subgroups relative to

open covers of X which is denoted by π

wtsp

(X, x

) (see [6]).

6. The n-path Spanier subgroup: We are inspired by [22] to deﬁne the n-path

Spanier subgroup. A path open cover of a pointed space (X, x

) is an open cover

V = {V

|α ∈ P (X, x

)} such that α(1) ∈ V

. Then we deﬁne eπ

(V, x

) as a subgroup

of π

(X, x

) which is spanned by all homotopy classes of the form

j=1

], where

for every 1 ≤ j ≤ n, α

(0) = x

and the n-loop v

lies in V

∈ V. We called this

subgroup the n-path Spanier subgroup with respect to the path open cover V. The

intersection of n-path Spanier subgroups relative to path open covers of X is called

the n-path Spanier subgroup and we denote it by eπ

(X, x

Remark 2.1. (i) By deﬁnition of n-Spanier subgroups and n-path Spanier subgroups,

it easy to see that eπ

(X, x

) ≤ π

(X, x

). Moreover, let V = {V

|β ∈ P (X, x

)} be a

path open cover of X and let x

∈ V

. If α be a small n-loop at x

in X, there exists

an n-loop f : I

→ X such that f(I

) ⊆ V

and [f] = [α], thus σ

([α]) = σ

([f]),

where σ

([α]) ∈ π

(X, x

) and σ

([f]) ∈ eπ

(V, x

). Hence π

(X, x

) ⊆ eπ

(V, x

)

for every path open cover V. Therefore, we have

(X, x

) ≤ eπ

(X, x

) ≤ π

(X, x

(ii) By [5, Proposition 4.6], [6, Proposition 3.15] and the above note we have the

following chain of subgroups of π

(X, x

) ≤ π

(X, x

) ≤ eπ

(X, x

) ≤ π

(X, x

)

≤ π

tsp

(X, x

) ≤ π

wtsp

(X, x

) ≤ kerΨ

where Ψ

is the canonical map from the n-th homotopy group to the n-th shape

homotopy groups (see [9]).

3. Some Topologies on Homotopy Groups

In order to review and introduce some topologies on the homotopy groups similar

to famous topologies of the fundamental group, we recall the notion of subgroup

topology of a neighbourhood family introduced in [8] especially a type of subgroup

topology with respect to a subgroup introduced in [21].

A collection Σ of subgroups of G is called a neighbourhood family if for any

H, K ∈ Σ, there is a subgroup S ∈ Σ such that S ⊆ H ∩ K. As a result of this

property, the collection of all left cosets of elements of Σ forms a basis for a topology

on G, which is called the subgroup topology determined by Σ and we denote it by

. The subgroup topology on a group G speciﬁed by a neighbourhood family was

deﬁned in [8, Section 2.5] and considered by some recent researchers such as [2, 23].

Since left translation by elements of G determine self-homeomorphisms of G, they

are homogeneous spaces. Bogley et al. [8] focused on some general properties of

subgroup topologies and by introducing the intersection S

= ∩{H | H ∈ Σ}, called

the inﬁnitesimal subgroup for the neighbourhood family Σ. They showed that the

closure of the element g ∈ G is the coset gS

Let H be a subgroup of a group G. Then we deﬁne Σ

as follows

= {K ⩽ G | H ⊆ K}.

It is easy to see that Σ

is a neighbourhood family. We consider the subgroup

topology on G determined by Σ

and denote it by G

. Note that the inﬁnitesimal

subgroup for the neighbourhood family Σ

is H. We have compared two subgroup

topologies on a group. If Σ and Σ

′

are two neighbourhood families on G such that

= G

′

, then S

= S

′

[21, Theorem 2.3]. Let Σ and Σ

′

be two neighbourhood

families on G such that S

≤ S

′

and S

∈ Σ then G

′

≼ G

. In particular, if

H ≤ K ≤ G, then G

≼ G

. Furthermore, in [21] we proved that G

is discrete if

and only if H = 1. Also, G

is indiscrete if and only if H = G. It is pointed out in [8]

that although the group G equipped with a subgroup topology may not necessarily

be a topological group, in general (it may not even be a quasitopological group),

because right translation maps by a ﬁxed element of G need not be continuous, but

it has some of properties of topological groups (for more details see Theorem 2.9 in

[8]). Wilkins [23, Lemma 5.4] showed that a group G with the subgroup topology

determined by a neighbourhood family Σ is a topological group when all subgroups in

Σ are normal. Moreover, it is proved in [2, Corollary 2.2] that a group equipped with

a subgroup topology is a topological group if and only if all right translation maps

are continuous. Using these facts we can easily see that G

is a topological group

if H is a normal subgroup of G and G/H is an abelian group. We vastly extended

these results in [21, Theorem 2.2], let G be a group and Σ be a neighbourhood family

on G such that S

∈ Σ, then G

is a topological group if and only if S

is a normal

subgroup of G. In particular, G

is a topological group if and only if H is a normal

subgroup of G.

Using the above subgroup topology and considering subgroups of the homotopy

groups mentioned in Section 2, we can introduce some topologies on the homotopy

groups π

(X, x

) such as the n-small subgroup topology, π

(X,x

)

(X, x

), the n-small

generated subgroup topology, π

(X,x

)

(X, x

), the n-path Spanier subgroup topol-

ogy, π

eπ

(X,x

)

(X, x

), the n-Spanier subgroup topology, π

(X,x

)

(X, x

), the n-thick

Spanier subgroup topology, π

tsp

(X,x

)

(X, x

), the n-weak thick Spanier subgroup

topology, π

wtsp

(X,x

)

(X, x

In the following, we review various well-known topologies that have been deﬁned

on the homotopy group π

(X, x

1. The whisker topology: Deﬁne

= {π

(i)π

(U, x

)| U is an open subset of X containing x

Then it is easy to see that Σ is a neighbourhood family on π

(X, x

). The whisker

topology on the nth homotopy group, π

(X, x

), of a pointed topological space (X, x

)

is the subgroup topology determined by the neighbourhood family Σ

which is

denoted by π

(X, x

) and by [4, Proposition 2.6] π

(X, x

) is a topological group

for n ≥ 2 (see [4]).

2. The compact-open quotient topology: Let (X, x

) be a pointed topolog-

ical space and Ω

(X, x

) denote the space of n-loops in X based at x

. There exists

the usual compact-open topology on Ω

(X, x

) which is generated by subbasis sets

⟨K, U⟩ = {α | α(K) ⊆ U} for compact K ⊆ I

and open U ⊆ X. By considering the

surjection map q

: Ω

(X, x

) → π

(X, x

), q(α) = [α] one can equip π

(X, x

) with

the quotient topology with respect to the map q

: Ω

(X, x

) → π

(X, x

) which is

denoted by π

qtop

(X, x

) (see [13, 14]).

3. The tau topology: In [7], it was observed that for any group with topology

G, there is a ﬁnest group topology on the group G, which is coarser than that of G.

The resulting topological group is denoted τ(G). It is often convenient to think of τ

as a functor which removes the smallest number of open sets from the topology of

G so that a topological group is obtained. The quasitopological group π

qtop

(X, x

)

with this topology is denoted by π

(X, x

4. The lim topology: Let U be an open neighborhood of x

and

U = {[f] | f ∈

Ω

(X, x

) and Imf ⊆ U}. Since π

(X, x

) is an abelian group, for n ≥ 2, the ﬁlter

base {

U} forms a fundamental system of neighborhoods of the identity element [c

and hence π

(X, x

) with this topology becomes a topological group, denoted by

lim

(X, x

) (see [14]).

5. The Spanier topology: Bahredar et al. in [6, Proposition 4.2] showed

that Σ

Span

= {π

(U, x

)| U is an open cover of X} is a neighbourhood family on

(X, x

) and they called the subgroup topology on π

(X, x

) determined by Σ

Span

the Spanier topology (see [6]).

6. The thick Spanier topology: Bahredar et al. in [6, Proposition 4.2] showed

that Σ

tSpan

= {π

tsp

(U, x

)| U is an open cover of X} is a neighbourhood family on

(X, x

) and they called the subgroup topology on π

(X, x

) determined by Σ

tSpan

the thick Spanier topology (see [6]).

7. The weak thick Spanier topology: Bahredar et al. in [6, Proposition 4.2]

showed that Σ

wtSpan

= {π

tsp

(U, x

)| U is an open cover of X} is a neighbourhood

family on π

(X, x

) and they called the subgroup topology on π

(X, x

) determined

by Σ

wtSpan

the weak thick Spanier topology (see [6]).

8. The path Spanier topology: It is easy to see that

pSpan

= {eπ

(V, x

)| V is a path open cover of X }

is a neighbourhood family on π

(X, x

). Therefore, we call the subgroup topology

on π

(X, x

) determined by Σ

pSpan

the path Spanier topology.

9. The shape topology: Let cov(X) be the directed set of pairs (U, U

) where

U is a locally ﬁnite open cover of X and U

is a distinguished element of U containing

. Here, cov(X) is directed by reﬁnement. Given (U, U

) ∈ cov(X) let N(U) be the

abstract simplicial complex which is the nerve of U. In particular, U is the vertex

set of U and the n vertices U

, ..., U

span an n-simplex if and only if ∩

i=1

= ∅.

The geometric realization |N(U)| is a polyhedron and thus π

(|N(U)|, U

) may be

regarded naturally as a discrete group, e.g. if it is given the quotient topology.

Given a pair (V, V

) which reﬁnes (U, U

), a simplicial map p

U V

: |N(V)| → |N(U)| is

constructed by sending a vertex V ∈ V to some U ∈ U for which V ⊆ U (in particular,

is mapped to U

) and extending linearly. The map p

U V

is unique up to homotopy

and thus induces a unique homomorphism p

U V

♯

: π

(|N(V)|, V

) → π

(|N(U)|, U

The inverse system (π

(|N(U)|, U

), p

U V

♯

, cov(X)) of discrete groups is the nth pro-

homotopy group and the limit ˇπ

(X, x

) (topologized with the usual inverse limit

topology) is the n-th shape homotopy group. The induced continuous homomorphism

♯

: π

(X, x

) → π

(|N(U)|, U

) satisﬁes p

♯

◦ p

U V

♯

= p

♯

whenever (V, V

) reﬁnes

(U, U

). Thus there is a canonical, continuous homomorphism Ψ

: π

(X, x

) →

ˇπ

(X, x

) to the n-th shape homotopy group, given by Ψ

([α]) = ([p

◦ α])

The shape topology on π

(X, x

) is the initial topology with respect to the ﬁrst

shape homomorphism Ψ

: π

(X, x

) → ˇπ

(X, x

). Let π

(X, x

) denote π

(X, x

)

equipped with the shape topology which is a topological group (see [9, Deﬁnition

3.1]).

10. The pseudometric topology: Let (X, d) be a path-connected metric space

and consider the uniform metric µ(α, β) = sup

t∈[0,1]

{d(α(t), β(t))} on Ω

(X, x

). Then

the function ρ : π

(X, x

) × π

(X, x

) → [0, ∞), ρ(a, b) = infµ(α, β)|α ∈ a, β ∈ b,

is a pseudometric on π

(X, x

) [9, Theorem 4.5]. Let π

met

(X, x

) denote the n-th

homotopy group equipped with the topology induced by the pseudometric ρ. Brazas

and Fabel called this topology the pseudometric topology (induced by d) (see [9]).

Some researchers have attempted to compare the above topologies as follows.

Ghane et al. in [14, p. 263] proved that π

qtop

(X, x

) is coarser than π

lim

(X, x

Babaee et al. in [4, p. 1441] mentioned that π

lim

(X, x

) coincides with the whisker

topology π

(X, x

). It is shown in [9, Proposition 3.2] that the shape topology

of π

(X, x

) is coarser than that of π

(X, x

). Note that by [6] one can show that

Span

(X, x

) is ﬁner than π

tSpan

(X, x

). By considering the deﬁnitions of π

qtop

(X, x

)

and π

(X, x

) it is easy to see that π

qtop

(X, x

) is ﬁner than π

(X, x

). The Spanier

topology is always ﬁner than the shape topology on π

(X, x

) [3, Remark 5.5].

Brazas and Fabel proved that if X is a compact metric space, then the topology

of π

met

(X, x

) is at least as ﬁne as that of π

(X, x

) [9, Proposition 5.2].

4. Comparison of Topologies on Homotopy Groups

In this section, using some properties of the subgroup topology presented in [8, 21]

and some results of other researchers, we intend to compare the various topologies

on homotopy groups mentioned in Section 3 with each other as much as possible.

In the following theorem, we compare the Spanier topology on homotopy groups

with the subgroup topology induced by the n-Spanier subgroup.

Theorem 4.1. (i) If H = π

(X, x

), then π

Span

(X, x

) ≼ π

(X, x

(ii) If π

Span

(X, x

) = π

(X, x

) for a subgroup K of π

(X, x

), then K = π

(X, x

Proof. (i) It is known that π

Span

(X, x

) has subgroup topology with respect to

Span

= {K ⩽ π

(X, x

) | K is an n-Spanier subgroup} (see [6]). Since H =

(X, x

), Σ

= {K ⩽ π

(X, x

) | π

(X, x

) ⊆ K} and π

(X, x

) = ∩

K∈Σ

Span

we have Σ

Span

⊆ Σ

. Hence π

Span

(X, x

) ≼ π

(X, x

(ii) It holds by [21, Theorem 2.3].

Remark 4.2. Bahredar et al. in [6] mentioned that if U is an open cover of X,

then π

(U, x

) ⊆ π

tsp

(U, x

) ⊆ π

wtsp

(U, x

). Also, it is easy to see that if V is a

path open cover of X, then eπ

(V, x

) ⊆ π

(V, x

). Thus similar to the proof of the

above theorem we have the following comparison of some topologies on the homotopy

groups.

(i) If H = π

tsp

(X, x

), then π

tSpan

(X, x

) ≼ π

(X, x

(ii) If π

tSpan

(X, x

) = π

(X, x

) for a subgroup K of π

(X, x

), then K = π

tsp

(X, x

(iii) If H = π

wtsp

(X, x

), then π

wtSpan

(X, x

) ≼ π

(X, x

(iv) If π

wtSpan

(X, x

) = π

(X, x

) for a subgroup K of π

(X, x

), then K = π

wtsp

(X, x

(v) If H = eπ

(X, x

), then π

pSpan

(X, x

) ≼ π

(X, x

(vi) If π

pSpan

(X, x

) = π

(X, x

) for a subgroup K of π

(X, x

), then K = eπ

(X, x

(vii) π

wtSpan

(X, x

) ≼ π

tSpan

(X, x

) ≼ π

Span

(X, x

) ≼ π

pSpan

(X, x

Remark 4.3. (i) In [6, Proposition 3.15], Bahredar et al. proved that for any pointed

space (X, x

), π

wtsp

(X, x

) ≤ kerΨ

, where Ψ

is the canonical map from the nth

homotopy group to the nth shape homotopy group. It is clear that kerΨ

≤ kerp

♯

On the other hand, Brazas and Fabel in [9] mentioned that the shape topology on

homotopy groups is generated by left cosets of subgroups of the form kerp

∗

. Thus,

we have the following result, which reﬁnes Corollary 5.7 from [9].

For any pointed space (X, x

) we have

(X, x

) ≼ π

wtSpan

(X, x

) ≼ π

tSpan

(X, x

) ≼ π

Span

(X, x

) ≼ π

pSpan

(X, x

(ii) If X is T

and paracompact space, then π

(X, x

) = π

tsp

(X, x

) = π

wtsp

(X, x

Thus π

wtsp

(X,x

)

(X, x

) = π

tsp

(X,x

)

(X, x

) = π

(X,x

)

(X, x

). Moreover, this iden-

tity holds if X is metrizable or path connected topological group (see [6, Theorem

3.11, Theorem 3.12]).

Deﬁnition 4.4. Let X be a topological space and H be a subgroup of π

(X, x

). Then

X is called n-semilocally H-connected at x

if there exists an open neighborhood U

of x

with π

(i)(π

(U, x

)) ≤ H.

Abdullahi et al. in [1, Theorem 4.2] proved that if H ≤ π

(X, x

), then X is

semilocally H-connented at x

if and only if H is open in π

(X, x

). Now we

generalize this result to homotopy groups as follows.

Theorem 4.5. Let H ≤ π

(X, x

), then X is n-semilocally H-connected at x

if and

only if H is an open subgroup of π

(X, x

Proof. Let X be n-semilocally H-connected at x

, then there is an open neighborhood

U of x

such that π

(i)(π

(U, x

)) ≤ H. By deﬁnition we have π

(i)(π

(U, x

)) ∈

, therefore H is open in π

(X, x

). Conversely, if H is an open subgroup of

(X, x

), then there exists an open neighborhood U of x

such that π

(i)(π

(U, x

))

≤ H. Hence X is n-semilocally H-connected at x

Using the above theorem and the deﬁnition of the whisker topology we have the

following result.

Corollary 4.6. Let H ≤ π

(X, x

), then X is n-semilocally H-connected at x

and only if π

(X, x

) ≼ π

(X, x

Let [α] ∈ ∩{π

(i)(π

(U, x

))| U is an open neighbourhood of x

}, then α has a

homotopic representative in any open neighbourhood of x

, that is, α is a small

n-loop at x

. Thus, the inﬁnitesimal subgroup of π

(X, x

) is equal to π

(X, x

Hence, π

(X, x

) ≼ π

(X,x

)

(X, x

). Note that the equality does not hold in general.

By considering the n-dimensional Hawaiian earring , HE

, at the origin θ, we have

(HE

,θ)

(HE

, θ) is discrete but π

(HE

, θ) is not discrete (see [4, Example 4.6]).

In the following we present an equivalent condition for the equality.

Corollary 4.7. Let (X, x

) be a pointed topological space, then

(X, x

≼ π

(X,x

)

(X, x

Also, we have X is n-semilocally π

(X, x

)-connected at x

if and only if π

(X, x

) =

(X,x

)

(X, x

). Moreover, if π

(X, x

) = π

(X, x

) for a subgroup K of π

(X, x

then K = π

(X, x

Proof. Since the inﬁnitesimal subgroup of π

(X, x

) is equal to π

(X, x

) we have

(X, x

) ≼ π

(X,x

)

(X, x

). Let X be n-semilocally π

(X, x

)-connected at x

Then by Corollary 4.6, π

(X,x

)

(X, x

) ≼ π

(X, x

). Thus we have π

(X, x

) =

(X,x

)

(X, x

). The converse is true by Corollary 4.6. By [21, Theorem 2.3] if

(X, x

) = π

(X, x

) for a subgroup K of π

(X, x

), then K = π

(X, x

Remark 4.8. (i) Note that by [9, Proposition 3.2] we have π

(X, x

) ≼ π

(X, x

Also, by [3, Lemma 3.10] we have π

(X, x

) ≼ π

Span

(X, x

). Note that by [3, Lemma

5.1] the equality holds if X is paracompact, Hausdorf and LC

n−1

space. A space X

is called LC

at x ∈ X if for every neighborhood U of x, there exists a neighborhood

V of x such that V ⊆ U and such that for all 0 ≤ k ≤ n (k < ∞ if n = ∞) every

map f : ∂∆

k+1

→ V extends to a map g : △

k+1

→ U (see [3, Deﬁnition 4.3]).

(ii) Note that π

qtop

(X, x

) is coarser than π

lim

(X, x

) (see [14, p. 263]). Also, by [4,

Page 1441] π

lim

(X, x

) coincides with the whisker topology π

(X, x

(iii) By [9, Proposition 4.13], π

qtop

(X, x

) and π

(X, x

) are at least as ﬁne as that

of π

met

(X, x

). Also, by [9, Corollary 5.12], π

met

(X, x

) ≼ π

Span

(X, x

(iv) Since π

(X, x

) is an abelian group for each n ≥ 2 by [21, Theorem 2.2] the

topologized homotopy group π

(X, x

) is a topological group for any subgroup H of

(X, x

),.

(v) In [21, Theorem 2.4] the authors proved that if H ≤ K ≤ G, then G

≼ G

Thus we have

wtsp

(X,x

)

(X, x

) ≼ π

tsp

(X,x

)

(X, x

) ≼ π

(X,x

)

(X, x

)

≼ π

eπ

(X,x

)

(X, x

) ≼ π

(X,x

)

(X, x

) ≼ π

(X,x

)

(X, x

Finally, we summarize all results of this section in order to compare various

topologies on the homotopy groups in the following diagram (note that A −→ B

means that A ≼ B).

(X,x

)

(X, x

)

(X,x

)

(X, x

)

(2)

lim

(X, x

) = π

(X, x

)

(1)

qtop

(X, x

)

(3)

(X, x

)

(4)

pSpan

(X, x

)

(5)

eπ

(X,x

)

(X, x

)

(6)

met

(X, x

)

(18)

(19)

;;

Span

(X, x

)

(7)

(9)

(X,x

)

(X, x

)

(8)

tSpan

(X, x

)

(10)

(12)

tsp

(X,x

)

(X, x

)

(11)

wtSpan

(X, x

)

(13)

(15)

wtsp

(X,x

)

(X, x

)

(14)

(X, x

)

(16)

(17)

]]

In the following, according to the enumeration in the above diagram, we give

references and complementary notes for each arrow.

(1) See Corollary 4.7. By Theorem 4.5 the equality holds if and only if X is

n-semilocally π

(X, x

)-connected at x

. Since π

(HE

, θ) = 1 the strict in-

equality holds for HE

at the origin θ (see [4, Example 4.6]).

(2) See Remark 4.8 (v). The equality holds if and only if π

(X, x

) = π

(X, x

)

by [21, Theorem 2.3].

(3) See Remark 4.8 (ii).

(4) See Deﬁnition 3. The equality holds if and only if π

qtop

(X, x

) is a topological

group. The strict inequality holds for space X in [12], because π

qtop

(X, p) is

not a topological group.

(5) By Remark 4.8 (i) if X is paracompact, Hausdorf and LC

n−1

space, then

pSpan

(X, x

) = π

Span

(X, x

) = π

(X, x

) ≼ π

(X, x

). We conjecture that

(X, x

) is ﬁner than π

pSpan

(X, x

) under mild conditions.

(6) See Remark 4.8 (v). The equality holds if and only if eπ

(X, x

) = π

(X, x

)

by [21, Theorem 2.3].

(7) See Remark 4.3 (i). By [21, Theorem 2.4] if eπ

(X, x

) = π

(X, x

) and

(X, x

) is open in π

Span

(X, x

), then the equality holds.Also, by [21, Theo-

rem 2.3] if the equality holds, then eπ

(X, x

) = π

(X, x

(8) See Remark 4.8 (v). The equality holds if and only if eπ

(X, x

) = π

(X, x

)

by [21, Theorems 2.3, 2.4].

(9) See Theorem 4.1. By [21, Theorem 2.4] if π

(X, x

) is open in π

Span

(X, x

then the equality holds.

(10) See Remark 4.3 (i). By [21, Theorem 2.4] if π

(X, x

) = π

tsp

(X, x

) and

tsp

(X, x

) is open in π

tSpan

(X, x

), then the equality holds. Also, by [21,

Theorem 2.3] if the equality holds, then π

(X, x

) = π

tsp

(X, x

(11) See Remark 4.8 (v). The equality holds if and only if π

(X, x

) = π

tsp

(X, x

)

by [21, Theorems 2.3, 2.4].

(12) See Remark 4.2 (i). By [21, Theorem 2.4] if π

tsp

(X, x

) is open in π

tSpan

(X, x

then the equality holds.

(13) See Remark 4.3 (i). By [21, Theorem 2.4] if π

wtsp

(X, x

) = π

tsp

(X, x

) and

wtsp

(X, x

) is open in π

wtSpan

(X, x

), then the equality holds. Also, by [21,

Theorem 2.3] if the equality holds, then π

wtsp

(X, x

) = π

tsp

(X, x

(14) See Remark 4.8 (v). The equality holds if and only if π

tsp

(X, x

) = π

wtsp

(X, x

)

by [21, Theorem 2.3, Theorem 2.4].

(15) See Remark 4.2 (iii). By [21, Theorem 2.4] if π

wtsp

(X, x

) is open in

wtSpan

(X, x

), then the equality holds.

(16) See Remark 4.3 (i).

(17) It holds for compact metric spaces by [9, Proposition 5.2]. If X is a path-

connected compact metric space, then the equality holds in the following two

case:

(i) if X is LC

n−1

(ii) if X = lim

←−

j∈N

, r

j+1,j

) is an inverse limit of ﬁnite polyhedra where the

bonding maps r

j+1,j

: X

j+1

→ X

are retractions (see [9, Theorem 1.1]).

(18) See Remark 4.8 (iii).

(19) See Remark 4.8 (iii).

In order to investigate further the above diagram, we raise some questions in

the following which we are interested in ﬁnding answers to them (the questions are

numbered to correspond with the respective arrow numbers in the diagram.).

(Q2) Is there a space X for which the strict inequality π

(X,x

)

(X, x

) ≺

(X,x

)

(X, x

) holds when n ≥ 2? Note that the strict inequality holds when

n = 1 for HA at b = 0 since π

(HA, b) = 1 and π

(HA, b) = π

(HA, b) ([1,

Example 3.12]).

(Q3) Is there a necessary and suﬃcient condition on X for the equality π

qtop

(X, x

) =

(X, x

) when n ≥ 2? Note that if X is locally path connected, then the

equality holds if and only if X is SLT at x

(see [18, Corollary 3.3]). Also, is

there a space X for which the strict inequality π

qtop

(X, x

) ≺ π

(X, x

) holds

when n ≥ 2? Note that the strict inequality holds when n = 1 for HE (see [2,

Example 3.25]).

(Q5) Is it true that π

pSpan

(X, x

) ≼ π

(X, x

Nasri et. al in [17, Theorem 2.1] proved that π

qtop

(X, x

)

∼

qtop

n−k

(Ω

(X, x

), e

where e

is a constant k-loop in X at x

. Also they presented the following

commutative diagram:

Ω

(X, x

)



Ω

n−k

(Ω

(X, x

), e

)



qtop

(X, x

)

∗

qtop

n−k

(Ω

(X, x

), e

)

where ϕ : Ω

(X, x

) → Ω

n−k

(Ω

(X, x

), e

) given by ϕ(f) = f

♯

is a homeomor-

phism with inverse g 7→ g

♭

in the sense of [20]. Since q is a quotient map, the ho-

momorphism ϕ

∗

is an isomorphism between quasitopological homotopy groups.

By the above diagram if we can prove that ϕ

∗

(eπ

(V, x

)) = eπ

n−1

′

, e

), then

we can conclude that π

pSpan

(X, x

) ≼ π

(X, x

(Q6) Is there a space X for which the strict inequality π

eπ

(X,x

)

(X, x

) ≺

(X,x

)

(X, x

) holds when n ≥ 2? Note that the strict inequality holds when

n = 1 for the space RX in [1, Example 2.5] (see [21, Example 2]).

(Q7) Is there a space X for which the strict inequality π

Span

(X, x

) ≺ π

pSpan

(X, x

)

holds?

(Q8) Is there a space X for which the strict inequality π

(X,x

)

(X, x

) ≺

eπ

(X,x

)

(X, x

) holds?

(Q9) Is there a space X for which the strict inequality π

Span

(X, x

) ≺ π

(X,x

)

(X, x

)

holds?

(Q10) Is there a space X for which the strict inequality π

tSpan

(X, x

) ≺ π

Span

(X, x

)

holds?

(Q11) Is there a space X for which the strict inequality π

tsp

(X,x

)

(X, x

) ≺

(X,x

)

(X, x

) holds?

(Q12) Is there a space X for which the strict inequality π

tSpan

(X, x

) ≺

tsp

(X,x

)

(X, x

) holds?

(Q13) Is there a space X for which the strict inequality π

wtSpan

(X, x

) ≺ π

tSpan

(X, x

)

holds?

(Q14) Is there a space X for which the strict inequality π

wtsp

(X,x

)

(X, x

) ≺

tsp

(X,x

)

(X, x

) holds?

(Q15) Is there a space X for which the strict inequality π

wtSpan

(X, x

) ≺

wtsp

(X,x

)

(X, x

) holds?

(Q16) Is there a necessary and suﬃcient condition on X for the equality π

(X, x

) =

wtSpan

(X, x

)? Is there a space X for which the strict inequality π

(X, x

) ≺

wtSpan

(X, x

) holds?

(Q17) Is there a compact metric space X for which the strict inequality π

(X, x

) ≺

met

(X, x

) holds?

(Q18) Is there a necessary and suﬃcient condition on X for the equality π

met

(X, x

) =

Span

(X, x

)? Is there a space X for which the strict inequality π

met

(X, x

) ≺

Span

(X, x

) holds?

(Q19) Is there a necessary and suﬃcient condition on X for the equality π

met

(X, x

) =

(X, x

)? Is there a space X for which the strict inequality π

met

(X, x

) ≺

(X, x

) holds?

Declaration of Interest Statement

The authors declare that they have no known competing ﬁnancial interests or per-

sonal relationships that could have appeared to inﬂuence the work reported in this

paper.

References

[1] M. Abdullahi Rashid, B. Mashayekhy, H. Torabi, S.Z. Pashaei, On subgroups of

topologized fundamental groups and generalized coverings, Bull. Iranian Math.

Soc., 43(7), 2349-2370, (2017).

[2] M. Abdullahi Rashid, N. Jamali, B. Mashayekhy, S.Z. Pashaei, H. Torabi, On

subgroup topologies on fundamental groups, Hacet. J. Math. Stat., 49(3), 935-

949, (2020).

[3] J. Aceti, J. Brazas, Elements of higher homotopy groups undetectable by poly-

hedral approximation, Paciﬁc J. Math., 322(2), 221-242, (2023).

[4] A. Babaee, B. Mashayekhy, H. Mirebrahimi, H. Torabi, M.A. Rashid, S.Z.

Pashaei, On topological homotopy groups and relation to Hawaiian groups,

Hacet. J. Math. Stat., 49(4), 1437-1449, (2020).

[5] A.A. Bahredar, N. Kouhestani, H. Passandideh, The n-dimensional Spanier

group, Filomat, 35(9), 3169-3182,(2021).

[6] A.A. Bahredar, N. Kouhestani, H. Passandideh, On (Thick, Weak Thick)Spanier

topology on nth homotopy group, Filomat, 38(12), 4381–4394,(2024).

[7] J. Brazas, The fundamental group as topological group, Topology Appl., 160

170-188 (2013).

[8] W.A. Bogley, A.J. Sieradski, Universal path spaces,

http://people.oregonstate.edu/~bogleyw/research/ups.pdf

[9] J. Brazas, P. Fabel, A natural pseudometric on homotopy groups of metric

spaces, Glasgow Math. J., 66, 162-174, (2024).

[10] J.S. Calcut, J.D. McCarthy, Discreteness and homogeneity of the topological

fundamental group, Topology Proc., 34, 339–349, (2009).

[11] P. Fabel, Multiplication is discontinuous in the Hawaiian earring group (with

the quotient topology), Bull. Pol. Acad. Sci. Math., 59 (1), 77–83, (2011).

[12] P. Fabel, Compactly generated quasitopological homotopy groups with discon-

tinuous multiplication, Topology Proc., 40, 303–309, (2012).

[13] F.H. Ghane, Z. Hamed, B. Mashayekhy and H. Mirebrahimi, Topological ho-

motopy groups, Bull. Belg. Math. Soc. Simon Stevin, 15, 455-464, (2008).

[14] F.H. Ghane, Z. Hamed, B. Mashayekhy and H. Mirebrahimi, On topological

homotopy groups of n-Hawaiian like spaces, Topology Proc., 36, 255-266, (2010).

[15] G .Naber, Topology, Geometry and Gauge ﬁelds, Foundations, Addison-Wesley

publishing company, U.S.A, INC, 1961.

[16] T. Nasri, B. Mashayekhy, H. Mirebrahimi, On quasitopological homotopy groups

of inverse limit spaces, Topology Proc., 46, 145-157, (2015).

[17] T. Nasri, H. Mirebrahimi, H. Torabi, Some results in quasitopological homotopy

groups, Ukrainian Math. J., 72(12), 1663-1668, (2020).

[18] S.Z. Pashaei, B. Mashayekhy, H. Torabi, M. Abdullahi Rashid, Small loop

transfer spaces with respect to subgroups of fundamental groups, Topology

Appl., 232, 242-255, (2017).

[19] H. Passandideh, F.H. Ghane and Z. Hamed, On the homotopy groups of sepa-

rable metric spaces, Topology Appl., 158, 1607-1614, (2011).

[20] J.J. Rotman, An Introduction to Algebraic Topology Springer-Verlag, GTM

119, New York, 1988.

[21] N. Shahami, B. Mashayekhy, Comparison of topologies on fundamental groups

with subgroup topology viewpoint, Math. Slovaca, 75(1), 189-204, (2025).

[22] H. Torabi, A. Pakdaman, B. Mashayekhy, On the Spanier groups and covering

and semicovering spaces, arXiv:1207.4394v1.

[23] J. Wilkins, The revised and uniform fundamental groups and universal covers

of geodesic spaces, Topology Appl., 160, 812–835, (2013).