# On Hawaiian homology groups

Hamid Torabi*, Hanieh Mirebrahimi* and Ameneh Babaee  * Department of Pure
Mathematics, Center of Excellence in Analysis on Algebraic Structures,
Ferdowsi University of Mashhad,
P.O.Box 1159-91775, Mashhad, Iran<EMAIL_ADDRESS><EMAIL_ADDRESS>(H. Mirebrahimi<EMAIL_ADDRESS>(A. Babaee)

###### Abstract.

In this paper, we introduce a kind of homology which we call Hawaiian homology
to study and classify pointed topological spaces. The Hawaiian homology group
has advantages of Hawaiian groups. Moreover, the first Hawaiian homology group
is isomorphic to the abelianization of the first Hawaiian group for path-
connected and locally path-connected topological spaces. Since Hawaiian
homology has concrete elements and abelian structure, its calculations are
more routine. Thus we use Hawaiian homology groups to compare Hawaiian groups,
and then we obtain some information about Hawaiian groups of some wild
topological spaces.

###### Key words and phrases:

Hawaiian group, singular homology, archipelago space, Higmann-complete group,
cotorsion group

###### 2010 Mathematics Subject Classification:

Primary 55Q05, 55Q20; Secondary 55P65, 55Q52

##  1\. Introduction and Motivation

Homology is a well-known useful tool of algebraic topology. Since homology has
concrete interpretation, different fields of science apply it as an algebraic
modelling of geometric properties to study their observations mathematically
[13]. Homology is a general way of associating a sequence of algebraic
objects, such as abelian groups or modules, with other mathematical objects
such as topological spaces. Homology groups were originally defined in
algebraic topology and then it is generalized in a wide variety of other
contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and
algebraic geometry.

The original motivation for defining homology groups was to distinguish shapes
by examining their holes. Homology group has a famous characteristic, the
Betti number, which is known as the various counting the number of holes of
the spaces, and by this fact, the Betti number helps to classify spaces. The
$n$th Betti number is defined as the rank of the $n$th homology group of the
given topological space.

There are many different homology theories, each of which has its advantages,
applications, and defects [1, 19]. A particular type of mathematical object,
such as a topological space or a group, may have one or more associated
homology theories. When the underlying object has a geometric interpretation
as topological spaces do, the $n$th homology group represents the behavior in
the dimension $n$. Most homology groups or modules may be formulated as
derived functors on appropriate abelian categories, measuring the failure of a
functor to be exact.

In Section 2, we define a Hawaiian homology similar to the singular homology
theory, with the difference whose simplexes are shrinking sequences of
singular simplexes tending to the base point. Thus, it is natural that
Hawaiian homology groups depend on the choice of the base point. More
precisely, the Hawaiian homology is a covariant functor from the category of
pointed topological spaces to the category of abelian groups. We see that
Hawaiian homology groups depend on the local behavior of the spaces at their
base points. This is a great difference between the Hawaiian homology and the
singular homology whose structure depends only on the homotopy kind of space,
while the base point has an essential role in calculating the Hawaiian
homology. In fact, the Hawaiian homology is an invariant of those homotopies
preserving the base point called pointed homotopies. We present some examples
to clarify the effects of choice of the base point and pointed homotopies.

Also, we investigate basic properties of the Hawaiian homology, including the
Hawaiian homology of the singletons, disjoint union, homotopic maps, and so
on. Then we present a close relation between the Hawaiian homology group and
Hawaiian group of topological spaces. In fact, the first Hawaiian homology
group equals the abelianization of the first Hawaiian group for any path-
connected and locally path-connected topological space. The $n$th Hawaiian
group of a pointed topological space was defined by Karimov and Repovš [15] as
the set of all pointed homotopy classes of continuous maps with a group
operation coming from the operation of $n$th homotopy groups. The $n$th
Hawaiian group, defined below, is a covariant functor from the category of
pointed topological spaces to the category of groups, denoted by
${\mathcal{H}}_{n}(X,x_{0})$, where $(X,x_{0})$ is a pointed topological
space.

######  Definition 1 ([15]).

Let $(X,x_{0})$ be a pointed space, let $[-]$ denote the class of pointed
homotopy, and let $n=1,2,\ldots$. Then the $n$th Hawaiian group of $(X,x_{0})$
is defined by $\mathcal{H}_{n}(X,x_{0})=\\{[f]:\
f:(\mathbb{HE}^{n},\theta)\to(X,x_{0})\\}$ with the multiplication induced by
$(f*g)|_{\mathbb{S}_{k}^{n}}=f|_{\mathbb{S}_{k}^{n}}\cdot
g|_{\mathbb{S}_{k}^{n}}$ ($k\in\mathbb{N}$) for any
$[f],[g]\in\mathcal{H}_{n}(X,x_{0})$, where $\cdot$ denotes the concatenation
of $n$-loops.

The Hawaiian functor has some advantages over other well-known covariant
functors from the category of topological spaces to the category of groups.
Unlike homotopy and homology groups, Hawaiian groups do not preserve free
homotopy equivalences. As an example, consider the cone space
$C(\mathbb{HE}^{1})$, where $\mathbb{HE}^{1}$ denotes the one-dimensional
Hawaiian earring. Since cone spaces are contractible, their homotopy,
homology, and cohomology groups are trivial [15], but the first Hawaiian group
of $C(\mathbb{HE}^{1})$ is uncountable [15]. Also, by using this functor, we
can study some local behaviors of spaces. For instance, if $X$ has a countable
local basis at $x_{0}$ and the $n$th Hawaiian group
${\mathcal{H}}_{n}(X,x_{0})$ is countable, then $X$ is locally $n$-simply
connected at $x_{0}$ (see [15, Theorem 2]). Moreover, let $X$ have a countable
local basis at $x_{0}$. Then $\mathcal{H}_{n}(CX,x)$ is trivial if and only if
$X$ is locally $n$-simply connected at $x_{0}$ and it is uncountable otherwise
[2, Corollaries 2.16 and 2.17]. Hence, unlike homotopy groups, Hawaiian groups
depend on the behavior of the space at the base point. In this regard, there
are path-connected spaces whose Hawaiian groups are not isomorphic at
different points, such as the $n$-dimensional Hawaiian earring, where $n\geq
2$ (see [2, Corollary 2.11]). In this paper, we show that the Hawaiian
homology groups have the advantages of Hawaiian groups, from the viewpoint of
homology. Moreover, Hawaiian homology groups have more concrete elements, and
also they can be computed by techniques of abelian groups (see theorem 4.2),
while there rarely exist similar techniques to calculate Hawaiian groups. By
using these facts, we use Hawaiian homology groups to study and classify
pointed topological spaces; for instance, the first Hawaiian homology group of
the Harmonic archipelago at the origin is not isomorphic to the first Hawaiian
homology group at any point else. Also since the first Hawaiian homology group
for any locally path-connected space is the abelianization of the first
Hawaiian group, it can help us to investigate the structure of Hawaiian
groups. Therefore, by this fact that the first Hawaiian homology groups are
not isomorphic at different points, the first Hawaiian groups of the Harmonic
archipelago are not isomorphic.

In Section 3, by employing Hawaiian homology groups, we compare Hawaiian
groups of some pointed spaces, for instance, the Harmonic archipelago,
Griffiths space, Hawaiian earrings, and so on. Moreover, we calculate Hawaiian
homology groups of the Harmonic archipelago and prove that the first Hawaiian
homology and the first singular homology groups of the Harmonic archipelago
are isomorphic. The Harmonic archipelago is an element of a class of spaces
called archipelago spaces constructed by attaching loops to some weak join of
spaces. Hawaiian earrings belong to the class of weak join spaces defined as
follows.

######  Definition 2 ([10, 6]).

The weak join of a family of spaces $\\{(X_{i},x_{i});i\in I\\}$, denoted by
$(X,x_{*})=\widetilde{\bigvee}_{i\in I}(X_{i},x_{i})$, is the underlying space
of wedge space $(\hat{X},\hat{x}_{*})=\bigvee_{i\in I}(X_{i},x_{i})$ with the
weak topology with respect to $X_{i}$’s, except at the common point. Every
open neighborhood in $X$ at $x_{*}$ is of the form $\bigcup_{i\in
F}U_{i}\cup\bigcup_{i\in I\setminus F}X_{i}$, where $U_{i}$ is an open
neighborhood at $x_{i}$ in $X_{i}\subset X$ and $F$ is a finite subset of $I$
(see [6, Section 2] and [10, p. 18]).

In [17, Definition 2.1], small $n$-Hawaiian loop was defined as a small map
from $\mathbb{HE}^{n}$ to any space $X$. All spaces are assumed to be first
countable. In this paper, by ${\widehat{\qquad}}^{p}$ we mean the $p$-adic
completion of a group and $P$ denotes the set of all primes.

##  2\. Hawaiian Homology groups

In this section, we define the Hawaiian homology as a new invariant from the
category of pointed topological spaces to the category of abelian groups.
First, we define the Hawaiian simplex in the natural way. That is, the
disjoint union of a countably infinite family of Euclidean simplexes whose
diameters tend to zero together with the origin as the cluster point.

######  Definition 3.

Let $\Delta_{k}^{n}=\Delta^{n}/k=\\{a\in\mathbb{R}^{n+1}|\
ka\in\Delta^{n}\\}$, for $k=1,2,\ldots$, where $\Delta^{n}$ denotes the
standard Euclidean $n$-simplex, and let $\Delta_{0}^{n}=\\{\theta\\}$, where
$\theta$ denotes the origin. The $n$-Hawaiian simplex is defined as follows:

$H\Delta^{n}=\bigcup_{k=0}^{\infty}\Delta^{n}_{k}.$

Moreover, an $n$-Hawaiian simplex in the given space $(X,x_{0})$ is a
continuous map $\sigma^{n}:(H\Delta^{n},\theta)\to(X,x_{0})$. The restriction
of $\sigma^{n}$ on the $k$th simplex $\Delta_{k}^{n}$ can be assumed as an
$n$-simplex in $X$, denoted by $\sigma_{k}^{n}=\sigma^{n}|_{\Delta_{k}^{n}}$.

Similar to the definition of singular homology, the group $S_{n}(X,x_{0})$ is
defined as the free abelian group generated by all $n$-Hawaiian simplexes in
$(X,x_{0})$. To define the boundary homomorphism
$\partial_{n}:S_{n}(X,x_{0})\to S_{n-1}(X,x_{0})$, we need to introduce face
maps. Let $k\in\mathbb{N}$. Consider
$\varepsilon_{i,k}^{n}:\Delta_{k}^{n-1}\to\Delta_{k}^{n}$ as the map defined
by the $i$th face-map on Euclidean $(n-1)$-simplex together with the
appropriate coefficient associated with $k\in\mathbb{N}$. Then let
$\varepsilon_{i}^{n}=\bigcup_{k=0}^{\infty}\varepsilon_{i,k}^{n}:\bigcup_{k=0}^{\infty}\Delta^{n-1}_{k}\to\bigcup_{k=0}^{\infty}\Delta^{n}_{k}$
be the map obtained by the union of all $\varepsilon^{n}_{i,k}$’s for
$k\in\mathbb{N}$ and the constant map for $k=0$. The mapping
$\varepsilon_{i}^{n}$ is continuous, because $\Delta_{k}^{n-1}$ is mapped into
$\Delta_{k}^{n}$ by a continuous mapping for each $k\in\mathbb{N}$ and then it
maps each convergent sequence to a convergent sequence. Then
$\varepsilon_{i}^{n}$ is called the $i$th face-map of the $n$-Hawaiian
simplex. Now the boundary map $\partial_{n}:S_{n}(X,x_{0})\to
S_{n-1}(x,x_{0})$ is defined by
$\partial_{n}(\sigma^{n})=\sum_{i=0}^{n}(-1)^{i}\sigma\circ\varepsilon_{i}^{n}$.
Then we have the following chain complex:

$\cdots\longrightarrow
S_{n}(X,x_{0})\mathop{\longrightarrow}^{\partial_{n}}S_{n-1}(X,x_{0})\mathop{\longrightarrow}^{\partial_{n-1}}\cdots\longrightarrow
S_{1}(X,x_{0})\mathop{\longrightarrow}^{\partial_{1}}S_{0}(X,x_{0})\mathop{\longrightarrow}^{\partial_{0}}0.$

######  Remark 1.

To define Hawaiian homology group, we must verify the equality
$\partial_{n}\partial_{n+1}=0$. It can be checked by the definition of
boundary homomorphism as follows:

$\displaystyle\partial_{n}\partial_{n+1}(\sigma^{n+1})$
$\displaystyle=\partial_{n}\big{(}\sum_{j=0}^{n+1}(-1)^{j}\sigma^{n+1}\circ\varepsilon_{j}^{n+1}\big{)}$
$\displaystyle=\sum_{i,j}(-1)^{i+j}\sigma^{n+1}\circ\varepsilon^{n+1}_{j}\circ\varepsilon_{i}^{n}$
$\displaystyle=\sum_{i,j}(-1)^{i+j}\bigcup_{k=0}^{\infty}\big{(}\sigma^{n+1}_{k}\circ\varepsilon^{n+1}_{j,k}\circ\varepsilon_{i,k}^{n}\big{)}$
$\displaystyle=\bigcup_{k=0}^{\infty}\sum_{i,j}(-1)^{i+j}\big{(}\sigma^{n+1}_{k}\circ\varepsilon^{n+1}_{j,k}\circ\varepsilon_{i,k}^{n}\big{)}$
$\displaystyle=\bigcup_{k=1}^{\infty}\partial_{n}^{s}\partial_{n+1}^{s}(\sigma^{n+1}_{k})\cup\sum_{i,j}(-1)^{i+j}\big{(}\sigma^{n+1}_{0}\circ\varepsilon^{n+1}_{j,0}\circ\varepsilon_{i,0}^{n}\big{)},$

where $\partial^{s}$ is defined similar to the boundary homomorphism of
singular complex; see [19, Theorem 4.6]. Therefore
$\partial_{n}\partial_{n+1}(\sigma^{n+1})$ equals zero because it is the union
of zero homomorphisms in singular homology for $k=1,2,\ldots$ [19, Theorem
4.6]. For $k=0$, since
$\sigma^{n+1}_{0}\circ\varepsilon^{n+1}_{j,0}\circ\varepsilon_{i,0}^{n}$
equals some constant value, namely $b$, we have the following equalities

$\displaystyle\sum_{i,j}(-1)^{i+j}\big{(}\sigma^{n+1}_{0}\circ\varepsilon^{n+1}_{j,0}\circ\varepsilon_{i,0}^{n}\big{)}$
$\displaystyle=(-1)^{0}b+(-1)^{1}b+(-1)^{2}b+\ldots+(-1)^{(n+1)+n}b$
$\displaystyle=b\big{(}1+(-1)+1+(-1)+\ldots+1+(-1)\big{)}$
$\displaystyle=b(0)=0.$

Now, we are able to define the Hawaiian homology group as follows.

######  Definition 4.

Let $(X,x_{0})$ be a pointed topological space. Then the $n$th Hawaiian
homology group of $(X,x_{0})$, denoted by $\mathbb{H}_{n}(X,x_{0})$, equals
the quotient group $\frac{Z_{n}(X,x_{0})}{B_{n}(X,x_{0})}$, where
$Z_{n}(X,x_{0})=\ker\partial_{n}$ and $B_{n}(X,x_{0})=\mathrm{im}\
\partial_{n+1}$. By $\mathrm{cls}\alpha=\alpha+B_{n}(X,x_{0})$ we mean the
conjugacy class of $\alpha$.

As a simple example, we compute the first Hawaiian homology group of the unit
circle.

######  Example 1.

Let $\mathbb{S}^{1}$ denote the circle with radius $1$ in the Euclidean space.
Then
$\mathbb{H}_{1}(\mathbb{S}^{1},a)=\frac{Z_{n}(\mathbb{S}^{1},a)}{B_{n}(\mathbb{S}^{1},a)}$.
Define $\psi:Z_{n}(\mathbb{S}^{1},a)\to\sum_{\aleph_{0}}H_{1}(\mathbb{S}^{1})$
by the rule
$\psi(\sigma^{n})=\\{\mathrm{cls}\sigma^{n}|_{\Delta_{k}^{n}}\\}_{k\in\mathbb{N}}$.
Since simplexes $\\{\sigma^{n}|_{\Delta_{k}^{n}}\\}_{k\in\mathbb{N}}$ tend to
be small, they are contained in a contractible subset, and then they may be
replaced with constant simplexes. Hence just a finite number of non-null-
homotopic simplexes remain. Therefore,
$\mathbb{H}_{1}(\mathbb{S}^{1},a)\cong\sum_{\aleph_{0}}H_{1}(\mathbb{S}^{1})\cong\sum_{\aleph_{0}}\mathbb{Z}$.

By Definition 4, for each pointed topological space, we correspond a sequence
of abelian groups. To define a functor $\mathbb{H}_{n}:Top_{*}\to Ab$, $n\geq
0$, from the category of pointed topological spaces to the category of abelian
groups, we need to correspond morphisms; that is, for each pointed map
$f:(X,x)\to(Y,y)$, a homomorphism
$\mathbb{H}_{n}(f):\mathbb{H}_{n}(X,x)\to\mathbb{H}_{n}(Y,y)$ exists that
satisfies

  * (i)

$\mathbb{H}_{n}(id_{X})=id_{\mathbb{H}_{n}(X,x)}$,

  * (ii)

$\mathbb{H}_{n}(f\circ g)=\mathbb{H}_{n}(f)\circ\mathbb{H}_{n}(g)$.

Similar to the definition of the functor of singular homology group [19, p.
66], we define $\mathbb{H}_{n}(f):\mathbb{H}_{n}(X,x)\to\mathbb{H}_{n}(Y,y)$
by $\mathrm{cls}z_{n}\mapsto\mathrm{cls}f_{\\#}(z_{n})$, where
$f_{\\#}:S_{n}(X,x)\to S_{n}(Y,y)$ is defined by $f_{\\#}(\sum
m_{\sigma}\sigma)=\sum m_{\sigma}(f\circ\sigma)$. The same argument as used in
[19, Theorem 4.10] implies that $\mathbb{H}_{n}(f)$ satisfies conditions (i)
and (ii), and then $\mathbb{H}_{n}$ is a functor. Then it is natural to know
if the Hawaiian homology functor preserves pointed homotopy mappings. There
are some differences between the singular homology group, as the most familiar
homology theory, and the Hawaiian homology group. The first one is the
preservation of homotopy mappings; for the singular homology group, it
suffices to be a free homotopy for inducing an isomorphism, but for the
Hawaiian homology group, if the homotopy mapping does not preserve the base
point, it does not necessarily induce an isomorphism (see Example 2).

Now it is natural to know if the Hawaiian homology functors satisfy the
homology axioms.

### Axioms of homology

In homology theory, there are five properties called axioms of homology or
sometimes called Eilenberg-Steenrood axioms, which consists of dimension
axiom, additivity, homotopy, exactness and excision. These axioms classify
different theories of homology and also help to calculate the homology group
of some spaces such as $n$-spheres. We see that the Hawaiian homology groups
do not satisfy all of these axioms. In the following we investigate homology
axioms for Hawaiian homology.

  * •

Dimension axiom for the Hawaiian homology group, similar to the singular
homology group, can be verified directly by calculating boundary homomorphism;
see [19, Theorem 4.12].

  * •

A modified version of the additivity axiom holds for Hawaiian homology group
as stated in Theorem 2.1. Moreover, Example 3 shows that classical additivity
axiom does not hold for Hawaiian homology.

  * •

Homotopy axiom holds if the homotopy mapping preserves the base point
(Proposition 1). Moreover, consider the cone over the $1$-dimensional Hawaiian
earring. Its $1$st Hawaiian homology group is not trivial (See Theorem 2.2 and
[2, Theorem 2.13]), while it is contractible.

  * •

The exactness and excision axioms hold for the Hawaiian homology by the
definition 4 and the proof is straightforward.

######  Proposition 1.

Let $f,g:(X,x_{0})\to(Y,y_{0})$ be pointed maps and let $F:f\simeq g$ rel
$\\{x_{0}\\}$. Then $\mathbb{H}_{n}(f)=\mathbb{H}_{n}(g)$ for $n\geq 0$.

###### Proof.

To prove the equality $\mathbb{H}_{n}(f)=\mathbb{H}_{n}(g)$, for every
$n$-Hawaiian simplex $\alpha$, we show that
$\mathbb{H}_{n}(f)(\mathrm{cls}\alpha)=\mathbb{H}_{n}(g)(\mathrm{cls}\alpha)$.
It is equivalent to $f_{\\#}(\alpha)-g_{\\#}(\alpha)$ being the boundary of an
$(n+1)$-Hawaiian simplex. By [19, Theorem 4.23], since $F:f\simeq g$,
$f^{s}_{\\#}(\alpha|_{\Delta^{n}_{k}})-g^{s}_{\\#}(\alpha|_{\Delta^{n}_{k}})$
equals the boundary of an $(n+1)$-simplex in $Y$, namely $\beta_{k}$ is
constructed by $F(\alpha|_{\Delta^{n}_{k}},-)$ for $k=0,1,2,\ldots$. Since $F$
is a pointed homotopy mapping, and $\mathbb{I}$ is compact, the images of
$\beta_{k}$’s tend to the point $y_{0}$ (for a detailed proof see [2, Theorem
2.13]). Then one can define the $(n+1)$-Hawaiian simplex
$\beta=\cup_{k=0}^{\infty}\beta_{k}$, where $\beta_{0}$ is the constant map at
the point $y_{0}$. Moreover, since
$f^{s}_{\\#}(\alpha|_{\Delta^{n}_{k}})-g^{s}_{\\#}(\alpha|_{\Delta^{n}_{k}})$
for each $k=0,1,\ldots$, $f_{\\#}(\alpha)-g_{\\#}(\alpha)$ equals the boundary
of $\beta$. ∎

A space $X$ is called semilocally strongly contractible at the point $x_{0}$
if there exists an open neighborhood $U$ of $x_{0}$ such that the inclusion
map $i:U\to X$ is homotopic to the constant map relative to the point $x_{0}$.
Since the image of any Hawaiian simplex is contained in $U$ except a finite
number of standard simplexes and $U$ can be contracted to $x_{0}$, Hawaiian
simplexes have no information more than finite standard simplexes. Example 1
calculates the Hawaiian homology group of the circle $\mathbb{S}^{1}$, and it
is formally generalized as follows.

######  Corollary 1.

Let $X$ be semilocally strongly contractible at the point $x_{0}$. Then

$\mathbb{H}_{n}(X,x_{0})\cong\sum_{\aleph_{0}}H_{n}(X).$

###### Proof.

We consider $H_{n}(X)$ as a subgroup of $\mathbb{H}_{n}(X,x_{0})$, where each
simplex vanishes except the $k$th one corresponded to the elements of
$H_{n}(X)$. The intersections vanish, because each factor is corresponded to a
unique $k\in\mathbb{N}$. It remains to verify that $\mathbb{H}_{1}(X,x_{0})$
is generated by $\sum_{\aleph_{0}}H_{n}(X)$. Let $\sigma$ be an $n$-Hawaiian
simplex in $X$ at $x_{0}$. Since $X$ is semilocally strongly contractible at
$x_{0}$, there exists a neighborhood $U$ of $x_{0}$ that is contractible in
$X$ at $x_{0}$. Since each $n$-Hawaiian simplex is a union of null-convergent
standard $n$-simplexes (see [2]), there is $K\in\mathbb{N}$ such that
$im(\sigma_{k})\subseteq U$ for $k\geq K$. Then $\bigcup_{k\geq K}\sigma_{k}$
is null-homotopic in $X$, and thus $\mathbb{H}_{n}(X,x_{0})$ is generated by
$\sum_{\aleph_{0}}H_{n}(X)$. ∎

We present a space in Example 2 to prove that a free homotopy equivalence does
not necessarily induce the isomorphism on Hawaiian homology groups.

######  Example 2.

Let $C(\mathbb{HE}^{1})$ be the cone over the one-dimensional Hawaiian
earring. Since the cones are contractible, the homotopy axiom implies that the
homology group must vanish, but it does not hold for the first Hawaiian
homology group. To prove this, consider the simple $1$-Hawaiian simplex
$\sigma$ whose image is the lowest Hawaiian earring. This Hawaiian simplex is
contained in $Z_{1}(C\mathbb{HE}^{1},\theta)=\ker\partial_{1}$, and also there
is no $2$-Hawaiian simplex whose boundary equals $\sigma$.

######  Theorem 2.1.

Let $X$ be semilocally path-connected at point $x_{0}$, let
$\\{X_{\lambda}\\}$ be the family of path components with $x_{0}\in
X_{\lambda_{0}}$, and let $n\geq 0$. Then

$\mathbb{H}_{n}(X,x_{0})\cong\big{(}\sum_{\lambda\neq\lambda_{0}}\sum_{\aleph_{0}}H_{n}(X_{\lambda})\big{)}\oplus\mathbb{H}_{n}(X_{\lambda_{0}},x_{0}).$
(2.1)

###### Proof.

First, note that for each finite sequence of standard simplexes, one may
construct a Hawaiian simplex by vanishing all terms of the sequence except the
finite number of given simplexes. Therefore,
$\sum_{\aleph_{0}}H_{n}(X_{\lambda})$ can be considered as a natural subgroup
of $\mathbb{H}_{n}(X,x_{0})$. Moreover, consider two elements
$\mathrm{cls}\sigma,\mathrm{cls}\beta\in\mathbb{H}_{n}(X_{\lambda_{0}},x_{0})$.
If $\mathrm{cls}\sigma=\mathrm{cls}\beta$, then $\sigma-\beta$ equals the
boundary of some $(n+1)$-Hawaiian simplexes in $X_{\lambda_{0}}$. Thus it
holds for $\sigma-\beta$ as an $n$-Hawaiian simplex in $X$. Then
$\mathrm{cls}\sigma=\mathrm{cls}\beta$ in $\mathbb{H}_{n}(X,x_{0})$. Hence,
$\mathbb{H}_{n}(X_{\lambda_{0}},x_{0})$ is a subgroup of
$\mathcal{H}_{n}(X,x_{0})$. Also, since $X$ is semilocally path-connected,
there exists a path-connected open neighborhood, namely $U$ of $X$ at $x_{0}$.
We show that the abelian group $\mathbb{H}_{n}(X,x_{0})$ is generated by the
family of subgroups
$\\{\sum_{\aleph_{0}}H_{n}(X_{\lambda})\\}_{\lambda\neq\lambda_{0}}\cup\\{\mathbb{H}_{n}(X_{\lambda_{0}},x_{0})\\}$.
To prove this, consider a Hawaiian simplex $\sigma$. Then there is a natural
number $K$ such that if $k\geq K$, then $im(\sigma(\Delta_{k}^{n}))\subseteq
U$. Note that since $U$ is path-connected, $U\subseteq X_{\lambda_{0}}$, and
then $im(\sigma|_{\bigcup_{k\geq K}\Delta_{k}^{n}})\subseteq X_{\lambda_{0}}$.
Therefore, $\sigma|_{\bigcup_{k\geq K}\Delta_{k}^{n}}$ can be considered as an
element of $\mathbb{H}_{n}(X_{\lambda_{0}},x_{0})$ and the other standard
simplexes $\Delta_{k}^{n}$, $k<K$, are generated by
$\sum_{\lambda\neq\lambda_{0}}\sum_{\aleph_{0}}H_{n}(X_{\lambda})$, as
desired. Since these subgroups have trivial intersections, because of the
path-connectedness of simplexes, the isomorphism holds. ∎

Now, we present an example to show that the isomorphism (2.1) differs from the
similar statement for the singular homology,

$H_{n}(X)\cong\sum_{\lambda}H_{n}(X_{\lambda}).$

######  Example 3.

Consider the space $X=\mathbb{HE}^{2}\dot{\cup}\mathbb{S}^{2}$ and the point
$a\in\mathbb{S}^{2}$. Then
$\mathbb{H}_{2}(X,a)\cong\sum_{\aleph_{0}}H_{2}(\mathbb{HE}^{2})\oplus\mathbb{H}_{2}(\mathbb{S}^{2},a)=\sum_{\aleph_{0}}\prod_{\aleph_{0}}\mathbb{Z}\oplus\sum_{\aleph_{0}}\mathbb{Z}$
[10]. On the other hand,
$\mathbb{H}_{2}(\mathbb{HE}^{2},\theta)\oplus\mathbb{H}_{2}(\mathbb{S}^{2},a)\cong\prod_{\aleph_{0}}\sum_{\aleph_{0}}\mathbb{Z}\oplus\sum_{\aleph_{0}}\mathbb{Z}$
and these two groups are not isomorphic [11].

By a well-known fact called Hurewicz theorem, for path-connected spaces, two
functors of homotopy and homology are connected. In fact, for a path-connected
space $X$, there exists a surjective homomorphism
$\phi_{s}:\pi_{1}(X,x_{0})\to H_{1}(X)$ whose kernel equals the commutator
subgroup of $\pi_{1}(X,x_{0})$. A similar relation exists between Hawaiian
homology and Hawaiian groups of pointed spaces. For each pointed space
$(X,x_{0})$, there exists a homomorphism
$\phi:\mathcal{H}_{1}(X,x_{0})\to\mathbb{H}_{1}(X,x_{0})$. The homomorphism
$\phi$ is surjective and its kernel equals the commutator subgroup if $X$ is
path-connected and locally path-connected at $x_{0}$.

######  Theorem 2.2.

Let $X$ be a path-connected space that is locally path-connected at $x_{0}$.
Then the homomorphism
$\phi:\mathcal{H}_{1}(X,x_{0})\to\mathbb{H}_{1}(X,x_{0})$ is surjective whose
kernel equals the commutator subgroup of $\mathcal{H}_{1}(X,x_{0})$.

###### Proof.

First, we correspond an element of $\mathbb{H}_{1}(X,x_{0})$ for each map
$f:(\mathbb{HE}^{1},\theta)\to(X,x_{0})$. Let
$\eta:\bigcup_{k=0}^{\infty}\Delta_{k}^{1}\to\mathbb{HE}^{1}$ maps each
$\Delta_{k}^{1}$ onto $\mathbb{S}_{k}^{1}$ homeomorphically except at the
vertices mapped to the base point $\theta$. Then $f\eta$ is a $1$-Hawaiian
simplex in $X$ at $x_{0}$. Also
$\partial(f\eta)=\bigcup_{k=0}^{\infty}f_{k}\eta_{k}(e_{1}^{k})-\bigcup_{k=0}^{\infty}f_{k}\eta_{k}(e_{0}^{k})=\bigcup_{k=0}^{\infty}\big{(}f_{k}\eta_{k}(e_{1}^{k})-f_{k}\eta_{k}(e_{0}^{k})\big{)}=\bigcup_{k=0}^{\infty}=0$,
where $\eta_{k}:\Delta_{k}\to\mathbb{S}_{k}^{1}$ is the restriction of $\eta$
to the $k$th standard $1$-simplex. Thus $f\eta\in im(\partial_{1})$. Hence one
can define $\phi:\mathcal{H}_{1}(X,x_{0})\to\mathbb{H}_{1}(X,x_{0})$ by
$\phi([f])=clsf\eta$.

To prove well-definedness, let $f\simeq g\ rel\\{\theta\\}$. Then $f_{k}\simeq
g_{k}\ rel\\{\theta\\}$ for each $k=0,1,\ldots$, where $f_{k}$ and $g_{k}$ are
the restrictions of $f$ and $g$, respectively, to the $k$th circle
$\mathbb{S}_{k}^{1}$. By the standard Hurewicz theorem,
$clsf_{k}\eta_{k}=clsg_{k}\eta_{k}$ for $k=0,1,\ldots$. Thus
$cls(f_{k}\eta_{k}-g_{k}\eta_{k})=0$, and then $f_{k}\eta_{k}-g_{k}\eta_{k}$
belongs to the image of $\partial$, the boundary of some linear composition of
standard $2$-simplex $\sum_{j=1}^{n}m_{j}\sigma^{j}_{k}$. By the proof of the
standard Hurewicz theorem [19, Lemma 4.26], this standard $2$-simplex is
constructed by the homotopy mapping $H_{k}:f_{k}\simeq g_{k}$ for
$k=1,2,\ldots$. Since $H_{k}$ is the restriction of the homotopy $H:f\simeq g$
to the $k$th circle, the image of $H_{k}$ is null-convergent, that is,
$im(H_{k})\subseteq U$ if $k\geq K$ for some $K\in\mathbb{N}$ corresponded to
the given open set $U$ containing $x_{0}$ (see [2, Definition 2.1]).
Therefore, $im(\cup_{j=0}^{n}\sigma^{j}_{k})$ is null-convergent and then
$\cup_{k=0}^{\infty}\sum_{j=1}^{n}m_{n}\sigma^{j}_{k}$, where the $0$-simplex
is the constant standard simplex, is a linear composition of $2$-Hawaiian
simplexes, whose boundary equals $f\eta-g\eta$. Hence $cls\ f-g=0$, and then
$cls\ f=cls\ g$. It implies that $\phi$ is well-defined.

Moreover, $\phi$ is a homomorphism. To prove this, consider two elements $[f]$
and $[g]$ in the group $\mathcal{H}_{1}(X,x_{0})$. Then the following sequence
of equalities holds:

$\displaystyle\phi([f][g])=\phi([f*g])=cls\ (f*g)\eta=cls\
\cup_{k=0}^{\infty}(f_{k}*g_{k})\eta_{k}.$

We must verify the equality

$cls\ \cup_{k=0}^{\infty}(f_{k}*g_{k})\eta_{k}=cls\
\cup_{k=0}^{\infty}f_{k}\eta_{k}+cls\ \cup_{k=0}^{\infty}g_{k}\eta_{k}.$

Equivalently, we must find a composition of $1$-Hawaiian simplexes whose
boundary equals
$\cup_{k=0}^{\infty}(f_{k}*g_{k})\eta_{k}-\cup_{k=0}^{\infty}f_{k}\eta_{k}-\cup_{k=0}^{\infty}g_{k}\eta_{k}$.
By the proof of the standard Hurewicz theorem [19, Theorem 4.27], there exists
a sequence of composition of standard $2$-simplexes, namely
$\sum_{j=1}^{n}m_{n}\sigma^{j}_{k}$ ($k\in\mathbb{N}$) constructed by the maps
$f_{k}*g_{k}$, $f_{k}$, and $g_{k}$ such that
$\partial(\sum_{j=1}^{n}m_{n}\sigma^{j}_{k})=(f_{k}*g_{k})\eta_{k}-f_{k}\eta_{k}-g_{k}\eta_{k}$.
It remains to verify that the sequence
$\\{\sum_{j=1}^{n}m_{n}\sigma^{j}_{k}\\}$ is null-convergent. Since the maps
$f_{k}*g_{k}$, $f_{k}$, and $g_{k}$ are null-convergent, so is
$\sum_{j=1}^{n}m_{n}\sigma^{j}_{k}$ ($k\in\mathbb{N}$). Therefore,
$\cup_{k=0}^{\infty}\sum_{j=1}^{n}m_{n}\sigma^{j}_{k}$ is a $2$-Hawaiian
simplex, and also its boundary equals
$\cup_{k=0}^{\infty}(f_{k}*g_{k})\eta_{k}-\cup_{k=0}^{\infty}f_{k}\eta_{k}-\cup_{k=0}^{\infty}g_{k}\eta_{k}=(f*g)\eta-f\eta-g\eta$.
Thus $cls\ (f*g)\eta=\mathrm{cls}\ f\eta+cls\ g\eta$, and then $\phi$ is a
homomorphism.

To prove surjectivity of the homomorphism $\phi$, we consider an element $cls\
\sigma$ of $\mathbb{H}_{1}(x,x_{0})$. By the restriction to the $k$th
$1$-simplex, denoted by $\sigma_{k}$, one can use the standard Hurewicz
theorem [19, Theorem 4.29], to obtain a sequence of maps $f_{k}$ defined from
$\mathbb{S}^{1}$ to $X$ such that $\phi_{s}([f_{k}])=cls\ \sigma_{k}$. If
$\sigma_{k}=\sum_{i=0}^{m}\sigma_{k}^{i}$, then by the proof of [19, Theorem
4.29], $f_{k}$ is defined by the concatenation of a sequence of simplexes and
some paths between the base point $x_{0}$ and the end points of simplexes.
Since $X$ is locally path-connected at $x_{0}$ having a countable local basis,
one can consider a sequence of nested open neighborhoods
$\\{U_{i}\\}_{i\in\mathbb{N}}$ of $x_{0}$ such that each element $U_{i}$ is
path-connected in $U_{i+1}$. Therefore, the sequence $f_{k}$ can be considered
to be null-convergent. Then define $f|_{\mathbb{S}_{k}^{1}}=f_{k}$. Since
$\\{f_{k}\\}$ is null-convergent, $f$ is continuous by [2, Lemma 2.2], and
also $\phi([f])=\sigma$. To calculate the kernel of $\phi$, again use the
proof of [19, Theorem 4.29], [2, Lemma 2.2], and locally path-connectedness at
$x_{0}$. ∎

By Theorem 2.2 and the first isomorphism theorem, it holds that

$\mathbb{H}_{1}(X,x_{0})\cong\frac{\mathcal{H}_{1}(X,x_{0})}{\big{(}\mathcal{H}_{1}(x,x_{0})\big{)}^{\prime}},$

where $\big{(}\mathcal{H}_{1}(x,x_{0})\big{)}^{\prime}$ denotes the commutator
subgroup of the given group $\big{(}\mathcal{H}_{1}(x,x_{0})\big{)}$. Then the
first Hawaiian homology group is isomorphic to the abelianization of the first
Hawaiian group for path-connected and locally path-connected pointed
topological spaces. This fact helps us to study Hawaiian groups of topological
spaces by Hawaiian homology.

##  3\. Higmann-complete Hawaiian groups

Herfort and Hojka proved that the fundamental group of archipelago spaces is
Higman-complete, introduced as a non-abelian form of cotorsion groups; for
more details, see [14, Definition 1]. In the rest of the paper, all the spaces
are assumed to be path-connected.

######  Definition 5 ([14]).

A group $G$ is Higman-complete if for any sequence $f_{1},f_{2},\ldots\in G$
and for a given sequence of words $w_{1},w_{2},\ldots$, there exists a
sequence $h_{1},h_{2},\ldots\in G$ such that all equations
$h_{i}=w_{i}(f_{i},h_{i+1})$ hold simultaneously.

The following lemma states that the first Hawaiian group of any space whose
loops are small, belongs to the class of Higman-complete groups. This fact
also holds for the fundamental group of archipelago spaces [14].

######  Lemma 3.1.

If all $1$-loops at the point $x\in X$ are small, then $\mathcal{H}_{1}(X,x)$
is Higmann-complete. If it is true for $n$-loops, $n\geq 2$, then
$\mathcal{H}_{n}(X,x)$ is cotorsion.

###### Proof.

Let all $1$-loops be small at $x\in X$. Assume that $f:\mathbb{HE}^{1}\to X$
is a $1$-Hawaiian loop at $x$. First, we show that $f$ is a small $1$-Hawaiian
loop. That is for each open neighborhood $U$ of $x$, there is a homotopic
representative $g\simeq f$ with image contained in $U$. Since $f$ is
continuous, there is $K\in\mathbb{N}$ such that
$im(f|_{\mathbb{S}_{k}^{1}})\subseteq U$ for $k\geq K$. Moreover, by the
assumption, all $f|_{\mathbb{S}_{k}^{1}}$’s are small. For $k<K$, define
$g_{k}$ as the homotopic representative $1$-loop of $f|_{\mathbb{S}_{k}^{1}}$
with image in $U$. Now put $g:\mathbb{HE}^{1}\to X$ as
$g|_{\mathbb{S}_{k}^{1}}=g_{k}$ for $k<K$, and
$g|_{\mathbb{S}_{k}^{1}}=f|_{\mathbb{S}_{k}^{1}}$ for $k\geq K$, which is
homotopic to $f$ by [2, Lemma 2.2]. In [14, Theorem 4], it was shown that
$\pi_{1}(X,x)$ is Higmann-complete, where all $1$-loops at $x$ are small. Now
by a similar argument, one can prove that $\mathcal{H}_{1}(X,x)$ is Higmann-
complete. Similarly, if all $n$-loops are small, $n\geq 2$, then
$\pi_{n}(X,x)$ and $\mathcal{H}_{n}(X,x)$ are Higmann-complete. Since
$\pi_{n}(X,x)$ and $\mathcal{H}_{n}(X,x)$ are abelian, by [14, Theorem 3],
they are cotorsion. ∎

Theorem 2.2 presents a version of Hurewicz theorem for Hawaiian groups and
Hawaiian homology groups. Now we use that theorem to obtain the following
corollary.

######  Corollary 2.

Let $X$ be locally path-connected at $x$, and let all $1$-loops at $x$ be
small. Then the first Hawaiian homology group $\mathbb{H}_{1}(X,x)$ is
cotorsion.

###### Proof.

Since all $1$-loops at $x$ are small, by Theorem 3.1, $\mathcal{H}_{1}(X,x)$
is Higmann-complete. Also by Theorem 2.2, the group $\mathbb{H}_{1}(X,x)$ is
the epimorphic image of the group $\mathcal{H}_{1}(X,x)$. Then it is Higmann-
complete by [14, Lemma 2]. Moreover, every abelian group is Higmann-complete
if and only if it is cotorsion [14, Theorem 3]. Therefore
$\mathbb{H}_{1}(X,x)$ is cotorsion. ∎

If all $n$-loops at a point are small, then both $n$th homotopy and $n$th
Hawaiian groups are Higman-complete. Note that the homotopy groups on path-
connected spaces do not depend on the choice of the base point. It implies
that $\pi_{n}(X,x)$ is Higmann-complete if all $n$-loops at some points of $X$
are small, but it is not true for Hawaiian groups. Some counterexamples are
investigated in Corollary 5, by applying the following lemma.

######  Lemma 3.2.

Let $X$ be semi-locally strongly contractible at $x_{0}$ and let $n\geq 2$. If
$H_{1}(X,x_{0})$ is not torsion, then neither $\mathbb{H}_{1}(X,x_{0})$ is
cotorsion, nor $\mathcal{H}_{1}(X,x_{0})$ is Higman-complete. Moreover, if
$\pi_{n}(X,x_{0})$ is not torsion, then $\mathcal{H}_{n}(X,x_{0})$ is not
cotorsion.

###### Proof.

Since $X$ is semi-locally strongly contractible at $x$, by Corollary 1, the
isomorphism $\mathbb{H}_{1}(X,x)\cong\sum_{\aleph_{0}}H_{1}(X)$ holds. If
$\mathbb{H}_{1}(X,x)$ is cotorsion, then $\sum_{\aleph_{0}}H_{1}(X)$ is
cotorsion. Thus by [12, Proposition 6.10], $H_{1}(X)$ is torsion, which is a
contradiction. By [2, Theorem 2.5],
$\mathcal{H}_{1}(X,x_{0})\cong\prod^{W}_{\aleph_{0}}\pi_{1}(X,x_{0})$. If
$\mathcal{H}_{1}(X,x_{0})$ is Higman-complete, then
$\prod^{W}_{\aleph_{0}}\pi_{1}(X,x_{0})$ is Higmann-complete. Thus by [14,
Theorem 3], its abelianization
$\mathbb{H}_{1}(X,x)\cong\sum_{\aleph_{0}}H_{1}(X)$ is cotorsion, which does
not hold. ∎

CW spaces are semilocally strongly contractible at any point, and then by
Lemma 3.2, the following corollary is obtained.

######  Corollary 3.

Let $X$ be a CW space with torsion-free first homology group. Then
$\mathcal{H}_{1}(X,x)$ is not Higman-complete for all $x\in X$. Moreover, if
$H_{n}(X)$ is torsion-free, then $\mathbb{H}_{n}(X,x)$ is not cotorsion for
$x\in X$.

If $\mathcal{H}_{n}(X,x)$ is Higman-complete for some $x\in X$, then so is
$\pi_{n}(X,x)$ as its epimorphic image, but the converse statement does not
hold. There are spaces such that their homotopy groups are Higman-complete but
not their Hawaiian groups neither Hawaiian homology groups; see the following
example.

######  Example 4.

Consider a Higman-complete torsion-free group, namely $\mathbb{Q}$, and
$X=K(\mathbb{Q},1)$ as the Eilenberg–MacLane space corresponded to
$\mathbb{Q}$. Then $H_{1}(X)\cong\pi_{1}(X)\cong\mathbb{Q}$, which is Higman-
complete because it is cotorsion. Moreover, since $X$ is semi-locally strongly
contractible at any point $x$,
$\mathcal{H}_{1}(X,x)\cong\sum_{\aleph_{0}}\pi_{1}(X,x)\cong\sum_{\aleph_{0}}\mathbb{Q}$
by [2, Theorem 2.5]. Since $X$ is a CW space and $H_{1}(X)$ is torsion-free,
by Corollary 3, $\mathcal{H}_{1}(X,x)$ is not Higman-complete for each $x\in
X$. Also,
$\mathbb{H}_{1}(X,x)\cong\sum_{\aleph_{0}}H_{1}(X)\cong\sum_{\aleph_{0}}\mathbb{Q}$
by Corollary 1, and then $\mathbb{H}_{1}(X,x)$ is not cotorsion for each $x\in
X$ by Corollary 3.

Some pseudomanifolds satisfy the conditions of Corollary 3; see [18, Theorem
8.2].

######  Corollary 4.

The first Hawaiian group of an orientable two-dimensional pseudomanifold $X$
is not Higman-complete. Also its first Hawaiian homology group
$\mathbb{H}_{1}(X,x)$ is not cotorsion for all $x\in X$.

Archipelago groups are the first examples of Higman-complete groups. They are
introduced as the fundamental groups of archipelago spaces.

The Harmonic archipelago is a non-locally Euclidean space, which was
introduced to be a counterexample for some statements. This space was
generalized to archipelago space [7]. Herfort and Hojka [14] computed the
singular homology groups of archipelago spaces. In this paper, we use a
similar argument to present Hawaiian homology groups of the archipelago
spaces.

######  Definition 6 ([7]).

Let $\\{(X_{i},x_{i})\\}_{i\in I}$ be a family of pointed spaces. Then the
archipelago space $\mathbb{A}=\mathbb{A}(\\{X_{i}\\}_{i\in I})$ on the given
family is defined as the mapping cone on the natural continuous bijection
$f:\bigvee_{i\in I}(X_{i},x_{i})\to\widetilde{\bigvee}_{i\in I}(X_{i},x_{i})$.
That is, the quotient space $C_{f}$ is obtained from $(X\times\mathbb{I})\cup
Y$ by identifying $(x,1/2)$ with $f(x)$ and $X\times\\{1\\}$ is contracted to
a point.

Archipelago spaces are examples of wild spaces. Free $\sigma$-groups are also
fundamental groups of some wild spaces [9, Theorem A.1]. Indeed free
$\sigma$-groups are not Higman-complete with behaves similar to free abelian
groups, which are not cotorsion.

######  Remark 2.

If $\\{G_{i}\\}_{I}$ is a family of groups such that $Ab(G_{i})$ is not
cotorsion for some $i\in I$, then the free product $\ast_{I}G_{i}$ and free
$\sigma$-product $\circledast_{I}G_{i}$ are not Higman-complete. In fact,
there are natural epimorphisms $\circledast_{I}G_{i}\to\prod_{I}G_{i}$ and
$\ast_{I}G_{i}\to\prod^{W}_{I}G_{i}$. Then $G_{i}$ and hence $Ab(G_{i})$ are
their epimorphic images for all $i\in I$. If any of $\circledast_{I}G_{i}$ or
$\ast_{I}G_{i}$ is Higman-complete, then so is $G_{i}$ for all $i\in I$ by
[14, Lemma 2]. Thus $Ab(G_{i})$ is cotorsion for all $i\in I$ by [14, Theorem
3].

By Lemma 3.2, Hawaiian groups of semi-locally strongly contractible spaces are
not Higman-complete. In the following result, we see that there are spaces,
such as the one-dimensional Hawaiian earring, that are not semi-locally
strongly contractible and their Hawaiian groups are Higman-complete.

######  Theorem 3.1.

Let $\\{X_{i}\\}_{i\in I}$ be a family of connected spaces, let
$X=\widetilde{\bigvee}_{I}X_{i}$, and let $x$ be the common point. If
$H_{1}(X_{i})$ is not cotorsion for some $i\in I$, then $\mathbb{H}_{1}(X,x)$
is not cotorsion, and moreover, $\pi_{1}(X,x)$ and $\mathcal{H}_{1}(X,x)$ are
not Higman-complete.

###### Proof.

Since $X_{i}$ is a retract of the space $X$, there exits a natural epimorphism
$\mathbb{H}_{1}(X,x)\to\mathbb{H}_{1}(X_{i},x_{i})$ induced by the Hawaiian
homology functor. Also By [9, Theorem A.1],
$\pi_{1}(X,x)\cong\circledast_{I}G_{i}$, where $G_{i}=\pi_{1}(X_{i},x_{i})$.
If $\mathcal{H}_{1}(X,x)$ is Higman-complete, then so is its epimorphic image
$H_{1}(X_{i})$ by [14, Lemma 2] for all $i\in I$. Thus, since $H_{1}(X_{i})$
is abelian by [14, Theorem 3], it must be cotorsion for all $i\in I$. ∎

Theorem 3.1 and Lemma 3.1 imply that one of the groups in the following
corollary is Higman-complete but not the other one. One can generalize
Griffiths space $\mathcal{G}(X_{i})$ over an arbitrary family of spaces
$\\{X_{i}\\}_{\mathbb{N}}$ as the wedge of two cones on
$\widetilde{\bigvee}_{\mathbb{N}}X_{i}$.

######  Corollary 5.

Let $\\{X_{i}\\}_{i\in I}$ be as in Theorem 3.1, let
$X=\widetilde{\bigvee}_{i\in I}$ be the week join space, let $A(X)$ be the
archipelago space, and let $\mathcal{G}(X)$ be the Griffiths space on $X$.
Then $\mathcal{H}_{1}(A(X),x_{*})\not\cong\mathcal{H}_{1}(X,x_{*})$, and also,
$\mathcal{H}_{1}(\mathcal{G}(X),x_{*})\not\cong\mathcal{H}_{1}(X,x_{*})$.

In the following theorem, we prove that for two points of the archipelago
spaces, the Hawaiian groups are not isomorphic. Thus archipelago spaces are
other examples of path-connected spaces whose Hawaiian groups are not
isomorphic at different points.

######  Theorem 3.2.

Let $\mathbb{A}=\mathbb{A}_{i\in I}(X_{i})$ be the archipelago space over the
family $\\{(X_{i},x_{i})\\}$. If $a\in\mathbb{A}$ is any point other than
common point $x$, then
$\mathcal{H}_{1}(\mathbb{A},x)\not\cong\mathcal{H}_{1}(\mathbb{A},a)$.

###### Proof.

By [14, Theorem 8],
$H_{1}(\mathbb{A})\cong\frac{\prod_{\aleph_{0}}\mathbb{Z}}{\sum_{\aleph_{0}}\mathbb{Z}}$,
which is torsion-free by [5]. Since $\mathbb{A}$ is semi-locally strongly
contractible at $a$, by Lemma 3.2, $\mathcal{H}_{1}(\mathbb{A},a)$ is not
Higman-complete. Also, since all $1$-loops at $x$ are small (see [14, Lemma
7]), by Lemma 3.1 $\mathcal{H}_{1}(X,x)$ is Higman-complete, and then it is
not isomorphic to $\mathcal{H}_{1}(\mathbb{A},a)$.

∎

Since the Griffiths space is not necessarily semi-locally strongly
contractible at $a$, Lemma 3.2 cannot be used.

######  Theorem 3.3.

If $a\in\mathcal{G}=\mathcal{G}(\\{X_{i}\\})$ is any point other than common
point $x$, then
$\mathcal{H}_{1}(\mathcal{G},x)\not\cong\mathcal{H}_{1}(\mathcal{G},a)$.

###### Proof.

By [8], $\mathcal{G}$ satisfies the premises of Lemma 3.1, and then
$\mathcal{H}_{1}(\mathcal{G},x)$ is Higman-complete. If $a\neq x_{*}$, then by
[3, Theorem 3.2], $\mathcal{H}_{1}(\mathcal{G},a)$ has
$\prod^{W}_{\aleph_{0}}\pi_{1}(\mathcal{G},a)$ as a direct factor. Therefore
$\sum_{\aleph_{0}}H_{1}(\mathcal{G})$ is the epimorphic image of
$\mathcal{H}_{1}(\mathcal{G},a)$, and by [14, Lemma 2 and Theorem 3], it is
cotorsion. By [12, Proposition 6.10], $H_{1}(\mathcal{G})$ should be torsion,
which contradicts with [16, p. 2]. ∎

For both archipelago and Griffiths spaces, by the same argument as Theorems
3.2 and 3.3, we can compare $L_{1}(\mathbb{A}(X_{i}))$ and
$L_{1}(\mathcal{G}(X_{i}))$ at different points. Then $L_{1}$ groups of both
of archipelago and Griffiths spaces are not isomorphic at different points
while they are path-connected spaces. It means that usual paths do not
transfer Hawaiian groups and $L_{n}$ groups isomorphically. In the following
theorem, we present an equivalent condition for paths to transfer Hawaiian
groups isomorphically. Some other conditions, namely $n$-SHLT introduced in
[4], are enough but not necessary. Recall that a path $\gamma$ from $x_{0}$ to
$x_{1}$ is called small $n$-loop transfer (abbreviated to $n$-SLT), if for
every open neighborhood $U$ of $x_{0}$, there exists an open neighborhood $V$
of $x_{1}$, such that for every $n$-loop
$\beta:(\mathbb{I}^{n},\dot{\mathbb{I}^{n}})\to(V,x_{1})$, there is an
$n$-loop $\alpha:(\mathbb{I}^{n},\dot{\mathbb{I}^{n}})\to(U,x_{0})$ being
homotopic to $\gamma_{\\#}^{-1}(\beta)$, where $\gamma_{\\#}$ induces the
homomorphism transfers $n$th homotopy group isomorphically along the points of
path $\gamma$ (see [20, p. 381]).

######  Theorem 3.4.

Let $X$ have local nested bases at $x_{0}$ and $x_{1}$. A path $\gamma$ from
$x_{0}$ to $x_{1}$ in $X$, induces the isomorphism
$\Gamma_{\gamma}:\mathcal{H}_{n}(X,x_{0})\to\mathcal{H}_{n}(X,x_{1})$ if and
only if $\gamma$ and $\gamma^{-1}$ are $n$-SLT paths preserving null-
convergent homotopies of $n$-loops.

###### Proof.

Let $\Gamma_{\gamma}$ induce the isomorphism. First, we show that $\gamma$ and
the inverse path $\gamma^{-1}$ are $n$-SLT paths. Let $\\{U_{m}\\}$ and
$\\{V_{m}\\}$ be local nested bases at $x_{0}$ and $x_{1}$, respectively.
Assume that $U$ is an open neighborhood of $x_{0}$ such that for each
$m\in\mathbb{N}$, there is an $n$-loop $\beta_{m}$ in $V_{m}$ at $x_{1}$
without any homotopic $n$-loop to $\gamma_{\\#}^{-1}(\beta_{m})$ in $U$.
Define $\beta:(\mathbb{HE}^{n},\theta)\to(X,x_{1})$ by
$\beta|_{\mathbb{S}_{m}^{n}}=\beta_{m}$. From [2, Lemma 2.2], $\beta$ is
continuous. Since $\Gamma_{\gamma}$ is an epimorphism, there is
$\alpha:(\mathbb{HE}^{n},\theta)\to(X,x_{0})$ such that
$\Gamma_{\gamma}([\alpha])=[\beta]$. Then
$\beta_{k}\simeq\gamma_{\\#}(\alpha|_{\mathbb{S}_{k}^{n}})$ or equivalently
$\gamma_{\\#}^{-1}(\beta|_{\mathbb{S}_{k}^{n}})\simeq\alpha_{k}=\alpha|_{\mathbb{S}_{k}^{n}}$.
Since $\alpha$ is continuous and $U$ is open, there is $K\in\mathbb{N}$ such
that if $k\geq K$, then $im(\alpha|_{\mathbb{S}_{k}^{n}})\subseteq U$. Because
of the definition of $\beta$, it is a contradiction. Therefore, $\gamma$ is an
$n$-SLT path. Similarly since $\Gamma_{\gamma}^{-1}$ is an epimorphism,
$\gamma^{-1}$ is an $n$-SLT path. Therefore, for each open neighborhood $U$ of
$x_{0}$, there is an open neighborhood $V$ of $x_{1}$ such that for any two
$n$-loops $\alpha$ and $\alpha^{\prime}$ in $U$, there are $n$-loops $\beta$
and $\beta^{\prime}$ in $V$ homotopic to $\gamma_{\\#}(\alpha)$ and
$\gamma_{\\#}(\alpha^{\prime})$, respectively. Let $\\{H_{k}\\}$ be a null-
convergent sequence of homotopies of $n$-loops. That is, there is an
increasing sequence of natural numbers, namely $\\{K_{m}\\}$, such that for
$K_{m}\leq k<K_{m+1}$, $im(H_{k})\subseteq U_{m}$. Since $\gamma^{-1}$ is an
$n$-SLT path, there are $n$-loops $\beta_{k}$ and $\beta_{k}^{\prime}$ with
$im(\beta_{k}),im(\beta^{\prime}_{k})\subseteq V_{m}$ such that
$\beta_{k}\simeq\gamma_{\\#}(H_{k}(-,0))$ and
$\beta_{k}^{\prime}\simeq\gamma_{\\#}(H_{k}(-,1))$ whenever $K_{m}\leq
k<K_{m+1}$. Define $f,f^{\prime}:(\mathbb{HE}^{n},\theta)\to(X,x_{0})$ and
$f,g:(\mathbb{HE}^{n},\theta)\to(X,x_{1})$ by
$f|_{\mathbb{S}_{k}^{n}}=H_{k}(-,0)$,
$f^{\prime}|_{\mathbb{S}_{k}^{n}}=H_{k}(-,1)$,
$g|_{\mathbb{S}_{k}^{n}}=\beta_{k}$, and
$g^{\prime}|_{\mathbb{S}_{k}^{n}}=\beta_{k}^{\prime}$. Now since $\\{H_{k}\\}$
is null-convergent, we have $[f]=[f^{\prime}]$ by [2, Lemma 2.2]. Also since
$\gamma$ induces a well-defined homomorphism,
$\Gamma_{\gamma}([f])=\Gamma_{\gamma}([f^{\prime}])$. Therefore,
$[g]=[g^{\prime}]$, or equivalently there is $G:g\simeq g^{\prime}$. Since
$G:\mathbb{HE}^{n}\times\mathbb{I}\to X$ is continuous and $\mathbb{I}$ is
compact, $G|_{\mathbb{S}_{k}^{n}\times\mathbb{I}}$ is null-convergent. Then
$\\{\beta_{k}\\}$ and $\\{\beta_{k}^{\prime}\\}$ are homotopic by a null-
convergent sequence of homotopies. Similarly since $\gamma^{-1}$ induces the
well-defined homomorphism $\Gamma_{\gamma}^{-1}$, one can show that
$\gamma^{-1}$ preserves null-convergent homotopies of $n$-loops. The proof of
the converse statement is the same as [17, Theorem 5.5]. ∎

##  4\. The first Hawaiian homology of Harmonic archipelago

In this section, we obtain the structure of the first Hawaiian homology of the
Harmonic archipelago spaces, up to isomorphism. In fact, the first Hawaiian
homology group and the first singular homology group of the Harmonic
archipelago spaces are isomorphic. First, we prove that it is locally free;
that is, all of its finitely generated subgroups are free.

######  Theorem 4.1.

Let $\mathbb{A}=\mathbb{A}(\\{X_{i}\\})$ be the archipelago space on
$\\{X_{i}\\}_{i\in\mathbb{N}}$, where each $X_{i}$ is locally strongly
contractible and let $\theta$ be the common point. If $\pi_{1}(X_{i})$ is free
for all $i\in I$ except a finite number, then
$\mathcal{H}_{1}(\mathbb{A},\theta)$ is locally free.

###### Proof.

Since $X_{i}$ is locally strongly contractible and first countable at $x_{i}$,
for $i\in\mathbb{N}$, there exists a nested local basis
$\\{V_{j}^{i}\\}_{j\in\mathbb{N}}$ at $x_{i}$ such that the inclusion mapping
$V_{j}^{i}\hookrightarrow V_{j-1}^{i}$ is null homotopic to $x_{i}$ in
$V_{j-1}^{i}$. Let $\\{U_{m}\\}_{m\in\mathbb{N}}$ be the local basis at
$x_{\ast}$ obtained by
$U_{m}=({\bigvee}_{i<m}V_{m}^{i})\vee\widetilde{\bigvee}_{i\geq m}X_{i}$.
Therefore, the inclusion map $U_{m+1}\hookrightarrow U_{m}$ is homotopic to
the map retracting each point onto $x_{i}$ for $i<m$ and identity for the
others. The topology of archipelago spaces implies that the natural basis at
the common point can be retracted onto the neighborhoods $\\{U_{m}\\}$.
Consider $M$ as the minimum number such that $\pi_{1}(X_{i})$ is free for all
$i\geq M$. Let $G=\langle\\{[f^{1}],[f^{2}],\ldots,[f^{m}]\\}\rangle$ be a
subgroup of $\mathcal{H}_{1}(\mathbb{A},\theta)$. Also let $w$ be a word such
that $w([f^{1}],[f^{2}],\ldots,[f^{l}])$ is the trivial element. Hence there
exists a homotopy mapping $H:w(f^{1},\ldots,f^{l})\simeq C_{\theta}$. Since
$H$ is continuous, for each $m\in\mathbb{N}$, there exists
$K_{m}\in\mathbb{N}$ such that $im(H|_{\mathbb{S}_{k}^{1}\times\mathbb{I}})$
can be considered homotopically as a subset of $U_{m+1}$ if $k\geq K_{m}$.
Then,
$im(w(f|_{\mathbb{S}_{k}^{1}}^{1},f|_{\mathbb{S}_{k}^{1}}^{2},\ldots,f|_{\mathbb{S}_{k}^{1}}^{l}))\subseteq
U_{m}$ for $k\geq K_{m}$. Thus
$r(w(f|_{\mathbb{S}_{k}^{1}}^{1},f|_{\mathbb{S}_{k}^{1}}^{2},\ldots,f|_{\mathbb{S}_{k}^{1}}^{l}))$
induces a word of $1$-loops at $\theta$ in $\widetilde{\bigvee}_{i\geq
m}X_{i}$ for $k\geq K_{m}$, where $r$ is the natural retraction onto the space
$\widetilde{\bigvee}_{i\geq m}X_{i}$. Also, $r\circ
H|_{\mathbb{S}_{k}^{1}\times\mathbb{I}}:r(w(f|_{\mathbb{S}_{k}^{1}}^{1},f|_{\mathbb{S}_{k}^{1}}^{2},\ldots,f|_{\mathbb{S}_{k}^{1}}^{l}))\simeq
c_{\theta}$ in $\widetilde{\bigvee}_{i\geq m}X_{i}$ for $k\geq K_{m}$. Since
$\pi_{1}(\widetilde{\bigvee}_{i\geq m}X_{i})$ is locally free, all $1$-loops
$f_{\mathbb{S}_{k}^{1}}^{j}$ are null-homotopic in $\widetilde{\bigvee}_{i\geq
m}X_{i}$ for $1\leq j\leq l$. Now by using [2, Lemma 2.2], one can obtain a
continuous homotopy to make $f^{j}$ null-homotopic. It implies that if a
finite word is trivial, then all of its generating elements are trivial, or
equivalently the group $\mathcal{H}_{1}(\mathbb{A},\theta)$ is locally free. ∎

In the following theorem, assume that each $X_{i}$ satisfies the assumption
$2\leq card\big{(}\pi_{1}(X_{i},x_{i})\big{)}\leq c$. Then we present the
structure of the first Hawaiian homology group of some archipelago spaces by
using the fact that this group is cotorsion.

######  Theorem 4.2.

Let $\mathbb{A}=\mathbb{A}(\\{X_{i}\\})$ be the archipelago space on
$\\{X_{i}\\}_{i\in\mathbb{N}}$, where each $X_{i}$ is locally strongly
contractible. Also let $\pi_{1}(X_{i})$ be free for all $i\in I$ except a
finite number. Then

$\mathbb{H}_{1}(\mathbb{A},x_{*})\cong\sum_{c}\mathbb{Q}\oplus\prod_{p\in
P}\widehat{\sum_{c}\mathbb{J}_{p}}^{p}\cong\frac{\prod_{\aleph_{0}}\mathbb{Z}}{\sum_{\aleph_{0}}\mathbb{Z}}.$

###### Proof.

Since all $1$-loops at $\theta$ are small, by Lemma 3.1,
$\mathcal{H}_{1}(\mathbb{A},x_{*})$ is Higman-complete. Then by [14, Lemma 2],
$Ab(\mathcal{H}_{1}(\mathbb{A},x_{*}))$ is Higman-complete. Thus by [14,
Theorem 3], $Ab(\mathcal{H}_{1}(\mathbb{A},x_{*}))$ is cotorsion. Moreover, by
Theorem 4.1, $\mathcal{H}_{1}(\mathbb{A},x_{*})$ is locally free. It was shown
that the abelianization of locally free group is torsion-free [14, Lemma 12].
Therefore, the group $Ab(\mathcal{H}_{1}(\mathbb{A},x_{*}))$ is a torsion-free
cotorsion group, and then it is algebraically compact. Also, it is torsion-
free algebraically compact, and then it is isomorphic to
$\sum_{m_{0}}\mathbb{Q}\oplus\prod_{p\in
P}\widehat{\sum_{m_{p}}\mathbb{J}_{p}}^{p}$ for some cardinalities $m_{0}$ and
$m_{p}$ ($p\in P$) [11, pp. 105,169]. Hence, it remains to obtain the
cardinalities $m_{0}$ and $m_{p}$ ($p\in P$). Moreover, since $X_{i}$ is
locally strongly contractible, it is locally path-connected, and then the
archipelago space $\mathbb{A}(X_{i})$ is locally path-connected. Therefore by
Theorem 2.2,

$\mathbb{H}_{1}(\mathbb{A},x_{*})\cong
Ab\big{(}\mathcal{H}_{1}(\mathbb{A},x_{*})\big{)}\cong\sum_{m_{0}}\mathbb{Q}\oplus\prod_{p\in
P}\widehat{\sum_{m_{p}}\mathbb{J}_{p}}^{p}.$

First, we show that $card\big{(}\mathbb{H}_{1}(\mathbb{A},x_{*})\big{)}\leq
c$. Thus $m_{0},m_{p}\leq c$, where $c$ is the continuum cardinality and $p$
is a prime. Let $X=\widetilde{\bigvee}_{i\in I}X_{i}$. One can generalize the
proof of [17, Theorem 2.8] for the homomorphism
$i_{*}:\mathcal{H}_{1}(X,x_{*}))\to\mathcal{H}_{1}(\mathbb{A},x_{*})$, induced
by the inclusion, to be an epimorphism. Therefore, we have
$card\big{(}\mathcal{H}_{1}(X,x_{*})\big{)}\geq
card\big{(}\mathcal{H}_{1}(\mathbb{A},x_{*})\big{)}$. By [2, Theorem 2.9],
$card\big{(}\mathcal{H}_{1}(X,x_{*})\big{)}\leq\prod_{i\in\mathbb{N}}card\big{(}\pi_{1}(X_{i},x_{i})\big{)}\leq
c^{\aleph_{0}}=c$. Thus
$card\big{(}\mathcal{H}_{1}(\mathbb{A},x_{*})\big{)}\leq c$. Moreover, for an
arbitrary pointed space $(X,x_{0})$, the group $\pi_{1}(X,x_{0})$ can be
considered as a subgroup of $\mathcal{H}_{1}(X,x_{0})$. It implies that
$card\big{(}\mathcal{H}_{1}(X,x_{0})\big{)}\geq
card\big{(}\pi_{1}(X,x_{0})\big{)}$. We know that
$card\big{(}\pi_{1}(\mathbb{A},x_{*})\big{)}=c$ and then
$card\big{(}\mathcal{H}_{1}(\mathbb{A},x_{*})\big{)}\geq c$. Therefore
$card\big{(}\mathcal{H}_{1}(\mathbb{A},x_{*})\big{)}=c$, and then we have
$card\big{(}\mathbb{H}_{1}(\mathbb{A},x_{*})\big{)}\leq c$, by Theorem 2.2.
Thus $m_{0},m_{p}\leq c$. Now by an epimorphism we show that $m_{0},m_{p}\geq
c$.

Since all $1$-loops at $\theta$ are small, by [4, Corollary 4.3], there is an
epimorphism
$\varphi:\mathcal{H}_{1}(\mathbb{A},\theta)\to\prod_{\aleph_{0}}\pi_{1}(\mathbb{A},\theta)$.
Also there is an isomorphism
$\pi_{1}(\mathbb{A},\theta)\to\frac{\times^{\sigma}_{\aleph_{0}}G}{*_{\aleph_{0}}G}$,
where $G$ is either $\mathbb{Z}$ or $\mathbb{Z}_{2}$ by [7, Theorem A]. Thus
there is an epimorphism
$\mathcal{H}_{1}(\mathbb{HA},\theta)\to\prod_{\aleph_{0}}\frac{\times^{\sigma}_{\aleph_{0}}G}{*_{\aleph_{0}}G}$.
This induces the epimorphism
$Ab\big{(}\mathcal{H}_{1}(\mathbb{A},\theta)\big{)}\to
Ab\big{(}\prod_{\aleph_{0}}\frac{\times^{\sigma}_{\aleph_{0}}G}{*_{\aleph_{0}}G}\big{)}$.
Moreover, for any family of groups $\\{G_{i}\\}_{i\in I}$, there is an
epimorphism $Ab\big{(}\prod_{i\in I}G_{i}\big{)}\to\prod_{i\in I}Ab(G_{i})$.
Since $\mathbb{H}_{1}(\mathbb{A},x_{*})\cong
Ab\big{(}\mathcal{H}_{1}(\mathbb{A},\theta)\big{)}$, there is an epimorphism
$\mathbb{H}_{1}(\mathbb{A},x_{*})\to\prod_{\aleph_{0}}Ab\big{(}\frac{\times^{\sigma}_{\aleph_{0}}G}{*_{\aleph_{0}}G}\big{)}$.
Also, $Ab\big{(}\frac{\times^{\sigma}_{\aleph_{0}}G}{*_{\aleph_{0}}G}\big{)}$
is isomorphic to
$\frac{\prod_{\aleph_{0}}\mathbb{Z}}{\sum_{\aleph_{0}}\mathbb{Z}}$ if $G$ is
either $\mathbb{Z}$ or $\mathbb{Z}_{2}$ by [14, Theorem 8]. Hence, we have an
epimorphism
$\mathbb{H}_{1}(\mathbb{A},x_{*})\to\prod_{\aleph_{0}}\frac{\prod_{\aleph_{0}}\mathbb{Z}}{\sum_{\aleph_{0}}\mathbb{Z}}$.
By [11, Corollary 42.2],
$\frac{\prod_{\aleph_{0}}\mathbb{Z}}{\sum_{\aleph_{0}}\mathbb{Z}}$ is
algebraically compact and it is isomorphic to
$\sum_{c}\mathbb{Q}\oplus\prod_{p\in P}\widehat{\sum_{c}\mathbb{J}_{p}}^{p}$
by [5]. Then there is an epimorphism
$\mathbb{H}_{1}(\mathbb{A},x_{*})\to\prod_{\aleph_{0}}\sum_{c}\mathbb{Q}\oplus\prod_{p\in
P}\widehat{\sum_{c}\mathbb{J}_{p}}^{p}$. Equivalently there is an epimorphism
$\sum_{m_{0}}\mathbb{Q}\oplus\prod_{p\in
P}\widehat{\sum_{m_{p}}\mathbb{J}_{p}}^{p}\to\prod_{\aleph_{0}}\bigg{(}\sum_{c}\mathbb{Q}\oplus\prod_{p\in
P}\widehat{\sum_{c}\mathbb{J}_{p}}^{p}\bigg{)}$. Therefore, $m_{0},m_{p}\geq
c$, and then $m_{0},m_{p}=c$ for $p\in P$. Hence
$\mathbb{H}_{1}(\mathbb{A},x_{*})\cong\sum_{c}\mathbb{Q}\oplus\prod_{p\in
P}\widehat{\sum_{c}\mathbb{J}_{p}}^{p}$, and by [11, Corollary 42.2],
$\mathbb{H}_{1}(\mathbb{A},x_{*})\cong\frac{\prod_{\aleph_{0}}\mathbb{Z}}{\sum_{\aleph_{0}}\mathbb{Z}}$.
∎

Herfort and Hojka [14] proved that the first singular homology group of any
archipelago space is isomorphic to
$\frac{\prod_{\aleph_{0}}\mathbb{Z}}{\sum_{\aleph_{0}}\mathbb{Z}}$. Now by
Theorem 4.2, the first singular homology and the first Hawaiian homology
groups are isomorphic for archipelago spaces. It does not hold for many
classes of spaces; for instance, see Example 4.

## Acknowledgment

This research was supported by a grant from Ferdowsi University of
Mashhad–Graduate Studies (No. 2/57128).

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