# Local structure of homogeneous $ANR$-spaces

V. Valov  Department of Computer Science and Mathematics, Nipissing
University, 100 College Drive, P.O. Box 5002, North Bay, ON, P1B 8L7, Canada
<EMAIL_ADDRESS>
###### Abstract.

We investigate to what extend finite-dimensional homogeneous locally compact
$ANR$-spaces have common properties with Euclidean manifolds. Specially, the
local structure of homogeneous $ANR$-spaces is described. Using that
description, we provide a positive solution of the problem whether every
finite-dimensional homogeneous metric $ANR$-compactum $X$ is dimensionally
full-valued, i.e. $\dim X\times Y=\dim X+\dim Y$ for any metric compactum $Y$.

###### Key words and phrases:

absolute neighborhood retracts, cohomological dimension, cohomology groups,
homogeneous spaces

###### 2000 Mathematics Subject Classification:

Primary 55M15; Secondary 55M10

The author was partially supported by NSERC Grant 261914-19.

##  1\. Introduction

By a _space_ we mean a locally compact separable metric space, and maps are
continuous mappings. Reduced Čech homology $H_{n}(X;G)$ and cohomology groups
$H^{n}(X;G)$ with coefficient from an abelian group $G$ are considered
everywhere below. A space is said to be homogeneous provided that, for every
two pints $x,y\in X$ there exists a homeomorphism $h$ mapping $X$ onto itself
with $h(x)=y$. Homogeneity of $X$ implies local homogeneity, that is for every
$x,y\in X$ there exists a homeomorphism $h$ mapping a neighborhood of $x$ onto
a neighborhood of $y$ such that $h(x)=y$.

One of the reasons to investigate the homogeneous $ANR$-spaces is the well-
known Bing-Borsuk [1] conjecture that every finite-dimensional homogeneous
metric $ANR$-compactum is an Euclidean manifold. According to Jacobsche [14],
a positive solution of that conjecture in dimension three implies the
celebrated Poincaré conjecture. J. Bryant and S. Ferry [4] announced the
existence of many counter-examples to that conjecture for every $n\geq 6$, but
to the best of author’s knowledge, the Bryant-Ferry paper is not published
yet. Despite that the status of the Bing-Borsuk conjecture is unknown, it is
still interesting to investigate the extend to which finite-dimensional
homogeneous $ANR$-spaces have common properties with Euclidean manifolds. We
show that homogeneous $ANR$-spaces have indeed some properties typical for
Euclidean manifolds.

The local structure of homogeneous $ANR$-compacta $X$ with a finite
cohomological dimension $\dim_{G}X=n$ was established in [29, Theorem 1.1],
where $G$ is a countable principal ideal domain with unity and $n\geq 2$. In
the present paper we improve the results in [29] by considering locally
compact homogeneous $ANR$-spaces and replacing the principal ideal domains
with countable groups. Moreover, we also provide some new properties of
homogeneous $ANR$’s or locally homogeneous $ANR$’s.

The next theorem describes the local structure of homogeneous $ANR$-spaces and
shows their similarity with Euclidean manifolds:

######  Theorem 1.1.

Let $X$ be a homogeneous connected $ANR$-space with $\dim_{G}X=n\geq 2$, where
$G$ is a countable group. Then every point $x\in X$ has a base
$\mathcal{B}_{x}$ of connected open sets $U\subset X$ each having a compact
closure and satisfying the following conditions:

  * (1)

$\rm{Int}\overline{U}=U$, the boundary $\mathrm{bd}\,\overline{U}$ of
$\overline{U}$ is connected and its complement in $X$ has exactly two
components;

  * (2)

$H^{n-1}(\mathrm{bd}\,\overline{U};G)\neq 0$, $H^{n-1}(\overline{U};G)=0$ and
$\overline{U}$ is an $(n-1)$-cohomology membrane spanned on
$\mathrm{bd}\,\overline{U}$ for any non-zero $\gamma\in
H^{n-1}(\mathrm{bd}\,\overline{U};G)$;

  * (3)

$\mathrm{bd}\,\overline{U}$ is an $(n-1,G)$-bubble.

Recall that for any nontrivial abelian group $G$ the Čech cohomology group
$H^{n}(X;G)$ is isomorphic to the group of pointed homotopy classes of maps
from $X$ to $K(G,n)$, where $K(G,n)$ is a $CW$-complex of type $(G,n)$, see
[17]. The cohomological dimension $\dim_{G}X$ is the largest number $n$ such
that there exists a closed set $A\subset X$ with $H^{n}(X,A;G)\neq 0$.
Equivalently, for a metric space $X$ we have $\dim_{G}X\leq n$ if and only if
for any closed pair $A\subset B$ in $X$ the homomorphism
$j_{B,A}^{n}:H^{n}(B;G)\to H^{n}(A;G)$, generated by the inclusion
$A\hookrightarrow B$, is surjective, see [10]. This means that every map from
$A$ to $K(G,n)$ can be extended over $B$. If $X$ is a finite-dimensional
space, then for every $G$ we have $\dim_{G}X\leq\dim_{\mathbb{Z}}X=\dim X$
[21]. On the other hand, there is a compactum $X$ [8] with $\dim X=\infty$ and
$\dim_{\mathbb{Z}}X=3$. A space $A$ is a $(k,G)$-bubble if $H^{k}(A;G)\neq 0$
but $H^{k}(B;G)=0$ for every closed proper subset $B$ of $A$.

The following property of homogeneous $ANR^{\prime}s$ is essential, see [1],
[2]: If $(K,A)$ is a pair of closed subsets of $X$ such that $K$ is an
$(n-1)$-cohomological membrane for some $\gamma\in H^{n-1}(A;G)$, then
$(K\setminus A)\cap\overline{X\setminus K}=\varnothing$. We call this property
the $n$-cohomology membrane property. Recall that $K$ is said to be an
$k$-cohomology membrane spanned on a closed set $A\subset K$ for an element
$\gamma\in H^{k}(A;G)$ if $\gamma$ is not extendable over $K$, but it is
extendable over every proper closed subset of $K$ containing $A$. Here,
$\gamma\in H^{k}(A;G)$ is not extendable over $K$ means that $\gamma$ is not
contained in the image $j_{K,A}^{k}\big{(}H^{k}(K;G)\big{)}$.

Everywhere below, $\mathcal{B}_{x}$ stands for the family of all open
neighborhoods of a point $x\in X$. We say that a space $X$ has an
$n$-dimensional $G$-obstruction at a point $x\in X$ [21] if there is
$W\in\mathcal{B}_{x}$ such that the homomorphism
$j^{n}_{U,W}:H^{n}(X,X\backslash U;G)\to H^{n}(X,X\backslash W;G)$ is
nontrivial for every $U\in\mathcal{B}_{x}$ with $U\subset W$. Kuzminov [21],
[22], proved that every compactum $X$ with $\dim_{G}X=n$ contains a compact
set $Y$ with $\dim_{G}Y=n$ such that $X$ has an $n$-dimensional
$G$-obstruction at any point of $Y$. According to the next theorem,
$n$-dimensional homogeneous spaces have an obstruction at every point and the
corresponding homomorphisms are surjective.

######  Theorem 1.2.

Let $X$ be a locally homogeneous $ANR$-space such that $\dim_{G}X=n$, where
$G$ is a countable group. Then the following holds:

  * (i)

$X$ has the $n$-cohomology membrane property;

  * (ii)

$X$ has an $n$-dimensional $G$-obstruction at every $x\in X$. Moreover, there
is $W\in\mathcal{B}_{x}$ such that the homomorphism $j_{U,V}^{n}$ is
surjective for any $U,V\in\mathcal{B}_{x}$ with $\overline{U}\subset
V\subset\overline{V}\subset W$.

Here is another common property of locally homogeneous $ANR$’s and Euclidean
manifolds.

######  Corollary 1.3.

Let $X$ be as in Theorem $1.2$. If $U\subset X$ is open and $f:U\to X$ is an
injective map, then $f(U)$ is also open in $X$.

One of the important questions concerning homogeneous $ANR$’s is whether every
finite-dimensional homogeneous $ANR$-compactum is dimensionally full-valued.
Recall that a locally compact space $X$ is dimensionally full-valued if $\dim
X\times Y=\dim X+\dim Y$ for any compact space $Y$. Let us note that there are
metric $ANR$-compacta which are not dimensionally full-valued, see [7] and
[9]. The question for the dimensional full-valuedness of homogeneous
$ANR$-compacta goes back to [3] and was also discussed in [5] and [13]. A
positive answer of that question for 3-dimensional spaces was given in [29].
Now, we provide a complete solution.

######  Theorem 1.4.

Let $X$ be a finite-dimensional locally homogeneous $ANR$-space. Then the
following holds:

  * (i)

$X$ is dimensionally full-valued;

  * (ii)

Providing $X$ is homogeneous and connected, every $x\in X$ has a neighborhood
$W_{x}$ such that $\mathrm{bd}\,\overline{U}$ is dimensionally full-valued for
all $U\in\mathcal{B}_{x}$ with $\overline{U}\subset W_{x}$.

##  2\. The cohomology membrane property

If, in the definition of the $n$-cohomology membrane property, we additionally
require $K$ to be a compact set contractible in a proper subset of $X$, then
we say that $X$ has the weak $n$-cohomology membrane property. We will see
that the proof of Theorem 1.1 is based mainly on that property. In this
section we prove that any locally homogeneous space $X$ with $\dim_{G}X=n$ has
the $n$-cohomology membrane property, and provide some implications of that
property.

A space $X$ is called a $(k,G)$-carrier of a nontrivial element $\gamma\in
H^{k}(X;G)$ if $j^{k}_{X,B}(\gamma)=0$ for every closed proper subset
$B\subset X$.

######  Lemma 2.1.

Let $A\subset X$ be a compact set and $\gamma$ be a non-zero element of
$H^{n}(A;G)$.

  * (i)

If $\gamma$ is not extendable over a compact set $P\subset X$ containing $A$,
then there exists an $n$-cohomology membrane $K\subset P$ for $\gamma$ spanned
on $A$;

  * (ii)

There is a closed set $B\subset A$ such that $B$ is a carrier of
$j^{n}_{A,B}(\gamma)$.

###### Proof.

$(i)$ Consider the family $\mathcal{F}$ of all closed subsets $F$ of $P$
containing $A$ such that $\gamma$ is not extendable over $F$. Let
$\\{F_{\tau}\\}$ be a decreasing subfamily of $\mathcal{F}$ and
$F_{0}=\bigcap_{\tau}F_{\tau}$. Suppose $\gamma$ is extendable to
$\widetilde{\gamma}\in H^{n}(F_{0};G)$. Then, considering $\widetilde{\gamma}$
as a map from $F_{0}$ into $K(G,n)$ and having in mind that $K(G,n)$ is an
absolute neighborhood for metrizable spaces [20], we can extend
$\widetilde{\gamma}$ over a neighborhood $W$ of $F_{0}$ in $P$. But that is
impossible since $W$ contains some $F_{\tau}$. Hence, by Zorn’s lemma,
$\mathcal{F}$ has a minimal element $K$ which is an $n$-cohomology membrane
for $\gamma$ spanned on $A$.

$(ii)$ Now, let $\mathcal{F}$ be the family of closed subsets $F$ of $A$ such
that $j^{n}_{A,F}(\gamma)\neq 0$ and $\\{F_{\tau}\\}$ be a decreasing
subfamily of $\mathcal{F}$ with $F_{0}=\bigcap_{\tau}F_{\tau}$. If
$\gamma_{0}=j^{n}_{A,F_{0}}(\gamma)=0$, then there is a homotopy
$H:F_{0}\times[0,1]\to K(G,n)$ connecting the constant map and $\gamma_{0}$.
Using again that $K(G,n)$ is an absolute neighborhood extensor for metric
spaces, we find a closed neighborhood $W$ of $F_{0}$ in $A$ and a homotopy
$\widetilde{H}:W\times[0,1]\to K(G,n)$ connecting $j^{n}_{A,W}(\gamma)$ and
the constant map. This is a contradiction because $W$ contains some
$F_{\tau}$. Hence, $\mathcal{F}$ has a minimal element $B$ which is a carrier
of $j^{n}_{A,B}(\gamma)$. ∎

We also need the following version of Effros’ theorem [11] for locally compact
spaces, see [18, Theorem 2.2]):

######  Theorem 2.2.

Let $X$ be a homogeneous space and $\rho$ be a metric on the one-point
compactification of $X$. Then for any $a\in X$ and $\varepsilon>0$ there
exists $\delta>0$ such that for every $x\in X$ with $\rho(x,a)<\delta$ there
exists a homeomorphism $h\colon X\to X$ with $h(a)=x$ and
$\rho(h(y),y)<\varepsilon$ for all $y\in X$.

######  Proposition 2.3.

Let $X$ be a homogeneous $ANR$-space with $\dim_{G}X=n$, where $G$ is an
arbitrary group. Then $H^{n}(P;G)=0$ for any proper compact set $P\subset X$.

###### Proof.

This was established in [30] in case $X$ is compact using the following
proposition, see [30, Proposition 2.3]: If $X$ is a metric compactum with
$\dim_{G}X=n$, $A\subset X$ is carrier of a nontrivial element of $H^{n}(X;G)$
and there is a map $f:X\to X$ homotopic to the identity $\rm{id}_{X}$ of $X$,
then $A\subset f(A)$. This statement remains true if $X$ is an $ANR$-space
with $\dim_{G}X=n$, $A$ is a carrier of a nontrivial element of $H^{n}(Q;G)$,
where $Q\subset X$ is compact, and there exists a homotopy $F:A\times[0,1]\to
X$ with $F(x,0)=x$ and $F(x,1)=f(x)$, $x\in A$.

Fix a metric $\rho$ on $X$ generating its topology and suppose there exists a
proper compact set $P\subset X$ with $H^{n}(P;G)\neq 0$. So, by Lemma 2.1(ii),
$P$ contains a compact set $K$ such that $K$ is a carrier for
$j^{n}_{P,K}(\alpha)$, where $\alpha\in H^{n}(P;G)$ is nontrivial. Take two
open sets $U,V\subset X$ containing $P$ and having compact closures in $X$
with $\overline{U}\subset V$. Since $X\in ANR$, there is an open cover
$\omega$ of $X$ such that any two $\omega$-close maps
$f_{1},f_{2}:\overline{U}\to X$ are homotopic. Restricting $\omega$ on
$\overline{V}$, we find $\varepsilon>0$ such that any maps
$f_{1},f_{2}:\overline{U}\to X$ with $\rho(f_{1}(x),f_{2}(x))<\varepsilon$,
$x\in\overline{U}$, are homotopic. Moreover, we can assume
$\varepsilon<\min\\{\rho(P,X\backslash U),\rho(\overline{U},X\backslash
V)\\}$. Next, take a point $a$ from the boundary $\rm{bd}\,K$ of $K$ in $X$
and a point $b\notin K$ with $\rho(a,b)<\delta$, where $\delta>0$ is the
Effros’ number corresponding to $\varepsilon$ and the point $a$. Accordingly,
there is homeomorphism $h:X\to X$ such that $h(a)=b$ and
$\rho(x,h(x))<\varepsilon$ for all $x\in X$. So, $h(K)\subset U$ and
$h(\overline{U})\subset V$. Since $h$ generates an isomorphism
$h^{*}:H^{n}(h(P);G)\to H^{n}(P;G)$, the set $A=h(K)$ is a carrier for
$j^{n}_{h(P),h(K)}(\beta)$, where $\beta=(h^{*})^{-1}(\alpha)$. Consider the
map $g=h^{-1}|\overline{U}:\overline{U}\to h^{-1}(\overline{U})$. Observe that
$\rho(g(x),x)<\varepsilon$ for all $x\in\overline{U}$. Hence, $g$ and the
identity $\rm{id}_{\overline{U}}$ on $\overline{U}$ are homotopic. In
particular, there is a homotopy $F:g(A)\times[0,1]\to X$ such that $F(x,0)=x$
and $F(x,1)=g(x)$, $x\in g(A)$. Therefore, we can apply the modification of
[30, Proposition 2.3] stated above to conclude that $A\subset g(A)$. Because
$b=h(a)\in A$, the last inclusion implies $b\in g(A)=K$, a contradiction. ∎

######  Remark.

Proposition $2.3$ remains true without homogeneity of $X$ provided $P$ is a
closed set contractible in $X$

This is true if $H^{n}(X;G)=0$ because $\dim_{G}X\leq n$. If $H^{n}(X;G)\neq
0$ this is also true. Indeed, since $\dim_{G}X\leq n$, every $\alpha\in
H^{n}(P;G)$ is extendable to $\widetilde{\alpha}\in H^{n}(X;G)$. On the other
hand $P$ is contractible in $X$, so every such $\widetilde{\alpha}$ is zero.

A compact pair $A\subset K$ is called a $k$-homological membrane spanned on
$A$ for a nontrivial element $\gamma\in{H}_{k}(A;G)$ if
$i^{k}_{A,K}(\gamma)=0$ but $i^{k}_{A,B}(\gamma)\neq 0$ for any proper closed
subset $B$ of $K$ containing $A$. Here, $H_{*}$ denotes the Čech homology and
$i^{k}_{A,K}:H_{k}(A;G)\to H_{k}(K;G)$ is the homomorphism induced be the
inclusion $A\hookrightarrow K$. According to Bing-Borsuk [1], if $A\subset K$
is a compact pair and $\gamma\in H_{k}(A;G)$ is a nontrivial element
homologous to zero in $K$, then there is a closed set $B\subset K$ containing
$A$ such that $A\subset B$ a $k$-homological membrane for $\gamma$ spanned on
$A$. Bing-Borsuk [1] considered the Vietoris homology which is isomorphic with
the Čech homology in the realm of metric compacta. So, homology membranes with
respect to Čech homology always exist in the class of metric compacta.

######  Proposition 2.4.

Any locally homogeneous $ANR$-space with $\dim_{G}X=n$, where $G$ is a
countable group, has the weak $n$-cohomological membrane property.

###### Proof.

Suppose there exists a compact pair $A\subset K$ such that $K$ is contractible
in a proper subset of $X$ and $K$ is an $(n-1)$-cohomological membrane for
some nontrivial $\gamma\in H^{n-1}(A;G)$, but $(K\setminus
A)\cap\overline{X\setminus K}\neq\varnothing$. Since $G$ is countable, the
homology group $H_{n-1}(Y;G^{*})$ is isomorphic to $H^{n-1}(Y;G)^{*}$ for any
metric compactum $Y$, see [16]. Here, $G^{*}$ and $H^{n-1}(Y;G)^{*}$ are the
character groups of $G$ and $H^{n-1}(Y;G)$, considered as discrete groups.
Then $K$ is an $(n-1)$-homological membrane spanned on $A$ for some nontrivial
$\beta\in H_{n-1}(A;G^{*})$, see the proof of [29, Proposition 2.1]. Hence,
following the proof of [1, Theorem 8.1], we can find a compact set $P\subset
X$ contractible in $X$ with $H_{n}(P;G^{*})\neq 0$. So, $H^{n}(P;G)\neq 0$,
which contradicts the remark after Proposition 2.3. ∎

We also need the following properties of cohomology membranes, see Corollary
2.2 and Lemma 2.4 from [29], respectively.

######  Lemma 2.5.

For any space $X$ with the weak $n$-cohomology membrane property the following
conditions hold:

  * (i)

If $K$ is an $(n-1)$-cohomology membrane spanned on a set $A\subset K$ for
some $\gamma\in H^{n-1}(A;G)$, where $K$ is a compactum contractible in a
proper subset of $X$, then $K\setminus A$ is a connected open subset of $X$;

  * (ii)

Let $A\subset P$ be a compact pair such that $P$ is contractible in a proper
subset of $X$ and there exists a non-zero $\gamma\in H^{n-1}(A;G)$ not
extendable over $P$. Then $A$ separates every set $\Gamma\subset X$ containing
$P$ as a proper subset.

Let $(X,\rho)$ be a metric space. We say that $\rho$ is convex if for each
$x,y\in X$ there exists an arc $A\subset X$ with end-points $x$ and $y$ such
that $A$ with the restriction of the metric $\rho$ is isometric to the
interval $[0,\rho(x,y)]$ in the real line (where the real line is considered
with its usual metric). According to [27], every connected locally connected
space admits a convex metric.

######  Proposition 2.6.

Let $\\{G_{i}\\}_{i=1}^{k}$ be a finite family of countable groups and $X$ be
a locally homogeneous $ANR$-space with $\dim_{G_{i}}X=n_{i}$. Then we have:

  * (1)

Every closed set $P\subset X$ with $\dim_{G_{i}}P=n_{i}$, $i\leq k$, has a
non-empty interior;

  * (2)

Every $x\in X$ has a neighborhood $W$ such that any connected
$U\in\mathcal{B}_{x}$ with $U=\rm{int}\overline{U}\subset W$ satisfies the
following for all $i$:

    * (i)

$H^{n_{i}}(\overline{U};G_{i})=0$ and
$H^{n_{i}-1}(\mathrm{bd}\,\overline{U};G_{i})\neq 0$ contains an element not
extendable over $\overline{U}$;

    * (ii)

$\overline{U}$ is an $(n_{i}-1)$-cohomology membrane spanned on
$\mathrm{bd}\,\overline{U}$ for any nontrivial $\alpha_{i}\in
H^{n_{i}-1}(\mathrm{bd}\,\overline{U};G_{i})$ not extendable over
$\overline{U}$.

###### Proof.

$(i)$ This was established in [29, Corollary 2.3] in case $X$ is compact
homogeneous $ANR$. For the covering dimension $\dim$ and locally homogeneous
$ANR$’s it was established in [25, Theorem A] (see also [24] for a property
stronger than locally homogeneity). Suppose $P\subset X$ is a closed set with
$\dim_{G_{i}}P=n_{i}$ for some $i$. Then $P$ is the union of countably many
compact sets $F_{j}$ each contractible in a proper subset of $X$. By the
countable sum theorem for $\dim_{G_{i}}$, we have $\dim_{G_{i}}F_{j}=n$ for at
least one $j$. So, we can assume that $P$ is a compact set contractible in a
proper subset of $X$. Since $\dim_{G_{i}}P=n_{i}$, there is a closed set
$A\subset P$ and an element $\gamma_{i}\in H^{n_{i}-1}(A;G_{i})$ not
extendable over $P$. Then, according to Lemma 2.1(i), there exists a closed
set $K\subset P$ such that $K$ is an $(n_{i}-1)$-cohomology membrane for
$\gamma_{i}$ spanned on $A$. Because $X$ has the weak $n_{i}$-cohomological
membrane property, $(K\setminus A)\cap\overline{X\setminus K}=\varnothing$,
which implies $K\setminus A$ is open in $X$. Finally, the inclusion
$K\setminus A\subset P$ completes the proof.

$(ii)$ Since $X$ is a countable union of compact sets each contractible in a
proper subset of $X$, as in the previous paragraph, for every $i$ there exits
a compact pair $A_{i}\subset K_{i}$ such that $\dim_{G_{i}}K_{i}=n_{i}$,
$K_{i}$ is an $(n_{i}-1)$-cohomology membrane for some $\gamma_{i}\in
H^{n_{i}-1}(A_{i};G_{i})$ spanned on $A_{i}$ and $K_{i}$ is contractible in a
proper subset of $X$. Then each $K_{i}\setminus A_{i}$ is a connected open set
in $X$, see Lemma 2.5(i). Let $x_{i}\in K_{i}\setminus A_{i}$ and let
$W_{i}\in\mathcal{B}_{x_{i}}$ with $\overline{W}_{i}\subset K_{i}\setminus
A_{i}$. Because of the local homogeneity of $X$, we can assume that all
$W_{i}$ are homeomorphic.

Let $U_{i}\in\mathcal{B}_{x_{i}}$ be connected with
$U_{i}=\rm{int}\overline{U}_{i}\subset W_{i}$. Such sets $U_{i}$ exist.
Indeed, since $X$ is locally connected, each its component of connectedness
$X_{c}$ is open, and according to [27], there is a convex metric generating
the topology of $X_{c}$. On the other hand, if $d$ is a convex metric, then
every open ball $B(x,\delta)=\\{y\in X_{c}:d(x,y)<\delta\\}$ is connected and
$\rm{int}\overline{B(x,\delta)}=B(x,\delta)$.

Since each $\overline{U}_{i}$ is contractible in a proper subset of $X$,
$H^{n_{i}}(\overline{U}_{i};G_{i})=0$. We claim that
$H^{n_{i}-1}(\mathrm{bd}\,\overline{U}_{i};G)\neq 0$ and contains an element
not extendable over $\overline{U}_{i}$. Indeed, $K_{i}\setminus U_{i}$ is a
proper closed subset of $K_{i}$ containing $A_{i}$. So, $\gamma_{i}$ can be
extended to $\widetilde{\gamma}_{i}\in H^{n_{i}-1}(K_{i}\setminus
U_{i};G_{i})$. Then $\gamma_{U_{i}}=j^{n_{i}-1}_{K_{i}\setminus
U_{i},\mathrm{bd}\,\overline{U}_{i}}(\widetilde{\gamma}_{i})$ is a non-zero
element of $H^{n_{i}-1}(\mathrm{bd}\,\overline{U}_{i};G)$ (otherwise
$\gamma_{i}$ would be extendable over $K_{i}$), and $\gamma_{U_{i}}$ is not
extendable over $\overline{U}_{i}$. Let show that $\overline{U}_{i}$ is an
$(n_{i}-1)$-cohomology membrane spanned on $\mathrm{bd}\,\overline{U}_{i}$ for
every $\alpha_{i}\in H^{{}_{i}-1}(\mathrm{bd}\,U_{i};G_{i})$ not extendable
over $\overline{U}_{i}$. By Lemma 2.1, for any such $\alpha_{i}$ there is an
$(n_{i}-1)$-cohomology membrane $K_{\alpha_{i}}\subset\overline{U}_{i}$ for
$\alpha_{i}$ spanned on $\mathrm{bd}\,\overline{U}_{i}$. Hence,
$(K_{\alpha_{i}}\setminus\mathrm{bd}\,\overline{U}_{i})\cap\overline{X\setminus
K_{\alpha_{i}}}=\varnothing$. In particular,
$K_{\alpha_{i}}\setminus\mathrm{bd}\,\overline{U}_{i}$ is open in $U_{i}$.
Thus, $K_{\alpha_{i}}=\overline{U}_{i}$, otherwise $U_{i}$ would be the union
of the non-empty disjoint open sets $U_{i}\setminus K_{\alpha_{i}}$ and
$(K_{\alpha_{i}}\setminus\mathrm{bd}\,\overline{U}_{i})$. Finally, since $X$
is locally homogeneous, every $x\in X$ has a neighborhood $W$ satisfying
condition $(2)$. ∎

######  Lemma 2.7.

Let $X$ be a locally homogeneous $ANR$-space and $x\in X$. If $G$ is a
countable group and $H^{n-1}(\mathrm{bd}\,\overline{U};G)\neq 0$ for all
sufficiently small neighborhoods $U\in\mathcal{B}_{x}$, then $\dim_{G}X\geq
n$.

###### Proof.

Suppose $\dim_{G}X\leq n-1$. Since the interior of $\mathrm{bd}\,\overline{U}$
is empty, $\dim_{G}\mathrm{bd}\,\overline{U}\leq n-2$ for all
$U\in\mathcal{B}_{x}$. This, according to the definition of cohomological
dimension, implies $H^{n-1}(\mathrm{bd}\,\overline{U};G)=0$, a contradiction.
∎

##  3\. Proof of Theorem 1.1

The next lemma was established in [6, Theorem 8] for locally connected
continua. The same proof works for locally connected and connected spaces $X$
by passing to the one-point compactification of $X$.

######  Lemma 3.1.

Let $X$ be a connected and locally connected space and
$\\{K_{\alpha}:\alpha\in\Lambda\\}$ be an uncountable collection of disjoint
continua such that for each $\alpha$ the set $X\setminus K_{\alpha}$ has more
than one component. Then there exists $\alpha_{0}\in\Lambda$ such that
$X\setminus K_{\alpha_{0}}$ has exactly two components.

The proof of Theorem 1.1 follows from Proposition 2.6 and Proposition 3.2
below.

######  Proposition 3.2.

Let $X$ be a homogeneous connected $ANR$-space with $\dim_{G}X=n\geq 2$, where
$G$ is a countable group. Then every point $x\in X$ has a basis
$\mathcal{B}_{x}$ of open connected sets $U$ each with a compact closure
satisfying the following conditions:

  * (i)

$H^{n-1}(\overline{U};G)=0$ and $X\setminus\mathrm{bd}\,\overline{U}$ has
exactly two components;

  * (ii)

$\mathrm{bd}\,\overline{U}$ is an $(n-1,G)$-bubble.

###### Proof.

By Proposition 2.6, every point $x\in X$ has a neighborhood $W_{x}$ with a
compact closure such any connected neighborhood $U=\rm{int}\overline{U}\subset
W_{x}$ of $x$ satisfies the following condition: $H^{n}(\overline{U};G)=0$,
$H^{n-1}(\mathrm{bd}\,\overline{U};G)$ contains elements $\gamma$ not
extendable over $\overline{U}$ and for each such $\gamma$ the set
$\overline{U}$ is an $(n-1)$-cohomology membrane for $\gamma$ spanned on
$\mathrm{bd}\,\overline{U}$. We assume also that $\overline{W}_{x}$ is a
connected set contractible in a proper subset of $X$ and there is
$\alpha_{x}\in H^{n-1}(\mathrm{bd}\,\overline{W}_{x};G)$ such that
$\overline{W}_{x}$ is an $(n-1)$-cohomology membrane for $\alpha_{x}$ spanned
on $\mathrm{bd}\,\overline{W}_{x}$. We fix $x\in X$ and let
$\mathcal{B}_{x}^{\prime}$ be the family of all open connected neighborhoods
$U$ of $x$ such that $U=\rm{int}\overline{U}$ and $\overline{U}$ is
contractible in $W_{x}$.

######  Claim 1.

For every $U\in\mathcal{B}_{x}^{\prime}$ there exists a non-zero
$\gamma_{U}\in H^{n-1}(\mathrm{bd}\,\overline{U};G)$ such that $\gamma_{U}$ is
extendable over $\overline{W}_{x}\setminus U$ and $\overline{U}$ is an
$(n-1)$-cohomology membrane for $\gamma_{U}$ spanned on
$\mathrm{bd}\,\overline{U}$.

Indeed, since $\overline{W}_{x}$ is an $(n-1)$-cohomology membrane for
$\alpha_{x}$ spanned on $\mathrm{bd}\,\overline{W}_{x}$, $\alpha_{x}$ is not
extendable over $\overline{W}_{x}$, but it is extendable over every closed
proper subset of $\overline{W}_{x}$ containing
$\mathrm{bd}\,\overline{W}_{x}$. So, $\alpha_{x}$ is extendable to an element
$\widetilde{\alpha}_{x}\in H^{n-1}(\overline{W}_{x}\setminus U;G)$ and the
element $\gamma_{U}=j_{\overline{W}_{x}\setminus
U,\mathrm{bd}\,\overline{U}}^{n-1}(\widetilde{\alpha}_{x})\in
H^{n-1}(\mathrm{bd}\,\overline{U};G)$ is not extendable over $\overline{U}$
(otherwise $\alpha_{x}$ would be extendable over $\overline{W}_{x}$), in
particular $\gamma_{U}\neq 0$. Therefore, according to the choice of $W_{x}$,
$\overline{U}$ is an $(n-1)$-cohomology membrane for $\gamma_{U}$ spanned on
$\mathrm{bd}\,\overline{U}$.

Let $\mathcal{B}_{x}^{\prime\prime}$ be the family of all
$U\in\mathcal{B}_{x}^{\prime}$ satisfying the following condition:
$\mathrm{bd}\,\overline{U}$ contains a continuum $F_{U}$ such that $X\setminus
F_{U}$ has exactly two components and $F_{U}$ is an $(n-1,G)$-carrier of
$j^{n-1}_{\mathrm{bd}\,\overline{U},F_{U}}(\gamma_{U})$.

######  Claim 2.

$\mathcal{B}_{x}^{\prime\prime}$ is a local base at $x$.

Since $X$ is arc connected and locally arc connected, there is a convex metric
$d$ generating the topology of $X$, see [27]. We fix
$W_{0}\in\mathcal{B}_{x}^{\prime}$ and for every $\delta>0$ denote by
$B(x,\delta)$ the open ball in $X$ with a center $x$ and a radius $\delta$.
There exists $\varepsilon_{x}>0$ such that $B(x,\delta)\subset W_{0}$ for all
$\delta\leq\varepsilon_{x}$. We already observed in the proof of Proposition
2.6 that any $B(x,\delta)$ is connected and
$\rm{int}(\overline{B(x,\delta)})=B(x,\delta)$. Moreover,
$\overline{B(x,\delta)}$ is contractible in $W_{x}$. Hence, all
$U_{\delta}=B(x,\delta)$, $\delta\leq\varepsilon_{x}$, belong to
$\mathcal{B}_{x}^{\prime}$. Consequently, by Claim 1, for every $\delta$ there
exists a non-zero $\gamma_{\delta}\in
H^{n-1}(\mathrm{bd}\,\overline{U}_{\delta};G)$ such that
$\overline{U}_{\delta}$ is an $(n-1)$-cohomology membrane for
$\gamma_{\delta}$ spanned on $\mathrm{bd}\,\overline{U}_{\delta}$ and
$\gamma_{\delta}$ is extendable over $\overline{W}_{x}\setminus U_{\delta}$.
By Lemma 2.1(ii), there exists a closed subset $F_{\delta}$ of
$\mathrm{bd}\,\overline{U}_{\delta}$ which is a carrier of
$\gamma^{*}_{\delta}=j^{n-1}_{\mathrm{bd}\,\overline{U}_{\delta},F_{\delta}}(\gamma_{\delta})$.
Since $n\geq 2$ and $F_{\delta}$ is a carrier of $\gamma^{*}_{\delta}\in
H^{n-1}(F_{\delta};G)$, $F_{\delta}$ is a continuum, see [29, Lemma 2.7]. Let
us show that the family $\\{F_{\delta}:\delta\leq\varepsilon_{x}\\}$ is
uncountable. Since the function $f\colon X\to\mathbb{R}$, $f(y)=d(x,y)$, is
continuous and $W_{0}$ is connected, $f(W_{0})$ is an interval containing
$[0,\varepsilon_{x}]$ and
$f^{-1}([0,\varepsilon_{x}))=B(x,\varepsilon_{x})\subset W_{0}$. So,
$f^{-1}(\delta)=\mathrm{bd}\,\overline{U}_{\delta}\neq\varnothing$ for all
$\delta\leq\varepsilon_{x}$. Hence, the family
$\\{F_{\delta}:\delta\leq\varepsilon_{x}\\}$ is uncountable and consist of
disjoint continua. Moreover, $\gamma^{*}_{\delta}$ is a non-zero element of
$H^{n-1}(F_{\delta};G)$ not extendable over $\overline{W}_{x}$ because $F_{U}$
is contractible in $\overline{W}_{x}$. Thus, by Lemma 2.5(ii), $F_{\delta}$
separates $X$. So, each $X\setminus F_{\delta}$ has at least two components.
Then, by Lemma 3.1, there exists $\delta_{0}\leq\varepsilon_{x}$ such that
$X\setminus F_{\delta_{0}}$ has exactly two components. Therefore,
$U_{\delta_{0}}=B(x,\delta_{0})\in\mathcal{B}_{x}^{\prime\prime}$ and it is
contained in $W_{0}$. This completes the proof of Claim 2.

Now, let $\widetilde{\mathcal{B}}_{x}$ be the family of all
$U\in\mathcal{B}_{x}^{\prime\prime}$ with
$H^{n-1}(\mathrm{bd}\,\overline{U};G)\neq 0$ such that both $U$ and
$X\setminus\overline{U}$ are connected.

######  Claim 3.

$\widetilde{\mathcal{B}}_{x}$ is a local base at $x$.

Let $U_{0}$ be an arbitrary neighborhood of $x$ such that $\overline{U}_{0}$
is contractible in $W_{x}$. We are going to find a member of
$\widetilde{\mathcal{B}}_{x}$ contained in $U_{0}$. To this end, let $\rho$ be
a metric generating the topology of the one-point compactification of $X$ and
let $\varepsilon=\rho(x,X\setminus U_{0})$. According to Theorem 2.2 there is
$\eta>0$ corresponding to $\varepsilon/2$ and the point $x$ (i.e., for every
$y\in X$ with $\rho(y,x)<\eta$, there exists a homeomorphism $h:X\to X$ with
$h(y)=x$ and $\rho(h(z),z)<\varepsilon/2$ for all $z\in X$). Now, choose a
connected neighborhood $W$ of $x$ with $\overline{W}\subset
B_{\rho}(x,\varepsilon/2)$ and $\rho-\rm{diam}(\overline{W})<\eta$. Finally,
take $U\in\mathcal{B}_{x}^{\prime\prime}$ such that $\overline{U}$ is
contractible in $W$. By Claim 2, there exists a continuum
$F_{U}\subset\mathrm{bd}\,\overline{U}$ such that $X\setminus F_{U}$ has
exactly two components and $F_{U}$ is an $(n-1,G)$-carrier for
$\gamma_{U}^{*}=j^{n-1}_{\mathrm{bd}\,\overline{U},F_{U}}(\gamma_{U})$. If
$F_{U}=\mathrm{bd}\,\overline{U}$, then $U$ is the desired member of
$\widetilde{\mathcal{B}}_{x}$. Indeed, since $X\setminus bd\overline{U}=U\cup
X\setminus\overline{U}$ with $U\cap(X\setminus\overline{U})=\varnothing$ and
$U$ is connected, then $X\setminus\overline{U}$ should be also connected
(recall that $X\setminus F_{U}$ has exactly two components).

Suppose that $F_{U}$ is a proper subset of $\mathrm{bd}\,\overline{U}$.
Because $F_{U}$ (as a subset of $\overline{U}$) is contractible in a compact
set $\Gamma\subset W$, $\gamma_{U}^{*}$ is not extendable over $\Gamma$. Thus,
we can apply Lemma 2.5(ii) to conclude that $F_{U}$ separates $\overline{W}$.
So, $\overline{W}\setminus F_{U}=V_{1}\cup V_{2}$ for some open, non-empty
disjoint subsets $V_{1},V_{2}\subset\overline{W}$. Since $U$ is a connected
subset of $\overline{W}\setminus F_{U}$, $U$ is contained in one of the sets
$V_{1},V_{2}$, say $U\subset V_{1}$. Hence,
$F_{U}\cup\overline{V}_{2}\subset\overline{W}_{x}\setminus U$. Since
$\gamma_{U}$ is extendable over $\overline{W}_{x}\setminus U$ (see Claim 1),
$\gamma_{U}^{*}$ is also extendable over $\overline{W}_{x}\setminus U$, in
particular $\gamma_{U}^{*}$ is extendable over $F_{U}\cup\overline{V}_{2}$. On
the other hand, $\gamma_{U}^{*}$ is not extendable over $\overline{W}$ because
$F_{U}$ is contractible in $\overline{W}$. The last fact, together with the
equality $(F_{U}\cup\overline{V}_{1})\cap(F_{U}\cup\overline{V}_{2})=F_{U}$,
yields that $\gamma_{U}^{*}$ is not extendable over
$F_{U}\cup\overline{V}_{1}$. Let
$\beta=j_{F_{U},F^{\prime}}^{n-1}(\gamma_{U}^{*})$, where
$F^{\prime}=\overline{V}_{1}\cap F_{U}$ (observe that
$F^{\prime}\neq\varnothing$ because $\overline{W}$ is connected). If
$F^{\prime}$ is a proper subset of $F_{U}$, then $\beta=0$ since $F_{U}$ is a
carrier for $\gamma_{U}^{*}$. So, $\beta$ would be extendable over
$\overline{V}_{1}$, which implies $\gamma_{U}^{*}$ is extendable over
$F_{U}\cup\overline{V}_{1}$, a contradiction. Therefore,
$F^{\prime}=F_{U}\subset\overline{V}_{1}$ and $\gamma_{U}^{*}$ is not
extendable over $\overline{V}_{1}$. Consequently, there exists an
$(n-1)$-cohomology membrane $P_{\beta}\subset\overline{V}_{1}$ for
$\gamma_{U}^{*}$ spanned on $F_{U}$. By Lemma 2.5(i), $V=P_{\beta}\setminus
F_{U}$ is a connected open set in $X$ whose boundary is the set
$F^{\prime\prime}=\overline{X\setminus
P_{\beta}}\cap\overline{P_{\beta}\setminus F_{U}}\subset F_{U}$. As above,
using that $\gamma_{U}^{*}$ is not extendable over $P_{\beta}$ and
$j_{F_{U},Q}^{n-1}(\gamma_{U}^{*})=0$ for any proper closed subset $Q\subset
F_{U}$, we can show that $F^{\prime\prime}=F_{U}$ and
$\mathrm{bd}\,\overline{V}=F_{U}$. Summarizing the properties of $V$, we have
that $\overline{V}$ is contractible in $W_{x}$ (because so is
$\overline{U}_{0}$), $V=\rm{int}(\overline{V})$ (because
$F_{U}=\mathrm{bd}\,\overline{V}$) and $V$ is connected. Moreover, since
$X\setminus F_{U}$ is the union of the open disjoint non-empty sets $V$ and
$X\setminus P_{\beta}$ such that $V$ is connected and $X\setminus F_{U}$ has
exactly two components, $X\setminus\overline{V}$ is also connected. Finally,
because $F_{U}$ is an $(n-1,G)$-carrier for the nontrivial $\gamma_{U}^{*}$,
$H^{n-1}(\mathrm{bd}\,\overline{V};G)\neq 0$. Thus, if $V$ contains $x$, then
$V$ is a member of $\widetilde{\mathcal{B}}_{x}$.

If $V$ does not contain $x$, we take a point $y\in V$ with $\rho(x,y)<\eta$.
This is possible because $V\subset\overline{W}$ and
$\rho-\rm{diam}(\overline{W})<\eta$. So, according to the choice of $\eta$,
there is a homeomorphism $h$ on $X$ such that $h(y)=x$ and
$\rho(z,h(z))<\varepsilon/2$ for all $z\in X$. Then $h(V)\subset U_{0}$ (this
inclusion follows from the choice of $\varepsilon$ and the fact that $h$ is
$(\varepsilon/2)$-close to the identity on $X$). So, $\overline{h(V)}$ is
contractible in $W_{x}$. Since the remaining properties from the definition of
$\widetilde{\mathcal{B}}_{x}$ are invariant under homeomorphisms, $h(V)$ is
the required member of $\widetilde{\mathcal{B}}_{x}$, which provides the proof
of Claim 3.

The next claim completes the proof of Proposition 3.2.

######  Claim 4.

$H^{n-1}(\overline{U};G)=0$ and $\mathrm{bd}\,\overline{U}$ is an
$(n-1,G)$-bubble for every $U\in\widetilde{\mathcal{B}}_{x}$.

Because each $\overline{U}$, $U\in\widetilde{\mathcal{B}}_{x}$, is
contractible in $\overline{W}_{x}$, any nontrivial $\gamma\in
H^{n-1}(\overline{U};G)$ cannot be extendable over $\overline{W}_{x}$. Since
$\overline{W}_{x}$ is contractible in a proper subset of $X$, by Lemma
2.5(ii), $\overline{U}$ would separate $X$ provided
$H^{n-1}(\overline{U};G)\neq 0$. On the other hand, each
$X\setminus\overline{U}$ is connected. Therefore, $H^{n-1}(\overline{U};G)=0$
for all $U\in\widetilde{\mathcal{B}}_{x}$. Suppose there exists a proper
closed subset $F\subset\mathrm{bd}\,\overline{U}$ and a nontrivial element
$\alpha\in H^{n-1}(F;G)$. Since $H^{n-1}(\overline{U};G)=0$, $\alpha$ is not
extendable over $\overline{U}$. Hence, there is an $(n-1)$-cohomology membrane
$K_{\alpha}\subset\overline{U}$ for $\alpha$ spanned on $F$. Therefore,
$(K_{\alpha}\setminus F)\cap\overline{X\setminus K_{\alpha}}=\varnothing$. In
particular, $K_{\alpha}\setminus F$ is open in $\overline{U}\setminus F$. On
the other hand, $K_{\alpha}\setminus F$ is also closed in
$\overline{U}\setminus F$. Because $\overline{U}\setminus F$ is connected
(recall that $U$ is a dense connected subset of $\overline{U}\setminus F$), we
obtain $K_{\alpha}=\overline{U}$. Finally, observe that any point from
$\mathrm{bd}\,\overline{U}\setminus F$ belongs to $(K_{\alpha}\setminus
F)\cap\overline{X\setminus K_{\alpha}}$, a contradiction. Therefore,
$\mathrm{bd}\,\overline{U}$ is an $(n-1,G)$-bubble. ∎

##  4\. Proof of Theorem 1.2 and Corollary 1.3

Proof of Theorem $1.2$. $(i)$ Suppose $K$ is an $(n-1)$-cohomology membrane
spanned on $A$ for some $\gamma\in H^{n-1}(A;G)$ and there is a point
$a\in(K\setminus A)\cap\overline{X\setminus K}$. Take $V\in\mathcal{B}_{a}$
with $\overline{V}\cap A=\varnothing$, where $\mathcal{B}_{a}$ is a local base
at $a$ satisfying the hypotheses of Proposition 2.6(2). Since $K\setminus V$
is a proper subset of $K$ containing $A$, $\gamma$ is extendable to
$\gamma^{*}\in H^{n-1}(K\setminus V;G)$. Then $\alpha_{V}=j^{n-1}_{K\setminus
V,K\cap\mathrm{bd}\,\overline{V}}(\gamma^{*})$ is not extendable over
$K\cap\overline{V}$ (otherwise $\gamma$ would be extendable over $K$). In
particular, $\alpha_{V}$ a non-zero element of
$H^{n-1}(K\cap\mathrm{bd}\,\overline{V};G)$. Because
$\mathrm{bd}\,\overline{V}$ has an empty interior in $X$,
$\dim_{G}\mathrm{bd}\,\overline{V}\leq n-1$, see Proposition 2.6(1).
Consequently, $\alpha_{V}$ can be extended to $\widetilde{\alpha}_{V}\in
H^{n-1}(\mathrm{bd}\,\overline{V};G)$. Moreover, since $\alpha_{V}$ is not
extendable over $K\cap\overline{V}$, $\widetilde{\alpha}_{V}$ is not
extendable over $\overline{V}$. Finally, using that $\overline{V}$ is an
$(n-1)$-cohomology membrane spanned on $\mathrm{bd}\,\overline{V}$ for
$\widetilde{\alpha}_{V}$ and, since $a\in(K\setminus
A)\cap\overline{X\setminus K}$ implies that
$\mathrm{bd}\,\overline{V}\cup(K\cap\overline{V})$ is a proper closed subset
of $\overline{V}$ containing $\mathrm{bd}\,\overline{V}$, we can find
$\beta\in H^{n-1}(\mathrm{bd}\,\overline{V}\cup(K\cap\overline{V});G)$
extending $\widetilde{\alpha}_{V}$. Thus, $\alpha_{V}$ is extendable over
$K\cap\overline{V}$, a contradiction. Therefore, $X$ has the $n$-cohomology
membrane property.

$(ii)$ We need the following result [22, Theorem 1]: For every metric
compactum $K$ with $\dim_{G}K=n$ there exist a closed set $Y\subset K$ and a
dense set $D\subset Y$ such that $\dim_{G}Y=n$ and $K$ has an $n$-dimensional
$G$-obstruction at every $y\in D$. In our situation we take a compact set
$K\subset X$ with $\dim_{G}K=n$ and find corresponding sets $D\subset Y\subset
K$. Since $\dim_{G}Y=n$, $\rm{int}Y\neq\varnothing$ (Proposition 2.6) and
there is $y\in D\cap\rm{int}Y$. Because $K$ has an $n$-dimensional
$G$-obstruction at $y$, we can find a neighborhood $W\subset K$ of $y$ such
that for any open in $K$ neighborhood $V$ of $y$ contained in $W$ the
homomorphism $H^{n}(K,K\setminus V;G)\to H^{n}(K,K\setminus W;G)$ is not
trivial. This implies that for every open in $K$ neighborhoods $U,V$ of $y$
with $\overline{U}\subset V$ the homomorphism $H^{n}(K,K\setminus U;G)\to
H^{n}(K,K\setminus V;G)$ is also nontrivial. Since $y\in\rm{int}Y$, we can
suppose that $W\in\mathcal{B}_{y}$ such that $\overline{W}$ is contractible in
$X$ and satisfying the hypotheses of Proposition 2.6(2). Then, by the excision
axiom, for every $U,V\in\mathcal{B}_{y}$ with $\overline{U}\subset
V\subset\overline{V}\subset W$, the groups $H^{n}(K,K\setminus U;G)$ and
$H^{n}(K,K\setminus V;G)$ are isomorphic to $H^{n}(X,X\setminus U;G)$ and
$H^{n}(X,X\setminus V;G)$, respectively. So, $j_{U,V}:H^{n}(X,X\setminus
U;G)\to H^{n}(X,X\setminus V;G)$ is a nontrivial homomorphism for all
$U,V\in\mathcal{B}_{y}$ with $\overline{U}\subset V\subset\overline{V}\subset
W$. Finally, because $X$ is locally homogeneous, it has an $n$-dimensional
$G$-obstruction at every $x\in X$.

To prove the second half of condition $(ii)$, let $U,V\in\mathcal{B}_{x}$ with
$\overline{U}\subset V\subset\overline{V}\subset W$.

######  Claim 5.

The homomorphism $j^{n-1}_{\overline{W}\setminus U,\overline{W}\setminus
V}:H^{n-1}(\overline{W}\setminus U;G)\to H^{n-1}(\overline{W}\setminus V;G)$
is surjective and $H^{n-1}(\overline{W}\setminus V;G)\neq 0$.

Indeed, consider the Mayer-Vietoris exact sequence below, where the
coefficient group $G$ is suppressed,

$\begin{CD}H^{n-1}(\overline{W}\backslash
U)@>{{\varphi}}>{}>H^{n-1}(\overline{V}\backslash U)\oplus
H^{n-1}(\overline{W}\backslash
V)@>{{\psi}}>{}>H^{n-1}(\rm{bd}\,\overline{V})...\end{CD}$

The maps $\varphi$ and $\psi$ are defined by
$\varphi(\gamma)=(j^{n-1}_{\overline{W}\backslash U,\overline{V}\backslash
U}(\gamma),j^{n-1}_{\overline{W}\backslash U,\overline{W}\backslash
V}(\gamma))$ and $\psi((\beta,\alpha))=j^{n-1}_{\overline{V}\backslash
U,\rm{bd}\,\overline{V}}(\beta)-j^{n-1}_{\overline{W}\backslash
V,\rm{bd}\,\overline{V}}(\alpha)$. For every $\alpha\in
H^{n-1}(\overline{W}\setminus V;G)$ the element
$\beta^{\prime}_{\alpha}=j^{n-1}_{\overline{W}\backslash
V,\rm{bd}\overline{V}}(\alpha)\in H^{n-1}(\rm{bd}\,\overline{V};G)$ is
extendable to an element $\beta_{\alpha}\in H^{n-1}(\overline{V}\setminus
U;G)$. Indeed, there are two possibilities: either $\beta^{\prime}_{\alpha}$
is extendable over $\overline{V}$ or it is not extendable over $\overline{V}$.
The first case obviously implies that $\beta^{\prime}_{\alpha}$ is extendable
over $\overline{V}\setminus U$. In the second case, by Proposition 2.6(2),
$\overline{V}$ is an $(n-1)$-cohomological membrane spanned on
$\rm{bd}\,\overline{V}$ for $\beta^{\prime}_{\alpha}$. Then, since
$\overline{V}\setminus U$ is a proper closed subset of $\overline{V}$,
$\beta^{\prime}_{\alpha}$ is extendable over $\overline{V}\backslash U$.
Hence, $\psi(\beta_{\alpha},\alpha)=0$ for any $\alpha\in
H^{n-1}(\overline{W}\setminus V;G)$. Consequently, there is
$\gamma_{\alpha}\in H^{n-1}(\overline{W}\backslash U;G)$ with
$\varphi(\gamma_{\alpha})=(\beta_{\alpha},\alpha)$. In particular,
$j^{n-1}_{\overline{W}\backslash U,\overline{W}\backslash
V}(\gamma_{\alpha})=\alpha$, which shows the surjectivity of
$j^{n-1}_{\overline{W}\backslash U,\overline{W}\backslash V}$. The
nontriviality of $H^{n-1}(\overline{W}\setminus V;G)$ follows from the Mayer-
Vietoris exact sequence

$\to H^{n-1}(\overline{V};G)\oplus H^{n-1}(\overline{W}\backslash V;G)\to
H^{n-1}(\rm{bd}\,\overline{V};G)\to H^{n}(\overline{W};G)\to$

Indeed, $\overline{W}$ being contractible in $X$ implies that
$H^{n}(\overline{W};G)=0$ (see the remark after Proposition 2.3). Hence, the
homomorphism $j^{n-1}_{\overline{V},\rm{bd}\overline{V}}$ would be surjective
provided $H^{n-1}(\overline{W}\backslash V;G)=0$. This would imply that every
non-trivial element of $H^{n-1}(\rm{bd}\,\overline{V};G)$ is extendable over
$\overline{V}$, which contradicts Proposition 2.6(2).

To complete the proof, consider the commutative diagram whose rows are parts
of exact sequences

$\begin{CD}H^{n-1}(\overline{W}\backslash
U;G)@>{{\delta_{U}}}>{}>H^{n}(\overline{W},\overline{W}\backslash
U;G)@>{{i_{U}}}>{}>H^{n}(\overline{W};G)\\\ @
VV{j^{n-1}_{\overline{W}\backslash U,\overline{W}\backslash
V}}V@V{}V{j_{U,V}^{\prime}}V@V{}V{\rm{id}}V\\\ H^{n-1}(\overline{W}\backslash
V;G)@>{{\delta_{V}}}>{}>H^{n}(\overline{W},\overline{W}\backslash
V;G)@>{{i_{V}}}>{}>H^{n}(\overline{W};G).\end{CD}$

Since $H^{n}(\overline{W};G)=0$, both $\delta_{U}$ and $\delta_{V}$ are
surjective. This, combined with the surjectivity of
$j^{n-1}_{\overline{W}\backslash U,\overline{W}\backslash V}$ and
nontriviality of $H^{n-1}(\overline{W}\setminus V;G)$, implies that
$j_{U,V}^{\prime}$ is also surjective. Finally, by the excision axiom the
groups $H^{n}(\overline{W},\overline{W}\backslash U;G)$ and
$H^{n}(\overline{W},\overline{W}\backslash V;G)$ are isomorphic to
$H^{n}(X,X\backslash U;G)$ and $H^{n}(X,X\backslash V;G)$, respectively.
Therefore, the homomorphism $j_{U,V}^{n}:H^{n}(X,X\backslash U;G)\to
H^{n}(X,X\backslash V;G)$ is surjective.

Proof of Corollary $1.3$. The conclusion of this corollary implies the well-
known invariance of domains property, which was established in [23] and [25],
respectively, for compact homogeneous and locally homogeneous $ANR$-spaces $X$
with $\dim X<\infty$. For homogeneous $ANR$-compacta $X$ with
$\dim_{G}X<\infty$, where $G$ is a countable principal ideal domain, it was
established in [29]. The arguments from [29] also work in our situation.

##  5\. Proof of Theorem $1.4$.

$(i)$ It suffices to show that the one-point compactification
$bX=X\cup\\{b\\}$ of $X$ is dimensionally full-valued. To this end, we use the
following result of Dranishnikov [7, Theorem 12.3-12.4]: If $Y$ is a finite-
dimensional $ANR$-compactum, then
$\dim_{\mathbb{Z}_{(p)}}Y=\dim_{\mathbb{Z}_{p}}Y$ for all prime $p$. Moreover,
$\dim_{\mathbb{Q}}Y\leq\dim_{G}Y$ for any group $G\neq 0$ and there exists a
prime number $p$ with $\dim
Y=\dim_{\mathbb{Z}_{(p)}}Y=\dim_{\mathbb{Z}_{p}}Y$. The proof presented in [7]
works also when $Y$ is the one-point compactification of an $ANR$-space. Here
$\mathbb{Z}_{p}=\mathbb{Z}/p\mathbb{Z}$ is the cyclic group and
$\mathbb{Z}_{(p)}=\\{m/n:n{~{}}\mbox{is not divisible
by}{~{}}p\\}\subset\mathbb{Q}$, where $\mathbb{Q}$ is the field of rational
numbers. We also consider the quotient group
$\mathbb{Z}_{p^{\infty}}=\mathbb{Q}/\mathbb{Z}_{(p)}$. It is well known [22]
that the so called Bockstein basis consists of the groups
$\sigma=\\{\mathbb{Q},\mathbb{Z}_{p},\mathbb{Z}_{(p)},\mathbb{Z}_{p^{\infty}}:p\in\mathcal{P}\\}$,
$\mathcal{P}$ is the set of all primes. For any group (not necessarily
countable) there exists a collection $\sigma(G)\subset\sigma$ such that
$\dim_{G}X=\sup\\{\dim_{H}X:H\in\sigma(G)\\}$ for any space $X$.

Let $X$ be a locally homogeneous $ANR$-space with $\dim X=n$. According to the
mentioned above Dranishnikov’s result, there exists $p\in\mathcal{P}$ with
$\dim_{\mathbb{Z}_{(p)}}bX=\dim_{\mathbb{Z}_{p}}bX=n$. If
$\dim_{\mathbb{Z}_{p^{\infty}}}bX=n$, we are done. Indeed, according to [7,
Lemma 2.6], $bX$ is $p$-regular, i.e.

$\dim_{\mathbb{Z}_{(p)}}bX=\dim{\mathbb{Z}_{p^{\infty}}}bX=\dim_{\mathbb{Z}_{p}}bX=\dim_{\mathbb{Q}}bX=n.$

Then, applying again Dranishnikov’s result [7, Theorem 12.3], we obtain
$\dim_{\mathbb{Q}}bX\leq\dim_{G}bX\leq n$ for any group $G\neq 0$. Hence,
$\dim_{G}bX=\dim bX=n$ for all nontrivial groups $G$, and by [21, Theorem 11],
$bX$ is dimensionally full-valued. Therefore, the next claim completes the
proof.

######  Claim 6.

If $\dim_{\mathbb{Z}_{(p)}}bX=\dim_{\mathbb{Z}_{p}}bX=n$ for some prime number
$p$, then $\dim{\mathbb{Z}_{p^{\infty}}}bX=n$.

Suppose $\dim{\mathbb{Z}_{p^{\infty}}}bX\leq n-1$. According to the Bockstein
inequalities (see, for example [7], or [21]), we have
$\dim_{\mathbb{Z}_{p}}bX\leq\dim{\mathbb{Z}_{p^{\infty}}}bX+1$, which in our
case implies
$\dim{\mathbb{Z}_{p^{\infty}}}bX=\dim{\mathbb{Z}_{p^{\infty}}}X=n-1$. Since
$\dim_{\mathbb{Z}_{p}}(bX)^{2}=2\dim_{\mathbb{Z}_{p}}X=2n$, using the next
equality (see [21])

$\dim_{\mathbb{Z}_{p^{\infty}}}(bX)^{2}=\max\\{2\dim_{\mathbb{Z}_{p^{\infty}}}bX,\dim_{\mathbb{Z}_{p}}(bX)^{2}-1\\},$

we obtain
$\dim{\mathbb{Z}_{p^{\infty}}}X^{2}=\dim{\mathbb{Z}_{p^{\infty}}}(bX)^{2}=2n-1$.
Then, by Proposition 2.6(2), for every $(x,y)\in X^{2}$ there are
neighborhoods $U_{x}$ and $V_{y}$ of $x$ and $y$ with compact closures such
that $W=U\times V\subset U_{x}\times V_{y}$ yields
$H^{2n-2}(\mathrm{bd}\,\overline{W};\mathbb{Z}_{{p^{\infty}}})\neq 0$ for any
$W\in\mathcal{B}_{(x,y)}$. Since
$\mathrm{bd}\,\overline{W}=(\overline{U}\times\mathrm{bd}\,\overline{V})\cup(\mathrm{bd}\,\overline{U}\times\overline{V})$,
we have the Meyer-Vietoris exact sequence (in all exact sequences below the
coefficient groups $Z_{p^{\infty}}$ are suppressed)

$H^{2n-3}(\Gamma_{U}\times\Gamma_{V})\to H^{2n-2}(\Gamma_{W})\to
H^{2n-2}(\overline{U}\times\Gamma_{V})\oplus
H^{2n-2}(\Gamma_{U}\times\overline{V})\to$

Here, $\Gamma_{U}=\mathrm{bd}\,\overline{U}$,
$\Gamma_{V}=\mathrm{bd}\,\overline{V}$ and
$\Gamma_{W}=\mathrm{bd}\,\overline{W}$. The Künneth formulas provide the
following exact sequences

$\sum_{i+j=2n-2}H^{i}(\overline{U})\otimes H^{j}(\Gamma_{V})\to
H^{2n-2}(\overline{U}\times\Gamma_{V})\to\sum_{i+j=2n-1}H^{i}(\overline{U})*H^{j}(\Gamma_{V}),$
$\sum_{i+j=2n-2}H^{i}(\Gamma_{U})\otimes H^{j}(\overline{V})\to
H^{2n-2}(\Gamma_{U}\times\overline{V})\to\sum_{i+j=2n-1}H^{i}(\Gamma_{U})*H^{j}(\overline{V})$

and

$\sum_{i+j=2n-3}H^{i}(\Gamma_{U})\otimes H^{j}(\Gamma_{V})\to
H^{2n-3}(\Gamma_{U}\times\Gamma_{V})\to\sum_{i+j=2n-2}H^{i}(\Gamma_{U})*H^{j}(\Gamma_{V}).$

Since $\dim_{Z_{p^{\infty}}}X=n-1$, by Proposition 2.6(2), we can assume that
the groups $H^{n-1+i}(\Gamma_{U};Z_{p^{\infty}})$,
$H^{n-1+i}(\overline{U};Z_{p^{\infty}})$, as well as the groups
$H^{n-1+i}(\Gamma_{V};Z_{p^{\infty}})$ and
$H^{n-1+i}(\overline{V};Z_{p^{\infty}})$ are trivial for all $U\subset U_{x}$,
$V\subset V_{y}$ and $i\geq 0$. Hence, the groups
$H^{2n-2}(\overline{U}\times\Gamma_{V};Z_{p^{\infty}})$,
$H^{2n-2}(\Gamma_{U}\times\overline{V};Z_{p^{\infty}})$ and
$H^{2n-3}(\Gamma_{U}\times\Gamma_{V};Z_{p^{\infty}})$ are also trivial, which
implies the triviality of the group $H^{2n-2}(\Gamma_{W};Z_{p^{\infty}})$, a
contradiction. Therefore, $\dim{\mathbb{Z}_{p^{\infty}}}bX=n$.

$(ii)$ To prove the second half of Theorem 1.4, denote by $\widehat{H}_{*}$
the exact homology developed in [26] for locally compact spaces. The
homological dimension $h\dim_{G}Y$ of a space $Y$ is the largest number $n$
such that $\widehat{H}_{n}(Y,\Phi;G)\neq 0$ for some closed set $\Phi\subset
Y$. According to [15] and [26], $h\dim_{G}Y$ is the greatest $m$ such that the
local homology group $\widehat{H}_{m}(Y,Y\setminus y;G)=\varinjlim_{y\in
U}\widehat{H}_{m}(Y,Y\setminus U;G)$ is not trivial for some $y\in Y$.
Moreover, for any field $F$ we have $h\dim_{F}Y=\dim_{F}Y$, see [15].
Therefore, since $\dim_{\mathbb{Q}}X=\dim X=n$, we have
$h\dim_{\mathbb{Q}}X=n$ and $\widehat{H}_{n}(X,X\setminus x;\mathbb{Q})\neq 0$
for all $x\in X$. This means that every $x\in X$ has a neighborhood $W$ such
that $\widehat{H}_{n}(X,X\setminus U;\mathbb{Q})\neq 0$ for all open sets
$U\subset W$ containing $x$. We assume that $W$ satisfies conditions $(1)-(3)$
from Theorem 1.1 such that $\overline{W}$ is contractible in a proper set in
$X$ and $H^{n-1}(\mathrm{bd}\,\overline{U};\mathbb{Z})\neq 0$ for all
neighborhoods $U$ of $x$ with $U\subset W$, see Proposition 2.6(ii). Let $V$
be a neighborhood of $x$ with $\overline{V}\subset W$.

######  Claim 7.

The pair $\overline{W}\backslash V\subset\overline{W}$ is an $(n-1)$-homology
membrane spanned on $\overline{W}\backslash V$ for any nontrivial $\gamma\in
H_{n-1}(\overline{W}\backslash V;\mathbb{Q})$.

Since $\widehat{H}_{n}(X,X\setminus V;\mathbb{Q})\neq 0$, by the excision
axiom $\widehat{H}_{n}(\overline{W},\overline{W}\setminus V;\mathbb{Q})\neq
0$. Consider the exact sequences (see [26] for the existence of such
sequences)

$\to\widehat{H}_{n}(\overline{W};\mathbb{Q})\to\widehat{H}_{n}(\overline{W},\overline{W}\backslash
V;\mathbb{Q})\to\widehat{H}_{n-1}(\overline{W}\backslash
V;\mathbb{Q})\to\widehat{H}_{n-1}(\overline{W};\mathbb{Q})$

and

$0\to\mathrm{Ext}(H^{n+1}(\overline{W};\mathbb{Z}),\mathbb{Q})\to\widehat{H}_{n}(\overline{W};\mathbb{Q})\to\mathrm{Hom}(H^{n}(\overline{W};\mathbb{Z}),\mathbb{Q})\to
0$
$0\to\mathrm{Ext}(H^{n}(\overline{W};\mathbb{Z}),\mathbb{Q})\to\widehat{H}_{n-1}(\overline{W};\mathbb{Q})\to\mathrm{Hom}(H^{n-1}(\overline{W};\mathbb{Z}),\mathbb{Q})\to
0.$

Since
$H^{n+1}(\overline{W};\mathbb{Z})=H^{n}(\overline{W};\mathbb{Z})=H^{n-1}(\overline{W};\mathbb{Z})=0$,
both $\widehat{H}_{n}(\overline{W};\mathbb{Q})$ and
$\widehat{H}_{n-1}(\overline{W};\mathbb{Q})$ are trivial. Therefore, the group
$\widehat{H}_{n-1}(\overline{W}\backslash V;\mathbb{Q})$ is nontrivial.
Moreover, the exact sequence

$\mathrm{Ext}(H^{n}(\overline{W}\backslash
V;\mathbb{Z}),\mathbb{Q})\to\widehat{H}_{n-1}(\overline{W}\backslash
V;\mathbb{Q})\to\mathrm{Hom}(H^{n-1}(\overline{W}\backslash
V;\mathbb{Z}),\mathbb{Q})\to 0$

shows that $\widehat{H}_{n-1}(\overline{W}\backslash V;\mathbb{Q})$ is
isomorphic to $\mathrm{Hom}(H^{n-1}(\overline{W}\backslash
V;\mathbb{Z}),\mathbb{Q})$ because $H^{n}(\overline{W}\backslash
V;\mathbb{Z})=0$ according to the remark after by Proposition 2.3.

To complete the proof of Claim 7, we need to show that if $B$ is a proper
closed subset of $\overline{W}$ containing $\overline{W}\backslash V$, then
$i^{n-1}_{\overline{W}\backslash V,B}(\gamma)\neq 0$ for every nontrivial
$\gamma\in H_{n-1}(\overline{W}\backslash V;\mathbb{Q})$. To this end,
consider the exact sequence

$0\to\mathrm{Ext}(H^{n}(B;\mathbb{Z}),\mathbb{Q})\to\widehat{H}_{n-1}(B;\mathbb{Q})\to\mathrm{Hom}(H^{n-1}(B;\mathbb{Z}),\mathbb{Q})\to
0.$

Since $B$ is contractible in $X$ (as a subset of $W$),
$H^{n}(B;\mathbb{Z})=0$. Hence, $\widehat{H}_{n-1}(B;\mathbb{Q})$ is
isomorphic to $\mathrm{Hom}(H^{n-1}(B;\mathbb{Z}),\mathbb{Q})$. Because
$B=\overline{W}\backslash\Gamma$ for some open set $\Gamma\subset V$, passing
to a smaller subset of $\Gamma$, we can assume that $\Gamma$ is a neighborhood
of some $y\in V$ such that $\overline{\Gamma}\subset V$ and $\Gamma$ satisfies
conditions $(1)-(3)$ from Theorem 1.1. Then, the arguments from the proof of
Claim 5 show that
$H^{n-1}(B;\mathbb{Z})=H^{n-1}(\overline{W}\backslash\Gamma;\mathbb{Z})$ is
not trivial and there is a surjective homomorphism $H^{n-1}(B;\mathbb{Z})\to
H^{n-1}(\overline{W}\backslash V;\mathbb{Z})$. Therefore, there exists an
injective homomorphism from $\widehat{H}_{n-1}(\overline{W}\backslash
V;\mathbb{Q})$ into $\widehat{H}_{n-1}(B;\mathbb{Q})$. Because we already saw
that $\widehat{H}_{n-1}(\overline{W};\mathbb{Q})=0$, this shows that
$\overline{W}\backslash V\subset\overline{W}$ is an $(n-1)$-homology membrane
spanned on $\overline{W}\backslash V$ for any nontrivial
$\gamma\in\widehat{H}_{n-1}(\overline{W}\backslash V;\mathbb{Q})$, but with
respect to the exact homology $\widehat{H}_{*}$. To complete the proof of
Claim 7, we observe that for every pair $A\subset Y$ of a space $Y$ and its
closed set $A$, any integer $k$ and a group $G$ there is an epimorphism
$T_{Y,A}^{k}:\widehat{H}_{k}(Y,A;G)\to H_{k}(Y,A;G)$, which is an isomorphism
when $G$ is a field, see [26, Theorem 4].

Suppose $U$ is a neighborhood of $x$ with $\overline{U}\subset V$. Since the
pair $\overline{W}\backslash V\subset\overline{W}$ is an $(n-1)$-homology
membrane spanned on $\overline{W}\backslash V$ for any nontrivial $\gamma\in
H_{n-1}(\overline{W}\backslash V;\mathbb{Q})$, according to [1],
$H_{n-1}(\mathrm{bd}\,\overline{U};\mathbb{Q})\neq 0$ and $\overline{U}$ is an
$(n-1)$-homology membrane spanned on $\mathrm{bd}\,\overline{U}$ for some
nontrivial $\gamma^{\prime}\in H_{n-1}(\mathrm{bd}\,\overline{U};\mathbb{Q})$.
Most important for us is the nontriviality of
$H_{n-1}(\mathrm{bd}\,\overline{U};\mathbb{Q})$ because it implies also the
nontriviality of $H_{n-1}(\mathrm{bd}\,\overline{U};\mathbb{Z})$, see [28,
Proposition 4.5]. Finally, since $\dim\mathrm{bd}\,\overline{U}=n-1$, [19,
Corollary] shows that $\mathrm{bd}\,\overline{U}$ is dimensionally full-
valued. $\Box$

Acknowledgments. The author would like to express his gratitude to A.
Dranishnikov, K. Kawamura, A. Koyama and E. Tymchatyn for their helpful
comments.

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