# On $Q$-manifolds bundles

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>and  J. West  Department of Mathematics, Cornell
University, 310 Malott Hall, Ithaca, New York 14853<EMAIL_ADDRESS>
###### Abstract.

We prove a homological characterization of $Q$-manifolds bundles over
$C$-spaces. This provides a partial answer to Question QM22 from [22].

###### Key words and phrases:

$ANR$-fibration, $C$-spaces, $Q$-bundles, homological $Z$-sets, $Z$-sets

###### 2010 Mathematics Subject Classification:

Primary 55M10, 55M15; Secondary 54F45, 54C55

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

##  1\. Introduction

The second author raised the question [22, QM22] whether there exists a Čech-
cohomology version of the fibred general position theory of Toruńczyk-West
[21] along the lines of Daverman-Walsh results [6] that detects the
$Q$-manifold bundles among the $ANR$-fibrations with compact fibers. In the
present paper we present a homological version of Toruńczyk-West’s [21]
characterization of $Q$-manifold bundles over $C$-spaces.

Following [21], a trivial $ANR$-fibration (resp., $AR$-fibration) is a map
$p:E\to B$ of paracompact Hausdorff spaces such that for some separable metric
$ANR$-space (resp., $AR$-space) $X$ there is a closed embedding
$i:E\hookrightarrow X\times B$ and a retraction $r:X\times B\to i(E)$ making
the diagram commute

---
$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{p}$$\textstyle{X\times
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$$\scriptstyle{\pi_{B}}$$\textstyle{i(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{B}}$$\textstyle{B}$

and a fiber preserving homotopy $H:X\times B\times\mathbb{I}\to X\times B$
from $r$ to $r_{b}\times\rm{id}_{B}$ for some $b\in B$, where $r_{b}:X\to
i(p^{-1}(b))$ is defined by $r_{b}(x,b)=r((x,b))$. A map $p:E\to B$ of
paracompact Hausdorff spaces is an $ANR$-fibration if there is an open cover
$\mathcal{U}$ of $B$ such that $p:p^{-1}(U)\to U$ is a trivial $ANR$-fibration
for each $U\in\mathcal{U}$. We call the cover $\mathcal{U}$ a local
trivialization of $p$. Note that every $U\in\mathcal{U}$ is also paracompact.
If the maps $r$ and $H$ may be taken proper, $p$ is called a proper
$ANR$-fibration. If each $X_{U}$ is locally compact, compact, or complete, we
say that $p:E\to B$ is a locally compact, compact, or complete
$ANR$-fibration. Every complete trivial $ANR$-fibration has the following
property: For every paracompact space $Z$, a closed subset $A\subset Z$ and
maps $f:A\to E$ and $g:Z\to B$ such that $p\circ f=g|A$ there is a map
$\widetilde{g}:V\to B$, where $V$ is a neighborhood of $A$ in $Z$, such that
$p\circ\widetilde{g}=g|V$ and $\widetilde{g}$ extends $f$. A map $p:E\to B$
with this property is said to be a locally soft map. A map $p$ is soft if in
the above definition $V$ is the whole space $Z$ [15]. Let us note that every
locally soft map is open.

Let $p:E\to B$ be an $ANR$-fibration and $A\subset B$. We say that $p^{-1}(A)$
has the fibred disjoint $n$-disks property if each pair of fiber preserving
maps $f,g:\mathbb{I}^{n}\times A\to p^{-1}(A)$ can be approximated arbitrary
closely by fiber preserving maps $f^{\prime},g^{\prime}:\mathbb{I}^{n}\times
A\to p^{-1}(A)$ with disjoint images, see [21]. Here, a map
$f:\mathbb{I}^{n}\times A\to p^{-1}(A)$ is said to be fiber preserving
provided $p(f(x,y))=y$ for all $(x,y)\in\mathbb{I}^{n}\times A$. The fibred
disjoint $2$-disks property is simply called fibred disjoint disks property.
If there is an open cover $\mathcal{U}$ of $B$ such that $p^{-1}(U)$ has the
fibred disjoint $n$-disks property for all $n$ and $U\in\mathcal{U}$, then $p$
satisfies the fibred general position (FGP)-property [21].

Recall that a closed set $F\subset X$ is said to be a $Z_{n}$-set in $X$ if
the set $C(\mathbb{I}^{n},X\setminus F)$ is dense in $C(\mathbb{I}^{n},X)$.
Note that if $X$ is a metric $LC^{n-1}$-space, then a closed set $F\subset X$
is $Z_{n}$-set iff for each at most $n$-dimensional metric compactum $Y$ the
set $\\{f\in C(Y,X):f(Y)\cap F=\varnothing\\}$ is dense in $C(Y,X)$, see [4].
We also say that a map $f:K\to X$ is a $Z_{n}$-map provided $f(K)$ is a
$Z_{n}$-set in $X$. For consistency with [19], instead of using the term
$Z_{\infty}$ to mean $Z_{n}$ for all n as is often done, we call
$Z_{\infty}$-sets and $Z_{\infty}$-maps, respectively, $Z$-sets and $Z$-maps.

There are homological analogues of $Z_{n}$-sets and $Z_{n}$-maps. A closed set
$F\subset X$ is called a homological $Z_{n}$-set in $X$ if the singular
homology groups $H_{k}(U,U\setminus F)$ are trivial for all open sets
$U\subset X$ and all $k\leq n$, see [2]. It can be shown that if $X$ does not
have isolated points then every homological $Z_{n}$-set in $X$ is nowhere
dense. The homological $Z_{n}$-property is finitely additive and hereditary
with respect to closed subsets [2]. A map $f:K\to X$ is a homological
$Z_{n}$-map provided the image $f(K)$ is a homological $Z_{n}$-set in $X$.
Homological $Z_{\infty}$-sets were considered in [6] under the name sets of
infinite codimension.

Combining [20, Corollary 3.3] and [2, Theorem 2.1], we have the following:

######  Proposition 1.1.

Let $X$ be an $LC^{n}$-space with $n\geq 2$. Then, a closed subset $A$ of $X$
is a $Z_{n}$-set in $X$ provided $A$ is an $LCC^{1}$ homological $Z_{n}$-set
in $X$. Equivalently, $A$ is a $Z_{n}$-set iff it is a $Z_{2}$-set and a
homological $Z_{n}$-set.

All function spaces in this paper, if not explicitly stated otherwise, are
equipped with the limitation topology, see [5] and [18]. Recall that a set
$U\subset C(X,E)$ is open with respect to the limitation topology if for every
$f\in U$ there is an open cover $\mathcal{V}$ of $Y$ such that $U$ contains
the set $B(f,\mathcal{V})=\\{g\in
C(X,E):g{~{}}\mbox{is}{~{}}\mathcal{V}-\mbox{close to}f\\}.$ Let $p:E\to B$ be
an $ANR$-fibration, $K\subset B$ and $Z$ an arbitrary space. We denote by
$C_{P}(Z\times K,p^{-1}(K))$ the set of all continuous fiber preserving maps
from $Z\times K$ into $p^{-1}(K)$. One can show that each $f\in C_{P}(Z\times
K,p^{-1}(K))$ is a perfect map provided $Z$ is compact.

##  2\. Preliminary results

######  Lemma 2.1.

Let $p:E\to B$ be a locally compact trivial $ANR$-fibration and $A\subset B$
be a closed set. Then for every space $K$ such that $K\times B$ is paracompact
the restriction map $q_{A}:C_{P}(K\times B,E)\to C_{P}(K\times A,p^{-1}(A))$,
$q_{A}(f)=f|(K\times A)$, is open.

###### Proof.

Let $W\subset C_{P}(K\times B,E)$ be an open set and $f\in W$. Then there
exists an open cover $\mathcal{U}$ of $E$ such that $B(f,\mathcal{U})\subset
W$. By [21, Proposition (A10)], there is an open cover $\mathcal{V}$ of $E$
such that any map $h\in C_{P}(K\times A,p^{-1}(A))$, which is
$\mathcal{V}$-close to $f|(K\times A)$, is extended to a map $\widetilde{h}\in
B(f,\mathcal{U})$. This means that the set $B(q_{A}(f),\mathcal{V})$ is
contained in $q_{A}(W)$. Since the interior of $B(q_{A}(f),\mathcal{V})$ in
$C_{P}(K\times A,p^{-1}(A))$ contains $q_{A}(f)$ (see [5]), $q_{A}(W)$ is open
in $C_{P}(K\times A,p^{-1}(A))$. ∎

In case $p:E\to B$ is a trivial locally compact fibration, we need another
description of the limitation topology on $C_{P}(K\times B,E)$. If $(X,d)$ is
the associated with $p$ separable metric $ANR$-space, we consider the pseudo-
metric $d_{E}$ on $E$ defined by $d_{E}(x,y)=d(\pi_{X}(x),\pi_{X}(y))$, where
$\pi_{X}:X\times B\to X$ is the projection.

######  Lemma 2.2.

Let $p:E\to B$ be a trivial $ANR$-fibration and $(X,d)$ be a locally compact
separable metric space associated with $p$. Then for any space $K$ a set
$U\subset C_{P}(K\times B,E)$ is open with respect to the limitation topology
iff for every $f\in U$ there is a continuous function $\varepsilon:E\to(0,1]$
such that $U$ contains the set

$\displaystyle B_{d_{E}}(f,\varepsilon)=\\{g\in C_{P}(K\times
B,E):d_{E}(f(y),g(y))<\varepsilon(f(y))\forall y\in K\times B\\}.$

###### Proof.

Suppose $U\subset C_{P}(K\times B,E)$ is open with respect to the limitation
topology and $f\in U$. Then there is an open cover $\mathcal{V}$ of $E$ such
that $U$ contains the set $B(f,\mathcal{V})$. Since $X\times B$ is
paracompact, so is $E$. Let $r:X\times B\to E$ be a fiber-preserving
retraction and $\mathcal{W}$ be a locally finite open cover of $X\times B$
which is star-refinement of the cover
$\widetilde{\mathcal{V}}=r^{-1}(\mathcal{V})$. We fix a countable open cover
$\\{X_{i}\\}_{i\geq 1}$ of $X$ such that each $X_{i}$ has a compact closure
and $\overline{X}_{i}\subset X_{i+1}$. Since $\overline{X}_{i}$ is compact,
every $b\in B$ has a neighborhood $O_{b}^{i}$ in $B$ such that for every
$x\in\overline{X}_{i}$ there is $W_{x}\in\mathcal{W}$ and a neighborhood
$O_{x}$ of $x$ in $X$ with $O_{x}\times O_{b}^{i}\subset W_{x}$. Choose a
bounded continuous pseudo-metric $\sigma_{i}\leq 1$ on $B$ such that the
family of all open balls $\\{B_{\sigma_{i}}(b,1):b\in B\\}$ refines the cover
$\\{O_{b}^{i}:b\in B\\}$. Let $\rho$ be the pseudo-metric on $X\times B$
defined by
$\rho((x,b),(x^{\prime},b^{\prime}))=d(x,x^{\prime})+\sigma(b,b^{\prime})$,
where
$\sigma(b,b^{\prime})=\sum_{i=1}^{\infty}\sigma_{i}(b,b^{\prime})/2^{i}$.
Define a function $\varepsilon$ on $E$ by

$\varepsilon(z)=\rho(z,E\backslash\rm{St}(z,\mathcal{W})).$

We can assume that $\varepsilon\leq 1$. Clearly $\varepsilon$ is continuous
and bounded, and it is positive. Indeed, suppose $\varepsilon(z)=0$ for some
$z=(x,b)\in E$, where $x\in X_{j}$. Then there is a sequence
$\\{z_{n}\\}\subset E\backslash\rm{St}(z,\mathcal{W})$, $z_{n}=(x_{n},b_{n})$,
such that $\rho(z,z_{n})<1/n$ for all $n$. Then $d(x,x_{n})<1/n$ and, since
$d$ is a metric on $X$, the sequence $\\{x_{n}\\}$ converges to $x$. We can
assume that each $x_{n}\in X_{j}$. On the other hand, $\sigma(b,b_{n})<1/n$
implies that $\sigma_{j}(b,b_{n})<2^{j}/n$ for all $n$. Hence, there exists
$n_{0}$ with $2^{j}/n<1$ for every $n\geq n_{0}$. Consequently, $b,b_{n}\in
O_{b^{*}}^{j}$ for any $n\geq n_{0}$ and some $b^{*}\in B$. According to the
construction of the neighborhoods $O_{b}^{j}$, we have $O_{x}\times
O_{b^{*}}^{j}\subset W_{x}$ for some $W_{x}\in\mathcal{W}$ and a neighborhood
$O_{x}$ of $x$. Now, take $m>n_{0}$ with $x_{m}\in O_{x}$, and observe that
$z,z_{m}\in O_{x}\times O_{b^{*}}^{j}\subset W_{x}$. Hence,
$z_{m}\in\rm{St}(z,\mathcal{W})$, a contradiction. Moreover, for any $z\in E$
the inequality $\rho(z,z^{\prime})<\varepsilon(z)$ implies that
$z,z^{\prime}\in W$ for some $W\in\mathcal{W}$. Hence,
$B_{\rho}(f,\varepsilon)\subset B(f,\rm{St}\mathcal{W})\subset
B(f,\mathcal{V})$. Because $\displaystyle
B_{d_{E}}(f,\varepsilon)=B_{\rho}(f,\varepsilon)$, we finally have
$\displaystyle B_{d_{E}}(f,\varepsilon)\subset U$.

To show the other implication of Lemma 2.2, let $\displaystyle
B_{d_{E}}(f,\varepsilon)\subset U$ for some positive continuous function
$\varepsilon$. For every $z\in E$ let

$W_{z}=\\{e\in
E:d_{E}(e,z)<\varepsilon(z)/4{~{}}\mbox{and}{~{}}\varepsilon(z)/2<\varepsilon(e)<3\varepsilon(z)/2\\}.$

Then, for the open cover $\mathcal{V}=\\{W_{z}:z\in E\\}$ of $E$ we have
$B(f,\mathcal{V})\subset\displaystyle B_{d_{E}}(f,\varepsilon)$. ∎

We need to consider another topology on function spaces $C(X,Y)$, the so
called source limitation topology [3]. The local base in that topology at
given $h:X\to Y$ consists of all sets

$\Lambda(h,\eta)=\\{h^{\prime}\in
C(X,Y):\rho(h(x),h^{\prime}(x))<\eta(x){~{}}\forall x\in X\\},$

where $\rho$ is a continuous pseudo-metric on $Y$ and $\eta$ is a continuous
and positive function on $X$. This topology, called sometimes the fine
topology, was initially introduced for metrizable spaces $Y$ (in this case
$\rho$ is a fixed compatible metric on $Y$), see [11], [14]. Clearly, the
source limitation topology is stronger than the limitation one. This topology
has the Baire property provided $X$ is paracompact and $Y$ is completely
metrizable, see [14].

######  Proposition 2.3.

Let $p:E\to B$ be a locally compact $ANR$-fibration. Then for any compact
space $K$ the space $C_{P}(K\times B,E)$ has the Baire property.

###### Proof.

According to [21, Proposition A9.4], there is a closed fiber-preserving
embedding of $E$ into $Q\times[0,\infty)\times B$. Let $C_{P}(K\times
B,Q\times[0,\infty)\times B)$ be the set of all fiber-preserving maps into
$Q\times[0,\infty)\times B$. Then $C_{P}(K\times B,E)$ is a closed subset of
$C_{P}(K\times B,Q\times[0,\infty)\times B)$ with respect to the limitation
topology. So, it suffices to show that $C_{P}(K\times
B,Q\times[0,\infty)\times B)$ equipped with the limitation topology has the
Baire property. Therefore, we can suppose that $E=X\times B$ and $p:E\to B$ is
the projection, where $X$ is a locally compact separable metric space with a
fixed complete metric $d$. Observe that the correspondence
$\theta:f\rightsquigarrow f_{X}=\pi_{X}\circ f$ between $C_{P}(K\times B,E)$
and $C(K\times B,X)$ is bijective. Since $C(K\times B,X)$ with the source
limitation topology has the Baire property, the proof is reduced to showing
that $\theta$ is a homeomorphism when $C_{P}(K\times B,E)$ carries the
limitation topology and $C(K\times B,X)$ is equipped with the source
limitation topology. Let show that both $\theta^{-1}$ and $\theta$ are
continuous.

Suppose $\\{f_{X}^{\alpha}\\}$ is a net in $C(K\times B,X)$ converging with
respect to the source limitation to $f_{X}=\pi_{X}\circ f$ for some $f\in
C_{P}(K\times B,E)$. To show that $\\{f^{\alpha}\\}$ converges to $f$, let $U$
be a neighborhood of $f$ in $C_{P}(K\times B,E)$. Then, by Lemma 2.2, there
exists a continuous positive function $\varepsilon$ on $E$ such that
$\displaystyle B_{d_{E}}(f,\varepsilon)\subset U$. Then there is $\alpha_{0}$
such that $d(f_{X}(z),f_{X}^{\alpha}(z))<\varepsilon(f(z))$ for each $z\in
K\times B$ and $\alpha\geq\alpha_{0}$. This implies $f^{\alpha}\in
B_{d_{E}}(f,\varepsilon)\subset U$ for all $\alpha\geq\alpha_{0}$. Hence,
$\theta^{-1}$ is continuous.

Suppose now that $\\{f^{\alpha}\\}$ converges to $f$ in $C_{P}(K\times B,E)$
with respect to the limitation topology, and let $\varepsilon^{\prime}$ be a
continuous positive function on $K\times B$. Since $f$ is a perfect map, we
find a continuous positive function $\varepsilon^{\prime\prime}$ on $f(K\times
B)$ such that $\varepsilon^{\prime\prime}(f(z))<\varepsilon^{\prime}(z)$,
$z\in K\times B$, (see the proof of Claim 4.2 below) and then extend
$\varepsilon^{\prime\prime}$ to a continuous positive function $\varepsilon$
on $E$ (this can be done because $E$ is paracompact and $f(K\times B)$ is
closed in $E$). The set $\displaystyle B_{d_{E}}(f,\varepsilon)$ is a
neighborhood of $f$ with respect to the limitation topology, see the proof of
Lemma 2.2. Therefore, there is $\alpha_{0}$ with $f^{\alpha}\in\displaystyle
B_{d_{E}}(f,\varepsilon)$ for all $\alpha\geq\alpha_{0}$. This means that
$d(f_{X}(z),f_{X}^{\alpha}(z))<\varepsilon(f(z))<\varepsilon^{\prime}(z)$ for
all $\alpha\geq\alpha_{0}$ and $z\in K\times B$. Hence, $\theta$ is
continuous. ∎

For a space $Y$ let $\mathcal{F}(Y)$ denote the family of all non-empty closed
subsets of $Y$. If $Y$ is a convex subspace of a linear space, then
$\mathcal{F}_{c}(Y)$ stands for the closed convex subsets of $Y$. A set-valued
map $\varphi:X\to\mathcal{F}(Y)$ is said to be lower semi-continuous, or
l.s.c., if $\varphi^{-1}(U)=\\{x\in X:\varphi(x)\cap U\neq\varnothing\\}$ is
open in $X$ for every open $U\subset Y$.

######  Proposition 2.4.

Let $p:E\to B$ be a locally compact $ANR$-fibration. Then the fibration
$p^{\prime}:E\times Q\to B$ has the FGP-property.

###### Proof.

Without loss of generality, we may assume that $p$ is a trivial fibration. We
need to show that every fiber preserving map $f:Q\times\\{1,2\\}\times B\to
E\times Q$ can be approximated in the limitation topology by maps $g\in
C_{P}(Q\times\\{1,2\\}\times B,E\times Q)$ such that $g(Q\times\\{1\\}\times
B)\cap g(Q\times\\{2\\}\times B)=\varnothing$. Here
$C_{P}(Q\times\\{1,2\\}\times B,E\times Q)$ is the set of all
$p^{\prime}$-preserving maps from $C(Q\times\\{1,2\\}\times B,E\times Q)$. To
this end, we fix $f\in C_{P}(Q\times\\{1,2\\}\times B,E\times Q)$ and an open
cover $\mathcal{U}$ of $E\times Q$. Let $f_{E},f_{Q}$ be the maps
$\pi_{E}\circ f$ and $\pi_{Q}\circ f$, where $\pi_{E}$ and $\pi_{Q}$ are the
projections of $E\times Q$ onto $E$ and $Q$, respectively. Since $E$ is
paracompact, so is $E\times Q$. Using that $Q$ is compact, every $e\in E$ has
a neighborhood $O_{e}$ such that for every $x\in Q$ there is $V\in\mathcal{U}$
with $O_{e}\times\\{x\\}\subset V$. As in the proof of Lemma 2.2, we can find
a bounded continuous pseudo-metric $\sigma$ on $E$ and a continuous bounded
positive function $\varepsilon$ on $E\times Q$ such that any $g\in
C_{P}(Q\times\\{1,2\\}\times B,E\times Q)$ is $\mathcal{U}$-close to $f$
provided $\rho(f(y),g(y)<\varepsilon(f(y))$ for all $y\in
Q\times\\{1,2\\}\times B$, where $\rho$ be the pseudo-metric on $E\times Q$
defined by
$\rho((e,x),(e^{\prime},x^{\prime}))=d(x,x^{\prime})+\sigma(e,e^{\prime})$
(here $d$ is the ordinary metric on $Q$).

We are going to show there is a function $g_{Q}:Q\times\\{1,2\\}\times B\to Q$
such that $g_{Q}(Q\times\\{1\\}\times B)\cap g_{Q}(Q\times\\{2\\}\times
B)=\varnothing$ and $d(f_{Q}(y),g_{Q}(y))<\varepsilon(f(y))$ for all $y\in
Q\times\\{1,2\\}\times B$. Since $\varepsilon$ is bounded, there is a
continuous extension $\widetilde{\varepsilon}:\beta(E\times Q)\to[0,\infty)$,
where $\beta(E\times Q)$ is the Čech-Stone extension of $E\times Q$. Let also
$\widetilde{f}:\beta(Q\times B)\times\\{1\\}\oplus\beta(Q\times
B)\times\\{2\\}\to\beta(E\times Q)$ and $\widetilde{\pi}_{Q}:\beta(E\times
Q)\to Q$ be the extensions of $f$ and $\pi_{Q}$. (We use $\oplus$ to denote
the disjoint union.) Then
$\widetilde{E}=\widetilde{\varepsilon}^{-1}(0,\infty)$ and
$Y=\widetilde{f}^{-1}(\widetilde{E})$ are locally compact and $\sigma$-compact
spaces. We represent $Y$ as the union of the sets
$K_{i}=Y_{i}\times\\{1\\}\oplus Y_{i}\times\\{2\\}$, where $Y_{i}$ is an
increasing sequence of compact subspaces of $\beta(Q\times B)$. Let
$\delta=\widetilde{\varepsilon}\circ\widetilde{f}:Y\to(0,\infty)$ and
$\widetilde{f}_{Q}=\widetilde{\pi}_{Q}\circ\widetilde{f}:Y\to Q$. We consider
the space $C(Y,Q)$ with the source limitation topology. Recall that the local
base in that topology at given $h:Y\to Q$ consists of all sets

$\Lambda(h,\eta)=\\{h^{\prime}\in
C(Y,Q):d(h(y),h^{\prime}(y))<\eta(y){~{}}\forall y\in Y\\},$

where $\eta$ is a continuous and positive function on $Y$. As we already
noted, this topology has the Baire property.

######  Claim 2.5.

All restriction maps $\theta_{i}:C(Y,Q)\to C(K_{i},Q)$,
$\theta_{i}(h)=h|K_{i}$, are surjective and open in the source limitation
topology.

Indeed, if $W\subset C(Y,Q)$ is open let $h^{\prime}\in\theta_{i}(W)$. Then
$h^{\prime}=\theta_{i}(h)$ for some $h\in W$ and there is $\eta\in
C(Y,(0,\infty)$ with $\Lambda(h,\eta)\subset W$. Consider the open in
$C(K_{i},Q)$ set

$\Lambda_{i}(h^{\prime},\eta/2)=\\{g\in
C(K_{i},Q):d(h^{\prime}y),g(y))<\eta(y)/2{~{}}\forall y\in K_{i}\\}.$

If $g\in\Lambda_{i}(h^{\prime},\eta/2)$, we define the lower semi-continuous
set valued map $\Phi:Y\to\mathcal{F}_{c}(Q)$, $\Phi(y)=\\{z\in
Q:d(z,h(y)\leq\eta(y)/2\\}$ if $y\not\in K_{i}$ and $\Phi(y)=g(y)$ if $y\in
K_{i}$. Then by Michael’s selection theorem [12], there is a continuous
function $\widetilde{g}\in
C(Y,Q{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1})}$
with $\widetilde{g}(y)\in\Phi(y)$ for all $y\in Y$. Obviously,
$\widetilde{g}\in\Lambda(h,\eta)\subset W$ and $\theta_{i}(\widetilde{g})=g$.
This means that $\Lambda_{i}(h^{\prime},\eta/2)\subset\theta_{i}(W)$. Hence,
each $\theta_{i}$ is open. Surjectivity of $\theta_{i}$ is obvious because $Q$
is an absolute retract for the normal spaces.

Now, for every $i$ let $G_{i}$ be the set of all maps $h\in C(K_{i},Q)$ such
that $h(Y_{i}\times\\{1\\})\cap h(Y_{i}\times\\{2\\})=\varnothing$. Using that
$K_{i}$ are compact, one can show that each $G_{i}$ is open and dense in
$C(K_{i},Q)$ (note that the source limitation topology on $C(K_{i},Q)$
coincides with the compact open topology). Hence,
$G=\bigcap_{i=1}^{\infty}\theta_{i}^{-1}(G_{i})$ is dense in $C(Y,Q)$ with
respect to the source limitation topology. Since
$\Lambda(\widetilde{f}_{Q},\delta)$ is open in that topology, there is
$\widetilde{g}_{Q}\in G\cap\Lambda(\widetilde{f}_{Q},\delta)$. Let
$g_{Q}=\widetilde{g}_{Q}|(Q\times\\{1,2\\}\times B)$ and define
$g:Q\times\\{1,2\\}\times B\to E\times Q$ by $g(y)=(f_{E}(y),g_{Q}(y))$.
Clearly, $g$ is $p^{\prime}$-preserving and
$\rho(f(y),g(y))=d(f_{Q}(y),g_{Q}(y))<\varepsilon(f(y))$ for all $y\in
Q\times\\{1,2\\}\times B$. This means that $g$ is $\mathcal{U}$-close to $f$.
Moreover, $\widetilde{g}_{Q}\in G$ implies $g(Q\times\\{1\\}\times B)\cap
g(Q\times\\{2\\}\times B)=\varnothing$. ∎

##  3\. Homological characterization of $Q$-manifold bundles over $C$-spaces

In this section we prove a homological characterization of $Q$-manifold
bundles over $C$-spaces. This provides a partial answer to Question QM22 from
[22].

The $C$-space property was originally defined by Haver [9] for compact metric
spaces. Addis and Gresham [1] reformulated Haver’s definition for arbitrary
spaces: A space $X$ has property $C$ if for every sequence
$\\{\mathcal{U}_{n}\\}$ of open covers of $X$ there exists a sequence
$\\{\mathcal{V}_{n}\\}$ of open disjoint families in $X$ such that each
$\mathcal{V}_{n}$ refines $\mathcal{U}_{n}$ and $\bigcup_{n\geq
1}\mathcal{V}_{n}$ is a cover of $X$. Every finite-dimensional paracompact
space, as well as every countable-dimensional (a countable union of finite-
dimensional sets) metric space, is a $C$-space [1], but there is a compact
metric $C$-space which is not countable-dimensional [16]. On the other hand,
normal $C$-spaces are weakly infinite-dimensional in the sense of Alexandroff
[1]. We say that $X$ is an hereditary $C$-space if every open subset of $X$ is
a $C$-space. For example, every paracompact $C$-space whose open sets are
$F_{\sigma}$ is an hereditary $C$-space.

Although not formulated in this form, Lemma 3.1 below was actually established
in [8, Proposition 3.1].

######  Lemma 3.1.

[8] Let $Y$ be a closed convex subset of a Banach space and $V\subset Y$ be
open. Suppose $X$ is a paracompact $C$-space and
$\varphi:X\rightsquigarrow\mathcal{F}_{c}(Y)$ is l.s.c. such that
$\varphi(x)\subset V$ for all $x\in X$. Then for every set-valued map
$\psi:X\to\mathcal{F}(V)$ with a closed graph (in $X\times V$) there is a map
$f:X\to V$ with $f(x)\in\varphi(x)\backslash\psi(x)$ for all $x\in X$ provided
each $\varphi(x)\cap\psi(x)$ is a $Z$-set in $\varphi(x)$.

######  Proposition 3.2.

Let $p:E\to B$ be a locally compact $ANR$-fibration with compact $Q$-manifold
fibers such that $B$ is a locally compact $C$-space. Then every $b\in B$ has a
basis of neighborhoods $U$ with compact closures such that the fibrations
$p|p^{-1}(\overline{U}):p^{-1}(\overline{U})\to\overline{U}$ have the FGP-
property.

###### Proof.

By Proposition 2.4, the fibration $E\times Q\to B$ has the FGP-property.
Hence, according to [21, Theorem 2.3], $p^{\prime}:E\times Q\to E\to B$ is a
$Q$-manifold bundle. Consequently, $p^{\prime\prime}:E\times Q\times[0,1)\to
E\to B$ is also a $Q$-manifold bundle. Let $\mathcal{U}$ be a simultaneous
local trivialization of $p$ and $p^{\prime}$, and hence of $p^{\prime\prime}$.
It is clear that the proof is reduced to show that for every compact set
$K\subset B$ which is contained in some $U\in\mathcal{U}$ the fibration
$p|p^{-1}(K):p^{-1}(K)\to K$ has the FGP-property. To this end, we can assume
that $p:E\to B$ is a trivial $ANR$-fibration over a compact space $B$ such
that both $p^{\prime}:E\times Q\to E\to B$ and $p^{\prime\prime}:E\times
Q\times[0,1)\to E\to B$ are trivial $Q$-manifold bundles. Therefore, we need
to show that for every $n$ every fiber-preserving map
$f:\mathbb{I}^{n}\times\\{1,2\\}\times B\to E$ can be approximated by fiber-
preserving maps $f^{\prime}:\mathbb{I}^{n}\times\\{1,2\\}\times B\to E$ such
that $f^{\prime}(\mathbb{I}^{n}\times\\{1\\})\cap
f^{\prime}(\mathbb{I}^{n}\times\\{2\\})=\varnothing$. To do this, take a
fiber-preserving homeomorphism

$h:E\times Q\times[0,1)\to B\times M,$

where $M$ is a $Q$-manifold homeomorphic to $p^{-1}(b)\times Q\times[0,1)$ for
all $b\in B$. Since $M$ is a product of the $Q$-manifolds $p^{-1}(b)\times Q$
and $[0,1)$, it can be embedded as an open subset of a copy $Q_{M}$ of the
Hilbert cube [13, Corollary 7.4.4(2)]. Since the domains of all function
spaces in the present proof are compact spaces, the limitation topology
coincides with the compact open topology. Therefore, all function spaces will
be considered with the compact open topology. We also consider the projection
$\pi_{E}:E\times Q\times[0,1)\to E$. Then $\pi_{E}$ generates the continuous
map

$\pi_{E}^{*}:C(\mathbb{I}^{n}\times\\{1,2\\},E\times Q\times[0,1))\to
C(\mathbb{I}^{n}\times\\{1,2\\},E),$

$\pi_{E}^{*}(w)=\pi_{E}\circ w$. Since,
$C(\mathbb{I}^{n}\times\\{1,2\\},E\times Q\times[0,1))$ is homeomorphic to the
product $C(\mathbb{I}^{n}\times\\{1,2\\},E)\times
C(\mathbb{I}^{n}\times\\{1,2\\},Q\times[0,1))$, $\pi_{E}^{*}$ is the
projection onto $C(\mathbb{I}^{n}\times\\{1,2\\},E)$. For every $b\in B$ let
$C(b)=C(\mathbb{I}^{n}\times\\{1,2\\},p^{-1}(b)\times Q\times[0,1))$ and
$C=\bigcup\\{C(b):b\in B\\}$ considered as a subspace of
$C(\mathbb{I}^{n}\times\\{1,2\\},E\times Q\times[0,1))$. Let also $L(b)$ be
the subspace of $C(b)$ consisting of all maps $g$ such that
$\pi_{E}^{*}(g)(\mathbb{I}^{n}\times\\{1\\})\cap\pi_{E}^{*}(g)(\mathbb{I}^{n}\times\\{2\\})\neq\varnothing$.
Actually, $L(b)$ is the product $L_{1}(b)\times
C(\mathbb{I}^{n}\times\\{1,2\\},Q\times[0,1))$, where $L_{1}(b)$ is the set of
all $g^{\prime}\in C(\mathbb{I}^{n}\times\\{1,2\\},p^{-1}(b))$ with
$g^{\prime}(\mathbb{I}^{n}\times\\{1\\})\cap
g^{\prime}(\mathbb{I}^{n}\times\\{2\\})\neq\varnothing$.

######  Claim 3.3.

$L(b)$ is a $Z$-set in $C(b)$ for each $b\in B$.

Since $L(b)$ is homeomorphic to the product $L_{1}(b)\times
C(\mathbb{I}^{n}\times\\{1,2\\},Q\times[0,1))$ and $C(b)$ is homeomorphic to
$C(\mathbb{I}^{n}\times\\{1,2\\},p^{-1}(b))\times
C(\mathbb{I}^{n}\times\\{1,2\\},Q\times[0,1))$, it suffices to show that
$L_{1}(b)$ is a $Z$-set in $C(\mathbb{I}^{n}\times\\{1,2\\},p^{-1}(b))$.
Obviously, $L_{1}(b)$ is closed in
$C(\mathbb{I}^{n}\times\\{1,2\\},p^{-1}(b))$. So, we need to show that every
map $u:Q\to C(\mathbb{I}^{n}\times\\{1,2\\},p^{-1}(b))$ is approximated by
maps $u^{\prime}:Q\to C(\mathbb{I}^{n}\times\\{1,2\\},p^{-1}(b))$ with
$u^{\prime}(Q)\cap L_{1}(b)=\varnothing$. Since $Q$ is compact, the
exponential map

$\Lambda_{b}:C(Q\times\mathbb{I}^{n}\times\\{1,2\\},p^{-1}(b))\to
C(Q,C(\mathbb{I}^{n}\times\\{1,2\\},p^{-1}(b))),$

defined by $\\{[\Lambda_{b}(v)](q)\\}(x)=v(q,x)$, $q\in Q$ and
$x\in\mathbb{I}^{n}\times\\{1,2\\}$, is a homeomorphism, see [7, Theorem
3.4.3]. So, $v=\Lambda_{b}^{-1}(u)\in
C(Q\times\mathbb{I}^{n}\times\\{1,2\\},p^{-1}(b))$ for every $u\in
C(Q,C(\mathbb{I}^{n}\times\\{1,2\\},p^{-1}(b)))$. Because $p^{-1}(b)$ is a
$Q$-manifold, $v$ can be approximated by maps $v^{\prime}\in
C(Q\times\mathbb{I}^{n}\times\\{1,2\\},p^{-1}(b))$ such that
$v^{\prime}(Q\times\mathbb{I}^{n}\times\\{1\\})\cap
v^{\prime}(Q\times\mathbb{I}^{n}\times\\{2\\})=\varnothing$. Therefore,
$u^{\prime}(Q)\cap L_{1}(b)=\varnothing$, where
$u^{\prime}=\Lambda_{b}(v^{\prime})$. This completes the proof of the claim.

Next, we consider the set $L=\bigcup\\{L(b):b\in B\\}\subset C$. It is easily
seen that $L$ is closed in $C$ and that $\widetilde{p}:C\to B$,
$\widetilde{p}(g)=b$ for all $g\in C(b)$, defines a continuous map. The fiber-
preserving homeomorphism $h$ provides a homeomorphism between the sets
$p^{-1}(b)\times Q\times[0,1)$ and $\\{b\\}\times M$ for every $b\in B$.
Consequently, $h$ generates a homeomorphism

$h^{*}:C(\mathbb{I}^{n}\times\\{1,2\\},E\times Q\times[0,1))\to
C(\mathbb{I}^{n}\times\\{1,2\\},B\times M)$

such that $h^{*}(g)\in C(\mathbb{I}^{n}\times\\{1,2\\},\\{b\\}\times M)$ for
all $g\in C(b)$. On the other hand, there is a natural homeomorphism between
$C(\mathbb{I}^{n}\times\\{1,2\\},B\times M)$ and the product
$C(\mathbb{I}^{n}\times\\{1,2\\},B)\times C(\mathbb{I}^{n}\times\\{1,2\\},M)$.
Therefore, we have the commutative diagram

$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{*}}$$\scriptstyle{\widetilde{p}}$$\textstyle{B\times
V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{B}}$$\textstyle{B}$

with $h^{*}(C(b))=\\{b\\}\times V$, where
$V=C(\mathbb{I}^{n}\times\\{1,2\\},M)$. Let us show that any fiber-preserving
map $f\in C_{P}(\mathbb{I}^{n}\times\\{1,2\\}\times B,E)$ can be approximated
by fiber-preserving maps $g:\mathbb{I}^{n}\times\\{1,2\\}\times B\to E$ such
that $g(\mathbb{I}^{n}\times\\{1\\}\times B)\cap
g(\mathbb{I}^{n}\times\\{2\\}\times B)=\varnothing$. To this end, fix a map
$f\in C_{P}(\mathbb{I}^{n}\times\\{1,2\\}\times B,E)$ and consider the
exponential homeomorphism

$\Lambda:C(\mathbb{I}^{n}\times\\{1,2\\}\times B,E)\to
C(B,C(\mathbb{I}^{n}\times\\{1,2\\},E)).$

Then $\Lambda(f)(b)\in C(\mathbb{I}^{n}\times\\{1,2\\},p^{-1}(b))$ for all
$b\in B$. Therefore, $\Lambda$ is a homeomorphism between
$C_{P}(\mathbb{I}^{n}\times\\{1,2\\}\times B,E)$ and the set $\Omega$
consisting of all $\\{l\in C(B,C(\mathbb{I}^{n}\times\\{1,2\\},E))$ with
$l(b)\in C(\mathbb{I}^{n}\times\\{1,2\\},p^{-1}(b))$ for every $b\in B$. We
identify $E$ with $E\times\\{\overline{0}\\}\times 0$ and each $p^{-1}(b)$
with $p^{-1}(b)\times\\{\overline{0}\\}\times 0$, where
$\overline{0}=(0,0,...)\in Q$. We also consider the retraction $r_{E}:E\times
Q\times[0,1)\to E\times\\{\overline{0}\\}\times\\{0\\}$ defined by
$r_{E}(z_{1},z_{2},z_{3})=(z_{1},\overline{0},0)$. This retraction generates a
continuous map

$r_{E}^{*}:C(\mathbb{I}^{n}\times\\{1,2\\},E\times Q\times[0,1))\to
C(\mathbb{I}^{n}\times\\{1,2\\},E\times\\{\overline{0}\\}\times\\{0\\}).$

Therefore, we identify $C(\mathbb{I}^{n}\times\\{1,2\\},E)$ with
$C(\mathbb{I}^{n}\times\\{1,2\\},E\times\\{\overline{0}\\}\times\\{0\\})$ and
every $C(\mathbb{I}^{n}\times\\{1,2\\},p^{-1}(b))$ with
$C(\mathbb{I}^{n}\times\\{1,2\\},p^{-1}(b)\times\\{\overline{0}\\}\times\\{0\\})$.
So, every $\Lambda(f)(b)$ can be considered as an element of $C(b)$. This
means that $\Lambda(f)$ is a map from $B$ to $C$. Moreover, one can show that
$\Lambda(f):B\to C$ is an embedding with
$\Lambda(f)(b)\in\widetilde{p}^{-1}(b)=C(b)$ for all $b\in B$. So is the
composition $u=h^{*}\circ\Lambda(f):B\to B\times V$ such that
$u(b)\in\pi_{B}^{-1}(b)$, $b\in B$.

######  Claim 3.4.

$f$ can be approximated by maps $g\in
C_{P}(\mathbb{I}^{n}\times\\{1,2\\}\times B,E)$ such that
$g(\mathbb{I}^{n}\times\\{1\\}\times B)\cap
g(\mathbb{I}^{n}\times\\{2\\}\times B)=\varnothing$.

Let $O_{\Lambda(f)}=\bigcap_{i=1}^{k}<P_{i},W_{i}>$ be a neighborhood of
$\Lambda(f)$ in the space
$C(B,C(\mathbb{I}^{n}\times\\{1,2\\},E\times\\{\overline{0}\\}\times\\{0\\}))$
with respect to the compact-open topology. Here, $P_{i}\subset B$ are compact
sets, $W_{i}$ are open in
$C(\mathbb{I}^{n}\times\\{1,2\\},E\times\\{\overline{0}\\}\times\\{0\\})$ and
$<P_{i},W_{i}>$ consists of all maps $l\in
C(B,C(\mathbb{I}^{n}\times\\{1,2\\},E\times\\{\overline{0}\\}\times\\{0\\}))$
such that $l(P_{i})\subset W_{i}$. The sets
$\widetilde{W}_{i}=(r_{E}^{*})^{-1}(W_{i})$ are open in
$C(\mathbb{I}^{n}\times\\{1,2\\},E\times Q\times[0,1))$. So are the sets
$h^{*}(\widetilde{W}_{i})$ in $C(\mathbb{I}^{n}\times\\{1,2\\},B\times M)$.
Hence, each $G_{i}=h^{*}(\widetilde{W}_{i})\cap(B\times V)$ is open in
$B\times V$. Since $\Lambda(f)(P_{i})\subset W_{i}$, $u(P_{i})\subset G_{i}$
for all $i$. Because $M$ is open in $Q_{M}$ and
$Y=C(\mathbb{I}^{n}\times\\{1,2\\},Q_{M})$ is homeomorphic to a closed convex
subset of the Banach space
$\big{(}C(\mathbb{I}^{n}\times\\{1,2\\},l_{2}),||.||\big{)}$,
$V=C(\mathbb{I}^{n}\times\\{1,2\\},M)$ is homeomorphic to an open sunset of
$Y$. Therefore, the continuous function $\alpha:B\to\mathbb{R}$,
$\alpha(b)=\rm{dist}(\pi_{V}(u(b)),Y\backslash V)$ is positive, where
$\pi_{V}:B\times V\to V$ is the projection. Using that $\\{P_{i}:i\leq k\\}$
is a finite family of compact sets in $B$ with $u(P_{i})\subset G_{i}$ and
each $\pi_{B}|u(P_{i}):u(P_{i})\to P_{i}$ being a homeomorphism, one can show
that there exists $m$ such that the sets $\displaystyle T_{b}=\\{(b,v)\in
B\times V:||v-\pi_{V}(u(b))||<\frac{\alpha(b)}{m}\\}$ are contained in $G_{i}$
for every $b\in P_{i}$ and $i\leq k$. Now, we consider the set-valued map

$\displaystyle\varphi:B\rightsquigarrow\mathcal{F}_{c}(Y),{~{}}\varphi(b)=\\{v\in
V:||v-\pi_{V}(u(b))||\leq\frac{\alpha(b)}{N}\\},$

where $N>m$. Obviously, $\\{b\\}\times\varphi(b)\subset G_{i}$ for all $b\in
P_{i}$, $i\leq k$. Moreover, $\varphi$ is lower semi-continuous. Observe also
that the set $h^{*}(L)$ is closed in $B\times V$ and $h^{*}(L(b))$ is closed
in $h^{*}(C(b))=\\{b\\}\times V$ for all $b\in B$. So, $\pi_{V}(h^{*}(L(b)))$
is a closed subset of $V$ homeomorphic to $h^{*}(L(b))$, and by Claim 3.3, it
is a $Z$-set in $V$. Hence, we have the set-valued map
$\psi:B\rightsquigarrow\mathcal{Z}(V)$, $\psi(b)=\pi_{V}(h^{*}(L(b)))$, where
$\mathcal{Z}(V)$ is the family of all $Z$-sets in $V$.

We claim that the graph $Gr(\psi)$ of $\psi$ is closed in $B\times V$. Indeed,
suppose $\\{(b_{\gamma},v_{\gamma})\\}\in Gr(\psi)$ is a net converging to
$(b_{0},v_{0})$ in $B\times V$. Since $h^{*}$ is a homeomorphism between $C$
and $B\times V$, there is a net $\\{g_{\gamma}\\}\subset C$ converging to some
$g_{0}\in C$ such that $g_{\gamma}\in L(b_{\gamma})$ and
$h^{*}(g_{\gamma})=(b_{\gamma},v_{\gamma})$. Since $L$ is closed in $C$,
$g_{0}\in L$. On the other hand, $\\{b_{\gamma}\\}$ converges to $b_{0}$
implies that $g_{0}\in C(b_{0})$. Hence $g_{0}\in L\cap C(b_{0})=L(b_{0})$.
Finally, $h^{*}(g_{0})=(b_{0},v_{0})\in Gr(\psi)$.

Since each set $O_{b}=\displaystyle\\{v\in
V:||v-\pi_{V}(u(b))||<\frac{\alpha(b)}{N}\\},{~{}}b\in B,$ is open in $V$,
$\psi(b)\cap O_{b}$ is a $Z$-set in $O_{b}$. This implies that
$\psi(b)\cap\varphi(b)$ is a $Z$-set in $\varphi(b)$ for each $b$. (This
follows because radial contraction toward $\pi_{V}(u(b)$ shows that the
boundary of $\phi(b)$ is a Z-set in $\phi(b)$, and as $\phi(b)\cap O_{b}$ is a
countable union of Z-sets of $C(b)$. Thus $\psi(b)$ is a countable union of
Z-sets of $\phi(b)$.) Therefore, we may apply Lemma 3.1 to find a map
$\phi:B\to V$ with $\phi(b)\in\varphi(b)\backslash\psi(b)$ for all $b\in B$.
Then the equality $u^{\prime}(b)=(b,\phi(b))$ provides a map $u^{\prime}:B\to
B\times V$ such that $u^{\prime}(P_{i})\subset G_{i}$ for all $i\leq k$ and
$u^{\prime}(B)\cap h^{*}(L)=\varnothing$. Hence, each
$(h^{*})^{-1}(u^{\prime}(b))$ belongs to $C(b)\backslash L$. Consequently, we
have a map $g^{\prime}:B\to C\backslash L$,
$g^{\prime}(b)=(h^{*})^{-1}(u^{\prime}(b))$, with
$g^{\prime}(P_{i})\subset\widetilde{W}_{i}$, $i\leq k$. The composition
$g^{\prime\prime}=r_{E}^{*}\circ g^{\prime}$ provides a map from $B$ into
$C_{P}(\mathbb{I}^{n}\times\\{1,2\\},E\times\\{\overline{0}\\}\times 0))$ such
that for every $b\in B$ we have:

  * (3)

$g^{\prime\prime}(b)\in
C(\mathbb{I}^{n}\times\\{1,2\\},p^{-1}(b)\times\\{\overline{0}\\}\times 0))$;

  * (4)

$g^{\prime\prime}(P_{i})\subset W_{i}$, $i\leq k$;

  * (5)

$g^{\prime\prime}(b)(\mathbb{I}^{n}\times\\{1\\})\cap
g^{\prime\prime}(b)(\mathbb{I}^{n}\times\\{2\\})=\varnothing$.

Condition $(4)$ means that $g^{\prime\prime}\in O_{\Lambda(f)}$. Therefore,
$g=\Lambda^{-1}(g^{\prime\prime})\in C(\mathbb{I}^{n}\times\\{1,2\\}\times
B,E)$ is a fiber-preserving approximation of $f$ such that
$g(\mathbb{I}^{n}\times\\{1\\}\times B)\cap
g(\mathbb{I}^{n}\times\\{2\\}\times B)=\varnothing$. This provides the proof
of Claim 3.4.

Therefore, $p$ has the FGP-property. ∎

######  Theorem 3.5.

Let $p:E\to B$ be a locally compact $ANR$-fibration with compact fibers such
that $B$ is a paracompact locally compact $C$-space. Then $p$ is a
$Q$-manifold bundle if and only for every $b\in B$ we have:

  * (1)

$p^{-1}(b)$ has the disjoint disks property;

  * (2)

$C(\mathbb{I}^{n},p^{-1}(b))$ contains a dense set of homological $Z_{n}$-maps
for each $n\geq 2$.

###### Proof.

Suppose $p$ is a $Q$-manifold bundle. Then there is a $Q$-manifold $M$ and an
open cover $\mathcal{U}$ of $B$ such that $p^{-1}(U)$ is homeomorphic to
$U\times M$ for all $U\in\mathcal{U}$. Because for every $U\in\mathcal{U}$ and
every $b\in U$ the fiber $p^{-1}(b)$ is a $Q$-manifold, conditions $(1)$ and
$(2)$ are satisfied.

Suppose now that $p$ satisfies conditions $(1)$ and $(2)$. Then each
$p^{-1}(b)$, $b\in B$, is a compact $ANR$ satisfying the disjoint disks
property and for every $n\geq 3$ the space $C(\mathbb{I}^{n},p^{-1}(b))$
contains a dense set of homological $Z_{n}$-maps. Therefore, by [10, Theorem
2.9], each fiber of $p$ is a $Q$-manifold. We choose a local trivialization
$\\{U_{\alpha}:\alpha\in A\\}$ of $p$ such that if $K\subset B$ is a compact
set contained in some $U_{\alpha}$, then the fibration
$p|p^{-1}(K):p^{-1}(K)\to K$ has the FGP-property, see Proposition 3.2.
Without loss of generality, we may assume that all $U_{\alpha}$ are
$F_{\sigma}$-subsets of $B$, hence $C$-spaces. According to [21, Theorem 2.3],
it suffices to show that each $p|p^{-1}(U_{\alpha})$ has the FGP-property.
Because every $U_{\alpha}$ is a locally compact $C$-space, we can assume that
$p$ is a trivial fibration such that any fibration $p|p^{-1}(K):p^{-1}(K)\to
K$, where $K\subset B$ is compact, has the FGP-property. Therefore, we need to
show that $p$ has the FGP-property. To this end, using the paracompactness of
$B$, for every $i$ choose an open discrete family $\gamma_{i}=\\{V_{s,i}:s\in
S\\}$ in $B$ such that $\bigcup_{i=1}^{\infty}\gamma_{i}$ is a cover of $B$
and each $\overline{V}_{s,i}$ is compact, see [7]. Then the family
$\overline{\gamma}_{i}=\\{\overline{V}_{s,i}:s\in S\\}$ is also discrete in
$B$ and $B_{i}=\bigcup\\{\overline{V}_{s,i}:s\in S\\}$ is closed in $B$. Since
for every $i$ the fibrations $p|p^{-1}(\overline{V}_{s,i})$, $s\in S$, have
the FGP-property, each $p|p^{-1}(B_{i})$ also has the FGP-property.
Consequently, for every $n$ and $i$ the set $\Theta_{n,i}$ of all $f\in
C_{P}(\mathbb{I}^{n}\times\\{1,2\\}\times B_{i},p^{-1}(B_{i}))$ with
$f(\mathbb{I}^{n}\times\\{1\\}\times B_{i})\cap
f(\mathbb{I}^{n}\times\\{2\\}\times B_{i})=\varnothing\\}$ is dense in
$C_{P}(\mathbb{I}^{n}\times\\{1,2\\}\times B_{i},p^{-1}(B_{i}))$.

######  Claim 3.6.

Each $\Theta_{n,i}$ is open in $C_{P}(\mathbb{I}^{n}\times\\{1,2\\}\times
B_{i},p^{-1}(B_{i}))$.

Indeed, let $f_{0}\in\Theta_{n,i}$. Using that $\overline{\gamma}_{i}$ is
discrete in $B$ and the pairs
$F_{s,i}^{1}=f_{0}(\mathbb{I}^{n}\times\\{1\\}\times\overline{V}_{s,i})$,
$F_{s,i}^{2}=f_{0}(\mathbb{I}^{n}\times\\{2\\}\times\overline{V}_{s,i})$ are
compact disjoint subsets of $p^{-1}(B_{i})$, we can find open subsets
$W_{s,i}^{j}$ and $G_{s,i}^{j}$ of $p^{-1}(B_{i})$, $j=1,2$, with the
following properties:

  * (6)

$F_{s,i}^{j}\subset W_{s,i}^{j}\subset\overline{W}_{s,i}^{j}\subset
G_{s,i}^{j}$, $j=1,2$;

  * (7)

For each $j=1,2$ the families $\beta_{i}^{j}=\\{W_{s,i}^{j}:s\in S\\}$ and
$\eta_{i}^{j}=\\{G_{s,i}^{j}:s\in S\\}$ are discrete in $p^{-1}(B_{i})$;

  * (8)

$G_{s,i}^{1}\cap G_{s,i}^{2}=\varnothing$ for each $s$ and $i$.

Now, let
$\mathcal{G}_{i}=\\{G_{s,i}^{1},G_{s,i}^{2},p^{-1}(B_{i})\setminus\overline{W}:s\in
S\\}$, where
$\overline{W}=\bigcup\\{\overline{W}_{s,i}^{1}\cup\overline{W}_{s,i}^{2}:s\in
S\\}$. Every $\mathcal{G}_{i}$ is an open cover of $p^{-1}(B_{i})$ such that
$f\in B(f_{0},\mathcal{G}_{i})$ implies
$f(\mathbb{I}^{n}\times\\{1\\}\times\\{b\\})\cap
f(\mathbb{I}^{n}\times\\{2\\}\times\\{b\\})=\varnothing$ for every $b\in
B_{i}$. Hence, $B(f_{0},\mathcal{G}_{i})\subset\Theta_{n,i}$. This completes
the proof of the claim because $B(f_{0},\mathcal{G}_{i})$ is a neighborhood of
$f_{0}$ with respect to the limitation topology, see [5].

Since each $q_{i}:C_{P}(\mathbb{I}^{n}\times\\{1,2\\}\times B,E)\to
C_{P}(\mathbb{I}^{n}\times\\{1,2\\}\times B_{i},p^{-1}(B_{i}))$ is open (see
Lemma 2.1), $q_{i}^{-1}(\Theta_{n,i})$ is open and dense in
$C_{P}(\mathbb{I}^{n}\times\\{1,2\\}\times B,E)$. According to Proposition
2.3, $C_{P}(\mathbb{I}^{n}\times\\{1,2\\}\times B,E)$ has the Baire property.
Therefore, the set $\Theta_{n}=\bigcap_{i=1}^{\infty}q_{i}^{-1}(\Theta_{n,i})$
is dense in $C_{P}(\mathbb{I}^{n}\times\\{1,2\\}\times B,E)$. Finally, observe
that $g\in\Theta_{n}$ implies $g(\mathbb{I}^{n}\times\\{1\\}\times\\{b\\})\cap
g(\mathbb{I}^{n}\times\\{2\\}\times\\{b\\})=\varnothing$ for every $b\in B$.
Therefore, $p$ has the FGP-property. ∎

##  4\. $ANR$-fibrations over $C$-spaces

The FGP property is necessary and sufficient for a locally compact $ANR$
fibration with compact fibers to be a $Q$-manifold bundle. In the present
section we provide another condition which still guarantees that any locally
compact $ANR$ fibration with compact fibers satisfying that condition is a
$Q$-manifold bundle provided the base is a $C$-space.

A closed set $A\subset E$ is said to be a fibred $Z_{n}$-set if the set of all
$f\in C_{P}(\mathbb{I}^{n}\times B,E)$ with $f(\mathbb{I}^{n}\times B)\cap
A=\varnothing$ is dense in $C_{P}(\mathbb{I}^{n}\times B,E)$. We also define
weak versions of fibred $Z_{n}$-sets: a closed set $A\subset E$ is a weak
fibred $Z_{n}$-set in $E$ (resp., weak fibred homological $Z_{n}$-set in $E$)
if $A\cap p^{-1}(b)$ is a $Z_{n}$-set (resp., homological $Z_{n}$-set) in
$p^{-1}(b)$ for all $b\in B$. Fibred $Z_{\infty}$-sets (resp., weak fibred
$Z_{\infty}$-sets or weak fibred homological $Z_{\infty}$-sets) will be called
fibred $Z$-sets (resp., weak fibred $Z$-sets or weak fibred homological
$Z$-sets). Finally, a map $f\in C_{P}(K\times B,E)$, where $K$ is a given
space, is fibred $Z$-map, weak fibred $Z$-map or weak fibred homological
$Z$-map if the image $f(K)$ has the corresponding property.

######  Theorem 4.1.

Let $p:E\to B$ be a locally compact $ANR$-fibration with compact fibers such
that $B$ is a $C$-space. Then every $b\in B$ has a basis of neighborhoods $U$
such that each weak fibred $Z$-set in $p^{-1}(U)$ is a fibred $Z$-set in
$p^{-1}(U)$.

###### Proof.

As in Proposition 3.2, where we were considering open $F_{\sigma}$ elements of
$\mathcal{U}$, the proof is reduced to the case when $p$ is a trivial
$ANR$-fibration such that both fibrations $p^{\prime}:E\times Q\to E\to B$ and
$p^{\prime\prime}:E\times Q\times[0,1)\to E\to B$ are trivial $Q$-manifold
bundles. Consequently, there is a fiber-preserving homeomorphism $h:E\times
Q\times[0,1)\to M\times B$, where $M$ is a $Q$-manifold homeomorphic to
$p^{-1}(b_{0})\times Q\times[0,1)$ for some $b_{0}\in B$ (actually, $M$ is
homeomorphic to $p^{-1}(b)\times Q\times[0,1)$ for all $b\in B$). We consider
$M$ as an open subset of a copy $Q_{M}$ of $Q$, and let $d$ be the usual
convex metric on $Q_{M}$. Suppose $A\subset E$ is a weak fibred $Z$-set in
$E$. We need to show that for every $n$, a map $f\in
C_{P}(\mathbb{I}^{n}\times B,E)$ and an open cover $\mathcal{U}$ of $E$ there
is a map $f^{\prime}\in C_{P}(\mathbb{I}^{n}\times B,E)$ such that
$f^{\prime}(\mathbb{I}^{n}\times B)\cap A=\varnothing$ and $f^{\prime}$ is
$\mathcal{U}$-close to $f$. To this end, fix such $f\in
C_{P}(\mathbb{I}^{n}\times B,E)$ and an open cover $\mathcal{U}$ of $E$. We
identify $E$ with $E\times\\{\overline{0}\\}\times\\{0\\}$, where
$\overline{0}=(0,0,...)\in Q$. Consider the retraction $r_{E}:E\times
Q\times[0,1)\to E\times\\{\overline{0}\\}\times 0$,
$r_{E}((z_{1},z_{2},z_{3}))=(z_{1},\overline{0},0)$. Under the above
identifications, $r_{E}$ is a fiber-preserving retraction and
$\mathcal{V}=h(r_{E}^{-1}(\mathcal{U}))$ is an open cover of $M\times B$.
Moreover, consider the map $\widetilde{f}:\mathbb{I}^{n}\times B\to M\times
B$, $\widetilde{f}(x,b)=h(f(x,b),\overline{0},0))$. Since $h$ is a fiber-
preserving homeomorphism, $\widetilde{f}(x,b)$ is of the form $(f_{M}(x,b),b)$
for each $(x,b)\in\mathbb{I}^{n}\times B$. Then, by Lemma 2.2, there is a
bounded continuous function $\varepsilon$ on $M\times B$ such that every map
from the set

$\\{g\in C_{P}(\mathbb{I}^{n}\times B,M\times
B):d_{E}(\widetilde{f}(y),g(y))<\varepsilon(\widetilde{f}(y))\forall
y\in\mathbb{I}^{n}\times B\\}$

is $\mathcal{V}$-close to $\widetilde{f}$, where $d_{E}$ is the continuous
pseudo-metric on $M\times B$ generated by $d$.

Since $M$ is open in $Q_{M}$, the function
$\eta(x,b)=d\big{(}\pi_{M}(\widetilde{f}(x,b)),Q_{M}\backslash M\big{)}$ is
positive and continuous on $\mathbb{I}^{n}\times B$ ($\pi_{M}:M\times B\to M$
is the projection. Let $\delta:\mathbb{I}^{n}\times B\to(0,\infty)$ be the
function defined by
$\delta(x,b)=\min\\{\eta(x,b),\varepsilon(\widetilde{f}(x,b))\\}$.

######  Claim 4.2.

There is a bounded continuous function $\alpha:B\to(0,\infty)$ such that
$\alpha(b)<\delta(x,b)$ for each $(x,b)\in\mathbb{I}^{n}\times B$.

Indeed, for every $b\in B$ let
$m_{b}=\min\\{\delta(x,b):x\in\mathbb{I}^{n}\\}$. Then each $m_{b}>0$ and
consider the set-valued map
$\theta:B\rightsquigarrow\mathcal{F}_{c}((0,\infty))$, $\theta(b)=(0,m_{b}]$.
This map is lower semi-continuous and, by [17, Theorem 6.2, p.116], $\theta$
has a continuous selection $\alpha$. Clearly, $\alpha$ has the required
property.

Define the set-valued map $\varphi:\mathbb{I}^{n}\times
B\to\mathcal{F}_{c}(Q_{M})$ by

$\varphi(x,b)=\\{u\in Q_{M}:d(f_{M}(x,b),u)\leq\alpha(b)\\}.$

Observe that each $\varphi(x,b)$ is a compact convex subset of $M$ because
$\alpha(b)<\eta(x,b)$ for all $(x,b)\in\mathbb{I}^{n}\times B$. Since $f_{M}$
and $\alpha$ are continuous, $\varphi$ is lower semi-continuous. Recall that
each $A(b)=A\cap p^{-1}(b)$ is a $Z$-set in $p^{-1}(b)$. So is $A(b)\times
Q\times[0,1)$ in the set $p^{-1}(b)\times Q\times[0,1)$. This implies that
$H(b)=h(A(b)\times Q\times[0,1))$ is a $Z$-set in $h(p^{-1}(b)\times
Q\times[0,1))$. On the other hand, $h(p^{-1}(b)\times
Q\times[0,1))=M\times\\{b\\}$. Therefore, $\psi(b)=\pi_{M}\big{(}H(b))$ is a
$Z$-set in $M$ for every $b\in B$.

######  Claim 4.3.

The set-valued map $\psi:B\rightsquigarrow\mathcal{F}(M)$ has a closed graph
and $\psi(b)\cap\varphi(x,b)$ is a $Z$-set in $\varphi(x,b)$ for all
$(x,b)\in\mathbb{I}^{n}\times B$.

To show that the graph $Gr(\psi)$ is closed in $B\times M$, take a net
$\\{(b_{\gamma},u_{\gamma})\\}$ in $Gr(\psi)$ converging to a point
$(b^{*},u^{*})\in B\times M$. Then for each $\gamma$ we have
$u_{\gamma}\in\psi(b_{\gamma})$, so $(u_{\gamma},b_{\gamma})\in h(A(b)\times
Q\times[0,1))$. This means that there exists $z_{\gamma}\in
A(b_{\gamma})\times Q\times[0,1)$ with
$h(z_{\gamma})=(u_{\gamma},b_{\gamma})$. Since the net
$\\{(u_{\gamma},b_{\gamma})\\}$ converges to $(u^{*},b^{*})$ in $M\times B$,
$\\{z_{\gamma}\\}$ converges to $z^{*}=h^{-1}(u^{*},b^{*})$ in $E\times
Q\times[0,1)$. Since $A\times Q\times[0,1)$ is closed in $E\times
Q\times[0,1)$ and contains all $z_{\gamma}$, $z^{*}$ belongs to $A\times
Q\times[0,1)$. On the other hand, $z^{*}\in p^{-1}(b^{*})\times Q\times[0,1)$
because $h$ is fiber-preserving. Hence, $z^{*}\in(A\times
Q\times[0,1))\cap(p^{-1}(b^{*})\times Q\times[0,1))$. The last intersection is
$A(b^{*})\times Q\times[0,1)$, so $h(z^{*})=(u^{*},b^{*})\in H(b^{*})$.
Therefore, $u^{*}\in\psi(b^{*})$, or equivalently, $(b^{*},u^{*})\in
Gr(\psi)$. This shows that $Gr(\psi)$ is closed in $B\times M$.

To prove the second part of Claim 4.3, for each $(x,b)\in\mathbb{I}^{n}\times
B$ consider the open in $M$ set $O(x,b)=\\{u\in
Q_{M}:d(f_{M}(x,b),u)<\alpha(b)\\}$. Because $\psi(b)$ is a $Z$-set in $M$,
$\psi(b)\cap O(x,b)$ is a $Z$-set in $O(x,b)$. Finally, since $\varphi(x,b)$
is the closure of $O(x,b)$, $\psi(b)\cap\varphi(x,b)$ is a $Z$-set in
$\varphi(x,b)$.

Since $\psi$ has a closed graph, so has the map $\psi^{\prime}:I^{n}\times
B\rightsquigarrow\mathcal{F}(M)$, $\psi^{\prime}(x,b)=\psi(b)$. Because
$\mathbb{I}^{n}\times B$ is a paracompact $C$-space, we can apply Lemma 3.1 to
obtain a map $g_{M}:\mathbb{I}^{n}\times B\to M$ with
$g_{M}(x,b)\in\varphi(x,b)\backslash\psi^{\prime}(x,b)$. Then $(g_{M}(x,b),b)$
defines a map $\widetilde{g}\in C_{P}(\mathbb{I}^{n}\times B,M\times B)$ with
$\widetilde{g}(x,b)\in M\times\\{b\\}\backslash H(b)$. So,
$g(x,b)\in(p^{-1}(b^{*})\backslash A(b))\times Q\times[0,1)$ for all
$(x,b)\in\mathbb{I}^{n}\times B$, where $g\in C_{P}(\mathbb{I}^{n}\times B\to
E\times Q\times[0,1))$ is the map defined by $g=h^{-1}\circ\widetilde{g}$.
Because
$d(f_{M}(x,b),g_{M}(x,b))\leq\alpha(b)<\varepsilon(\widetilde{f}(x,b))$,
$\widetilde{f}$ and $\widetilde{g}$ are $\mathcal{V}$-close maps.
Consequently, $(f(x,b),\overline{0},0)$ and $g(x,b)$ are
$r_{E}^{-1}(\mathcal{U})$-close for each $(x,b)$. This implies that $f$ is
$\mathcal{U}$-close to the map $f^{\prime}\in C_{P}(\mathbb{I}^{n}\times B\to
E)$, $f^{\prime}=r_{E}\circ g$. Finally, observe that
$g(x,b)\in(p^{-1}(b^{*})\backslash A(b))\times Q\times[0,1)$ yields
$f^{\prime}(x,b)\in p^{-1}(b^{*})\backslash A(b)$ for all $(x,b)$. Hence,
$f^{\prime}(\mathbb{I}^{n}\times B)\cap A=\varnothing$. This completes the
proof of Theorem 4.1. ∎

######  Theorem 4.4.

Let $p:E\to B$ be a locally compact $ANR$-fibration with compact fibers. Then
$p$ is a $Q$-manifold bundle provided the following holds:

  * (*)

Every point of $B$ has a basis consisting of $C$-spaces $U$ such that each
$C_{P}(\mathbb{I}^{n}\times U,p^{-1}(U))$, $n\geq 1$, contains a dense set of
weak fibred $Z$-maps.

###### Proof.

We take a local trivialization $\mathcal{U}$ of $p$ such that each
$U\in\mathcal{U}$ is a $C$-space and for every $n$ the space
$C_{P}(\mathbb{I}^{n}\times U,p^{-1}(U))$ contains a dense set of weak fibred
$Z$-maps. Moreover, by Theorem 4.1, we can also assume that each
$U\in\mathcal{U}$ satisfies the additional condition:

  * •

Every weak fibred $Z$-set in $p^{-1}(U)$ is a fibred $Z$-set in $p^{-1}(U)$.

Let show that each $p_{U}=p|p^{-1}(U)$, $U\in\mathcal{U}$, has the FGP-
property. It suffices to prove that for every $n$ every weak fibred $Z$-map
$f:\mathbb{I}^{n}\times U\to p^{-1}(U)$ can be approximated by fiber-
preserving maps $f^{\prime}:\mathbb{I}^{n}\times U\to p^{-1}(U)$ such that
$f^{\prime}(\mathbb{I}^{n}\times U)\cap f(\mathbb{I}^{n}\times
U)=\varnothing$. This follows from the fact that the set
$A=f(\mathbb{I}^{n}\times U)$ is a fibred $Z$-set in $p^{-1}(U)$ because $f$
is weak fibred $Z$-map. ∎

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