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Abstract. In this paper, the concept of cellularity is extended to spaces which are not compact. The extension is called quasicellularity, where a quasicell is a regular neighborhood of a tree. This general extension splits into two concepts, strong quasicellularity and weak quasicellularity. Strong quasicellularity implies weak quasicellularity, but the reverse is not true. In the course of extending cellularity, an extension of the UV^{4 } property is defined, called strong UV^{4}, or SUV^{4}, and it is shown that SUV^{4} is a topological property. It is demonstrated that a strongly quasicellular set embedded in an nmanifold, n$4, has the SUV^{4 }property and satisfies a strong quasicellularity criterion. In like manner, a weakly quasicellular set embedded in an nmanifold, n$4, satisfies part of a weak quasicellularity criterion.
This research is a condensation of the first half of the author's Ph.D. dissertation, written at the University of Georgia under the direction of Professor R. B. Sher. I would like to express my thanks to Professor Sher for his aid. His advice and criticism have been invaluable.
AMS 1970 Subject Classifications. Primary 57A60, 57C99;
Secondary 57C40.
MSC 2000 Subject Classifications. Primary 57N60, 57Q99; Secondary 57Q40,
57Q65.
Key words and phrases. Cellular, cellularity criterion, properly S^{k}inessential, quasicell, quasicellular, quasicellularity criterion, quasitrivial, strong quasicellularity criterion, strongly quasicellular, SUV^{4}, tree, UV^{4}, weak quasicellularity criterion, weakly quasicellular.
NOTATION
UE and Int(U) symbolize the topological interior of
U.
ME_{i+1} symbolizes the topological
interior of M_{i+1} (the apparent sequence of the
superscript and subscripts is purely an artifact of the HTML).
( €_{i})_{i=1}^{4}
symbolizes an operation or group of indexed entities ranging from the index of 1
to the index of 4 (the apparent sequence
of the superscript and subscripts is purely an artifact of the HTML).
J^{+ }symbolizes the set of positive integers.
ú^{1} symbolizes the real number
line.
S^{n} symbolizes the Euclidean nsphere.
M^{n} and X^{x} symbolize ndimensional
and xdimensional objects, respectively.
F^{i} and G^{i}, where F and G
are maps, symbolize indexed maps, not ndimensional objects.
F_{0} and F_{1}, where F is a
homotopy, F:XHIàY,
symbolize F(X,0) and F(X,1), respectively.
MB symbolizes the boundary of
B.
>> symbolizes an onto map.
Cl(X) symbolizes the topological closure of X.
K symbolizes
the complex closure of K.
FY symbolizes the Freudenthal
compactification of a space Y.
EY symbolizes the set of ends of Y.
` symbolizes a collapse.
1 INTRODUCTION.
The concept of cellularity is a wellknown and useful concept. McMillan has shown [14, Theorem 1] that a UV^{4} compactum lying in the interior of a PL nmanifold, n$5, which satisfies his cellularity criterion is piecewise linearly cellular. It is also true that a cellular set is a UV^{4} compactum and satisfies the cellularity criterion if it lies in the interior of an nmanifold, n$3.
In this work, we extend the concept of cellularity to objects which are not compact. We define a quasicell to be a regular neighborhood of a tree, and use this as an analog of a cell. Because of certain differences between compact and noncompact objects, the extension splits into two concepts, strong quasicellularity and weak quasicellularity. We show that the concept of weak quasicellularity is properly contained in the concept of strong quasicellularity.
In the course of extending cellularity, we also define an extension of the UV^{4 } property, called strong UV^{4}, or SUV^{4}, and show that SUV^{4} is a topological property.
We shall attempt to maintain standard notations and usage whenever possible. Dugundji [7] and Hudson [10] will be our references for general background material. We shall thus assume familiarity with the concepts of simplexes, complexes, polyhedrons, joins, triangulations, subdivisions, PL, combinatorial and topological manifolds, and simplicial and PL maps and homeomorphisms. We distinguish between topological closure and complex closure by Cl(K) and K, respectively.
Most of the spaces with which we will be concerned will be locally compact metric spaces. DIST(x,y) will refer to the distance from x to y using some (perhaps not explicitly named) metric. Some concepts obviously require a metric, even if none has been specified. For instance, the εneighborhood of a set X, N(X; ε), for ε>0, is the set of points having distance from X less than or equal to ε. Likewise, we need a metric for the concept of εhomotopy: If H:XHIàY, where I=[0, 1], is a homotopy with ε>0 given, and for each p X, H({p}HI) has diameter less than ε, then H is an εhomotopy.
The work below comes from the first half of [8]; however, there is additional background material in the Appendix of [8] that is not included. This material includes proofs on infinite regular neighborhoods, proper maps, mapping cylinders, general position, and engulfing theorems.
2 SUV^{4}.
2.1 DEFINITION. A map f:XàY of topological spaces is proper if f^{ l}(C) is compact whenever C is compact.
2.2 DEFINITION. A tree is a connected, simply connected, locallyfinite lcomplex.
2.3 DEFINITION. Suppose N is an nmanifold. If there is a PL nmanifold N'dE^{n} which is a closed regular neighborhood of some tree, and if there is a homeomorphism h from N' onto N, then N is called an nquasicell. If the dimension is obvious, or not relevant, N will be referred to simply as a quasicell.
2.4 DEFINITION. Suppose M is a topological space and X is a subset of M. If, for each closed neighborhood U of X, there is a closed neighborhood V of X contained in UE, a tree T, a map f:VàT, a proper map g:TàU, and a proper homotopy H:VHIàU such that H_{0} is the inclusion and H_{l }=gBf, the we say that X has the strong UV^{4} property in M; this being the case, we say that X has property SUV^{4} in M.
2.5 REMARK. Since H_{1} is proper, f must be proper also. In [18], Sher uses a slightly modified version of SUV^{4}. He requires that f be onto T, which, along with gBf being proper, implies that g is proper. If X is connected, these two definitions coincide. In any case, if X satisfies the definition of [18], it satisfies Definition 2.4. If X is connected and satisfies Definition 2.4, then there is a single component V' of V containing X. Then f(V') is connected and is a tree. Let H':V'HIàU be defined by H'=HV'. If g'=gf(V'), then H'_{1}=g'BfV' and fV':V'>>f(V') is onto. Since V' is closed, then H' is proper and X satisfies the definition of [18].
2.6 DEFINITION. Suppose M is a topological space and X is a compact subset of M. If, for each open neighborhood U of X, there is an open neighborhood V of X contained in U such that V is homotopic in U to a point, then we say that X has the UV^{4} property in M.
2.7 REMARKS. (a) It follows easily from the above definitions that a
space having UV^{4} or SUV^{4} must necessarily be connected.
(b) If M is a normal space and X is a compact subset of M
with the SUV^{4} property in M, then X has the UV^{4} property in M.
Obviously, properties UV^{4} and SUV^{4} depend on the ambient space M as well as the set X. For example, it can be shown that the "topologists' sin 1/x  curve" has property UV^{4}, and hence SUV^{4}, in any manifold M, but not in itself. However, McMillan has shown [15, Theorem 2, p. 20] that UV^{4 }is a property which is topologically invariant with respect to embeddings in metrizable ANRs. The following theorem is the analog for SUV^{4}.
2.8 THEOREM. The property SUV^{4} is topologically invariant with respect to embeddings in locally compact metric ANRs.
Proof. Suppose M and M^{*} are locally compact ANRs and XdM, X^{*}dM^{*} are closed sets such that X has SUV^{4} in M and there is a homeomorphism onto, h:X>>X^{*}, and suppose U^{*} is a given closed neighborhood of X^{*}.
Since M is an ANR and UE^{*} is open, UE^{*} is an ANR and we may extend h to H on a neighborhood U of X into U^{*}, H:UàU^{*}, HX=h. We may also choose U so that H is proper by means of [2, Lemma 3.2].
Figure 1. H:UàU^{*}
Now, since X has property SUV^{4} in M, there exists a closed neighborhood V of X lying in UE, a tree T, a map f:VàT, a proper map g:TàU, and a proper homotopy K:VHIàU such that K_{0} is the inclusion and K_{1}=gBf.
Figure 2. f:VàT and g:TàU
Let U_{1}^{*} be a neighborhood of X^{*} lying in UE^{*} and H^{*}:U_{1}^{*}àU be a proper map with H^{*}X^{*}=h^{1}. Let U_{2}^{*}=H^{*1}(VE), which is an open neighborhood of X^{*} in U_{1}^{*}. Let ε:U^{*}à(0,4) be any map. Since UE^{*} is an ANR and HBH^{*}U_{2}^{*}:U_{2}^{*}àU^{*} is a welldefined map which agrees with hBh^{1}=1_{U*}X^{*} on X^{*}, by [9, Theorem 1.1, Chapter IV] there exists a closed neighborhood V^{*} and an εhomotopy L:V^{*}HIàU^{*} with L_{0} the inclusion and L_{1}=HBH^{*}V^{*}. By AII.5 of [8], ε may be chosen so that L is proper.
Figure 3. H^{*}:U_{1}^{*}àU
Now define T^{*}=T, f^{*}=fBH^{*}V^{*},
g^{*}=HBg, and K^{*}:V^{*}HIàU^{*}
by
:L_{2t},
0#t#1/2
K^{*}_{t}= ;
<HBK_{2t1}BH^{*}V^{*}, 1/2#t#1.
We have that V^{*} is a closed neighborhood of X^{*} lying in UE^{*}, T^{*} is a tree, f^{*} is a map from V^{*} to U^{*}, g^{*} is a map from T^{*} to U^{*}, and since g and H are proper, g^{*} is proper. We have that K^{*} is a homotopy (since L_{1}=HBH^{*}V^{*}=HBK_{0}BH^{*}V^{*}) and that K is proper since L, H, K, H^{*} all are proper. We also have that K^{*}_{0}=L_{0} is the inclusion on V and K^{*}_{1}=HBK_{1}BH^{*}V^{*}=HBgBfBH^{*}V^{*}=g^{*}Bf^{*}. Therefore X^{*} has SUV^{4} in M^{*}. €
Since all the ambient spaces with which we shall concern ourselves will be locally compact metric ANRs (usually manifolds), we shall speak of a set X having SUV^{4}, rather than the embedding of X into a space having the SUV^{4} property.
2.9 REMARK. It can easily be shown that if A is an AR embedded as a closed subset of a locally compact metric ANR, T is a tree, and F:AHIàA is a proper homotopy with F_{0} the inclusion and F_{1}(A)dT, then A has SUV^{4}.
The following theorem gives an equivalent definition for a closed subset of an nmanifold, n$3, having SUV^{4}.
2.10 THEOREM. Suppose M is a PL nmanifold, n$3, and X is a closed subset of M lying in ME. Then X has SUV^{4} in M if and only if for each closed neighborhood U of X, there is a PL submanifold V, with a nonempty boundary, such that XdVEdVdUE, a tree T embedded as a closed PL subset of M in UE, and a proper PL homotopy H:VHIàU with H_{0} the inclusion, H_{1}(V)=T and H(VHI)dUE.
Proof. Suppose X has SUV^{4}. Let U' be a closed neighborhood of X contained in UE. There exists a closed neighborhood V' of X in U', a tree T', a map f:V'àT', a proper map g:T'àU', and a proper homotopy H':V^{'}HIàU' with H'_{0} the inclusion and H'_{1}=gBf. Since X is connected, let VdVE' be a connected PL submanifold of M, with nonempty boundary, whose interior contains X. Since T' is locally compact, M is hausdorff, and f is proper, it follows that f is closed. Then T"=f(V)dT' is a closed tree.
Since n$3, gT" is proper, and DIM T"=1, then there is a proper
homotopy L:T"HIàU'
with L_{0}=gT" and L_{1} a PL embedding. Define H:VHIàU'
by
:H'_{2t},
0#t#1/2
H_{t}= ;
<L_{2t1}BfV, 1/2#t#1.
We have that H:VHIàU'dUE is a homotopy (since H'_{1}V=[gf(V)]BfV=L_{0}BfV)
and is proper since H'_{t}V, fV and L are all proper. By construction,
H0 is the inclusion,
H_{1}(V)=L_{1}(f(V))=L_{1}(T")=T,
and H(VHI)dU'dUE. Using general position, we can homotop H
to a PL homotopy in general position, staying fixed on the 0 and 1 levels, so
that we have the first half of the theorem.
The second half of the theorem follows trivially. €
3 QUASITRIVIALITY.
3.1 DEFINITION. A set X contained in the interior of an nmanifold is said to be quasitrivial in M if X is contained in a quasicell in ME.
3.2 DEFINITION. Let K and L be topological spaces and f:KàL
a continuous function. Let (KHI)+L
be the free union of (KHI) and L, with
topology such that a set is open in (KHI)+L
if and only if its intersections with both KHI
and L are open in KHI and L, respectively.
Let R be the equivalence relation on (KHI)+L
generated by (x,1) ~ f(x) for all x in K.
Let P be the projection
P:(KHI)+Là[(KHI)+L]/R.
We shall say f is the attaching map of KHI
to L; [(KHI)+L]/R is written M_{f},
and is called the topological mapping cylinder of f. We shall regard K as
a subset of M_{f} by identifying K with KH{0}
and noting that (KH{0})/R is still KH{0}.
We shall regard L as a subset of M_{f} since L/R is still L.
It can be shown that if K and L are locally finite simplicial complexes and f:KàL is a proper simplicial map, then M_{f} can be triangulated as a locally finite simplicial complex C_{f} with a subdivision K' of K and L and subcomplexes and C_{f}`L. See [5] for a discussion of the compact case and [8] for a generalization to the noncompact case. We take our theory of noncompact regular neighborhoods and collapses from [17].
The following theorem is quite similar to Lemmas 39 and 49 of [22, Chapter VII]. We refer the reader to this source for definitions of piping and TRAIL in the compact case and for discussions of these topics.
3.3 THEOREM. Let M be a PL nmanifold, X^{x} a closed PL subspace of M in ME, nx$1, T a tree embedded as a closed PL subspace of M in ME, and F:XHIàM a proper PL homotopy such that F(XHI)dME, F_{0} is the inclusion and F_{1}(X)dT. Then there are closed PL subspaces Y^{y} and Z^{z} of M lying in ME such that XdY`Z with y#x+1 and 0<z#max{2xn+2,1}.
Proof. If x=m1 or x=m2, the conclusion of the theorem follows immediately by letting Y=Z=X. Therefore we assume mx$3.
Since F is proper, we may triangulate M and XHI so that F is simplicial. We use the homotopy F to obtain g:C_{F1}àM, a proper PL map in general position with g(C_{F1})dME, gX the inclusion on X, and gT the inclusion on T. (See Section 4 of [8] for details.) Let X^{x1} be the (x1)skeleton of X and let f=F_{1}X^{x1}:X^{x1}àT. We apply a noncompact version of Zeeman's Piping lemma [22, Lemma 48] to the triple X^{x}, C_{f}^{x}dC_{F1}^{x+1}. We obtain a proper PL map g':C_{F1}àM such that g'(C_{F1})dME, g'X^{x}ÈC_{f}^{x}=gX^{x}ÈC_{f}^{x}, and a PL subspace J_{1}dC_{F1}^{x+1} such that
(i) S(g')dJ_{1} (S(g') is the singular set of g')
(ii) DIM J_{1}#2xn+2
(iii) DIM(C_{f}ÇJ_{1})#2xn+1
(iv) C_{F1}`C_{f}ÈJ_{1}`C_{f}.
Define Y=g'(C_{F1}). Then DIM Y#x+1,
Y is closed, YdME, and XdY.
By (i) and (iv), Y=g'(C_{F1})`g'(C_{f}ÈJ_{1}).
Let b be a simplicial collapse applicable to C_{f}
such that b:C_{f}`T.
We have C_{f}`TÈTRAIL_{b}(J_{1}ÇC_{f}).
By (i) again, we have Y`g'(C_{f}ÈJ_{1})
and using (i) once more, and the collapse of the last sentence we have
g'(C_{f}ÈJ_{1})`g'(TÈTRAIL_{b}(J_{1}ÇC_{f})ÈJ_{1}).
Let Z=g'(TÈTRAIL_{b}(J_{1}ÇC_{f})ÈJ_{1}).
Since g'(T)dZ and DIM TRAIL_{b}(J_{1}ÇC_{f})#2xn+2,
then DIM Z#max{2xn+2,1}, and 0<DIM Z. We
now have XdY`Z,
Z is closed, and ZdME. Note that if x<(n1)/2, then 2xn+2<1 so
that DIM J_{1}#0 and DIM TRAIL_{b}(J_{1}ÇC_{f})#0.
Since Y`Z using the collapse of C_{F1}
to T, each component of C_{F1} contains a
component of T, and T is connected, then C_{F1} is
connected and Y and Z must be connected. Hence the 0skeleton of Z, i.e., from J_{1},
and TRAIL_{b}(J_{1}ÇC_{f}),
must be in T, so that Zg'(T)=T. Thus Y`T. €
We now use Theorem 3.3 to obtain sufficient conditions for a subspace of a manifold to be quasitrivial. The following is a noncompact analog of [13, Lemma 3].
3.4 THEOREM. Suppose {M_{i}}_{i=1}^{4} is a sequence of PL nmanifolds such that each M_{i} is a closed PL subspace of M_{i+1} lying in ME_{i+1}, and for i=1, 2, ... there is a tree T_{i} embedded as a closed PL subspace of M_{i+1} and a proper homotopy F^{i}:M_{i}HIàM_{i+1} such that F^{i}(M_{i}HI)dME_{i+1}, F^{i}_{0} is the inclusion on M_{i}, and F^{i}_{1}(M_{i})dT_{i}. Then if i=1, 2, ... and X^{x} is a closed PL subspace of with nx$3, X^{x} is quasitrivial in M_{i+x+1}.
Proof. We begin the inductive proof of the theorem by considering the case x=0. Hence, let i be a positive integer and X be a discrete set of points in M_{i}. Let f denote F^{i}XHI. We may suppose that f is a general position map in general position with respect to T_{i}. It follows, since DIM(XHI)=DIM T_{i}=1, DIM M_{i+1}$3, and F is proper, that F is an embedding and that F(XHI)ÈT_{i} is a PL tree embedded as a closed subspace of M_{i+1} lying in ME_{i+1}. Letting C be a regular neighborhood in ME_{i+1} of F(XHI)ÈT_{i}, then C is a quasicell, XdCE and it follows that X is quasitrivial in M_{i+1}.
Now suppose, inductively, that the conclusion of the theorem holds for every
positive integer i and for all nonnegative integers x#a1,
where n(a1)$4. Consider X^{a}dM_{i},
and let F denote F^{i}X. Then
by Theorem 3.3, there exist closed PL subspaces Y^{y} and Z^{z} of M_{i+1} lying in
ME_{i+1} such that XdY`Z
with 0<z#max{2an+2,1}.
Then z#2an+2=a(na)+2#a3+2=a1
or if n=4, a=1, z=1. In the first case, by
the inductive assumption, Z is quasitrivial in M_{i+1+z+1}. But
i+1+z+1#i+2+2an+2=i+a+1+(an+3)#i+a+1.
It follows that M_{i+1+z+1}dM_{i+a+1},
and hence, that Z is quasitrivial in M_{i+a+1}.
But Y`Z, so Y is quasitrivial in M_{i+a+1}
and, since XdY, X is quasitrivial in M_{i+a+1}.
This completes the induction and leaves only the special case n=4,
a=1. The last three sentences of the proof
of Theorem 3.3 show how to complete the proof for this case. Since 2an+2<1, Y`Z=T_{i},
which is quasitrivial in M_{i+1}dM_{i+2}.
Thus X is quasitrivial in M_{i+a+1}=M_{i+2}. €
3.5 DEFINITION. Suppose M is a PL nmanifold, XdME and U is a neighborhood of X in M. A sequence of triples {H_{i1},T_{i},F^{i}}_{i=1}^{4} is said to be a quasispecial sequence for X in U provided:
(i) H_{i+1} and T_{i+1} are closed PL subspaces of M lying in HE_{i}, i=0, 1, 2, ... ,
(ii) H_{i1} is a PL nmanifold with MH_{i1}¹Æ, and T_{i} is a tree, i=1, 2, ... ,
(iii) F^{i+1}:H_{i+1}HIàM is a proper PL homotopy with F^{i+1}(H_{i+1}HI)dHE_{i}, F_{0}^{i+1} the inclusion on H_{i+1}, and F_{1}^{i+1}(H_{i+1})=T_{i+1}, i=0, 1, 2, ... ,
(iv) H_{i} is a closed neighborhood of X, i=1, 2, ... , and X=Ç_{i=1}^{4}H_{i}, and
(v) if Y is a closed PL subspace of H_{i+1} and DIM Y#n3, then Y is quasitrivial in H_{i}, i=1, 2, ... .
If for each closed neighborhood U of X, there is a quasispecial sequence for X in U, then we say that quasispecial sequences for X in M exist strongly. If there is a quasispecial sequence for X in M and if each quasicell N with XdNE there is a quasispecial sequence for X in N, then we say that quasispecial sequences for X in M exist weakly.
3.6 REMARKS. (a) In requirement (iii) we define F^{i+1} as
having range M; since the image of F^{i+1} is contained in HE_{i} and we shall always be
dealing with situations in which H_{i} is a closed subspace of M, i=0,
1, ... , we shall often regard H_{i} as the range of F^{i+1}.
(b) If quasispecial sequences for a set X in a manifold M
exist strongly, then X must be connected, since each T_{i} is connected.
The following result connects the notions of quasispecial sequences and the property SUV^{4}.
3.7 THEOREM. Let M be a PL nmanifold, n$3. Then a closed subset X of M lying in ME has SUV^{4} if and only if quasispecial sequences for X in M exist strongly.
Proof. Suppose quasispecial sequences for X in M exist strongly. We shall show that for each closed neighborhood U of X, there is a PL submanifold V of M, with nonempty boundary, such that XdVEdVdUE, a tree T embedded as a closed PL subset of U in UE, and a proper PL homotopy F:VHIàU with F_{0} the inclusion on V, F_{1}(V)=T and F(VHI)dUE. It then follows from Theorem 2.10 that X has property SUV^{4} in M.
Let H_{0} and H_{1} be PL submanifolds of M lying in UE, given by the assumption, with F^{1}, T_{1} the homotopy and tree, respectively for H_{1} into H_{0}. By Remark 3.6 (b) we may assume that H_{1} is connected. The required V is H_{1}, the required homotopy is F^{1} and the tree is F_{1}^{1}(H_{1})=T_{1}.
Suppose X has SUV^{4}. Let ε_{1}, ε_{2}, ... be a sequence of positive real numbers with lim_{i®4 } ε_{i}=0 and let N(x: ε_{i}) be the closed ε_{i}neighborhood of X in M. Let U be a given closed neighborhood of X in M. We shall construct a quasispecial sequence for X in U.
Let V_{0} be a closed PL manifold neighborhood of X in U. Let U_{1}=V_{0}ÇN(x: ε_{1}) and let V_{1} be a connected PL submanifold of M with nonempty boundary such that XdVE_{1}dV_{1}dUE_{1}, T_{1} be a tree embedded as a closed PL subset of U_{1} in UE_{1}, and G':V_{1}HIàU_{1}dV_{0} be a proper homotopy with G'_{0} the inclusion on V_{1}, G'_{1}(V_{1})=T_{1}' and G'(V_{1}HI)dUE_{1}dVE_{0}.
Let U_{2}=V_{1}ÇN(x: ε_{2}) and choose V_{2}, G^{2} and T_{2}' as above, where G^{2}(V_{2}HI)dUE_{2}dVE_{1}. Inductively, having obtained V_{i}, G^{i}, T_{i}', define U_{i+1}=V_{i}ÇN(x: ε_{i+1}) and choose V_{i+1}, G^{i+1} and T_{i+1}' as above. The sequence of triples {(V_{i1}, T_{i}', G^{i})}_{i=1}^{4} satisfies properties (i) through (iv) of the definition of a quasispecial sequence for X in U. Now let H_{0}=V_{0} and for i=1, 2, ..., let H_{i}=V_{1+i(n2)}, T_{i}=T_{1+i(n2)}' and F^{i}=G^{1+i(n2)}V_{1+i(n2)}HI. Then by the comment above and by Theorem 3.4 {(H_{i1}, T_{i}, F^{i})}_{i=1}^{4} is a quasispecial sequence for X in U. €
4 QUASICELLULARITY.
A closed set X lying in the interior of an nmanifold M is said to be cellular if there exists a sequence {Ci}_{i=1}^{4} of ncells such that MÉC_{1}ÉCE_{1}ÉC_{2}ÉCE_{2}É ... ÉÇ_{i=1}^{4}Ci=X. For a summary of results concerning this concept, see [15].
We would like to define an analogous concept which need not require compactness and in which cells are replaced by quasicells. Unfortunately, we encounter problems in dealing with such objects which do not occur with cellular sets. One such problem involves the fact that a cellular set is compact; hence in the interior of any neighborhood of the cellular set, there is an ncell whose interior contains the cellular set. This is not true in general for the nested intersection of quasicells.
Suppose X is the xaxis in ú^{2}. Then X is a tree embedded very nicely in ú^{2}. Certainly we would like X to be "quasicellular" when this term is defined. If we were given only one sequence of quasicells having X as its intersection, we would not know whether of not X has the property of having a quasicell neighborhood lying in the interior of any given neighborhood of X. For instance, suppose {N_{i}}_{i=1}^{4}, where N_{i}={(x, y)0ú^{2}  y#1/i}, is the one given sequence (Figure 4),
Figure 4. X and {N_{i}}_{i=1}^{4}
and define U={(x, y)0ú^{2}  xy#1 and y#1} (Figure 5). Clearly there is no positive integer i so that N_{i}dUE. Thus the property of a "quasicellular" set having arbitrarily close quasicell neighborhoods must be included in the definition if it is to be required.
Figure 5. X, {N_{i}}_{i=1}^{4} and U
4.1 DEFINITION. A closed set X contained in the interior of an nmanifold M is said to be strongly quasicellular if for each closed neighborhood U of X, there is a quasicell N such that XdNEdNdUE.
4.2 REMARK. It is easily seen that if X is strongly quasicellular in an nmanifold M, then X is connected.
We also define a weaker form of "quasicellularity."
4.3 DEFINITION. A closed set X contained in the interior of an nmanifold M is said to be weakly quasicellular if there exists a sequence of quasicells {N_{i}}_{i=1}^{4}, N_{i+1}dNE_{i}, X=Ç_{i=1}^{4}N_{i}, and if for each quasicell Q containing X in its interior, there is such a sequence with N_{1}dQE.
4.4 REMARKS. (a) If X is a set contained in the interior of a
topological nmanifold, n$5, and X has SUV^{4}, then some open neighborhood of X has a PL
structure by Kirby and Siebenmann [11]. Since our interest is in "close
neighborhoods" of such sets, we shall assume that if XdME^{n}, n$5, has SUV^{4}, then M has a PL structure.
(b) If X is a strongly or weakly quasicellular set lying in
the interior of a topological nmanifold, n$5, then some neighborhood of X has a PL structure
trivially. As above, if XdME^{n}, n$5, and X is strongly or weakly quasicellular, we
will assume that M has a PL structure.
4.5 THEOREM. Let X be contained in the interior of a PL nmanifold M, n$2. If X is strongly quasicellular, then X is weakly quasicellular. However, if X is weakly quasicellular, it need not be strongly quasicellular.
4.6 EXAMPLE. The "topologists' sin 1/xcurve" minus an end point of the "closure line segment," when embedded as a closed subset of ú^{2}dú^{n} is weakly quasicellular in ú^{n}, but not strongly quasicellular in ú^{n}. In Figure 6, X is the curve (including the heavy closure line, but minus the point p). U is a neighborhood that does not include the elliptical areas in each of the bends of the curve.
Figure 6. "Topologists' sin 1/xcurve" minus an end point
Figure 7 illustrates the objects that are used in showing that X is weakly quasicellular. Each T_{i} is a tree that coincides with X through the first i bends, then connects with the closure line at the point p. Each A_{i} is a neighborhood of T_{i} that also includes the bends of X that aren't in T_{i}. Each A_{i} is contained in U through the first i bends, but "bridges" the holes thereafter.
Figure 7. Sketch of logic demonstrating weak quasicellularity
The demonstration that X is not strongly quasicellular involves using this U and is described in [8].
We will now examine some properties of quasicells and strongly and weakly quasicellular sets.
4.7 DEFINITION. Suppose X is a compact topological space, YdZdW are topological spaces, and J^{+} is the space consisting of the positive integers with the discrete topology. A proper map g:XHJ^{+}àY is said to be properly Xinessential in Z if there exists a proper map G:(XHJ^{+})HIàZ such that for each x in X and j in J^{+}, G(x, j, 0)=g(x, j), and for each j in J^{+}, G(XH{j})H{1} is a constant map. If, in addition, A is a subset of W and there is a map G as above such that G[(XHJ^{+})HI]ÇA=f, then we say that g is properly Xinessential in Z missing A.
4.8 LEMMA. Let X be a strongly quasicellular set in the nmanifold M. Suppose N_{1} is a quasicell in M containing X in its interior, P is an (n3)dimensional polyhedron and g:PHJ^{+}àN_{1}X is properly Pinessential in N_{1}. Then g is properly Pinessential in N_{1} missing X.
Proof. Since g:PHJ^{+}àN_{1} is a proper map and g(PHJ^{+})ÇX=f, NE_{1}g(PHJ^{+}) is a neighborhood of X. Thus, since X is strongly quasicellular in M, there exists a quasicell N_{2} such that XdNE_{2}dN_{2}dNE_{1}g(PHJ^{+}). We regard N_{2} as a piecewise linear manifold and, as such, there exists a closed polyhedral tree TdNE_{2} such that N_{2}`T.
By hypothesis, there exists a proper homotopy, F^{1}:(PHJ^{+})HIàN_{1}
such that F^{1}_{0}=g and, for each j in J^{+}, F^{1}_{1}PH{j}
is a constant map. Let Y=(F^{1})^{1}(NE_{2}) and Y_{0}=YÇ((PHJ^{+})H{1}).
Then, as an open subset of (PHJ^{+})HI,
Y has a PL structure and, by the aforementioned properties of F^{1}, Y_{0}
is a closed polyhedral subspace of Y. Also YÇ((PHJ^{+})H{0})=f.
Let
ε:(PHJ^{+})HIà(0, 4) be a map and let d:Yà(0, 4) be defined by d(y)=min{ε(y),DIST(y,(PHJ^{+})HIY)},
for all y in Y. Now, using general position we obtain a PL map F':YàNE_{2} in general position such
that for all y0Y, DIST(F'(y),F^{1}(y))<d(y)
and, for all y0Y_{0}, F'(y)=F^{1}(y). Define the
function F^{2}:(PHJ^{+})HIàN_{1}
by
:F'(y),
if y0Y
F^{2 }(y)=;
<F^{1}(y), if yÏY.
By our choice of d, F^{2} is continuous and, we may choose ε so that since DIST(F^{2}(y), F^{1}(y))<ε(y) for all y0(PHJ^{+})HI, F^{2} is proper. Also, since YÇ((PHJ^{+})H{0})=f, F^{2}_{0}=F^{1}_{0}=g and, since F'Y_{0}:Y_{0}àN_{2}=F^{1}Y_{0}, F^{2}_{1}PH{j} is a constant map for all j0J^{+}. Now, by applying a general position argument again (to shift F'(Y) off of T), we may further suppose that F^{2}((PHJ^{+})HI)ÇT=f. (It is here that the hypothesis DIM P#n3 is used.)
Let N_{3} be a regular neighborhood of T in NE_{2} such that N_{3}ÇF^{2}((PHJ^{+})HI)=f.
Then, as in the case of compact regular neighborhoods [10, Corollary 2.16.2], it
follows that N_{2}NE_{3}=Cl(N_{2}N_{3})
can be identified with MN_{2}H[0,
1], with MN_{2}H{0}
and MN_{2} identified. The map
F":Cl(N_{1}N_{3})àN_{1}
defined, via this identification, by
:z,
if z0N_{1}N_{2}
F^{"}(z)=;
<(x,
0), if z=(x,
t)0Cl(N_{2}N_{3})
is proper. Thus the map G:(PHJ^{+})HIàN_{1}
defined by G(x, j, t)=F"(F^{2}(x, j, t)) is proper. It follows readily
that G_{0}=g, G_{1}PH{j}
is a constant map for all j0J^{+}, and that N_{2}Ç((PHJ^{+})H{0})=f.
The latter implies that XÇ((PHJ^{+})H{0})=f
and the proof is complete. €
4.9 REMARK. Suppose XdME^{n}, n$6, and X is strongly or weakly quasicellular. Then we may as well assume that X is strongly or weakly PL quasicellular. That is, we may assume that the quasicells obtained from the respective definitions are actually piecewise linear subsets of M (cf. Remarks 4.4 (a), (b)). (See the fourth theorem on page 1 of [12].)
4.10 REMARK. It is easily seen that if the image of g and the image of the homotopy F' are in NE_{1}, then the image of G can be required to be in NE_{1}X.
4.11 DEFINITION. Suppose X is a set contained in the topological space Y. We say that X satisfies the strong quasicellularity criterion (in Y), if for each closed neighborhood U of X, there is a closed neighborhood V of X lying in UE such that each proper map g:S^{k}HJ^{+}àVX which is properly S^{k}inessential in V is properly S^{k}inessential in U missing X, for k=0 or 1.
4.12 REMARK. (a) McMillan [15, p.3] defines the cellularity
criterion for a compactum X embedded in a space Y as the property that for
each open set U containing X, there is an open set V such that XdVdU
and each loop in VX is nullhomotopic in UX.
(b) It is easy to see that if X is a compactum with property
UV^{4} in the normal space Y and satisfying the
strong quasicellularity criterion in Y, then X satisfies the cellularity
criterion in Y. To see this, let U be an open set containing X, let U' be a
closed neighborhood of X such that U'dU,
and let V be the closed neighborhood given by the strong quasicellularity
criterion for the neighborhood U'. Since X has property UV^{4} in Y, there exists an open neighborhood W of
X such that XdVE and W is inessential in V. Let
l:S^{1}àWX
be a loop. Then, since W is inessential in V, l
is nullhomotopic in V. Hence, by our choice of V,
l is nullhomotopic in U'X, and hence in UX.
(c) Suppose X lies in the interior of an nmanifold M, has
SUV^{4} and satisfies the strong quasicellularity
criterion. If U is a neighborhood of X in M, then there is a quasispecial
sequence for X in U {H_{i1},T_{i},F^{i}}_{i=1}^{4}
with the nullhomotopy property: for each i=0, 1, 2, ... and each proper
map g:S^{k}HJ^{+}àH_{i+1}X
which is properly S^{k}inessential in H_{i+1}, g is properly S^{k}inessential
in H_{i} missing X, k=0 or 1.
(d) Suppose X is closed and connected and satisfies the
strong quasicellularity criterion in the locally path connected space Y and U
is a closed neighborhood of X in Y. Then UX is path connected if U is path
connected.
(e) It is easy to see that if X is a closed subset of the
space Y having property SUV^{4} in Y and satisfying the strong
quasicellularity criterion in Y, then if U is a simply connected closed
neighborhood of X in Y, UX is simply connected.
(f) If S={H_{i1},T_{i},F^{i}}_{i=1}^{4}
is a quasispecial sequence for a closed subset X of a PL nmanifold, and S has
the nullhomotopy property, then there is a subsequence {H'_{i1},T'_{i},F'^{i}}_{i=1}^{4}
of S with the additional property that if l
is a loop in H'_{i+1}X, then is nullhomotopic in H'_{i}X,
and if a:S^{0}àH'_{i+1}X
is a map, then a is nullhomotopic in H'_{i}X,
for i=1, 2, ... .
4.13 THEOREM. Suppose X is a set contained in the interior of a PL nmanifold, n$4, and X is strongly quasicellular in M. Then X has SUV^{4} and X satisfies the strong quasicellularity criterion.
Proof. Let U be a closed neighborhood of X and V a quasicell such that XdVEdVdUE. Let V be triangulated so that there is a closed tree T embedded as a subcomplex of V in VE with V as the simplicial neighborhood of T and V`T. This collapse defines a proper PL homotopy H:VHIàU with H_{0} the inclusion on V, H_{1}(V)=T and H(VHI)dUE. Thus X has SUV^{4}.
Let U be a closed neighborhood of X and V a quasicell such that XdVEdVdUE. Let k=0 or 1 and let g:S^{k}HJ^{+}àVX be a proper map which is properly S^{k}inessential in V. By Lemma 4.8, g is properly S^{k}inessential in V missing X; hence g is properly S^{k}inessential in U missing X; and X satisfies the strong quasicellularity criterion in M. €
4.14 THEOREM. Suppose X is a set contained in the interior of a PL nmanifold, n$4, and X is weakly quasicellular in M, then quasispecial sequences for X in M exist weakly.
Proof. By the definition of weak quasicellularity there is a nested sequence of quasicells closing down on X. This satisfies the condition of a quasispecial sequence for X in M existing. For each quasicell containing X in its interior, we have such a sequence, and so we also have a quasispecial sequence for X in this quasicell. €
4.15 DEFINITION. If X is contained in a manifold M and there is a quasispecial sequence for X in M with the nullhomotopy property, and for each quasicell Q with X in its interior there is such a quasispecial sequence in Q, then we say that X satisfies the weak quasicellularity criterion.
4.16 REMARK. The weak quasicellularity criterion subsumes the requirement that quasispecial sequences for X in M exist weakly.
We shall now consider some examples. We mentioned earlier that it seemed fitting that the xaxis in ú^{2} should be quasicellular. Now we can say more.
4.17 EXAMPLE. Any tree T embedded as a closed subcomplex of some locallyfinite triangulation of a PL nmanifold M, n$1, is strongly quasicellular.
However, not every embedding of a tree in a manifold need be strongly quasicellular.
4.18 EXAMPLE. For each n$4, there is an arc embedded in ú^{n} which is not strongly quasicellular. Let E be a set homeomorphic to [0, 1] in ú^{n}, n$3, such that ú^{n}E is not simply connected [3, Theorem 3F]. If E satisfies the strong quasicellularity criterion then, by and earlier remark, E satisfies the cellularity criterion, and hence, for n$3 by [15] is cellular, a contradiction. Hence E does not satisfy the strong quasicellularity criterion and, by Theorem 4.13, if n$4, E is not strongly quasicellular.
4.19 EXAMPLE. For each n$4, there is a closed topological line in
ú^{n} which is not strongly quasicellular. Let Edú^{n}
be the arc of Example 4.18 with
ú^{n}E not simply connected. Since is
ú^{n1}/EHú^{1}
homeomorphic to
ú^{n}, [1], then
L=E/EHú^{1}dú^{n1}/EHú^{1}
is a topological line in
ú^{n}. To see that
ú^{n}L is not simply connected, we simply note that (ú^{n1}E)Hú^{1}
is not simply connected. By a previous remark, L does not satisfy the strong
quasicellularity criterion in
ú^{n}, and hence L is not strongly quasicellular in
ú^{n}.
We now show that certain pathological sets are strongly quasicellular.
4.20 EXAMPLE. For n$4, there is an everywhere wild, noncompact, strongly
quasicellular set in
ú^{n}. By judicious use of Rushing's theorems and lemmas in [16],
we obtain an everywhere wild (n3)dimensional cellular set F in
ú^{n1} and see that FHú^{1}
is everywhere wild in
ú^{n}. Let U be a closed neighborhood of FHú^{1}
in
ú^{n}. For each i$0, define
ε_{i}>0 so that N(FH[i,
i+1];
ε_{i})dUE and so that for i>j,
ε_{i}<ε_{j}
and lim_{i®4}
ε_{i}=0. Similarly, for each i<0,
define d_{i}$0 so that N(FH[i1,
i]; d_{i})dUE, so that for i>j,
d_{i}>d_{j} and lim_{i®4} d_{i}=0. Since F is cellular, define for each i$0, N_{i} to be an (n1)cell so that
N_{i}dN(F;
ε_{i})ÇN(F;
d_{i})dú^{n1}
and so that for i>j, N_{i}dN_{j}.
For i$0, N_{i}H[i,
i+1]ÇN_{i+1}H[i+1,
i+2]=N_{i+1}, and N_{i}H[i1,
i]ÇN_{i+1}H[i2,
i1]=N_{i+1}, we have that Q=È_{i=0}^{4}[N_{i}H[i,
i+1]ÈN_{i}H[i1,
i]] is an nquasicell lying in UE containing X in its interior. (Note
that it is easy to see that Q may be triangulated so that it collapses to pHú^{1}
for some point p in F).
4.21 REMARK. Suppose XdME has property SUV^{4} and satisfies the strong quasicellularity
criterion. Then, if U is a closed neighborhood of X, there exists a closed
neighborhood V of X lying in U such that for every v0VX, the inclusioninduced homomorphism
i_{*}:π_{1}(VX,v)àπ_{1}(UX,v)
is trivial. The converse is not true in general, unless X is compact.
However, the strong quasicellularity criterion can be stated in algebraic
terms. E. M. Brown [4] has defined, for noncompact polyhedra K, the groups π_{m}(K,a),
where a:[0, 4)àK
is a proper map. In terms of these groups (when they exist) we can give the
following equivalent statement of the strong quasicellularity criterion in the
PL manifold M:
Suppose a polyhedron XdME has property SUV^{4}. Then X satisfies the strong
quasicellularity criterion if and only if for each closed neighborhood U of X
there exists a closed polyhedral neighborhood V of X such that
(1) the inclusion induced
homomorphism i_{*}:π_{1}(VX,v)àπ_{1}(UX,v)
is trivial for all v0VX, and (if X is noncompact),
(2) the inclusion induced
homomorphism π_{1}(i):π_{1}(VX,a)àπ_{1}(UX,a)
is trivial for all proper maps a:[0, 4)àV
with a([0, 4))dVX
CONTINUATION: "QuasiCellularity Criteria."
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