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Abstract. It is demonstrated that a set embedded in a manifold (with certain technical conditions) is strong quasicellularity if and only if it has the SUV^{4} property and satisfies a strong quasicellularity criterion. Moreover, a locally compact, finitedimensional metric space has the SUV^{4} property with respect to metric ANRs if and only if it embeds as a strongly quasicellular subset of some (high dimensional) manifold. In like manner, a set embedded in an nmanifold (with technical conditions) is weakly quasicellular if it satisfies a weak quasicellularity criterion.
This research is a condensation of the second 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 57C30, 57C40.
MSC 2000 Subject Classifications. Primary 57N60, 57Q99; Secondary 57Q30,
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.
In this work we show that our extensions of cellularity [5] were the correct ones in that we are able to prove McMillan type theorems [8 Theorem 1] for strong quasicellularity and for weak quasicellularity. In fact, our extension of UV^{4} to SUV^{4} shows an even closer analogy to the compact case. McMillan remarks [9, p 21] that, for finite dimensional continua X, X0UV^{4} if and only if X has cellular embeddings into some (high dimensional) Euclidean space. We show in Theorem 3.12 that, for locally compact, finite dimensional metric spaces X, X0SUV^{4} if and only if X has a strongly quasicellular embedding into some (high dimensional) manifold. B. J. Ball and R. B. Sher have developed a theory of proper shape [1] and Sher has used this [13, Theorem 3.1] to show that, for X a locally compact metrizable space, X0SUV^{4} if and only if X has the proper shape of a tree.
The work below comes from the second half of [6]; however, there is additional background material in the Appendix of [6] that is not included. This material includes proofs on infinite regular neighborhoods, proper maps, mapping cylinders, general position, and engulfing theorems. Some of the material on engulfing theorems is repeated in section 5, below.
2 PRELIMINARY DEFINITIONS AND NOTATION.
We wish to recall some definitions and notation from [5] for the reader's convenience.
We will continue to use cl(K) to mean the topological closure of a set K and K to mean the complex closure, that is the smallest complex containing K. We will continue to use Dugundji [4] and Hudson [7] for general background material and most of the spaces we deal with will be assumed to be locally compact metric spaces.
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 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.
2.5.1 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.
2.5 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}¹f, 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. We say that {H_{i1},T_{i},F^{i}}_{i=1}^{4} has the nullhomotopy property if 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.
2.6 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.
The following theorem [5, Theorem 2.10] gives an equivalent definition for the case that M is a PL nmanifold, n$3.
2.7 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.
We will also use the following theorem [5, Theorem 3.7].
2.8 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.
2.9 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.
2.10 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.
3 STRONG QUASICELLULARITY AND SUV^{4}.
3.1 LEMMA. Suppose X is a closed subset of a PL nmanifold M, X lies in ME and DIM X#n1. Further suppose that V is a closed neighborhood of X in M and g:J^{+}àV is a proper map. Then there exists a proper map G:J^{+}HIàV such that G_{0}=g and G(i,1)ÏX, for all i0J^{+}.
Proof. Clearly, we may suppose that for each i0J^{+}, g(i)0VE; for otherwise, we would simply define G on these points which are not carried into VE to be the constant homotopy. Let {e_{i}}_{i=1}^{4} be a sequence of positive numbers converging to zero. For each i0J^{+}, let N_{i}dVE be a PL nball whose interior contains g(i) and whose diameter is less than e_{i}. Since DIM X#n1, NE_{i}Ç(VX)¹f, for all i0J^{+}. Since NE_{i} is arcconnected, there exists a path a_{i}:[0,1]àNE_{i} such that a_{i}(0)=g(i) and a_{i}(1)0VX. Then, since g is proper and {e_{i}}_{i=1}^{4}®0, the map G:J^{+}HIàV defined by G(i,t)=a_{i}(t) is proper. Since G_{0}=g and G(i,1)=a_{i}(1)0VX, this completes the proof. €
3.2 LEMMA. Suppose that U is a locally compact topological space, VdU, T is a tree embedded as a closed subset of U, and F:VHIàU is a proper map such that F_{0} the inclusion and F_{1}(V)=T. Then, if P is a compact, connected space and g:PHJ^{+}àV is a proper map, g is properly Pinessential in U.
Proof. We shall define a proper map H:(PHJ^{+})HIàV such that H_{0}=g and, for each j0J^{+}, H_{1}PH{j} is a constant map. We first define G on (PHJ^{+})H[0, 1/2] by G_{t}=F_{2t}Bg, for t0[0, 1/2]. Since P is compact and connected, G_{1/2}(PH{j}) is a compact subtree of T, for each j0J^{+}, and as such is contractible. Thus we have that the map G_{1/2}PH{j} is homotopic to a constant map in G_{1/2}(PH{j}). We shall call this homotopy r^{j}, with r_{0}^{j}=G_{1/2}PH{j}. Let R:(PHJ^{+})H[1/2, 1]àT be the map defined by R_{t}PH{j}=r_{2t1}^{j}, for t0[1/2, 1]. We note that for each j0J^{+}, the set {kG_{1/2}(PH{j})ÇG_{1/2}(PH{k})¹f} is finite. It follows from this that R is proper. Since R_{1/2}=G_{1/2}, we may define H:(PHJ^{+})HIàU by H_{t}=G_{t}, t0[0, 1/2] and H_{t}=G_{t}, t0[1/2, 1]. Since H is proper, H_{0}=g, and H_{1}PH{j} is a constant map, for each j0J^{+}, then we are through. €
3.3 DEFINITION. Suppose M is a PL nmanifold, V is a given open set in M, and U is a given closed set in M containing MV in its interior. Let P denote an arbitrary closed kdimensional PL subspace of M, and Q a closed PL subspace of P with QdV. We say locally finite kcomplexes in M can be pulled into V in U if for each such P and Q there is a proper homotopy H:PH[0, 1]àM such that H_{0} is the inclusion on P, H_{1}(P)dV, H_{t} is the inclusion on QÈ[(MU)ÇP] for each t0[0, 1], H(PQÇUH[0, 1]dU, and there is a map d:Pà(0, 4) with DIST(H_{1}(p), MV)>d(p), for each point p0P.
3.4 LEMMA. Let M be a PL nmanifold and let X be a closed, connected subset of ME with DIM X#n1. Let U be a closed neighborhood of X in M such that a quasispecial sequence for X in U with the nullhomotopy property exists. Then locally finite 2complexes in M can be pulled into MX in U.
Proof. We first note that [M(MX)]dUE, so that the conclusion makes sense according to the definition. Now let P be any closed 2dimensional PL subspace of M and Q be any closed PL subspace of P in MX.
Let {H_{i1},T_{i},F^{i}}_{i=1}^{4} be a quasispecial sequence for X in U with the nullhomotopy property. Since X is connected, we may assume that each H_{i} is connected. We shall only be concerned with the first five manifolds, H_{0}, H_{1}, H_{2}, H_{3}, and H_{4}. Let R be a triangulation of M with P, Q and H_{i}, i=0, 1, 2, 3, 4, triangulated as subcomplexes of R.
We wish to produce a proper homotopy G:PH[0, 1]àM
such that G_{0} is the inclusion on P, G_{1}(P)dMX,
G_{t} is the inclusion on QÈ[(MU)ÇP]
for each t0[0, 1], G(PQÇUH[0, 1]dU,
and so that there is a map d:Pà(0, 4)
with DIST(G_{1}(p), X)>d(p), for each point p0P.
We will define G
(a) on (the vertices of P)H[0, 1],
(b) on (the 1simplexes of P)H[0, 1], and
(c) on (the 2simplexes of P)H[0, 1].
(a) Let V={v_{1}, v_{2}, v_{3}, ...} be the set of vertices of P. We define G:VHIàM by defining G(v_{i}, t)=v_{i}, for all t0[0, 1], if v_{i} is a vertex of RH_{4}ÈQ. Let {v^{~}_{1}, v^{~}_{2}, v^{~}_{3}, ...} be the set of vertices of P not in RH_{4}ÈQ. Define g:J^{+}àH_{3} by g(j)=v^{~}_{j}. Lemma 3.1 gives us a proper map G:J^{+}HIàH_{4} such that G_{0}=g and G(i, 1)ÏX, for all i0J^{+}. We define G(v^{~}_{i}, t)=G(i, t), for all i0J^{+}.
(b) Let S={s_{1}, s_{2}, s_{3}, ...} be the set of 1simplexes of P. We define G:SHIàM by defining G(x, t)=x, for all t0[0, 1] and x0s_{i} if s_{i}0RH_{4}ÈQ. Note that if s_{i}0RH_{4}ÈQ, then so are its vertices, so that G, here defined, agrees with its definition on VHI. Let {s^{~}_{1}, s^{~}_{2}, s^{~}_{3}, ...} be the 1simplexes of P not in RH_{4}ÈQ. Define G_{0}s^{~}_{i} to be the inclusion on s^{~}_{i}, for each i0J^{+}. Now for each i0J^{+}, Ms^{~}_{i} is a pair of vertices, and G(Ms^{~}_{i}H{1})dH_{4}X. For each i0J^{+}, let h_{i} be a homeomorphism from S^{0} onto Ms^{~}_{i}. Define F:S^{0}HJ^{+}àH_{4} by F(z, i)=G(h_{i}(z), 1). Then since G(s^{~}_{i}H{0}ÈMs^{~}_{i}H[0, 1])dH_{4} and G is proper, F is properly S^{0}inessential in H_{4}. By the nullhomotopy property, F is properly S^{0}inessential in H_{3} missing X. It follows from this that G can be properly extended to (È_{i=1}^{4}s^{~}_{i})H{1}, where the extension carries (È_{i=1}^{4}s^{~}_{i})H[0, 1] into H_{3}X. In a natural way (cf. the construction of F above), G determines a proper map S^{1}HJ^{+} from into H_{3}. Since S^{1} is compact and connected, such a map is properly S^{1}inessential in H_{2} by Lemma 3.2. It follows from this that we may properly extend G to È_{i=1}^{4}(s^{~}_{i}H[0, 1]), where the extension carries È_{i=1}^{4}(s^{~}_{i}H[0, 1]) into H_{2}.
(c) Let T={t_{1}, t_{2}, t_{3}, ...} be the set of 2simplexes of P. We define G:THIàM by defining G(x, t)=x, for all t0[0, 1] and x0t_{i} if t_{i}0RH_{4}ÈQ. Note that if t_{i}0RH_{4}ÈQ, then so are the 1simplexes and vertices which are its faces, so that G, here defined, agrees with its definition on SHI. Let {t^{~}_{1}, t^{~}_{2}, t^{~}_{3}, ...} be the 2simplexes of P not in RH_{4}ÈQ. Define G_{0}t^{~}_{i} to be the inclusion on t^{~}_{i}, for each i0J^{+}. Now for each i0J^{+}, Mt^{~}_{i} is identifiable with S^{1} and G(Mt^{~}_{i}H{1})dH_{3}X. In a natural way (cf. the construction of F above), G determines a proper map S^{1}HJ^{+} from into H_{3}X which is properly S^{1}inessential in H_{2}. By the nullhomotopy property, this map is properly S^{1}inessential in H_{1} missing X. It follows from this that G can be properly extended to (È_{i=1}^{4}t^{~}_{i})H{1}, where the extension carries (È_{i=1}^{4}t^{~}_{i})H[0, 1] into H_{1}X. We now have the proper map G defined on È_{i=1}^{4}M(t^{~}_{i}H[0, 1]). In a natural way again, G determines a proper map S^{2}HJ^{+} from into H_{1}. Since S^{2} is compact and connected, such a map is properly S^{2}inessential in H_{0} by Lemma 3.2. It follows from this that we may properly extend G to È_{i=1}^{4}(t^{~}_{i}H[0, 1]), where the extension carries È_{i=1}^{4}(t^{~}_{i}H[0, 1]) into H_{0}.
We now have our desired homotopy G:PH[0, 1]àM such that G_{0} is the inclusion on P, G_{1}(P)dMX, G_{t} is the inclusion on QÈ[(MU)ÇP] for each t0[0, 1], and G(PQÇUH[0, 1]dU. Since X is closed and G is continuous, we may define a map d:Pà(0, 4) with DIST(G_{1}(p), X)>d(p), for each point p0P. €
The properties we are concerned with are topological in nature. We do, however, use a metric defined on whichever manifold is our ambient space. Some of our proofs (e.g., the following lemma and many of the results of Section 4) will require certain properties of that metric which are not available for every metric which might be defined on our manifold. One property we sometimes require is that the closed eball, e>0, about a point in the manifold be compact. This means that if a closed subset is noncompact, it has infinite diameter. We have this, for example, if the metric for the manifold is complete [4, Chapter XIV, Theorem 2.3]. Requiring that the metric for our manifold be complete is not a great restriction, since every locally compact metric space has a complete metric [4, Chapter XIV, Corollary 2.4]. We shall henceforth assume that our manifolds are equipped with such metrics.
3.5 DEFINITION. We say that a space Y is kULC at the closed subset X, if for each e>0, there is a d>0 such that for each point x0X and map g:S^{k}àN(x; d), g is enullhomotopic in Y.
3.6 DEFINITION. Let (M, r) be a metric space with P and U subspaces. We say that P tends to U if given e>0, there is a compact subset CdP such that r(x, U)<e, for each x0PC.
The following lemma is analogous to Lemma 3.1 with the requirement that X be (n1)dimensional replaced by X tends to MX and M is 0ULC at X. It is easy to see that if X is (n1)dimensional, then X tends to MX and M is 0ULC at X.
3.7 LEMMA. Suppose X is a closed subset of a PL nmanifold M, X lies in ME, X tends to MX and M is 0ULC at X. If V is a closed neighborhood of X in M and g:J^{+}àV is a proper map, then there exists a proper map G:J^{+}HIàV such that G_{0}=g and G(i,1)ÏX, for all i0J^{+}.
Proof. We write M as an expanding union of compact sets
C_{1}dCE_{2}dC_{2}dCE_{3}d
... dÈ_{i=1}^{4}C_{i}=M
where C_{1} is chose arbitrarily and, for i>1, C_{i} is chose
inductively by the following method. Suppose C_{i1} has been chosen.
Using the 0ULC condition of the hypothesis, let d_{i}>0
be such that both d_{i}<1/i and such that for
each point x0X, any map g:S^{0}àN(x;
d) is (1/i)nullhomotopic. Now choose C_{i}
so large that if xÏC_{i}, then DIST(x, C_{i1})>1
and, using the hypothesis that X tends to MX, so large that if xÏC_{i},
then N(x; d_{i})ÇMX¹f.
Now consider the proper map g:J^{+}àV. If g(j)ÏX, for some j0J^{+}, we let G(j, t)=g(j), for all t0[0, 1]. Thus, for the rest of the argument, we assume that g(j)0X.
If j is such that g(j)0C_{2}, pick p_{j}0VX so that p_{j} is in the component of V containing g(j). Define GjH[0, 1]:jH[0, 1]àV so that G_{0}(j)=g(j) and G_{1}(j)=p_{j}.
For i>1, let j0J^{+} be such that g(j)0C_{i+1}C_{i}. Since g(j)ÏC_{i}, there is a point p_{j} in (MX)ÇN(g(j); d_{i}). By the 0ULC condition, we may define G^{~j}:jH[0, 1]àN(g(j); d_{i}) so that G^{~j}_{0}(j)=g(j) and G^{~j}_{1}(j)=p_{j}. Since g(j)ÏC_{i}, then DIST(g(j), C_{i1})>1; and since d_{i}<1/i<1, we have G^{~j}(jH[0, 1])ÇC_{i1}=f. We now let q_{j} be the right hand endpoint of the component of (G^{~j})^{1}(X) containing (j,0). (We regard {j}H[0, 1] as running from left to right, with (j,0) on the left.) Since G^{~j} is continuous and V is a neighborhood of X, there is a neighborhood of q_{j} in {j}H[0, 1] contained in (G^{~j})^{1}(V). Let r_{j} be a point in this neighborhood not in (G^{~j})^{1}(X). Define G^{j}:jH[0, 1]àV by G^{j}(j,t)=G^{~j}(j,t^{.}r_{j}).
We now define the map G:J^{+}HIàV such that G{j}H[0, 1]=G^{j}. It follows that G_{0}=g and G(j,1)ÏX, for all j0J^{+}. To see that G is proper, let C be a compact subset of M. then let C_{i} be such that CdC_{i}. Since g is proper, the subset of J^{+}, K=G^{1}(C_{i+1}), is finite. Since G has been constructed so that for jÏK, G(jH[0, 1])ÇC_{i}=f, then for j0K, G(jH[0, 1])ÇC=f. Thus G^{1}(C) is compact, G is proper, and the argument is complete. €
The following lemma is the analog of Lemma 3.4, just as the previous lemma was the analog of Lemma 3.1. Also, just as was the case for the previous lemma and Lemma 3.1, the following lemma subsumes Lemma 3.4.
3.8 LEMMA. Let M be a PL nmanifold and let X be a closed, connected subset of ME such that X tends to MX and M is 0ULC at X. Let U be a closed neighborhood of X in M such that a quasispecial sequence for X in U with the nullhomotopy property exists. Then locally finite 2complexes in M can be pulled into MX in U.
Proof. The proof is the same as that of Lemma 3.4, except that where in the proof of Lemma 3.4 an appeal to Lemma 3.1 is made, here the appeal is made to Lemma 3.7. €
We now state and prove our main theorem.
3.9 THEOREM. Let M be a PL nmanifold, n$6, and let X be a closed, connected subset of ME such that M is 0ULC at X and X tends to MX. Then, X is strongly quasicellular if and only if X has SUV^{4} and satisfies the strong quasicellularity criterion.
Proof. The proof of necessity is given by [5, Theorem 4.13].
We will now show the sufficiency of the conditions. For each closed neighborhood U of X, we must find a quasicell N such that XdNEdNdUE.
Let H_{0} and H_{1} be the first two manifolds of a quasispecial sequence for X in U with the nullhomotopy property. Let R be a triangulation of M with H_{0} and H_{1} as subcomplexes. Let R^{2} be the 2skeleton of R. We use H_{1} as the U of Lemma 3.8 to obtain a proper homotopy G:R^{2}H[0, 1]àM such that G_{0} is the inclusion on R^{2}, G_{1}(R^{2})dMX, G_{t} is the inclusion on (MH_{1})ÇR^{2} for each t0[0, 1], G(R^{2}ÇH_{1}H[0, 1])dH_{1}, and so that there is a map d:R^{2}à(0, 4) with DIST(G_{1}(p), X)>d(p), for each p0R^{2}. If we let {A_{p}} be {G(pH[0, 1])} for all points p in 2complexes in M and all homotopies G pulling these 2complexes into MX in H_{2}, we see that we may apply the Infinite Radial Engulfing Theorem (5.2, below) to obtain an engulfing isotopy G':MH[0, 1]àM such that G'_{0} is the identity, G'_{t}R^{2}ÇH_{1}=1R^{2}ÇH_{1}, R^{2}dG'_{1}(MX), and G'_{t}cl(MH_{1})=1cl(MH_{1}). This last result is achievable since G moves no point which is in cl(MH_{1}) and moves no point into cl(MH_{1}) from HE_{1} and since G' is picked so that points are moved close to the movement of points by G.
Now define K to equal R^{2}ÇH_{1}dG'_{1}(MX)ÇG'_{1}(H_{1}) which equals G'_{1}(H_{1}X). Let L be the complementary complex of K in H_{1}. Since DIM L#n3, there is a quasicell N* in HE_{0} with LdNE*, by the definition of quasispecial sequences. Since G'_{1}(X) does not intersect K, there is an ambient isotopy, fixed outside of H_{0}, G":MH[0, 1]àM which pushes N* along the join structure from L to K far enough so that G'_{1}(X)dG"_{1}(NE*). Define N=(G'_{1})^{1}BG"_{1}(N*). Then XdNEdNdHE_{0}dUE, and the proof is complete. €
3.10 REMARK. Not every set having SUV^{4} is quasicellular as shown by [5, Example 4.19]. However, the following shows that if X is a locally compact, finite dimensional metric space with SUV^{4} (with respect to metric ANRs), then X embeds in some Euclidean space as a strongly quasicellular set.
3.11 THEOREM. If X is a closed subset of E^{n} with SUV^{4}, then X is strongly quasicellular in E^{n+3}, for n$3.
Proof. Since n+3$6, since DIM X<n+3 supplies that X tends to E^{n+3}X and E^{n+3} is 0ULC at X, and since SUV^{4} is a topological property, so that X has SUV^{4} in E^{n+3}, Then Theorem 3.9 shows that we need only demonstrate that X satisfies the strong quasicellularity criterion to have that X is strongly quasicellular.
Suppose that U is a closed neighborhood of X in E^{n+3}. Let V be any closed neighborhood of X in E^{n+3} lying in UE. Now let g:S^{k}HJ^{+}àVX be a map such that there is a proper homotopy G:(S^{k}HJ^{+})HIàV with G_{0}=g and G_{1}S^{k}H{j} a constant map, for j0J^{+} and k=0 or 1. We may suppose, without loss of generality, that G is in general position with respect to E^{n}. Thus, since DIM(G[S^{k}HJ^{+}HI])#2, then E^{n}ÇG[S^{k}HJ^{+}HI]=f. Thus g is properly S^{k}inessential in U missing X, k=0 or 1, and the proof is complete. €
3.12 COROLLARY. Suppose X is a locally compact, finite dimensional, metric space. Then X has SUV^{4} with respect to metric ANRs if and only if X embeds as a strongly quasicellular subset of some PL nmanifold, n$6.
Proof. Since X is a locally compact, finite dimensional metric space, it embeds in E^{n}, for some n$3. Theorem 3.11 thus completes the proof in one direction. [5, Theorem 4.13] completes the proof for the other direction. €
We now state the following theorem of R. B. Sher [13, Theorem 3.1] to complete the analogy with the compact case mentioned in the introduction (Sh_{p} is the proper shape function).
3.13 THEOREM. Suppose X is a locally compact metrizable space. Then X0SUV^{4} if and only if there exists a tree T such that Sh_{p}X=Sh_{p}T.
4 STRONG QUASICELLULARITY AND WEAK QUASICELLULARITY.
In section 3 we presented one McMillan type theorem connecting strong quasicellularity with SUV^{4} and the strong quasicellularity criterion. Our use of the Infinite Radial Engulfing Theorem required certain technical conditions. Rushing [11] has other infinite engulfing theorems which yield similar theorems with other technical conditions. We state these here. We also give McMillan type theorems (of one direction only) for weak quasicellularity.
4.1 DEFINITION. Let M be a PL manifold and U an open subset of M. Then
we say that (M, U) is pconnected if and only if for each integer i, 0
£ i £ p, and each map
f:(D^{i}, MD^{i})à(M,
U)
there exists a homotopy
F:(D^{i}, MD^{i})HIà(M,
U)
such that for each t0[0, 1], F_{t} is a map
of the pair (D^{i}, MD^{i})
into the pair (M, U), F_{0}=f, and F_{1}(D^{i})dU.
4.2 DEFINITION. We call a manifold M uniformly locally pconnected, pULC, if given e>0, there is a d>0 for which every map f:S^{p}àM such that the diameter of f(S^{p}) is less than d is enullhomotopic. If M is pULC for 0 £ p £ k, the we say M is ULC^{k}.
4.3 DEFINITION. Let M be a PL nmanifold, n$2. A set XdME is said to thin down in M, if for each locally finite triangulation R of M, the 2skeleton R^{2} of R tends to MX.
4.4 LEMMA. Suppose M is a PL nmanifold, n$2. Then a Set XdME tends to MX if and only if X thins down in M.
Proof. Suppose X tends to MX and R is any locally finite triangulation of M. We shall show that R^{2} tends to MX. Let e>0 be given. Then there is a compact set C such that if x is a point of XC, then DIST (x, MX)<e. Now let p be a point of R^{2}C. If p0X, we have p0XC, so DIST (p, MX)<e. If pÏX, then p0MX, so that DIST (p, MX)=0.
Suppose X thins down in M. Let e>0 be given. Let R be a locally finite triangulation of M so that MESH R<e/2. Let C be a compact set such that if p0R^{2}C, then DIST (p, MX)<e/2. Let C' be the smallest subcomplex of R containing C and let C" be the closed star of C' in R. It is clear that C" is compact. Let x be a point of XC" and let Q be the set {qq0R^{2} and DIST(x, q)<e/2}. Now Q is not empty since there is a simplex s in R with x in the interior (rel s) of s, and since the diameter of s is less than e/2, then sÇR^{2}dQ. Also, there is a point p in Q with p0R^{2}C', for if x0sE and (sÇR^{2})dC', then s must be in C", but xÏC". Now R^{2}CÉR^{2}C', so that if p0R^{2}C and DIST (p, MX)<e/2, we have DIST (x, MX)<e. €
4.5 THEOREM. Let M be a PL nmanifold, n$5, and let X be a subset of ME which tends to MX. Suppose that for each open neighborhood U of X there is an open neighborhood V of X lying in U such that V is ULC^{2} and VX is ULC^{1}. Then X is strongly quasicellular if and only if X is closed, has SUV^{4} and satisfies the strong quasicellularity criterion.
Proof. The proof of necessity is given by [5, Theorem 4.13].
Suppose then that XdME is closed, has property SUV^{4} in M, and satisfies the strong
quasicellularity criterion. Let U be any closed neighborhood of X. Let H_{i},
i=1, 2, 3, 4, and 5, be connected PL submanifolds of M so that
(1) XdHE_{i+1}dH_{i+1}dHE_{i}dH_{i}UE, i=1, 2, 3, or 4,
(2) there exists a proper PL homotopy F^{i+1}:H_{i+1}H[0, 1]àH_{i},
with F_{0}^{i+1} the inclusion, F^{i+1}(H_{i+1}H[0, 1])dHE_{i}, and F_{1}^{i+1}(H_{i+1})=T_{i+1}
a closed tree, i=1, 2, 3, or 4,
(3) MH_{i}¹f,
i=1, 2, 3, 4, or 5,
(4) if YdH_{i+1}
is a polyhedron and DIM Y£n3, then Y is
quasitrivial in H_{i}, i=1, 2, 3, or 4,
(5) each map g:S^{k}àH_{i+1}X
is nullhomotopic in HE_{i}X, i=1, 2, 3, or 4, and k=0 or
1. This last condition derives from the strong quasicellularity criterion [5,
Remark 4.12 (f)].
Let R be a locally finite triangulation of M with H_{i} i=1, 2, 3, 4, or 5, as subcomplexes and let R^{2} be the 2skeleton of R. By Lemma 4.4 we know that R^{2} tends to MX.
Let V be an open subset of HE_{5} such that XdV, V is ULC^{2}, and VX is ULC^{1}. Using the first of Rushing's infinite engulfing theorems (Theorem 5.4, below). Let e'_{t}:VH[0, 1]àV be an ambient isotopy which extends by the identity to all of M, for which e'_{1}(MX) contains all of R^{2} except some compact subset contained in e'_{1}(V)=V. Let L be a compact subcomplex of R^{2} which contains R^{2}Çe'_{1}(X) and such that R^{2}Lde'_{1}(MX).
We now wish to apply Stallings' Engulfing Theorem with (L, HE_{2}Ç(R^{2}L), e'_{1}(HE_{2}X), HE_{2}) replacing (P, Q, U, M) (see Theorem 5.3, below, for the variation of the theorem used here). We must show that the pair p=(HE_{2}, HE_{2}X)»(e'_{1}(HE_{2}), e'_{1}(HE_{2}X))=(HE_{2}, e'_{1}(HE_{2}X)) is 2connected.
Since H_{5} is arc connected, if
f:(D^{0}, MD^{0})à(HE_{2}, HE_{2}X)
is a map, with f(D^{0})dX,
then there is an arc in H_{5} from f(D^{0})
to a point in H_{5}X. Thus p is 0connected.
Now consider any map
f:(D^{1}, MD^{1})à(HE_{2}, HE_{2}X).
Suppose f(D^{1})dH_{5}.
Since f( MD^{1})
is a pair of points in H_{5}X, there is a map g:D^{1}àHE_{4}X with gMD^{1}=fMD^{1}.
Since F_{1}^{4}[f(D^{1})Èg(D^{1})]
is a continuum in T_{4}, f is homotopic (rel MD^{1})
in HE_{3} to g. In the general case,
cover f^{1}(X) with a finite number of disjoint arcs s_{1},
s_{2}, s_{3},
... s_{k} so close to f^{1}(X) that
each f(s_{i})dH_{5}.
By the special case fs_{i} is homotopic (rel Ms_{i})
in HE_{3} to a path in HE_{4}X. Putting these homotopies
together, we see that f is homotopic (rel [D^{1}È_{i=1}^{k}sE_{i}]) in HE_{2} to a path in HE_{2}X. so p
is 1connected.
Now consider any map
f:(D^{2}, MD^{2})à(HE_{2}, HE_{2}X).
Suppose f(MD^{2})dH_{4}X
and f(D^{2})dH_{3}.
Then we have a map g:D^{2}àHE_{3}X with gMD^{2}=fMD^{2}.
Since F_{1}^{3}[f(D^{2})Èg(D^{2})]
is a continuum in T_{3}, f is homotopic (rel MD^{2})
in HE_{2} to g. In the general case,
cover f^{1}(X) with the interiors of a finite number of punctured
polyhedral 2cells, t_{1},
t_{2}, t_{3},
... t_{m} which do not meet MD^{2}
and such that to f(t_{i})dH_{5},
i=1,2, ...m. Let S be a triangulation of D^{2}
so that the t_{i}s are subcomplexes of S. Let
A=È_{i=1}^{m}(1skeleton
t_{i}) and let B=D^{2}È_{i=1}^{m}tE_{i}. By the 0connectivity of H_{5},
we have a homotopy (rel M(È_{i=1}^{m}t_{i})),
(since f^{1}(X)dINT(È_{i=1}^{m}t_{i}))
of the 0skeleton of A in H_{5} to H_{5}X. We use the homotopy
extension property for polyhedral pairs [14, Corollary 5, p. 118] to obtain a
homotopy from AH[0, 1] into H_{5} with the restriction
to the 0skeleton agreeing with the original homotopy. We now use the special
case of the proof of the 1connectivity of , (since the 0skeleton of AH{1} is carried into H_{5}X), to obtain
a homotopy φ:AH[0, 1]àHE_{3} such that
φ_{0}=fA,
φ_{1}(A)dHE_{4}X and
φ_{t}AÇB=fAÇB
(AÇB=M(È_{i=1}^{m}t_{i})).
Using the homotopy extension property again, we extend
φ to Φ:(È_{i=1}^{m}t_{i})HI with Φ_{0}=fÈ_{i=1}^{m}t_{i}, Φ_{1}(A)dHE_{4}X and
Φ_{t}AÇB=fAÇB,
t0[0, 1]. We now extend Φ
to all of D^{2}HI by defining Φ_{t}B=fB,
for each t0[0, 1]. Let r
be a 2simplex of È_{i=1}^{m}t_{i},
then Φr is
a map from (r, Mr)
to (H_{3}, H_{4}X). By applying the special case, we see that Φ
is homotopic (rel AÇB) in HE_{2} to a map G:D^{2}àHE_{2}X. Thus f is homotopic (rel
B, implying rel MD^{2})
to G in HE_{2}, and p
is 2connected.
We have, from Stallings' Engulfing Theorem, an ambient isotopy e^{2}:HE_{2}H[0, 1]àH_{2} and a compact set EdHE_{2} such that Lde_{1}^{2}(e'_{1}(HE_{2}X)) and e_{t}^{2}(HE_{2}E)È(HE_{2}ÇR^{2}L)=1(HE_{2}E)È(HE_{2}ÇR^{2}L), so that we may extend e_{t}^{2} by the identity on MHE_{2}.
Now define K to be the complex H_{2}ÇR^{2}de_{1}^{2}(e'_{1}(HE_{2}X)). Let J be the complementary complex of K in H_{2} and let N* be a quasicell in HE_{1} with JdNE*. As in the proof of Theorem 3.9, let e^{3}:MH[0, 1]àM be an ambient isotopy, fixed outside of H_{1}, which engulfs e_{1}^{2}(e'_{1}(X)) with e_{1}^{3}(NE*). Define N to be [(e'_{1})^{1}B(e_{1}^{2})^{1}Be_{1}^{3}](N*). €
Prior to proving the third of the theorems about strong quasicellularity, we give the definition of a property used in the second of Rushing's infinite engulfing theorems. This property is used to reduce the ULC conditions.
4.6 DEFINITION. Let M be a PL nmanifold, U an open subset of M, and P
a PL subspace of dimension k in M. We say that most of P can be pulled
through M into U by a short homotopy H:(PA)HIàM,
where cl(A) is a compact subset of P, if
(1) H(p, 0)=p, for all p0PA,
(2) H(p, 1) is in U, for all p0PA,
and
(3) given e>0, there is a compact
set BdP such that DIAM(H(pH[0, 1]))<e, for p0PB.
4.7 THEOREM. Let M be a PL nmanifold, n$5, and let X be a subset of ME which tends to MX. Suppose that for each open neighborhood U of X there is an open neighborhood V of X lying in U such that V is ULC^{6n}, VX is ULC^{5n}, and for each locally finite triangulation R of M, most of the 2skeleton R^{2}ÇV can be pulled through V into VX by a short homotopy. Then X is strongly quasicellular if and only if X is closed, has SUV^{4} and satisfies the strong quasicellularity criterion.
Proof. Repeat the proof of Theorem 4.5, replacing the use of the first of Rushing's infinite engulfing theorems with the second (5.5, below). €
4.8 REMARK. If n$7 in Theorem 4.7 then the ULC conditions disappear. The resulting theorem is very much like Theorem 3.9. The 0ULC at X condition for M is replaced by the short homotopy condition. It is possible that the short homotopy can be obtained from SUV^{4} and strong quasicellularity in somewhat the same manner as the homotopy pulling locally finite 2complexes into MX in U is obtained in Lemma 3.8. However, some other condition, such as 0ULC at X may be required.
We now state and prove three theorems for sufficiency conditions for weak quasicellularity.
4.9 THEOREM. Let M be a PL nmanifold, n$6, and let X be a closed, connected subset of ME such that M is 0ULC at X and X tends to MX. Then X is weakly quasicellular if X satisfies the weak quasicellularity criterion.
Proof. By the definition of the weak quasicellularity criterion (2.10), there exists a quasispecial sequence {H_{i1},T_{i},F^{i}}_{i=1}^{4} for X in M having the nullhomotopy property. Then the proof of sufficiency in Theorem 3.9 gives a quasicell N with XdNEdNdHE_{0}dME.
We need only show that for each quasicell Q containing X in its interior, there is a sequence of quasicells {Q_{i}}_{i=1}^{4} lying in Q with XdQE_{i+1}dQ_{i+1}dQE_{i} and X=Ç_{i=1}^{4}Q_{i}. Since we already have the existence of the quasicell N, this will complete the proof.
Write M as the union of a countable number of compact PL nmanifolds, M_{1}dME_{2}dM_{2}dME_{3}d ... dÈ_{i=1}^{4}M_{i}=M. Let {e_{i}}_{i=1}^{4} be a decreasing sequence of positive numbers converging to zero.
There exists a quasispecial sequence for X in Q having the nullhomotopy property, so we may choose Q_{1}dQE as N was chosen for M. Inductively, suppose Q_{j} has been defined. Let {H_{i1},T_{i},F^{i}}_{i=1}^{4} be a quasispecial sequence for X in Q_{j} with the nullhomotopy property. Let H_{i} have large enough subscript so that H_{i}ÇM_{j}dN(X;e_{j}), and as before, there exists a quasicell Q_{j+1} containing X in its interior with Q_{j+1}dHE_{i}. Let {Q_{i}}_{i=1}^{4} be the sequence of quasicells so defined.
It is clear that XdQE_{i+1}dQ_{i+1}dQE_{i}, for all i. Let p be a point in MX, and let e=DIST(p, X). Let e_{i}<e and M_{j} be such that p0M_{j}. Then p is not in Q_{i+j}. Thus X=Ç_{i=1}^{4}Q_{i}, and the proof is complete. €
4.10 THEOREM. Let M be a PL nmanifold, n$5, and let X be a closed, connected subset of ME which tends to MX. Suppose that for each open neighborhood U of X there is an open neighborhood V of X lying in U such that V is ULC^{2} and VX is ULC^{1}. Then X is weakly quasicellular if X satisfies the weak quasicellularity criterion.
Proof. Let {H_{i1},T_{i},F^{i}}_{i=1}^{4} be a quasispecial sequence for X in M having the nullhomotopy property. The proof of sufficiency in Theorem 4.5 gives a quasicell N with XdNEdNdHE_{0}dME. The argument in the proof of Theorem 4.9 finishes the proof. €
4.11 THEOREM. Let M be a PL nmanifold, n$5, and let X be a closed, connected subset of ME which tends to MX. Suppose that for each open neighborhood U of X there is an open neighborhood V of X lying in U such that V is ULC^{6n}, VX is ULC^{5n}, and for each locally finite triangulation R of M, most of the 2skeleton R^{2}ÇV can be pulled through V into VX by a short homotopy. Then X is weakly quasicellular if X satisfies the weak quasicellularity criterion.
Proof. Repeat the proof of Theorem 4.10, replacing the use of the first of Rushing's infinite engulfing theorems with the second (5.5, below). €
4.12 REMARK. If n$7 in Theorem 4.11, we have a theorem without the ULC conditions, as in Remark 4.8.
It should be noted that some of our results are not as strong as they might be. For instance, the conditions X tends to MX, V is ULC^{2}, and VX is ULC^{1} in Theorems 3.9, 4.5, 4.7, 4.9, 4.10, and 4.11 are conditions on the metric of the ambient manifold M. Since quasicellularity is a topological concept, it seems natural to suspect that these are conditions which allow us to prove our results in this particular manner, not necessarily conditions required for proof of these results. Obviously, the theorems can be strengthened by weakening the hypotheses by requiring only that there be an ambient isotopy of the ambient manifold so that the image of X under the ambient isotopy has the required conditions. This is a somewhat clumsy requirement, so it is stated here as a remark, rather than being included in each theorem.
For example, consider X in Figure 1, illustrated in ú^{3}, and its analogs in ú^{n}. The bumps have constant height and their indentations have constant depth (x_{3}direction) and a constant width (x_{2}direction), but have decreasing length (x_{1}direction). X does not tend to ú^{3}X and there is no V that is 0ULC. If we alternate the bumps with bumps that have decreasing widths, there is no V that is 1ULC, with VX either 0ULC or 1ULC. Despite these problems, there is an ambient isotopy between either version of X and X' shown in Figure 2. X' does tend to ú^{3}X and for each closed neighborhood U of X, there is a closed neighborhood V of X in U such that V is ULC^{2} and VX is ULC^{0}. Clearly both X and X' are strongly quasicellular, yet only for analogs of X' may we apply any of our results to show this.
Figure 1. Strongly quasicellular, but not provable by these theorems
Figure 2. Strongly quasicellular, with ambient isotopy to Figure 1
There are other questions that could lead to further research.
For example, the number of ends of a quasicell is welldefined and equals the number of ends of a tree of which it is a regular neighborhood. Is the number of ends of a quasicell well defined? If so, is there a relation between the number of ends of the defining quasicells and the number of ends of the quasicellular set? Figures 3 and 4 give examples of strongly quasicellular sets and quasicells that might appear in sequences used to define the sets. The symbol e_{i}(Y) refers to the i^{th} end of Y, for whatever space Y is.
Figure 3. Strongly quasicellular set and containing quasicell
Figure 4. A different strongly quasicellular set and containing quasicell
A question arises concerning compactification. If a quasicellular set is compactified by adding a point at each end, this same compactification of the quasicells defining the set gives rise to a "pinched" cell definition of the set. When can these cells be "unpinched" so that the set is cellular? On the other hand, suppose a set is cellular and certain points are removed. If the resulting set and cells minus points are embedded as closed subsets of some manifold, when can quasicellularity be achieved? Notice, in Figures 5 and 6, that the cells minus points are neighborhoods of the ends, not necessarily quasicells.
Figure 5. A cellular set and containing cell
Figure 6. A cellular set minus a point and containing neighborhood
In this example, X is cellular in S^{2} and N is a cell in S^{2} containing X in its interior. Removing an endpoint of X yields X' as a closed subset of ú^{2} and N' as a neighborhood of infinity containing X' in its interior.
5 APPENDIX OF ENGULFING THEOREMS.
Our first engulfing theorem is the Infinite Radial Engulfing Theorem. Bing defines radial engulfing and proves a series of theorems [2]. He also mentions the possibility of infinite engulfing and points out techniques which might be useful [2, Modification 5, p. 7]. Our theorem is a generalization of his radial engulfing in codimension four from section 3 of [2]. We first give a modification of Definition 3.3 which increases the generality of the Infinite Radial Engulfing Theorem in some situations. Our use here of the theorem does not require this generality, however. The proof of our theorem and many ancillary lemmas and definitions are found in [6].
5.1 DEFINITION. Suppose M is a PL nmanifold, U is an open subset of M, and {A_{a}} is a collection of sets in M. We say locally finite kcomplexes in M^{n} can be pulled into U along {A_{a}} in M if, for each closed PL subspace P^{k} of M and closed set QdP^{k} such that QdU, there is a proper homotopy H:PH[0, 1]àM such that H_{0} is the inclusion, H_{1}(P)dU, H_{t} is the inclusion on Q, for each p0P, H(pH[0, 1]) lies in an element of {A_{a}}, and there is a map d:Pà(0, 4) with DIST (H_{1}(p), MU)>d(p), for each p0P.
5.2 THEOREM. Suppose U is an open subset of a PL nmanifold M, P is a closed subspace of M, Q is a closed PL subspace of P lying in U and R=cl(PQ) is rdimensional, r£n4. If {A_{a}} is a collection of subsets of M such that locally finite rcomplexes in M can be pulled into U along {A_{a}}, then for each map e:Rà(0, 4), there exists an engulfing isotopy G:MH[0, 1]àM such that G_{0}=1, G_{t}Q=1Q, RdG_{1}(U) and there is a function z:Mà(0, 4), depending on e, such that for each y0M, there exist r+1 or fewer elements of {A_{a}} such that G(yH[0, 1]) lies in the z(y)neighborhood of the union of these r+1 elements.
Our next three engulfing theorems are all to be found in Rushing [11]. They are stated for PL manifolds without boundary; however, if a PL manifold has boundary, its interior is a manifold without boundary and the conclusions of these theorems allow for the engulfing isotopies to be extended by the identity on the boundary of such a manifold.
The first of these theorems is Rushing's version of Stallings' Engulfing Theorem.
5.3 THEOREM. Let M be a PL nmanifold without boundary, U an open
subset of M, P^{k} a finite polyhedron in M of dimension k£n3
and Q^{q}dU a (possibly infinite)
polyhedron of dimension q£n3 such that (cl(Q)Q)ÇP=f.
Let (M, U) be kconnected. Then there is a compact set EdM
and an ambient isotopy e_{t} of M such that Pde_{t}(U)
and
e_{t}(ME)ÈQ=1(ME)ÈQ.
In the statements of Rushing's Infinite Engulfing Theorems, M^{n} is a connected PL nmanifold without boundary, U is an open set in M, P^{k} is an infinite polyhedron of dimension k£n3, which is contained in M (P is not necessarily closed in M) and Q^{q} is a (possibly infinite) polyhedron of dimension q£n3 such that (cl(Q)Q)ÇP=f and (cl(P)P)ÇQ=f. The symbol "4^{~}" denotes a closed subset of M(PÈQ) containing (cl(P)P)È(cl(Q)Q).
Our next theorem appears in [11] as the corollary to the Infinite Engulfing Theorem 1.
5.4 THEOREM. Suppose that M4^{~}
is ULC^{k}, U4^{~} is ULC^{k1}
and P tends to U. Then most of P^{k} can be engulfed by U staying fixed
on Q, in the following sense:
Given a compact set CdP
and e>0, there exists and ambient isotopy e_{t}
of M^{n} such that e_{t}(MN(PC; e))ÈQ=1(MN(PC; e))ÈQ
and such that e_{1}(U) contains all of P except some compact subset.
Furthermore, for each d>0, there exists a compact
subset KdM4^{~}
such that e_{t}MK is a disotopy.
The last theorem is Rushing's Infinite Engulfing Theorem 1 of [11].
5.4 THEOREM. Suppose that M4^{~} is ULC^{max(k,q)+kn+2}, U4^{~} is ULC^{max(k,q)+kn+2}. Also suppose that most of P^{k} can be pulled through M4^{~} into U4^{~} by a short homotopy. Then most of P^{k} can be engulfed by U in the same sense as in Theorem 5.4
CONTINUATION: "More about Property SUV^{4} and Strong QuasiCellularity."
REFERENCES
1. Ball, B. J. and R. B. Sher, "A Theory of Proper Shape for Locally Compact Metric Spaces," Fund. Math., 86 (1974), 164192.
2. Bing, R. H., "Radial Engulfing," in Conference on the Topology of Manifolds, Prindle, Weber, and Schmidt (Boston), (1968), 118.
3. Borsuk, K., "Fundamental Retracts and Extensions of Fundamental Sequences," Fund. Math., 64 (1969), 5585.
4. Dugundji, J., Topology, Allyn and Bacon, Inc., (Boston), 1966.
5. Hartley, D. S., III, "Fundamentals of QuasiCellularity," unpublished.
6. _____________, QuasiCellularity in Manifolds, Dissertation, University of Georgia, Athens, GA, 1973 (Ref # A 515428, Dec 27 1973, University Microfilms, 300 North Zeeb Rd, Ann Arbor, MI 48106).
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8. McMillan, D. R., Jr., "A Criterion for Cellularity in a Manifold," Ann. of Math. (2) 79(1964), 327337.
9. ___________, "UV Properties and Related Topics," Mimeographed notes.
10. Mardesic, S., "Retracts in Shape Theory," Glasnik Mat. Ser. III 6 (26), (1971), 153163.
11. Rushing, T. B., "Infinite Engulfing," preprint. Possibly published as "A summation of results of infinite engulfing," Proceedings of the University of Oklahoma Topology Conference Dedicated to Robert Lee Moore (Norman, Okla., 1972), Univ. of Oklahoma, Norman, Oklahoma, 1972, pp. 284–293.
12. Scott, A., "Infinite Regular Neighborhoods," J. London Math. Soc. 42 (1967), 245253.
13. Sher, R. B., "Property SUV^{4} and Proper Shape Theory," Trans. Amer. Math. Soc. 190 (1974), 345356.
14. Spanier, E. H., Algebraic Topology, McGrawHill Book Company, (San Francisco), 1966.
15. Stallings, J., "The PiecewiseLinear Structure of Euclidean Space," Proc. Cambridge Philos. Soc. 58 (1962), 481488.
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