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Abstract. Heretofore the use of property SUV^{4} and strong quasicellularity has been restricted to closed subsets of manifolds. These concepts are extended to include nonclosed subsets for the purpose of examining the problems: When is the compactification of an SUV^{4} set UV^{4}? When is the compactification of a strongly quasicellular set cellular? What sets may be removed from a UV^{4} set to obtain an SUV^{4} set? What sets may be removed from a cellular set to obtain a strongly quasicellular set? Partial answers are given to these questions.
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, quasicell, quasicellular, strongly quasicellular, SUV^{4}, tree, UV^{4}.
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 concepts of SUV^{4} and strong quasicellularity were introduced in [3] as extensions of UV^{4} and cellularity to the noncompact case. Further connections among these concepts were also given in [3]. The dissertation [3] was divided into two papers [4 and 5], for publication. In both of these papers, strong quasicellularity was defined only for closed subsets of manifolds; and while the definition of property SUV^{4} did not explicitly require that the set in question be closed, in the case that the ambient topological space is a manifold, it was tacitly assumed that the set would be closed. In this paper, we shall examine subsets of manifolds which are not necessarily closed and identify some conditions under which these sets satisfy extended definitions of property SUV^{4} and strong quasicellularity. We use some algebraic tools developed by E. M. Brown [2] and R. B. Sher [9], which are still in genesis. As a result, our work is restricted to simplicial complexes in many cases. We also require that all spaces be locally compact metrizable spaces.
2 UV^{4} AND SUV^{4}.
2.1 DEFINITION. Suppose Y is a topological space and XdY. Then X is said to have property UV^{4} for each open neighborhood U of X in Y, there is an open neighborhood V of X in U such that V is contractible in U.
We have defined property SUV^{4} in [4] for general topological spaces; however, we shall use a more convenient definition, which is equivalent when the ambient space is a manifold.
2.2 DEFINITION. A tree is a connected, simply connected, locallyfinite lcomplex.
2.3 DEFINITION. A map f:XàY is proper if f^{ l}(C) is compact whenever C is compact.
2.4 DEFINITION. Suppose M is a PL nmanifold n$3, and X is a closed subset of M lying in ME. Then X is said to have property SUV^{4} in M if for each closed neighborhood U of X in M, there is a PL submanifold V, with nonempty boundary, such that XdVEdVdUE, a tree T embedded as a closed PL subset of M lying in UE, and a proper PL homotopy H:VHIàU with H_{0} the inclusion, H_{l }(V)=T and H(VHI)dUE.
It has been shown [8] that property UV^{4} is a property that is topologically invariant with respect to embeddings in metrizable ANRs. It has also been shown [4] that property SUV^{4} is a property which is topologically invariant with respect to embeddings (as a closed subset) in locally compact metric ANRs. The connections between UV^{4} and SUV^{4} are the following: If X0SUV^{4}, then X0UV^{4}. If X is compact, then X0UV^{4} if and only if X0SUV^{4}. There do exist noncompact UV^{4} spaces that are not SUV^{4} (ú^{2} for instance).
From our definition of SUV^{4}, it is clear that even if a set X has property SUV^{4} when embedded as a closed subset of a manifold, if it is embedded as a nonclosed subset of a manifold, there need not be a submanifold V for each closed neighborhood U such that XdVEdVdUE (cf. [0, 4)dú^{1}dú^{n} and [0, 1)dú^{1}dú^{n}). However, the following theorem [9, Theorem 4.3] identifies (in terms of homotopy groups and Brown's proper homotopy groups [2]) those connected locally finite simplicial complexes that may have property SUV^{4}. For these spaces, we take the following theorem to be the definition of SUV^{4} for nonclosed sets in manifolds.
2.5 THEOREM. Suppose K is a connected locallyfinite simplicial complex. Then K0SUV^{4} if and only if
(i) for each n0J^{+}, π_{n}(K)=0, and
(ii) for each n0J^{+}c{4} and end [a] of K, π_{n}(K,a)=0.
2.6 COROLLARY. Suppose L is a compact, connected, locallyfinite simplicial complex. Then L0UV^{4}, if and only if for each n0J^{+}, π_{n}(L)=0.
We now define the property CC^{n}. This property has been previously defined by Lacher [6] under the name SUV^{n}, which we obviously can't use. We use the "CC" since lCC is McMillan's cellularity criterion [7], which he names property CC [8].
2.7 DEFINITION. Suppose Y is a topological space and XdY. If for each open neighborhood U of X in Y, there is an open neighborhood V of X in U such that any map f:S^{j}àVX can be extended to a map g:B^{J+1}àUX, then we say X has property jCC in Y and write (XdY)0CC^{n}. If X has property jCC in Y for each 0#j#n, then we say X has property CC^{n} in Y and write (XdY)0CC^{n}. If X has property jCC in Y for each nonnegative integer j, then we say X has property CC^{ω} in Y. If for each open neighborhood U of X in Y, there is an open neighborhood V of X in U such that VX is contractible in UX, then we say X has property CC^{4} in Y.
2.8 THEOREM. If (XdY)0CC^{4}, then (XdY)0CC^{ω}.
We are now in a position to show that certain noncompact subsets of certain compact UV^{4} sets are SUV^{4} sets.
2.9 THEOREM. Suppose L is a compact, connected, locallyfinite simplicial complex and L0UV^{4}. If C is a compact connected PL subspace of L, K=LC is connected, C0UV^{4}, and (CdL)0CC^{4}, then K0SUV^{4}.
Proof. We first show that π_{n}(K)=0, for each N0J^{+}. Suppose f:S^{n}àK is a map. Then f(S^{n})dL and since π_{n}(L)=0, then f extends to g:B^{n+1}àL. We may assume that g is a PL map. We use C0UV^{4} and (CdL)0CC^{4} to get nconnectivity by a standard argument [7, 5] and thus homotop g (rel MB^{n+1}) off of C. Hence π_{n}(K)=0.
We now consider π_{n}(K,a), for each n0J^{+}c{4} and each end [a] of K. Suppose [a] is an end of K, and α:S_{n}àK is a representative of an element of π_{n}(K,a), n0J^{+}c{4}. Now let U_{1}=L and V_{1}dU_{1} be chosen using (CdL)0CC^{4} . We write K as an expanding union of compact subcomplexes of a locally finite triangulation of K, K=c_{i=1}^{4}_{ }Q_{i}, Q_{i}dQE_{i+1}. Given that j0J^{+} and that V_{j} has been defined, let Q_{k} be such that KV_{j}dQE_{k}. Let U_{j+1}=LQ_{k}, and choose V_{j+1} using (CdL)0CC^{4}. Now for any j0J^{+} only a finite number of images of spheres in S^{n} are to be found not entirely inside V_{j}. If we define a homotopy from α to αN, where αN is a map from ([0,4) union a finite number of spheres) to K, by homotoping each sphere in V_{j}C (j the largest possible subscript) to a point in U_{j}C, then we find this homotopy is proper. Examine the image of the homotopy of any sphere: it is outside some U_{j}, so that all the spheres inside V_{j} have homotopies missing the homotopy of the given sphere. Thus only finitely many hit the given sphere's homotopy. Since π_{n}(K)=0 we may properly homotop αN to a map αNN: [0,4)àK. Since both of these homotopies may be chosen so as to agree on [0,4), α is 0 in π_{n}(K,a) and so π_{n}(K,a)=0. Hence K0SUV^{4}. €
2.10 COROLLARY. Suppose L is a compact, connected, locallyfinite simplicial complex and L0UV^{4}. If C is a compact PL subspace of L, K=LC is connected, and each component of C has UV^{4} and CC^{4} in L, then K0SUV^{4}.
The techniques of the proof of Theorem 2.9 can also be used for situations in which LóUV^{4}. For instance, we delete the requirement for (CdL)00CC, and require B_{n}(LC)=0 and we can handle such situations as S^{1} minus a point, as in Figure 1.
Figure 1. S^{1} minus a point.
We now consider the problem in reverse: Suppose K0SUV^{4}. If L is a compactification of K, when is L0UV^{4}? It is clear that if K has more than one end and L is the onepoint compactification of K, then L is not simply connected and thus LóUV^{4}. For this reason, we turn to the Freudenthal end point compactification, FK.
2.11 THEOREM. Suppose K0SUV^{4}, then L=FK0UV^{4}.
Proof. By [9, Theorem 3.1] there is a tree T such that Sh_{p}K=Sh_{p}T. Then ShFK=ShFT and, since FT is a compact metric absolute retract, FT0UV^{4}. Hence FK0UV^{4}. €
3 CELLULARITY AND STRONG QUASICELLULARITY.
3.1 DEFINITION. A closed set X lying in the interior of an nmanifold M is said to be cellular in M if there exists a sequence {Ci}_{i=1}^{4}_{ }of ncells such that MeC_{1}eCE_{1}eC_{2}eCE_{2}e...e1 _{i=1}^{4}_{ }C_{i}=X.
3.2 DEFINITION. Suppose N is an nmanifold. If there is a PL nmanifold N^{1}dE^{n} which is a regular neighborhood of some tree embedded as a closed PL subset of E^{n}, and if there is a homeomorphism h:N^{1}>>N then N is called an nquasicell. If the dimension is obvious, or not relevant, N will be referred to simply as a quasicell.
3.3 DEFINITION. A closed set X lying in the interior of an nmanifold M is said to be strongly quasicellular in M if for each closed neighborhood U of X, there is an nquasicell N such that XdNEdNdUE.
We now wish to extend the definition of strong quasicellularity. As with SUV^{4} we wish to consider the possibility of X not being a closed subset of the ambient manifold. We also note that when we are dealing with strong quasicellularity in the PL category, we mean for the quasicells to be PL subspaces.
3.4 DEFINITION. We say that a set X lying in the interior of an nmanifold M is strongly quasicellular in M if there is a closed set C lying in the interior of M such that C1X=N, CeCl(X)X, MC is an nmanifold and X is strongly quasicellular in MC in the sense of Definition 3.3.
We now are in a position to consider when a compactification of a strongly quasicellular set is strongly quasicellular and what sets may be removed from a cellular set to yield a strongly quasicellular set. In the following theorem, we give a partial answer to the second question.
3.5 THEOREM. Suppose X is a locallyfinite simplicial complex embedded as a cellular subset of the PL nmanifold M, n$6. Suppose that C is a closed PL subspace of X, Y=XC is connected, each component of C has UV^{4} and CC^{4} in X, and MC is a PL nmanifold. If MC is 0ULC at Y, Y tends to MX and for each closed neighborhood U of Y, there is a closed polyhedral neighborhood V of Y lying in UE such that
(i) the inclusion inducedhomeomorphism i_{*}: π_{1}(VX,v)àπ_{1}(UX,v) is trivial for all v0VX, and
(ii) the inclusioninduced homeomorphism π_{1}(i): π_{1}(VX,a)àπ_{1}(UX,a) is trivial for all proper maps a:[0,4)àVC with a([0,4))dVX, then Y is strongly quasicellular in M.
Proof. Since X is cellular, X0UV^{4}. By Theorem 2.10, Y0SUV^{4}. By [4, Remark 4.21], Y satisfies the strong quasicellularity criterion in MC. Thus Theorem 3.9 of [5] gives us that Y is strongly quasicellular in MC. This satisfies our new definition and Y is strongly quasicellular in M. €
The following theorem gives a partial answer to the first question. We use the notation FY for the Freudenthal compactification of a space Y, and EY for the set of ends of Y.
3.6 THEOREM. Suppose X is a closed subset of the PL nmanifold M^{n}, n$5, such that XdM and X is strongly quasicellular in M. Suppose FM is a PL nmanifold, each end of M contains no more than one end of X, and EX1E(MM)=N. Then FX is cellular in FM.
Proof. Since each end of M contains no more than one end of X, and since EX1E(MM)=N, then the construction of FM yields FX as a subset of FM, lying in Int(FM). By [4, Theorem 4.13], X has SUV^{4} in M. By [9, Theorem 3.1], X has the proper shape of a tree. Hence FX has the shape of a point and has UV^{4} in FM.
Suppose U is an open neighborhood of FX in FM. Let U be an open neighborhood of FX in UN that is homotopic to a point in U through a homotopy H. Let N be a quasicell neighborhood of X in M1UN. Let EM and EX be the ends of M and X respectively. Since X is closed in M, EXdEM. For each end e in EX, let V_{e} be a closed polyhedral neighborhood of e in UN such that for e…eN, V_{e}1V_{e}_{N}=i. Let l:S_{1}àInt(Nc(c_{e}V_{e})) be a map. Let NN and VN_{e} be chosen in a similar manner to N and V_{e} so that FXdInt(NNc(c_{e}VN_{e})), l(S^{1})1(NNc(c_{e}VN_{e}))=i, and so that H(l(S^{1})HI)1(c_{e}VN_{e})=i. (To achieve this last condition, we must have H(l(S^{1})HI)1EX=i. Since EX is a discrete set of points in FM, this may be obtained using general position.) By [4, Lemma 4.8], H may be modified to miss X and thus FX. We now have that FX satisfies the cellularity criterion in FM. Hence, by [7, Theorem 1], FX is cellular in FM. €
We now increase the generality of Theorem 3.6 to include the case for the extended definition of strong quasicellularity.
3.7 THEOREM. Suppose X is a subset of the PL nmanifold M^{n}, n$5, such that XdME and X is strongly quasicellular in M. If FM is a PL nmanifold, if the closed set C in Definition 3.4 can be given C=c_{i=1}^{k}Q_{i}, where each Q_{i} is a flat q_{i}cell, 0#q_{i}#n, Q_{i}1Q_{j}=N, for i… j, if each end of M contains no more than one end of XcC, and if E(XcC)1E(MM)=N, then F(XcC) is cellular in FM.
Proof. By the nature of C, there is a homotopy H: MHIàM such that each H_{t} is a homeomorphism off of C, H_{0} is the identity and H_{1}(Q_{i}) is a point, for each 1#i#k. Thus H_{1}(C) is a discrete collection of points. Since applying the Freudenthal compactification to H_{1}(MC) with regard to only those ends produced by removing H_{1}(C) from H_{1}(M) yields H_{1}(M) again, then by Theorem 3.6, F(H_{1}(XcC)) is cellular in F(H_{1}(M)). From [1, Section 4], we have that each H_{t}:MàM extends uniquely to a map G_{t}:(FM,EM)à(F(H_{t}(M)),E(H_{t}(M))) with (F(XcC),E(XcC))à(F(H_{t}(XcC)),E(H_{t}(XcC))). Since EM=E(H_{t}(M)) and E(XcC)=E(H_{t}(XcC)), this family of maps forms a homotopy constant on the ends. Since G_{t} is a homeomorphism off of F(XcC), by the nature of the cellularity criterion and property UV^{4}, then F(H_{1}(XcC)) having UV^{4} and satisfying the cellularity criterion implies that F(XcC) has UV^{4} and satisfies the cellularity criterion. Again by [7, Theorem 1], F(XcC) is cellular in FM. €
REFERENCES
1. Ball, B. J. and R. B. Sher, "A Theory of Proper Shape for Locally Compact Metric Spaces," Fund. Math., to appear.
2. Brown, E. M., "Proper Homotopy Theory in Simplicial Complexes," preprint. Possibly published as "On the Proper Homotopy Type of Simplicial Complexes," in Topology Conference, LNmaths 375 (1974) Springer, pp 4146.
3. Hartley, D. S., III, 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).
4. Hartley, D. S., III, "Fundamentals of QuasiCellularity," unpublished.
5. Hartley, D. S., III, "QuasiCellularity Criteria," unpublished.
6. Lacher, R. C., "CellLike Mappings," I, Pacific J. Math. 30(1969), 717713.
7. McMillan, D. R., Jr., "A Criterion for Cellularity in a Manifold," Ann. of Math. (2)79(1964), 327337.
8. McMillan, D. R., Jr., "UV^{4} Properties and Related Topics," Mimeographed notes.
9. Sher, R. B., "Property SUV^{4} and Proper Shape Theory," Trans. Amer. Math. Soc. 190 (1974), 345356.
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