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HARTLEY CONSULTING
Solving
Complex Operational and Organizational Problems

PUBS: More about Property SUV4 and Strong Quasi-Cellularity

Dean S. Hartley III

 

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Project Metadata Keywords
Label Name Other Year DurationYrs
Client University of Georgia Math Deparment none NonProfit
Dates 1974 0.5
Employer University of Georgia
Partner N/A
Pubs "More about Property SUV-infinity and Strong Quasi-Cellularity" author 1974
Science, Math and Medicine
Topology

Abstract. Heretofore the use of property SUV4 and strong quasi-cellularity has been restricted to closed subsets of manifolds. These concepts are extended to include non-closed subsets for the purpose of examining the problems: When is the compactification of an SUV4 set UV4? When is the compactification of a strongly quasi-cellular set cellular? What sets may be removed from a UV4 set to obtain an SUV4 set? What sets may be removed from a cellular set to obtain a strongly quasi-cellular 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, quasi-cell, quasi-cellular, strongly quasi-cellular, SUV4, tree, UV4.


NOTATION

UE and Int(U) symbolize the topological interior of U.
MEi+1 symbolizes the topological interior of Mi+1 (the apparent sequence of the superscript and subscripts is purely an artifact of the HTML).
( i)i=14 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.
1 symbolizes the real number line.
Sn symbolizes the Euclidean n-sphere.
Mn and Xx symbolize n-dimensional and x-dimensional objects, respectively.
Fi and Gi, where F and G are maps, symbolize indexed maps, not n-dimensional objects.
F0 and F1, where F is a homotopy, F:XHIY, 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 SUV4 and strong quasi-cellularity were introduced in [3] as extensions of UV4 and cellularity to the non-compact 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 quasi-cellularity was defined only for closed subsets of manifolds; and while the definition of property SUV4 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 SUV4 and strong quasi-cellularity. 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 UV4 AND SUV4.

2.1 DEFINITION. Suppose Y is a topological space and XdY. Then X is said to have property UV4 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 SUV4 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, locally-finite l-complex.

2.3 DEFINITION. A map f:XY is proper if f -l(C) is compact whenever C is compact.

2.4 DEFINITION. Suppose M is a PL n-manifold n$3, and X is a closed subset of M lying in ME. Then X is said to have property SUV4 in M if for each closed neighborhood U of X in M, there is a PL submanifold V, with non-empty boundary, such that XdVEdVdUE, a tree T embedded as a closed PL subset of M lying in UE, and a proper PL homotopy H:VHIU with H0 the inclusion, Hl (V)=T and H(VHI)dUE.

It has been shown [8] that property UV4 is a property that is topologically invariant with respect to embeddings in metrizable ANRs. It has also been shown [4] that property SUV4 is a property which is topologically invariant with respect to embeddings (as a closed subset) in locally compact metric ANRs. The connections between UV4 and SUV4 are the following: If X0SUV4, then X0UV4. If X is compact, then X0UV4 if and only if X0SUV4. There do exist non-compact UV4 spaces that are not SUV4 (2 for instance).

From our definition of SUV4, it is clear that even if a set X has property SUV4 when embedded as a closed subset of a manifold, if it is embedded as a non-closed subset of a manifold, there need not be a submanifold V for each closed neighborhood U such that XdVEdVdUE (cf. [0, 4)d1dn and [0, 1)d1dn). 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 SUV4. For these spaces, we take the following theorem to be the definition of SUV4 for non-closed sets in manifolds.

2.5 THEOREM. Suppose K is a connected locally-finite simplicial complex. Then K0SUV4 if and only if

(i) for each n0J+, πn(K)=0, and

(ii) for each n0J+c{4} and end [a] of K, πn(K,a)=0.

2.6 COROLLARY. Suppose L is a compact, connected, locally-finite simplicial complex. Then L0UV4, if and only if for each n0J+, πn(L)=0.

We now define the property CCn. This property has been previously defined by Lacher [6] under the name SUVn, which we obviously can't use. We use the "CC" since l-CC 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:SjV-X can be extended to a map g:BJ+1U-X, then we say X has property j-CC in Y and write (XdY)0CCn. If X has property j-CC in Y for each 0#j#n, then we say X has property CCn in Y and write (XdY)0CCn. If X has property j-CC in Y for each non-negative integer j, then we say X has property CCω in Y. If for each open neighborhood U of X in Y, there is an open neighborhood V of X in U such that V-X is contractible in U-X, then we say X has property CC4 in Y.

2.8 THEOREM. If (XdY)0CC4, then (XdY)0CCω.

We are now in a position to show that certain non-compact subsets of certain compact UV4 sets are SUV4 sets.

2.9 THEOREM. Suppose L is a compact, connected, locally-finite simplicial complex and L0UV4. If C is a compact connected PL subspace of L, K=L-C is connected, C0UV4, and (CdL)0CC4, then K0SUV4.

Proof. We first show that πn(K)=0, for each N0J+. Suppose f:SnK is a map. Then f(Sn)dL and since πn(L)=0, then f extends to g:Bn+1L. We may assume that g is a PL map. We use C0UV4 and (CdL)0CC4 to get n-connectivity by a standard argument [7, 5] and thus homotop g (rel MBn+1) off of C. Hence πn(K)=0.

We now consider πn(K,a), for each n0J+c{4} and each end [a] of K. Suppose [a] is an end of K, and α:SnK is a representative of an element of πn(K,a), n0J+c{4}. Now let U1=L and V1dU1 be chosen using (CdL)0CC4 . We write K as an expanding union of compact subcomplexes of a locally finite triangulation of K, K=ci=14 Qi, QidQEi+1. Given that j0J+ and that Vj has been defined, let Qk be such that K-VjdQEk. Let Uj+1=L-Qk, and choose Vj+1 using (CdL)0CC4. Now for any j0J+ only a finite number of images of spheres in Sn are to be found not entirely inside Vj. 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 Vj-C (j the largest possible subscript) to a point in Uj-C, then we find this homotopy is proper. Examine the image of the homotopy of any sphere: it is outside some Uj, so that all the spheres inside Vj have homotopies missing the homotopy of the given sphere. Thus only finitely many hit the given sphere's homotopy. Since πn(K)=0 we may properly homotop αN to a map αNN: [0,4)K. Since both of these homotopies may be chosen so as to agree on [0,4), α is 0 in πn(K,a) and so πn(K,a)=0. Hence K0SUV4.

2.10 COROLLARY. Suppose L is a compact, connected, locally-finite simplicial complex and L0UV4. If C is a compact PL subspace of L, K=L-C is connected, and each component of C has UV4 and CC4 in L, then K0SUV4.

The techniques of the proof of Theorem 2.9 can also be used for situations in which LUV4. For instance, we delete the requirement for (CdL)00-CC, and require Bn(L-C)=0 and we can handle such situations as S1 minus a point, as in Figure 1.

Figure 1. S1 minus a point.

We now consider the problem in reverse: Suppose K0SUV4. If L is a compactification of K, when is L0UV4? It is clear that if K has more than one end and L is the one-point compactification of K, then L is not simply connected and thus LUV4. For this reason, we turn to the Freudenthal end point compactification, FK.

2.11 THEOREM. Suppose K0SUV4, then L=FK0UV4.

Proof. By [9, Theorem 3.1] there is a tree T such that ShpK=ShpT. Then ShFK=ShFT and, since FT is a compact metric absolute retract, FT0UV4. Hence FK0UV4.


3 CELLULARITY AND STRONG QUASI-CELLULARITY.

3.1 DEFINITION. A closed set X lying in the interior of an n-manifold M is said to be cellular in M if there exists a sequence {Ci}i=14 of n-cells such that MeC1eCE1eC2eCE2e...e1 i=14 Ci=X.

3.2 DEFINITION. Suppose N is an n-manifold. If there is a PL n-manifold N1dEn which is a regular neighborhood of some tree embedded as a closed PL subset of En, and if there is a homeomorphism h:N1-->>N then N is called an n-quasi-cell. If the dimension is obvious, or not relevant, N will be referred to simply as a quasi-cell.

 3.3 DEFINITION. A closed set X lying in the interior of an n-manifold M is said to be strongly quasi-cellular in M if for each closed neighborhood U of X, there is an n-quasi-cell N such that XdNEdNdUE.

We now wish to extend the definition of strong quasi-cellularity. As with SUV4 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 quasi-cellularity in the PL category, we mean for the quasi-cells to be PL subspaces.

3.4 DEFINITION. We say that a set X lying in the interior of an n-manifold M is strongly quasi-cellular in M if there is a closed set C lying in the interior of M such that C1X=N, CeCl(X)-X, M-C is an n-manifold and X is strongly quasi-cellular in M-C in the sense of Definition 3.3.

We now are in a position to consider when a compactification of a strongly quasi-cellular set is strongly quasi-cellular and what sets may be removed from a cellular set to yield a strongly quasi-cellular set. In the following theorem, we give a partial answer to the second question.

3.5 THEOREM. Suppose X is a locally-finite simplicial complex embedded as a cellular subset of the PL n-manifold M, n$6. Suppose that C is a closed PL subspace of X, Y=X-C is connected, each component of C has UV4 and CC4 in X, and M-C is a PL n-manifold. If M-C is 0-ULC at Y, Y tends to M-X 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 induced-homeomorphism i*: π1(V-X,v)π1(U-X,v) is trivial for all v0V-X, and

(ii) the inclusion-induced homeomorphism π1(i): π1(V-X,a)π1(U-X,a) is trivial for all proper maps a:[0,4)V-C with a([0,4))dV-X, then Y is strongly quasi-cellular in M.

Proof. Since X is cellular, X0UV4. By Theorem 2.10, Y0SUV4. By [4, Remark 4.21], Y satisfies the strong quasi-cellularity criterion in M-C. Thus Theorem 3.9 of [5] gives us that Y is strongly quasi-cellular in M-C. This satisfies our new definition and Y is strongly quasi-cellular 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 n-manifold Mn, n$5, such that XdM and X is strongly quasi-cellular in M. Suppose FM is a PL n-manifold, 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 SUV4 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 UV4 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 quasi-cell 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 Ve be a closed polyhedral neighborhood of e in UN such that for eeN, Ve1VeN=i. Let l:S1Int(Nc(ceVe)) be a map. Let NN and VNe be chosen in a similar manner to N and Ve so that FXdInt(NNc(ceVNe)), l(S1)1(NNc(ceVNe))=i, and so that H(l(S1)HI)1(ceVNe)=i. (To achieve this last condition, we must have H(l(S1)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 quasi-cellularity.

3.7 THEOREM. Suppose X is a subset of the PL n-manifold Mn, n$5, such that XdME and X is strongly quasi-cellular in M. If FM is a PL n-manifold, if the closed set C in Definition 3.4 can be given C=ci=1kQi, where each Qi is a flat qi-cell, 0#qi#n, Qi1Qj=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: MHIM such that each Ht is a homeomorphism off of C, H0 is the identity and H1(Qi) is a point, for each 1#i#k. Thus H1(C) is a discrete collection of points. Since applying the Freudenthal compactification to H1(M-C) with regard to only those ends produced by removing H1(C) from H1(M) yields H1(M) again, then by Theorem 3.6, F(H1(XcC)) is cellular in F(H1(M)). From [1, Section 4], we have that each Ht:MM extends uniquely to a map Gt:(FM,EM)(F(Ht(M)),E(Ht(M))) with (F(XcC),E(XcC))(F(Ht(XcC)),E(Ht(XcC))). Since EM=E(Ht(M)) and E(XcC)=E(Ht(XcC)), this family of maps forms a homotopy constant on the ends. Since Gt is a homeomorphism off of F(XcC), by the nature of the cellularity criterion and property UV4, then F(H1(XcC)) having UV4 and satisfying the cellularity criterion implies that F(XcC) has UV4 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 41-46.

3. Hartley, D. S., III, Quasi-Cellularity 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 Quasi-Cellularity," unpublished.

5. Hartley, D. S., III, "Quasi-Cellularity Criteria," unpublished.

6. Lacher, R. C., "Cell-Like Mappings," I, Pacific J. Math. 30(1969), 717-713.

7. McMillan, D. R., Jr., "A Criterion for Cellularity in a Manifold," Ann. of Math. (2)79(1964), 327-337.

8. McMillan, D. R., Jr., "UV4 Properties and Related Topics," Mimeographed notes.

9. Sher, R. B., "Property SUV4 and Proper Shape Theory," Trans. Amer. Math. Soc. 190 (1974), 345-356.


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