Page Last Updated: Tuesday, 19 January 2016 11:02 EDT, (c) 1973, 2007, 2008, 2016

## PUBS: Quasi-Cellularity Criteria

### Dean S. Hartley III

Note: If SUV-infinity appears as SUV4, you may not have the font "WP MathA" installed. You may download the WP MathA and WP MathA Extended fonts with control+click on the following links: WPHV06NA.TTF and  WPHV07NA.TTF. (Or you may find them at http://instruct1.cit.cornell.edu/courses/fontfix.htm.) Save the files to your Fonts folder (probably C:\Windows\Fonts). If you cannot save them there, save them somewhere else. Then right click on the file and choose "install." You may have to reboot for these to be applied by Windows.

 Label Name Other Year DurationYrs Client University of Georgia Math Deparment none NonProfit Dates 1973 0.5 Employer University of Georgia Partner N/A Pubs "Quasi-Cellularity Criteria" author 1974
 Science, Math and Medicine Topology

Abstract. It is demonstrated that a set embedded in a manifold (with certain technical conditions) is strong quasi-cellularity if and only if it has the SUV4 property and satisfies a strong quasi-cellularity criterion. Moreover, a locally compact, finite-dimensional metric space has the SUV4 property with respect to metric ANRs if and only if it embeds as a strongly quasi-cellular subset of some (high dimensional) manifold. In like manner, a set embedded in an n-manifold (with technical conditions) is weakly quasi-cellular if it satisfies a weak quasi-cellularity 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 Sk-inessential, quasi-cell, quasi-cellular, quasi-cellularity criterion, quasi-trivial, strong quasi-cellularity criterion, strongly quasi-cellular, SUV4, tree, UV4, weak quasi-cellularity criterion, weakly quasi-cellular.

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.
ú1 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: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 quasi-cellularity and for weak quasi-cellularity. In fact, our extension of UV4 to SUV4 shows an even closer analogy to the compact case. McMillan remarks [9, p 21] that, for finite dimensional continua X, X0UV4 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, X0SUV4 if and only if X has a strongly quasi-cellular 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, X0SUV4 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, locally-finite l-complex.

2.3 DEFINITION. Suppose N is an n-manifold. If there is a PL n-manifold N'dEn 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 n-quasi-cell. If the dimension is obvious, or not relevant, N will be referred to simply as a quasi-cell.

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 X-inessential 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 X-inessential in Z missing A.

2.5.1 DEFINITION. A set X contained in the interior of an n-manifold is said to be quasi-trivial in M if X is contained in a quasi-cell in ME.

2.5 DEFINITION. Suppose M is a PL n-manifold, XdME and U is a neighborhood of X in M. A sequence of triples {Hi-1,Ti,Fi}i=14 is said to be a quasi-special sequence for X in U provided:

(i)    Hi+1 and Ti+1 are closed PL subspaces of M lying in HEi, i=0, 1, 2, ... ,

(ii)   Hi-1 is a PL n-manifold with MHi-1¹f, and Ti is a tree, i=1, 2, ... ,

(iii)  Fi+1:Hi+1HIàM is a proper PL homotopy with Fi+1(Hi+1HI)dHEi, F0i+1 the inclusion on Hi+1, and F1i+1(Hi+1)=Ti+1, i=0, 1, 2, ... ,

(iv)  Hi is a closed neighborhood of X, i=1, 2, ... , and X=Çi=14Hi, and

(v)   if Y is a closed PL subspace of Hi+1 and DIM Y#n-3, then Y is quasi-trivial in Hi, i=1, 2, ... .

If for each closed neighborhood U of X, there is a quasi-special sequence for X in U, then we say that quasi-special sequences for X in M exist strongly. If there is a quasi-special sequence for X in M and if each quasi-cell N with XdNE there is a quasi-special sequence for X in N, then we say that quasi-special sequences for X in M exist weakly. We say that {Hi-1,Ti,Fi}i=14 has the null-homotopy property if for each i=0, 1, 2, ... and each proper map g:SkHJ+àHi+1-X which is properly Sk-inessential in Hi+1, g is properly Sk-inessential in Hi 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 H0 is the inclusion and Hl =gBf, the we say that X has the strong UV4 property in M; this being the case, we say that X has property SUV4 in M.

The following theorem [5, Theorem 2.10] gives an equivalent definition for the case that M is a PL n-manifold, n\$3.

2.7 THEOREM. Suppose M is a PL n-manifold, n\$3, and X is a closed subset of M lying in ME. Then X has SUV4 in M if and only if for each closed neighborhood U of X, there is a PL submanifold V, with a non-empty 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 H0 the inclusion, H1(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 n-manifold, n\$3. Then a closed subset X of M lying in ME has SUV4 if and only if quasi-special 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 quasi-cellularity 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:SkHJ+àV-X which is properly Sk-inessential in V is properly Sk-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 quasi-special sequence for X in M with the null-homotopy property, and for each quasi-cell Q with X in its interior there is such a quasi-special sequence in Q, then we say that X satisfies the weak quasi-cellularity criterion.

3 STRONG QUASI-CELLULARITY AND SUV4.

3.1 LEMMA. Suppose X is a closed subset of a PL n-manifold M, X lies in ME and DIM X#n-1. 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 G0=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 {ei}i=14 be a sequence of positive numbers converging to zero. For each i0J+, let NidVE be a PL n-ball whose interior contains g(i) and whose diameter is less than ei. Since DIM X#n-1, NEiÇ(V-X)¹f, for all i0J+. Since NEi is arc-connected, there exists a path ai:[0,1]àNEi such that ai(0)=g(i) and ai(1)0V-X. Then, since g is proper and {ei}i=14®0, the map G:J+HIàV defined by G(i,t)=ai(t) is proper. Since G0=g and G(i,1)=ai(1)0V-X, 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 F0 the inclusion and F1(V)=T. Then, if P is a compact, connected space and g:PHJ+àV is a proper map, g is properly P-inessential in U.

Proof. We shall define a proper map H:(PHJ+)HIàV such that H0=g and, for each j0J+, H1|PH{j} is a constant map. We first define G on (PHJ+)H[0, 1/2] by Gt=F2tBg, for t0[0, 1/2]. Since P is compact and connected, G1/2(PH{j}) is a compact subtree of T, for each j0J+, and as such is contractible. Thus we have that the map G1/2|PH{j} is homotopic to a constant map in G1/2(PH{j}). We shall call this homotopy rj, with r0j=G1/2|PH{j}. Let R:(PHJ+)H[1/2, 1]àT be the map defined by Rt|PH{j}=r2t-1j, for t0[1/2, 1]. We note that for each j0J+, the set {k|G1/2(PH{j})ÇG1/2(PH{k})¹f} is finite. It follows from this that R is proper. Since R1/2=G1/2, we may define H:(PHJ+)HIàU by Ht=Gt, t0[0, 1/2] and Ht=Gt, t0[1/2, 1]. Since H is proper, H0=g, and H1|PH{j} is a constant map, for each j0J+, then we are through.

3.3 DEFINITION. Suppose M is a PL n-manifold, V is a given open set in M, and U is a given closed set in M containing M-V in its interior. Let P denote an arbitrary closed k-dimensional PL subspace of M, and Q a closed PL subspace of P with QdV. We say locally finite k-complexes 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 H0 is the inclusion on P, H1(P)dV, Ht is the inclusion on QÈ[(M-U)ÇP] for each t0[0, 1], H(P-QÇUH[0, 1]dU, and there is a map d:Pà(0, 4) with DIST(H1(p), M-V)>d(p), for each point p0P.

3.4 LEMMA. Let M be a PL n-manifold and let X be a closed, connected subset of ME with DIM X#n-1. Let U be a closed neighborhood of X in M such that a quasi-special sequence for X in U with the null-homotopy property exists. Then locally finite 2-complexes in M can be pulled into M-X in U.

Proof. We first note that [M-(M-X)]dUE, so that the conclusion makes sense according to the definition. Now let P be any closed 2-dimensional PL subspace of M and Q be any closed PL subspace of P in M-X.

Let {Hi-1,Ti,Fi}i=14 be a quasi-special sequence for X in U with the null-homotopy property. Since X is connected, we may assume that each Hi is connected. We shall only be concerned with the first five manifolds, H0, H1, H2, H3, and H4. Let R be a triangulation of M with P, Q and Hi, 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 G0 is the inclusion on P, G1(P)dM-X, Gt is the inclusion on QÈ[(M-U)ÇP] for each t0[0, 1], G(P-QÇUH[0, 1]dU, and so that there is a map d:Pà(0, 4) with DIST(G1(p), X)>d(p), for each point p0P. We will define G
(a) on (the vertices of P)H[0, 1],
(b) on (the 1-simplexes of P)H[0, 1], and
(c) on (the 2-simplexes of P)H[0, 1].

(a) Let V={v1, v2, v3, ...} be the set of vertices of P. We define G:VHIàM by defining G(vi, t)=vi, for all t0[0, 1], if vi is a vertex of R-H4ÈQ. Let {v~1, v~2, v~3, ...} be the set of vertices of P not in R-H4ÈQ. Define g:J+àH3 by g(j)=v~j. Lemma 3.1 gives us a proper map G:J+HIàH4 such that G0=g and G(i, 1)ÏX, for all i0J+. We define G(v~i, t)=G(i, t), for all i0J+.

(b) Let S={s1, s2, s3, ...} be the set of 1-simplexes of P. We define G:SHIàM by defining G(x, t)=x, for all t0[0, 1] and x0si if si0R-H4ÈQ. Note that if si0R-H4ÈQ, 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 1-simplexes of P not in R-H4ÈQ. Define G0|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~iH{1})dH4-X. For each i0J+, let hi be a homeomorphism from S0 onto Ms~i. Define F:S0HJ+àH4 by F(z, i)=G(hi(z), 1). Then since G(s~iH{0}ÈMs~iH[0, 1])dH4 and G is proper, F is properly S0-inessential in H4. By the null-homotopy property, F is properly S0-inessential in H3 missing X. It follows from this that G can be properly extended to (Èi=14s~i)H{1}, where the extension carries (Èi=14s~i)H[0, 1] into H3-X. In a natural way (cf. the construction of F above), G determines a proper map S1HJ+ from into H3. Since S1 is compact and connected, such a map is properly S1-inessential in H2 by Lemma 3.2. It follows from this that we may properly extend G to Èi=14(s~iH[0, 1]), where the extension carries Èi=14(s~iH[0, 1]) into H2.

(c) Let T={t1, t2, t3, ...} be the set of 2-simplexes of P. We define G:THIàM by defining G(x, t)=x, for all t0[0, 1] and x0ti if ti0R-H4ÈQ. Note that if ti0R-H4ÈQ, then so are the 1-simplexes 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 2-simplexes of P not in R-H4ÈQ. Define G0|t~i to be the inclusion on t~i, for each i0J+. Now for each i0J+, Mt~i is identifiable with S1 and G(Mt~iH{1})dH3-X. In a natural way (cf. the construction of F above), G determines a proper map S1HJ+ from into H3-X which is properly S1-inessential in H2. By the null-homotopy property, this map is properly S1-inessential in H1 missing X. It follows from this that G can be properly extended to (Èi=14t~i)H{1}, where the extension carries (Èi=14t~i)H[0, 1] into H1-X. We now have the proper map G defined on Èi=14M(t~iH[0, 1]). In a natural way again, G determines a proper map S2HJ+ from into H1. Since S2 is compact and connected, such a map is properly S2-inessential in H0 by Lemma 3.2. It follows from this that we may properly extend G to Èi=14(t~iH[0, 1]), where the extension carries Èi=14(t~iH[0, 1]) into H0.

We now have our desired homotopy G:PH[0, 1]àM such that G0 is the inclusion on P, G1(P)dM-X, Gt is the inclusion on QÈ[(M-U)ÇP] for each t0[0, 1], and G(P-QÇUH[0, 1]dU. Since X is closed and G is continuous, we may define a map d:Pà(0, 4) with DIST(G1(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 e-ball, e>0, about a point in the manifold be compact. This means that if a closed subset is non-compact, 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 k-ULC at the closed subset X, if for each e>0, there is a d>0 such that for each point x0X and map g:SkàN(x; d), g is e-null-homotopic 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 x0P-C.

The following lemma is analogous to Lemma 3.1 with the requirement that X be (n-1)-dimensional replaced by X tends to M-X and M is 0-ULC at X. It is easy to see that if X is (n-1)-dimensional, then X tends to M-X and M is 0-ULC at X.

3.7 LEMMA. Suppose X is a closed subset of a PL n-manifold M, X lies in ME, X tends to M-X and M is 0-ULC 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 G0=g and G(i,1)ÏX, for all i0J+.

Proof. We write M as an expanding union of compact sets
C1dCE2dC2dCE3d ... dÈi=14Ci=M
where C1 is chose arbitrarily and, for i>1, Ci is chose inductively by the following method. Suppose Ci-1 has been chosen. Using the 0-ULC condition of the hypothesis, let di>0 be such that both di<1/i and such that for each point x0X, any map g:S0àN(x; d) is (1/i)-null-homotopic. Now choose Ci so large that if xÏCi, then DIST(x, Ci-1)>1 and, using the hypothesis that X tends to M-X, so large that if xÏCi, then N(x; di)ÇM-X¹f.

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)0C2, pick pj0V-X so that pj is in the component of V containing g(j). Define G|jH[0, 1]:jH[0, 1]àV so that G0(j)=g(j) and G1(j)=pj.

For i>1, let j0J+ be such that g(j)0Ci+1-Ci. Since g(j)ÏCi, there is a point pj in (M-X)ÇN(g(j); di). By the 0-ULC condition, we may define G~j:jH[0, 1]àN(g(j); di) so that G~j0(j)=g(j) and G~j1(j)=pj. Since g(j)ÏCi, then DIST(g(j), Ci-1)>1; and since di<1/i<1, we have G~j(jH[0, 1])ÇCi-1=f. We now let qj 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 qj in {j}H[0, 1] contained in (G~j)-1(V). Let rj be a point in this neighborhood not in (G~j)-1(X). Define Gj:jH[0, 1]àV by Gj(j,t)=G~j(j,t.rj).

We now define the map G:J+HIàV such that G|{j}H[0, 1]=Gj. It follows that G0=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 Ci be such that CdCi. Since g is proper, the subset of J+, K=G-1(Ci+1), is finite. Since G has been constructed so that for jÏK, G(jH[0, 1])ÇCi=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 n-manifold and let X be a closed, connected subset of ME such that X tends to M-X and M is 0-ULC at X. Let U be a closed neighborhood of X in M such that a quasi-special sequence for X in U with the null-homotopy property exists. Then locally finite 2-complexes in M can be pulled into M-X 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 n-manifold, n\$6, and let X be a closed, connected subset of ME such that M is 0-ULC at X and X tends to M-X. Then, X is strongly quasi-cellular if and only if X has SUV4 and satisfies the strong quasi-cellularity 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 quasi-cell N such that XdNEdNdUE.

Let H0 and H1 be the first two manifolds of a quasi-special sequence for X in U with the null-homotopy property. Let R be a triangulation of M with H0 and H1 as subcomplexes. Let R2 be the 2-skeleton of R. We use H1 as the U of Lemma 3.8 to obtain a proper homotopy G:R2H[0, 1]àM such that G0 is the inclusion on R2, G1(R2)dM-X, Gt is the inclusion on (M-H1)ÇR2 for each t0[0, 1], G(R2ÇH1H[0, 1])dH1, and so that there is a map d:R2à(0, 4) with DIST(G1(p), X)>d(p), for each p0R2. If we let {Ap} be {G(pH[0, 1])} for all points p in 2-complexes in M and all homotopies G pulling these 2-complexes into M-X in H2, 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|R2ÇH1=1|R2ÇH1, R2dG'1(M-X), and G't|cl(M-H1)=1|cl(M-H1). This last result is achievable since G moves no point which is in cl(M-H1) and moves no point into cl(M-H1) from HE1 and since G' is picked so that points are moved close to the movement of points by G.

Now define K to equal R2ÇH1dG'1(M-X)ÇG'1(H1) which equals G'1(H1-X). Let L be the complementary complex of K in H1. Since DIM L#n-3, there is a quasi-cell N* in HE0 with LdNE*, by the definition of quasi-special sequences. Since G'1(X) does not intersect K, there is an ambient isotopy, fixed outside of H0, 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)-1BG"1(N*). Then XdNEdNdHE0dUE, and the proof is complete.

3.10 REMARK. Not every set having SUV4 is quasi-cellular as shown by [5, Example 4.19]. However, the following shows that if X is a locally compact, finite dimensional metric space with SUV4 (with respect to metric ANRs), then X embeds in some Euclidean space as a strongly quasi-cellular set.

3.11 THEOREM. If X is a closed subset of En with SUV4, then X is strongly quasi-cellular in En+3, for n\$3.

Proof. Since n+3\$6, since DIM X<n+3 supplies that X tends to En+3-X and En+3 is 0-ULC at X, and since SUV4 is a topological property, so that X has SUV4 in En+3, Then Theorem 3.9 shows that we need only demonstrate that X satisfies the strong quasi-cellularity criterion to have that X is strongly quasi-cellular.

Suppose that U is a closed neighborhood of X in En+3. Let V be any closed neighborhood of X in En+3 lying in UE. Now let g:SkHJ+àV-X be a map such that there is a proper homotopy G:(SkHJ+)HIàV with G0=g and G1|SkH{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 En. Thus, since DIM(G[SkHJ+HI])#2, then EnÇG[SkHJ+HI]=f. Thus g is properly Sk-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 SUV4 with respect to metric ANRs if and only if X embeds as a strongly quasi-cellular subset of some PL n-manifold, n\$6.

Proof. Since X is a locally compact, finite dimensional metric space, it embeds in En, 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 (Shp is the proper shape function).

3.13 THEOREM. Suppose X is a locally compact metrizable space. Then X0SUV4 if and only if there exists a tree T such that ShpX=ShpT.

4 STRONG QUASI-CELLULARITY AND WEAK QUASI-CELLULARITY.

In section 3 we presented one McMillan type theorem connecting strong quasi-cellularity with SUV4 and the strong quasi-cellularity 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 quasi-cellularity.

4.1 DEFINITION. Let M be a PL manifold and U an open subset of M. Then we say that (M, U) is p-connected if and only if for each integer i, 0 £ i £ p, and each map
f:(Di, MDi)à(M, U)
there exists a homotopy
F:(Di, MDi)HIà(M, U)
such that for each t0[0, 1], Ft is a map of the pair (Di, MDi) into the pair (M, U), F0=f, and F1(Di)dU.

4.2 DEFINITION. We call a manifold M uniformly locally p-connected, p-ULC, if given e>0, there is a d>0 for which every map f:SpàM such that the diameter of f(Sp) is less than d is e-null-homotopic. If M is p-ULC for 0 £ p £ k, the we say M is ULCk.

4.3 DEFINITION. Let M be a PL n-manifold, n\$2. A set XdME is said to thin down in M, if for each locally finite triangulation R of M, the 2-skeleton R2 of R tends to M-X.

4.4 LEMMA. Suppose M is a PL n-manifold, n\$2. Then a Set XdME tends to M-X if and only if X thins down in M.

Proof. Suppose X tends to M-X and R is any locally finite triangulation of M. We shall show that R2 tends to M-X. Let e>0 be given. Then there is a compact set C such that if x is a point of X-C, then DIST (x, M-X)<e. Now let p be a point of R2-C. If p0X, we have p0X-C, so DIST (p, M-X)<e. If pÏX, then p0M-X, so that DIST (p, M-X)=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 p0R2-C, then DIST (p, M-X)<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 X-C" and let Q be the set {q|q0R2 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 R2dQ. Also, there is a point p in Q with p0R2-C', for if x0sE and (R2)dC', then s must be in C", but xÏC". Now R2-CÉR2-C', so that if p0R2-C and DIST (p, M-X)<e/2, we have DIST (x, M-X)<e.

4.5 THEOREM. Let M be a PL n-manifold, n\$5, and let X be a subset of ME which tends to M-X. Suppose that for each open neighborhood U of X there is an open neighborhood V of X lying in U such that V is ULC2 and V-X is ULC1. Then X is strongly quasi-cellular if and only if X is closed, has SUV4 and satisfies the strong quasi-cellularity criterion.

Proof. The proof of necessity is given by [5, Theorem 4.13].

Suppose then that XdME is closed, has property SUV4 in M, and satisfies the strong quasi-cellularity criterion. Let U be any closed neighborhood of X. Let Hi, i=1, 2, 3, 4, and 5, be connected PL submanifolds of M so that
(1) XdHEi+1dHi+1dHEidHi-UE, i=1, 2, 3, or 4,
(2) there exists a proper PL homotopy Fi+1:Hi+1H[0, 1]àHi, with F0i+1 the inclusion, Fi+1(Hi+1H[0, 1])dHEi, and F1i+1(Hi+1)=Ti+1 a closed tree, i=1, 2, 3, or 4,
(3) MHi¹f, i=1, 2, 3, 4, or 5,
(4) if YdHi+1 is a polyhedron and DIM Y£n-3, then Y is quasi-trivial in Hi, i=1, 2, 3, or 4,
(5) each map g:SkàHi+1-X is null-homotopic in HEi-X, i=1, 2, 3, or 4, and k=0 or 1. This last condition derives from the strong quasi-cellularity criterion [5, Remark 4.12 (f)].

Let R be a locally finite triangulation of M with Hi i=1, 2, 3, 4, or 5, as subcomplexes and let R2 be the 2-skeleton of R. By Lemma 4.4 we know that R2 tends to M-X.

Let V be an open subset of HE5 such that XdV, V is ULC2, and V-X is ULC1. 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(M-X) contains all of R2 except some compact subset contained in e'1(V)=V. Let L be a compact subcomplex of R2 which contains R2Çe'1(X) and such that R2-Lde'1(M-X).

We now wish to apply Stallings' Engulfing Theorem with (L, HE2Ç(R2-L), e'1(HE2-X), HE2) 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=(HE2, HE2-X)»(e'1(HE2), e'1(HE2-X))=(HE2, e'1(HE2-X)) is 2-connected.

Since H5 is arc connected, if
f:(D0, MD0)à(HE2, HE2-X)
is a map, with f(D0)dX, then there is an arc in H5 from f(D0) to a point in H5-X. Thus p is 0-connected.

Now consider any map
f:(D1, MD1)à(HE2, HE2-X).
Suppose f(D1)dH5. Since f( MD1) is a pair of points in H5-X, there is a map g:D1àHE4-X with g|MD1=f|MD1. Since F14[f(D1)Èg(D1)] is a continuum in T4, f is homotopic (rel MD1) in HE3 to g. In the general case, cover f-1(X) with a finite number of disjoint arcs s1, s2, s3, ... sk so close to f-1(X) that each f(si)dH5. By the special case f|si is homotopic (rel Msi) in HE3 to a path in HE4-X. Putting these homotopies together, we see that f is homotopic (rel [D1-Èi=1ksEi]) in HE2 to a path in HE2-X. so p is 1-connected.

Now consider any map
f:(D2, MD2)à(HE2, HE2-X).
Suppose f(MD2)dH4-X and f(D2)dH3. Then we have a map g:D2àHE3-X with g|MD2=f|MD2. Since F13[f(D2)Èg(D2)] is a continuum in T3, f is homotopic (rel MD2) in HE2 to g. In the general case, cover f-1(X) with the interiors of a finite number of punctured polyhedral 2-cells, t1, t2, t3, ... tm which do not meet MD2 and such that to f(ti)dH5, i=1,2, ...m. Let S be a triangulation of D2 so that the tis are subcomplexes of S. Let A=Èi=1m(1-skeleton ti) and let B=D2-Èi=1mtEi. By the 0-connectivity of H5, we have a homotopy (rel M(Èi=1mti)), (since f-1(X)dINT(Èi=1mti)) of the 0-skeleton of A in H5 to H5-X. We use the homotopy extension property for polyhedral pairs [14, Corollary 5, p. 118] to obtain a homotopy from AH[0, 1] into H5 with the restriction to the 0-skeleton agreeing with the original homotopy. We now use the special case of the proof of the 1-connectivity of , (since the 0-skeleton of AH{1} is carried into H5-X), to obtain a homotopy φ:AH[0, 1]àHE3 such that φ0=f|A, φ1(A)dHE4-X and φt|AÇB=f|AÇB (AÇB=M(Èi=1mti)). Using the homotopy extension property again, we extend φ to Φ:(Èi=1mti)HI with Φ0=f|Èi=1mti, Φ1(A)dHE4-X and Φt|AÇB=f|AÇB, t0[0, 1]. We now extend Φ to all of D2HI by defining Φt|B=f|B, for each t0[0, 1]. Let r be a 2-simplex of Èi=1mti, then Φ|r is a map from (r, Mr) to (H3, H4-X). By applying the special case, we see that Φ is homotopic (rel AÇB) in HE2 to a map G:D2àHE2-X. Thus f is homotopic (rel B, implying rel MD2) to G in HE2, and p is 2-connected.

We have, from Stallings' Engulfing Theorem, an ambient isotopy e2:HE2H[0, 1]àH2 and a compact set EdHE2 such that Lde12(e'1(HE2-X)) and et2|(HE2-E)È(HE2ÇR2-L)=1|(HE2-E)È(HE2ÇR2-L), so that we may extend et2 by the identity on M-HE2.

Now define K to be the complex H2ÇR2de12(e'1(HE2-X)). Let J be the complementary complex of K in H2 and let N* be a quasi-cell in HE1 with JdNE*. As in the proof of Theorem 3.9, let e3:MH[0, 1]àM be an ambient isotopy, fixed outside of H1, which engulfs e12(e'1(X)) with e13(NE*). Define N to be [(e'1)-1B(e12)-1Be13](N*).

Prior to proving the third of the theorems about strong quasi-cellularity, 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 n-manifold, 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:(P-A)HIàM, where cl(A) is a compact subset of P, if
(1) H(p, 0)=p, for all p0P-A,
(2) H(p, 1) is in U, for all p0P-A, and
(3) given e>0, there is a compact set BdP such that DIAM(H(pH[0, 1]))<e, for p0P-B.

4.7 THEOREM. Let M be a PL n-manifold, n\$5, and let X be a subset of ME which tends to M-X. Suppose that for each open neighborhood U of X there is an open neighborhood V of X lying in U such that V is ULC6-n, V-X is ULC5-n, and for each locally finite triangulation R of M, most of the 2-skeleton R2ÇV can be pulled through V into V-X by a short homotopy. Then X is strongly quasi-cellular if and only if X is closed, has SUV4 and satisfies the strong quasi-cellularity 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 0-ULC at X condition for M is replaced by the short homotopy condition. It is possible that the short homotopy can be obtained from SUV4 and strong quasi-cellularity in somewhat the same manner as the homotopy pulling locally finite 2-complexes into M-X in U is obtained in Lemma 3.8. However, some other condition, such as 0-ULC at X may be required.

We now state and prove three theorems for sufficiency conditions for weak quasi-cellularity.

4.9 THEOREM. Let M be a PL n-manifold, n\$6, and let X be a closed, connected subset of ME such that M is 0-ULC at X and X tends to M-X. Then X is weakly quasi-cellular if X satisfies the weak quasi-cellularity criterion.

Proof. By the definition of the weak quasi-cellularity criterion (2.10), there exists a quasi-special sequence {Hi-1,Ti,Fi}i=14 for X in M having the null-homotopy property. Then the proof of sufficiency in Theorem 3.9 gives a quasi-cell N with XdNEdNdHE0dME.

We need only show that for each quasi-cell Q containing X in its interior, there is a sequence of quasi-cells {Qi}i=14 lying in Q with XdQEi+1dQi+1dQEi and X=Çi=14Qi. Since we already have the existence of the quasi-cell N, this will complete the proof.

Write M as the union of a countable number of compact PL n-manifolds, M1dME2dM2dME3d ... dÈi=14Mi=M. Let {ei}i=14 be a decreasing sequence of positive numbers converging to zero.

There exists a quasi-special sequence for X in Q having the null-homotopy property, so we may choose Q1dQE as N was chosen for M. Inductively, suppose Qj has been defined. Let {Hi-1,Ti,Fi}i=14 be a quasi-special sequence for X in Qj with the null-homotopy property. Let Hi have large enough subscript so that HiÇMjdN(X;ej), and as before, there exists a quasi-cell Qj+1 containing X in its interior with Qj+1dHEi. Let {Qi}i=14 be the sequence of quasi-cells so defined.

It is clear that XdQEi+1dQi+1dQEi, for all i. Let p be a point in M-X, and let e=DIST(p, X). Let ei<e and Mj be such that p0Mj. Then p is not in Qi+j. Thus X=Çi=14Qi, and the proof is complete.

4.10 THEOREM. Let M be a PL n-manifold, n\$5, and let X be a closed, connected subset of ME which tends to M-X. Suppose that for each open neighborhood U of X there is an open neighborhood V of X lying in U such that V is ULC2 and V-X is ULC1. Then X is weakly quasi-cellular if X satisfies the weak quasi-cellularity criterion.

Proof. Let {Hi-1,Ti,Fi}i=14 be a quasi-special sequence for X in M having the null-homotopy property. The proof of sufficiency in Theorem 4.5 gives a quasi-cell N with  XdNEdNdHE0dME. The argument in the proof of Theorem 4.9 finishes the proof.

4.11 THEOREM. Let M be a PL n-manifold, n\$5, and let X be a closed, connected subset of ME which tends to M-X. Suppose that for each open neighborhood U of X there is an open neighborhood V of X lying in U such that V is ULC6-n, V-X is ULC5-n, and for each locally finite triangulation R of M, most of the 2-skeleton R2ÇV can be pulled through V into V-X by a short homotopy. Then X is weakly quasi-cellular if X satisfies the weak quasi-cellularity 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 M-X, V is ULC2, and V-X is ULC1 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 quasi-cellularity 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 ú3, and its analogs in ún. The bumps have constant height and their indentations have constant depth (x3-direction) and a constant width (x2-direction), but have decreasing length (x1-direction). X does not tend to ú3-X and there is no V that is 0-ULC. If we alternate the bumps with bumps that have decreasing widths, there is no V that is 1-ULC, with V-X either 0-ULC or 1-ULC. Despite these problems, there is an ambient isotopy between either version of X and X' shown in Figure 2. X' does tend to ú3-X and for each closed neighborhood U of X, there is a closed neighborhood V of X in U such that V is ULC2 and V-X is ULC0. Clearly both X and X' are strongly quasi-cellular, yet only for analogs of X' may we apply any of our results to show this.

Figure 1. Strongly quasi-cellular, but not provable by these theorems

Figure 2. Strongly quasi-cellular, with ambient isotopy to Figure 1

There are other questions that could lead to further research.

For example, the number of ends of a quasi-cell is well-defined and equals the number of ends of a tree of which it is a regular neighborhood. Is the number of ends of a quasi-cell well defined? If so, is there a relation between the number of ends of the defining quasi-cells and the number of ends of the quasi-cellular set? Figures 3 and 4 give examples of strongly quasi-cellular sets and quasi-cells that might appear in sequences used to define the sets. The symbol ei(Y) refers to the ith end of Y, for whatever space Y is.

Figure 3. Strongly quasi-cellular set and containing quasi-cell

Figure 4. A different strongly quasi-cellular set and containing quasi-cell

A question arises concerning compactification. If a quasi-cellular set is compactified by adding a point at each end, this same compactification of the quasi-cells 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 quasi-cellularity be achieved? Notice, in Figures 5 and 6, that the cells minus points are neighborhoods of the ends, not necessarily quasi-cells.

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 S2 and N is a cell in S2 containing X in its interior. Removing an endpoint of X yields X' as a closed subset of ú2 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 n-manifold, U is an open subset of M, and {Aa} is a collection of sets in M. We say locally finite k-complexes in Mn can be pulled into U along {Aa} in M if, for each closed PL subspace Pk of M and closed set QdPk such that QdU, there is a proper homotopy H:PH[0, 1]àM such that H0 is the inclusion, H1(P)dU, Ht is the inclusion on Q, for each p0P, H(pH[0, 1]) lies in an element of {Aa}, and there is a map d:Pà(0, 4) with DIST (H1(p), M-U)>d(p), for each p0P.

5.2 THEOREM. Suppose U is an open subset of a PL n-manifold M, P is a closed subspace of M, Q is a closed PL subspace of P lying in U and R=cl(P-Q) is r-dimensional, r£n-4. If {Aa} is a collection of subsets of M such that locally finite r-complexes in M can be pulled into U along {Aa}, then for each map e:Rà(0, 4), there exists an engulfing isotopy G:MH[0, 1]àM such that G0=1, Gt|Q=1|Q, RdG1(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 {Aa} 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 n-manifold without boundary, U an open subset of M, Pk a finite polyhedron in M of dimension k£n-3 and QqdU a (possibly infinite) polyhedron of dimension q£n-3 such that (cl(Q)-Q)ÇP=f. Let (M, U) be k-connected. Then there is a compact set EdM and an ambient isotopy et of M such that Pdet(U) and
et|(M-E)ÈQ=1|(M-E)ÈQ.

In the statements of Rushing's Infinite Engulfing Theorems, Mn is a connected PL n-manifold without boundary, U is an open set in M, Pk is an infinite polyhedron of dimension k£n-3, which is contained in M (P is not necessarily closed in M) and Qq is a (possibly infinite) polyhedron of dimension q£n-3 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 M-4~ is ULCk, U-4~ is ULCk-1 and P tends to U. Then most of Pk 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 et of Mn such that et|(M-N(P-C; e))ÈQ=1|(M-N(P-C; e))ÈQ and such that e1(U) contains all of P except some compact subset. Furthermore, for each d>0, there exists a compact subset KdM-4~ such that et|M-K is a d-isotopy.

The last theorem is Rushing's Infinite Engulfing Theorem 1 of [11].

5.4 THEOREM. Suppose that M-4~ is ULCmax(k,q)+k-n+2, U-4~ is ULCmax(k,q)+k-n+2. Also suppose that most of Pk can be pulled through M-4~ into U-4~ by a short homotopy. Then most of Pk can be engulfed by U in the same sense as in Theorem 5.4

CONTINUATION: "More about Property SUV4 and Strong Quasi-Cellularity."

REFERENCES

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