Some topics concerning homeomorphic parameterizations

Publ. Mat. 45 (2001), 3–67 SOME TOPICS CONCERNING HOMEOMORPHIC PARAMETERIZATIONS Stephen Semmes Abstract In this survey, we consider several questions pertaining to homeomorphisms, including criteria for their existence in certain circumstances, and obstructions to their existence. Contents 1. Wildness and tameness phenomena 4 2. Contractable open sets 8 2.1. Some positive results 11 2.2. Ends of manifolds 16 3. Interlude: looking at infinity, or looking near a point 16 4. Decomposition spaces, 1 18 4.1. Cellularity, and the cellularity criterion 24 5. Manifold factors 26 6. Decomposition spaces, 2 29 7. Geometric structures for decomposition spaces 32 7.1. A basic class of constructions 32 7.2. Quotient spaces can be topologically standard, but geometrically tricky 36 7.3. Examples that are even simpler topologically, but still nontrivial geometrically 43 8. Geometric and analytic results about the existence of good coordinates 47 8.1. Special coordinates that one might consider in other dimensions 50 9. Nonlinear similarity: Another class of examples 54 10. Doing pretty well with spaces which may not have nice coordinates 55 References 58 The author was partially supported by the National Science Foundation. 4 S. Semmes 1. Wildness and tameness phenomena Consider the following question. Let nbe a positive integer, and let Kbe a compact subset of Rn.IfKis homeomorphic to the unit interval [0,1], is there a global homeomorphism from Rnonto itself which maps Kto a straight line segment? (1.1) If n= 1, then Kitself is a closed line segment, and the answer is “yes”. When n= 2, the answer is also “yes”, but this is more complicated, and is more in the spirit of the Sch¨onflies theorem in the plane. See [Moi], especially Chapter 10. When n≥3, the answer to the question above can be “no”. An arc K is said to be “tame” (or flat) when a homeomorphism does exist as in (1.1), and “wild” when it does not exist. See [Moi] for some examples of wild arcs in R3. Smooth arcs are always tame, as are polygonal arcs, i.e., arcs made up of finitely many straight line segments. For these one can take the corresponding homeomorphism to be smooth or piecewise-linear as well. (Compare with Theorem 1 on p. 134 of [Moi], for instance.) In order for an arc to be wild, some amount of infinite processes are needed. A simple closed curve in R3might be smooth or polygonal and still knotted, so that there does not exist a homeomorphism of R3onto itself which maps the curve onto a standard circle (inside a standard 2-dimensional plane in R3). There are many well-known examples of this, like the trefoil knot. Thus, for a closed curve, one defines “wildness” in a slightly differently way, in terms of the existence of local flattenings, for instance. This turns out to be compatible with the case of arcs (for which there is no issue of knottedness), and there are some other natural variants of this. Here is another basic example, for sets of higher dimension. Suppose that γis a simple closed curve in R3, which is a polygonal curve, and which represents the trefoil knot. Consider the cone over γ, which gives a 2-dimensional polyhedron in R4, and which is in fact piecewise-linearly equivalent to a standard 2-dimensional cell. One can show that this embedding of the 2-cell is not locally flat at the cone point, i.e., it cannot be straightened out to agree with a standard (geometrically flat) embedding by a homeomorphism defined on a neighborhood in R4of the cone point. Similar phenomena occur for codimension-2 embeddings in Rn for all n≥4, as in Example 2.3.2 on p. 59–60 of [Rus1]. Some Topics Concerning Homeomorphisms 5 This phenomenon is special to codimension 2, however. Specifically, a piecewise-linear embedding of a k-dimensional piecewise-linear manifold into Rnis locally topologically flat if n−k= 2 (or if k= 1 and n=3, as before). See Theorem 1.7.2 on p. 34 of [Rus1]. In the context of piecewise-linear embeddings, one can also look for local flattenings which are piecewise-linear. A similar remark applies to other categories of mappings. We shall not pursue this here. Wild embeddings of cells and spheres (and other manifolds) exist in Rnfor all n≥3, and for all dimensions of the cells and spheres (from 1ton−1). This includes embeddings of cells and spheres which are not equivalent to piecewise-linear embeddings in codimension 2. We shall mostly consider here issues of existence of topological flattenings or local flattenings, and embeddings which are not normally given as piecewise-linear. See [Bin6], [Bur], [BurC], [Can1], [Dave1], [Dave2], [Edw1], [Moi], [Rus1], [Rus2] for more information, and for further references. Let us also mention that embeddings, although wild, may still enjoy substantial good behavior. For instance, they may be bilipschitz, so that distances are not increased or decreased by more than a bounded factor, or quasisymmetric, in the sense of [TukV]. Roughly speaking, the latter means that relative distances are not distorted too much, rather than distances themselves, as for a bilipschitz mapping. See [Geh1], [LuuV], [V¨ai2] for some basic results in these directions. As another version of wildness for embeddings, imagine that one has a compact set Cin some Rn, and that Cis homeomorphic to the usual middle-thirds Cantor set. Can one move Cto a subset of a straight line in Rn, through a homeomorphism from Rnonto itself? When n= 1 this is automatically true. It is also true when n=2; see [Moi], especially Chapter 13. In higher dimensions it is not true in general, as is shown by a famous construction of Antoine (“Antoine’s necklaces”). See Chapter 18 of [Moi] and [Bla]. How can one tell when a set is embedded wildly or not? As a simple case, let us consider Cantor sets. If Cis a compact subset of Rnwhich lies in a line and is homeomorphic to the Cantor set, and if nis at least 3, then the complement of Cin Rnis simply-connected. This is not hard to see. Basically, if one takes a loop in the complement of Cand fills it with a disk in Rn, and if that disk happens to run into C, then one can make small perturbations of the disk to avoid intersecting C. The complement of Cis also simply-connected if there is a global homeomorphism from Rnonto itself which maps Cinto a line. This is 6 S. Semmes merely because the homeomorphism itself permits one to reduce to the previous case. However, Antoine’s necklaces have the property that their complements are not simply-connected. See [Moi], [Bla]. Note that the homology of the complement of a compact set in Rnis controlled through the intrinsic topology of the set itself, as in Alexander duality [Spa]. In particular, while the complement of an Antoine’s necklace may not be simply-connected, its 1-dimensional homology does vanish. Versions of the fundamental group play an important role for wildness and taming in general, and not just for Cantor sets. For this one may not take (or want to take) the fundamental group of the whole complementary set, but look at more localized forms of the fundamental group. A specific and basic version of this is the following. Suppose that Fis a closed set inside of some Rn. Given a point p∈F,andaloopγin Rn\Fwhich lies close to p, one would like to know whether it is possible to contract γtoapointinRn\Fwhile staying in a small neighborhood of p. This second neighborhood of pmight not be quite as small the first one; a precise statement would say that for every >0 there is a δ>0 so that if γlies in B(p, δ)∩(Rn\F), then γcan be contracted to a point in B(p, )∩(Rn\F). This type of condition is satisfied by standard embeddings of sets into Rn, like Cantor sets, cells, and spheres, at least when the dimension of the set is different from n−2. For a point in R2, or a line segment in R3, etc., one would get Zfor the corresponding localized fundamental group of the complement of the set. (In the case of a line segment in R3, one should restrict one’s attention to points pin the interior of the segment for this.) Conversely, there are results which permit one to go backwards, and say that localized fundamental group conditions for the complement like these (localized simple-connectedness conditions in particular) lead to tameness of a given set, or other kind of “standard” (non-wild) behavior. See [Bin5], [Bin6], [Bin8], [Bur], [BurC], [Can1], [Can2], [Dave1], [Dave2], [Edw1], [Moi], [Qui1], [Qui2], [Rus1], [Rus2] for more information about localized fundamental groups and their role in wildness phenomena and taming theorems (and for related matters and further references). One might wonder why π1and localized versions of it play such an important role. Some basic points behind this are as follows. For homology (or cohomology), one often has good information from data in the given situation through standard results in algebraic topology, like duality theorems. In circumstances with suitable simple-connectivity, Some Topics Concerning Homeomorphisms 7 one can pass from information about homology to information about homotopy (in general dimensions), as in the Hurewicz and Whitehead theorems. See [Bre1], [Spa]. Another basic point concerns the effect of stabilization. A wild embedding of a set into some Rncan become tame when viewed as an embedding into an Rmwith m>n(m=n+1 in particular). The same is true for knotting. For instance, a smooth loop may be knotted in R3, but when viewed as a subset of R4, it is always unknotted. This is easy to see in explicit examples (like a trefoil knot). For some simple and general results about wild embeddings in Rn becoming tame in a larger Rm, see Proposition 4 on p. 84 of [Dave2], and the corollaries on p. 85 of [Dave2]. These involve a famous device of Klee. In concrete examples, one can often see the taming in a largerdimensional space directly, and explicitly. Examples of wild sets are often made with the help of various linkings, or something like that, and in a higher-dimensional space one can disentangle the linked parts. This can be accomplished by taking individual pieces and pulling them into a new dimension, moving them around freely there, and then putting them back into the original Rnin a different way. This simplifying effect of stabilization also fits with the role of localized versions of fundamental groups indicated before. Let Fbe a closed set inside of some Rn, and imagine that one has a loop γin Rn\Fwhich lies in a small ball centered at a point in F. In the condition that was discussed earlier, one would like to contract γto a point in the complement of F, while remaining in a small ball. If one thinks of γand Fas being also inside Rn+1, then it is easy to contract γto a point in the complement of Fin Rn+1, while remaining in a small ball. Specifically, one can first translate γinto a parallel copy of Rninside of Rn+1, i.e., into Rn×{a}for some a= 0 rather than Rn×{0}in Rn+1 (using the obvious identifications). This parallel copy is then disjoint from F, and one can contract the loop in a standard way. Note that this argument works independently of the behavior of F. Using taming theorems based on localized fundamental group conditions, and considerations like those in the previous paragraph, one can get stronger results on tameness that occurs from stabilization than the ones on p. 84–85 in [Dave2] mentioned above. More precisely, instead of needing kextra dimensions in some cases, it is enough to go from Rnto Rn+1. Compare with the bottom of p. 390 and the top of p. 391 in [Dave1], and the references indicated there. (Compare also with the remarks on p. 452 of [Can1].) 8 S. Semmes 2. Contractable open sets Fix a positive integer n. If Uis a nonempty contractable open subset of Rn,is Unecessarily homeomorphic to the open unit ball in Rn? (2.1) For the record, to say that Uis contractable means that the identity mapping on Uis homotopic to a constant, through (continuous) mappings from Uinto itself. In particular, the homotopy and homology groups of U(of positive dimension) would then be trivial, just as for an n-dimensional ball. When n= 1, the answer to the question in (2.1) is “yes”. In this case, Uis either the whole real line, an open segment in the real line, or an open ray. Each of these is easily seen to be homeomorphic to the interval (−1,1), which is the unit ball in this case. If n= 2, then the answer to the question in (2.1) is “yes” again. This is a well-known fact, and we shall return to it later, in Section 8. Starting in dimension 3, the answer to the question in (2.1) is “no”. We shall say something about examples for this in a moment, but let us first ask ourselves the following: how might one be able to tell that a given contractable open set in Rnis not homeomorphic to an n-dimensional ball? Here again a localized version of the fundamental group is important. If n≥3, then a necessary condition for a set Uto be homeomorphic to an n-dimensional ball is that Ube “simply connected at infinity”. Roughly speaking, this means that if one takes a closed loop γout near infinity in U, then it should be possible to contract γto a point, while staying out near infinity too (although perhaps not as much as γitself is). Here is a more formal definition. For this we also include “connectedness at infinity” as a first part. Definition 2.2. Let Ube an open set in Rn, or a topological space more generally. (Normally one might at least ask that Ube locally compact.) Uis connected at infinity if for each compact set K0⊆Uthere is a larger compact set L0⊆Usuch that every pair of points in U\L0is contained in a connected set which is itself contained in U\K0. (One can define “arcwise connectedness at infinity” in a similar manner. The two notions are equivalent under assumptions of local arcwise connectedness, and for topological manifolds in particular.) Some Topics Concerning Homeomorphisms 9 Uis simply-connected at infinity if it is connected at infinity, and if for every compact set K1⊆Uthere is a larger compact set L1⊆Uso that if γis an arbitrary closed loop in U\L1(i.e., an arbitrary continuous mapping from the unit circle S1into U\L1), then γis homotopic to a constant through continuous mappings from the circle into U\K1. If Uis the unit ball in Rn,n≥3, then Uis simply-connected at infinity. Indeed, let B(0,r) denote the open ball in Rnwith center 0 and radius r, and let B(0,r) denote the corresponding closed ball. Then every compact subset of B(0,1) is contained in B(0,r) for some r<1. For each r<1, B(0,r) is a compact subset of B(0,1), and B(0,1)\B(0,r) is connected when n≥2, and simply-connected when n≥3. This is because B(0,1)\B(0,r) is homeomorphic to Sn−1×(r, 1), and Sjis connected when j≥1, and simply-connected when j≥2. The property of being simply-connected at infinity is clearly preserved by homeomorphisms. Thus to get a contractable open set Uin Rnwhich is not homeomorphic to an n-dimensional ball, it suffices to choose Uso that it is not simply-connected at infinity. If Uis contractable, then it is simply-connected itself in particular. If Uis simply-connected, connected at infinity, and not simply-connected at infinity, then it means that there is a compact set K⊆Uand loops γin Uwhich lie as far towards infinity as one would like (i.e., in the complement of any given compact subset of U) such that (a) γcan be contracted to a point in U, and (b) γcannot be contracted to a point in U\K. To put it another way, these loops γcan be contracted to points in U, but in doing this one always has to pass through at least one element of the compact set K. A mechanism for having this happen for a set Ucontained in R3is given by the construction of “the Whitehead continuum” [Whit]. (See also [Dave2], [Kir].) Here is an outline of the procedure. Start with a standard smooth “round” solid torus Tin R3. Here T should be a compact set, i.e., it should contain its boundary. Next one chooses another smooth solid torus T1inside T. More precisely, T1should lie in the interior of T. One chooses T1in a particular way, which can be imagined as follows. (Pictures can be found on p. 68 of [Dave2] and p. 82 of [Kir].) First take a “small” solid torus in T, small enough to be contained in a topological ball in T. One can think of grabbing hold of this small solid torus at two ends, and then stretching them around the “hole” in the larger torus T. One stretches them around the two different sides of the hole in T. To get T1, these two ends should hook around each other on the other side of the hole. In other 10 S. Semmes words, one might imagine having the two ends of the small solid torus from before, stretched around opposite sides of T, and then passing one across the other, until they do not touch any more, but are clasped together, like two hooks, or two links in a chain. The configuration looks locally like two hooks or links clasped together, but in fact one has two ends of the single solid torus T1, wrapped around the hole in T. If T1is chosen in this way, then it has the following two basic properties. The first is that it is homotopically trivial in T. That is, the identity mapping on T1is homotopic to a constant mapping through (continuous) mappings from T1into T. This follows exactly the description above; in making the homotopy, one is allowed to stretch or move T1around as much as one like, and one is allowed to have different parts of (images of) T1cross each other in T. To put it a bit differently, the mappings being deformed are not required to be injective. The second property is that T1is not “isotopically trivial”. This means that one cannot continuously deform T1through an isotopy of T into a set which lies in a ball contained in T. In effect, this means that one cannot continuously deform T1inside Tin such a way that T1ends up in a ball in T, and so that the deformations do not ever cross each other (unlike the homotopy in the previous paragraph). If one could get T1inside a ball in T, then one could continue the deformation to get an isotopy into an arbitrarily small ball. One would not ask for shrinking T1to a point here, because this is automatically prevented by injectivity (independent of clasping or not). This explains how Tand T1should be chosen. Since T1is a 3-dimensional smooth solid torus in its own right, one can repeat the process to get another smooth solid torus T2contained in it, and in fact contained in the interior of T1. In other words, since Tand T1are both smooth solid tori, they are diffeomorphic to each other in particular, and this can be used to make precise the idea of “repeating the process”. Specifically, if φ:T→T1is such a diffeomorphism, then one can take T2to be φ(T1). One then repeats the process indefinitely, getting smooth solid tori Tj for j=1,2,... such that Tj+1 is contained in the interior of Tjfor each j, and so that Tj+1 is arranged in Tjin the same way as T1is arranged in T. Now let Wbe the intersection of all these solid tori Tj. This gives a nonempty compact set in R3. We can think of Was lying inside of S3, and then take U=S3\W. One can also rotate this around so that U actually lies in R3. One can show that Uis contractable, but not simplyconnected at infinity. See [Dave2], [Kir], [Whit] for more information. (For the purposes of looking at U, the complement of W, it can be Some Topics Concerning Homeomorphisms 11 convenient to use a modestly different description of the construction, in which one builds Uup from smaller pieces in an “increasing” manner, analogous to the “decreasing” construction for Wabove.) Although Uis not homeomorphic to a 3-dimensional ball in this case, the Cartesian product of Uwith a nonempty open interval is homeomorphic to a 4-dimensional ball. This is attributed to Arnold Shapiro in [Bin3]; see also Section 10 of [Bin4] and [Kir]. This is analogous to the effect of stabilization before, in Section 1. In particular, one can check directly that taking the Cartesian product with the interval gets rid of the problem that Uitself has with simple-connectivity at infinity. Beginning in dimension 4, there are contractable open sets in Rn which are not topological n-balls, and which have the additional feature that their closures are compact manifolds with boundary. This last does not work in dimension 3, and, for that matter, the complement of the Whitehead continuum in S3cannot be realized as the interior of a compact manifold with boundary, whether or not this compact manifold should occur as the closure of the set in S3. The reason is that if such a compact manifold did exist, its boundary would be a 2-dimensional surface with the homology of the 2-sphere. We shall say more about this in a moment. In this case the boundary would have to be homeomorphic to the 2-sphere. This would contradict the failure of simple-connectivity at infinity for the original space, since S2is simply-connected. The difference with n≥4 is that the boundary can be a homology (n−1)-sphere (i.e., a manifold with the same homology as Sn−1) which is not simply-connected. The interior then fails to be simply-connected at infinity again, and is not homeomorphic to an n-ball in particular. For some related information and references concerning these examples in dimensions greater than or equal to 4, see [Dave2], including the top of p. 94, and the discussion on p. 103–104. 2.1. Some positive results. For dimensions n≥4, it is known that every contractable topological manifold Mwhich is simply-connected at infinity is homeomorphic to Rn. See [Sta] for n≥5, and Corollary 1.2 on p. 366 of [Fre] for n= 4. A related reference is [McMZ]. Actually, [Sta] is stated for the piecewise-linear category; one can go from there to the topological category via [KirS]. The four-dimensional result does not work in the smooth or piecewise-linear categories (which are equivalent in dimension 4), because of the existence of “fake R4’s” (smooth manifolds homeomorphic to R4, but not diffeomorphic to it). Concerning the latter, see [FreQ] (p. 122 in particular) and [Kir] (Chapter XIV). 18 S. Semmes As a special case, imagine now that  Mis a finite polyhedron of dimension n. Let Ldenote the codimension-1 link of qin  M.ThusLis an (n−1)-dimensional finite polyhedron, and  Mlooks locally at qlike a cone over L. In order for  Mto be an n-dimensional topological manifold in a neighborhood of q, the link Lshould be fairly close to a standard (n− 1)-sphere. In particular, it is not hard to see that Lshould be homotopyequivalent to Sn−1. This implies that Lshould be connected and simplyconnected, under our assumption that nis at least 3. In fact, in the case where  Mis a finite polyhedron, the connectedness and simple-connectedness of the link Laround qare equivalent to the local connectivity and simple-connectivity conditions for  M\{q}near q indicated above, with (3.3) and (3.4). This is not hard to see, and it is also rather nice. To put it a bit differently, imagine that one starts with the class of finite polyhedra, and then tries to go to more general contexts of topological spaces. The local connectivity and simple-connectivity conditions for  M\{q}at qas described above provide a way to capture the information in the connectedness and simple-connectedness of the codimension-1 link at qin the case where  Mis a polyhedron, in a manner that makes sense for arbitrary topological spaces, without special structure as one has for finite polyhedra. 4. Decomposition spaces, 1 Let nbe a positive integer, and let Kbe a nonempty compact subset of Rn. One could also consider general manifolds instead of Rnhere, but we shall generally stick to Euclidean spaces for simplicity. The main ideas come up in this case anyway. Imagine shrinking Kto a single point, while leaving the rest of Rn alone, and looking at the topological space that results. This can be defined more formally as follows. Let us write Rn/K for the set which consists of the points in Rnwhich do not lie in K, together with a single point which corresponds to Kitself. In other words, this is where we shrink Kto a single point. This set can be given a topology in a standard way, so that a subset Uof Rn/K is open if and only if its inverse image back in Rnis open. Here “inverse image” uses the automatic quotient mapping Rnto Rn/K. (In concrete terms, the inverse image of Uin Rnmeans the set of points in Rnwhich correspond to elements of U, where one includes all points in Kif the element of Rn/K associated to Klies in U.) Some Topics Concerning Homeomorphisms 19 This type of quotient Rn/K is a special case of a “decomposition space”. We shall discuss the general situation further in Section 6, but this special case already includes a lot of interesting examples and phenomena. Now let us consider the following question: Given Kas above, when is Rn/K a topological manifold?(4.1) This is really a special case of the situation in Section 3. For this it is better to use Sninstead of Rn, so that Sn/K —defined in the same manner as above— is equivalent to the one-point compactification of Sn\K. Let us consider some basic examples. If Kconsists of only a single point, then Rn/K is automatically the same as Rnitself, and there is nothing to do. If Kis a finite set with more than one element, then it is easy to see that Rn/K is not a manifold. If we let qdenote the point in Rn/K which corresponds to K, then (Rn/K)\{q}does not enjoy the local connectedness property that it should if Rn/K were a manifold at q, as in (3.3) in Section 3. More precisely, this local connectedness property for the complement of {q}would be necessary only when n≥2. When n= 1, one does not have to have this local connectedness condition, but then (Rn/K)\{q}would have too many local components near qfor Rn/K to be a manifold at q. (That is, there would be more than 2 such local components.) Now suppose that Kis a straight line segment in Rn. In this event, Rn\Kis homeomorphic to Rnagain. This is not hard to check. This would also work if Kwere a standard rectangular cell of higher dimension in Rn. More generally, this works if Kis a tame cell in Rn, meaning the image of a standard rectangular cell under a homeomorphism of Rnonto itself. This follows automatically from the case of standard rectangular cells. However, if one merely assumes that Kis homeomorphic to a standard rectangular cell, then it is not necessarily true that Rn/K is a manifold! This is another aspect of wild embeddings, from Section 1. We shall say more about this as this section goes on. A concrete example is given by taking Kto be a copy of the Fox-Artin wild arc in R3. (Compare with [Fer].) Note that we are not saying that Rn/K is always not a manifold when Kis wildly embedded. The converse is true, that Kmust be wildly embedded when Rn/K is not a manifold (and Kis a topological 20 S. Semmes cell). This is just a rephrasal of the remark above, that Rn/K is a manifold when Kis a tamely embedded cell. Here is a slightly more foolish example, which one might view as a generalization of the earlier comments about the case where Kis a finite set with more than a single point. Imagine now that Kis a copy of the j-dimensional sphere Sj,1≤j≤n−1. For this let us use a standard, smooth, round sphere; it is not a matter of wildness that we want to consider. In this case Rn/K is never a topological manifold. If j=n−1, then Rn/K is homeomorphic to the union of Rnand an n-sphere, with the two meeting at a single point. This point is the one that corresponds to Kin Rn/K. Let us denote this point by qagain, as above. In this case (Rn/K)\{q}does not have the right local-connectedness property at q in order for Rn/K to be a manifold, as in (3.3) in Section 3. If j=n−2, then one runs into trouble with local simple-connectivity of (Rn/K)\{q}at q, as in (3.4) in Section 3. For this one might think about the special case where n= 3, so that Kis a standard circle in R3. It is easy to take small loops in R3\K, lying close to K, which are nonetheless linked with K. These loops then project down into (R3/K)\{q}, where they can be as close to the point qas one likes, but they are never contractable in (R3/K)\{q}at all, let alone in small neighborhoods of q(as in (3.4) in Section 3). This is the same as saying that these loops are not contractable inside of Rn\K, which is equivalent to (Rn/K)\{q}. When j<n−2, then one has similar obstructions to Rn/K being a manifold, but in terms of the failure of higher-dimensional forms of local connectedness of (Rn/K)\{q}(using homology or homotopy). This is analogous to the cases already described, when j=n−1orn−2. We shall say more about this soon, but for the moment let us go on to some other matters. For this example, where Kis taken to be a standard j-dimensional sphere, note that Rn/K itself is locally contractable at q. This is as opposed to connectedness properties of (Rn/K)\{q}, and it is analogous to what happens in the case of finite polyhedra. Specifically, for finite polyhedra one always has local contractability, but the behavior near a given point of punctured neighborhoods around that point is another matter. The latter is connected to the behavior of the codimension-1 link of the polyhedron around the given point, as in Section 3. In the present case, where we have Rn/K with Ka standard round j-dimensional sphere, one can see the local contractability of Rn/K at Some Topics Concerning Homeomorphisms 21 the point q(corresponding to K) as follows. In Rn, one can take a tubular neighborhood of K, which is homeomorphic to the Cartesian product of the j-sphere Kand an (n−j)-dimensional ball. This neighborhood can be contracted onto Kin a simple way, and this leads to the local contractability of Rn/K at q. Now let us consider the case of the Whitehead continuum, from Section 2. We should not really say the Whitehead continuum here, as there is some flexibility in the construction, which can lead to the resulting set Wnot being pinned down completely. This ambiguity will not really cause trouble for us here, and we can work with any compact set Win R3which is obtained as in the procedure described in Section 2. The set Whas the feature of being cell-like, as in the following definition. Definition 4.2 (Cell-like sets).A compact set Kin Rnis said to be cell-like if Kcan be contracted to a point inside of any neighborhood U of itself in Rn. Compare with [Dave2], especially p. 120. That the Whitehead continuum Wis cell-like is not hard to see from the construction of W,as the intersection of a decreasing sequence of solid tori with certain properties. Specifically, for this the key point is that the &th solid torus can be contracted to a point inside the previous one. If Kis a topological cell, then Kis contractable to a point inside of itself, without using the extra bit of room provided by a small neighborhood of itself. This is also independent of the way that Kmight be embedded into some Rn, i.e., wildly or tamely. For W,itisnot true that R3/W is a topological manifold. If we let q denote the point in R3/W corresponding to W, then (R3/W)\{q}is not locally simply-connected at q(in the sense of the condition in Section 3, around (3.4)). In concrete terms, this means that there are loops in R3\W(which is the same as (R3/W)\{q}) which lie as close to Was one likes (in their entirety), but which cannot be contracted to a point in R3\Wwhile remaining reasonably close to W. These loops can be described concretely, as meridians in the solid tori whose intersection gives W. The loop from the solid torus Tjcan be filled with a disk inside Tj, but not without crossing the smaller torus Tj+1, or any of its successors. This comes back to the way that each T+1 is “clasped” inside of T. See [Dave2], [Kir] for more information (including Proposition 9 on p. 76 of [Dave2]). In any event, the failure of the local simple-connectivity of (R3 /W)\{q} at qis equivalent to S3\Wnot being simply-connected at infinity, as in 22 S. Semmes Section 2. This also follows the discussion in Section 3, and the comment just after (4.1). This case is quite different from the one of embedding round spheres in Rn, as discussed before. More precisely, let us compare the situation with Wand the example before where Kis a standard circle inside of R3. For the latter, there are loops in R3\Kwhich lie as close to K as one wants, and which are not contractable to a point in R3\Kat all, let alone in a neighborhood of K.ForW, one has that S3\Wis contractable (as mentioned in Section 2), and this implies that R3\Wis simply-connected. (This is a straightforward exercise.) Thus these loops near Wcan be contracted to a point in R3\W, if one allows oneself to go away from Wfor the contraction. Here is another aspect of this. Although one has these loops in R3\W which lie near Wbut cannot be contracted to a point in R3\Wwhile staying near W, these loops can be made homologically trivial in R3\W while staying near W. That is, one can fill the loops with surfaces inside R3\Wwhile staying close to W, if one allows the surfaces to have handles (rather than simply being a disk, as in the case of homotopic triviality). This is something that one can easily see from the pictures (as in [Dave2], [Kir]). The basic idea is that one can fill the loops with disks, where the disks stay close to W, but also pass through W(and so are not in R3\W). However, one can avoid the intersection with Wby cutting out a couple of small holes in the disk, and attaching a handle to them which goes along the boundary of the solid torus in the next generation of the construction. Then Wwill stay inside this next solid torus, throughout the rest of the construction, and this surface gives a way of filling the loop without intersecting W(or being forced to go far away from it). This kind of filling by surfaces does not work in the case where we take Kto be a standard circle in R3. In this situation, we have loops in R3\Kwhich lie close to K, and which are linked homologically with the circle K. In other words, the linking number of the loop with Kis nonzero, and this linking number is a homological invariant which would vanish if the loop could be filled with a surface without intersecting K. (For more about “linking numbers”, see [BotT], [Bre1], [Fla], [Spa].) Here is another feature of W, which distinguishes it from ordinary circles in R3(or spheres in Rnmore generally). Let us think of Wnow as lying in R4rather than R3, through the inclusion of R3in R4by taking the fourth coordinate to be 0. For R4, we have that R4/W is a topological manifold (homeomorphic to R4). The basic point behind this is the following. In the realization Some Topics Concerning Homeomorphisms 23 of Was the intersection of a decreasing sequence of solid tori in R3, the &th solid torus was always “clasped” in the previous one (as in Section 2, and [Dave2], [Kir]). In R4, the extra dimension provides a lot of extra room, in such a way that this “clasping” is not really present any more. If Tis a solid torus which is embedded and clasped inside of another solid torus Tin R3, one can “unclasp” Tin R4by lifting one end up, bringing it around the hole in T, and leaving the other end alone. This is a standard observation, and it is analogous to the way that knots in R3become unknotted in R4. In other words, this procedure gives a way to make a deformation of R4, in which the solid torus Tis mapped to a set of small diameter, while not moving points some distance away at all. By contrast, back in R3, it is not possible to make an isotopy which shrinks Tto a set of small diameter, while leaving the points in the complement of the larger solid torus fixed. This is exactly because of the way that Tis “clasped” in T, so that it cannot be “unclasped” by an isotopy in T. When one has the extra dimension in R4, one can “undo” the clasping, by lifting one end up and moving it around, as indicated above. Once one has this kind of “shrinking” in R4, one can use this to show that R4/W is homeomorphic to R4. One can do this directly, using shrinking homeomorphisms like this, and combinations of them, to make a mapping from R4to itself which shrinks Wto a point while remaining injective (and continuous) everywhere else. One puts homeomorphisms like this on top of each other, and deeper and deeper in the construction of W, until Witself is shrunk all the way to a point. The various solid tori Tjin the construction, of which Wis the intersection, are made smaller and smaller in this process. The trick is to do this without shrinking everything, so that the mapping that results remains a homeomorphism on the complement of W. This idea of shrinking can be given a general form, and is discussed in detail in [Dave2]. See also [Edw2], [Kir]. By contrast, let us consider the case of a circle Kin R3. If one views Kas a subset of R4in the same way, then R4/K is still not a topological manifold. This follows from our earlier discussion about circles and spheres of higher dimensions inside of Rnin general. One also does not get a manifold by replacing R4with Rmfor larger m’s. Notice, however, that there is a kind of “improvement” that occurs in adding dimensions in this way. If Kis a circle in R3, and if qdenotes the point in R3/K which corresponds to K, then (R3/K)\{q}is not locally simply-connected at q. For that matter, (R3/K)\{q}∼ =R3\K is not simply-connected at all. When one considers Kas a subset of 24 S. Semmes R4, and asks analogous questions for R4/K (or R4\K), then there is no longer any trouble with simple-connectivity. The basic underlying problem continues, though, in the form of 2-dimensional connectivity. This is not hard to see. Similarly, if one views Kas a subset of Rnfor larger n, then the trouble with connectivity in lower dimensions goes away, but (n−2)-dimensional connectivity still does not work. With the Whitehead continuum we are more fortunate. The problem with local simple-connectivity goes away when we proceed from R3 to R4, and difficulties with higher-dimensional connectivity do not then arise in their place. One should not be too surprised about this, since the Whitehead continuum is cell-like, while circles or spheres of higher dimension are not at all cell-like. In other words, with circles or spheres (and their complements in Rn), there is some clear and simple nontrivial topology around, while the Whitehead continuum is much closer to something like a standard cell, which causes less trouble. 4.1. Cellularity, and the cellularity criterion. Now let us look at some general notions and results, concerning the possibility that Rn/K be a topological manifold (and, in fact, homeomorphic to Rn). Definition 4.3 (Cellularity).A compact set Kin Rn(or, more generally, an n-dimensional topological manifold) is said to be cellular if it can be realized as the intersection of a countable family of sets Bi, where each Biis a topological n-cell (or, equivalently, homeomorphic to the closed unit ball in Rn), and if each Bi+1 is contained in the interior of the preceding Bi. Compare with [Dave2], especially p. 35, [Edw2], and p. 44 of [Rus1]. Alternatively, a compact set Kis cellular if and only if any neighborhood of Kcontains an open set which contains Kand is homeomorphic to the standard n-dimensional ball. Theorem 4.4. Let Kbe a compact subset of Rn. Then Rn/K is a topological manifold if and only if Kis cellular in Rn. In this case, Rn/K is homeomorphic to Rn. See Exercise 7 on p. 41 of [Dave2] for the first assertion, and Proposition 2 on p. 36 of [Dave2] for the second one. (Concerning the latter, see Section 5 in [Dave2] too. Note that some of the notation in Exercise 7 on p. 41 in [Dave2] is explained in the statement of Proposition 2 on Some Topics Concerning Homeomorphisms 25 p. 36 of [Dave2].) See also [Edw2], especially the theorem on p. 114, and p. 44ff of [Rus1]. For the record, let us mention the following. Proposition 4.5. Let Kbe a compact subset of Rn.IfKis cellular, then Kis cell-like. Conversely, if nis equal to 1or 2, then Kis cellular if it is cell-like. The fact that cellularity implies cell-likeness follows easily from the definitions. When n= 1, the converse is very simple, since connectedness implies that a set is an interval, and hence cellular. In R2, the argument uses special features of plane topology. See Corollary 4C on p. 122 of [Dave2]. In higher dimensions, cell-like sets need not be cellular. Examples are given by Whitehead continua, and some wild embeddings of cells. However, there is an exact characterization of cellular sets among cell-like sets, which is the following. Basically, the point is to include the same kind of localized simple-connectivity of Rn\Karound Kas discussed before. Theorem 4.6. Let Kbe a compact set in Rn, with n≥3. Then Kis cellular inside of Rnif and only if (a) it is cell-like, and (b) for every open neighborhood Uof Kin Rnthere is another open neighborhood Vof Kso that every continuous mapping from S1into V\Kcan be contracted to a point inside of U\K. This characterization of cellularity is stated in Theorem 5 on p. 145 of [Dave2]. This uses also the definition of the cellularity criterion given on p. 143 of [Dave2]. When n≥4, this result works for subsets of general n-dimensional topological manifolds, and not just Rn. When n=3, there is trouble with the general case of manifolds, related to the 3-dimensional Poincar´e conjecture being unsettled; if the cellularity criterion holds for general manifolds, then the 3-dimensional Poincar´e conjecture would follow, as discussed on p. 145 of [Dave2]. See Theorem 1.11 on p. 373 of [Fre] concerning the 4-dimensional case, and [McM] and Section 4.8 of [Rus1] for dimensions 5 and higher. Corollary 4.7. Let Kbe a compact subset of Rn,n≥3.IfKis celllike in Rn, then K×{0}is cellular in Rn+1. See Corollary 5A on p. 145 of [Dave2]. The main point behind the derivation of Corollary 4.7 from Theorem 4.6 is that by passing to a Euclidean space of one higher dimension, potential trouble with local simple-connectedness of the complement of Kgoes away. This fits with basic examples, and the Whitehead continuum in particular. 26 S. Semmes Let us note the following simple converse to Corollary 4.7. Lemma 4.8. Suppose that Kis a compact subset of Rn.IfK×{0}is cellular in Rn+1, then Kis cell-like in Rn. Indeed, if K×{0}is cellular in Rn+1, then it is also cell-like in Rn+1, as in Proposition 4.5. It is easy to check that cell-likeness for K×{0} in Rn+1 implies cell-likeness for Kinside Rn, just by the definitions. (Thus cell-likeness, unlike cellularity, is not made more feasible by the extra room of extra dimensions.) This implies Lemma 4.8. For concrete examples of cell-like sets, often the cellularity in higherdimensional spaces, as in Corollary 4.7, can be seen in fairly direct and simple terms. The room from the extra dimensions makes it easy to move pieces of the set apart, without the claspings, knottings, etc., which occurred originally. Some aspects of this came up earlier, concerning Whitehead continua. Note that the localized simple-connectivity conditions that are used here are a bit different from those employed in the context of taming theorems, as in Section 1. To make this precise, let Kbe a compact subset of some Rn. The conditions that come up in the present section involve the behavior of Rn\K, localized around K(the whole of K). That is, one looks at the behavior of Rn\Kwithin arbitrarily-small neighborhoods of Kin Rn. In the context of Section 1, one would look at the behavior of Rn\Knear individual points in K. To put it another way, here one seeks to contract loops in Rn\K that are close to Kto points, while staying close to K. In the context of Section 1, one looks at small loops in Rn\Knear K, and tries to contract them to points in the complement of Kwhile staying in small balls, and not just staying near K. 5. Manifold factors Let Wbe a Whitehead continuum, constructed through a decreasing sequence of solid tori in R3, as in Section 2. Theorem 5.1. If R3/W is defined as in Section 4, then (R3/W)×R is homeomorphic to R4. In particular, (R3/W)×Ris a topological manifold, even though R3/W itself is not. Thus R3/W is a manifold factor. The fact that (R3/W)×Ris homeomorphic to R4is given as Corollary 3B on p. 84 of [Dave2]. See also [AndR], [Kir]. Note that the existence of a homeomorphism from (R3/W)×Ronto R4is not the same as the observation mentioned in Section 4, that Some Topics Concerning Homeomorphisms 27 R4/(W×{0}) is homeomorphic to R4. In considering (R3/W)×R, one is in effect taking R4, and then shrinking each copy W×{u}of Wto a point, where uruns through all real numbers. For R4/(W×{0}), one shrinks only a single copy of Wto a point. Although the construction is more complicated for (R3/W)×Rthan for R4/(W×{0}), there are some common aspects. As before, one of the main points is that the solid tori in R3which are “clasped” (inside of other solid tori) become unclasped in R4. With the extra dimension in R4, one can pick up one end of one of these tori, bring it around, and then lay it down again, so that the clasping is undone. For the present situation with (R3/W)×R, one performs this kind of action for all of the copies W×{u}of Wat once, u∈R, rather than just a single copy. (Compare also with Section 6, and the general notion of decomposition spaces mentioned there.) In Section 2, it was mentioned that S3\Wis a contractable open set which is not homeomorphic to a 3-ball (because it is not simplyconnected at infinity), and that (S3\W)×Ris homeomorphic to a 4-dimensional open ball. (See [Bin3], [Bin4], [Kir].) This result is similar in some ways to Theorem 5.1, but the conclusions are not quite the same either. In this vein, let us make the following observation. As usual, denote by qthe (singular) point in R3/W that corresponds to W. Let us write L for the subset of (R3/W)×Rgiven by {q}×R.ThusLis homeomorphic to a line. Using a homeomorphism from (R3/W)×Rto R4, one gets an embedding of Linto R4. It is not hard to see that any such embedding of Linto R4has to be wild. Just as R3\Wis not locally simply-connected near W,ifLdenotes the image of Lin R4by an embedding as above, then R4\L is not locally simply-connected near L. (Note that R4\L is homeomorphic to (R3\W)×R, by construction.) This ensures that L is wild in R4, no matter what homeomorphism from (R3/W)×Ronto R4one might use, since ordinary straight lines in R4do not behave in this way. One can also make local versions of this argument, to show that Lis locally wild in the same manner. If one were to want to pass from a homeomorphism from (R3/W)×R onto R4in Theorem 5.1 to a homeomorphism from (S3\W)×Ronto R4, then in particular one could be lead to try to figure out something about what happens when one deletes Lfrom R4. Conversely, if one wanted to go in the other direction, one might have to figure out something about 34 S. Semmes of something like dilations, translations, rotations, and reflections. In other words, except for a uniform scale factor, one might hope that the geometry does not have to change. Normally this will not be the case. Some amount of bending or twisting, etc., will (in general) be involved, and needed, to accommodate the kind of topological behavior that is present. This includes linking, clasping, or things like that. For the purpose of choosing a geometry that might fit with a given decomposition space, however, one can modify the usual Euclidean metric so that the embeddings involved in the basic “rule” do have the kind of behavior indicated above, i.e., a constant scale factor together with an isometry. The scale factors should be less that 1, to reflect the shrinking that is supposed to take place for the decomposition spaces (even at a purely topological level). It is not hard to see that one can make deformations of geometry like this. One can do this in a kind of direct and “intrinsic” way, defining metrics on Rnwith suitable properties. One can also do this through embeddings of the decomposition spaces into higher-dimensional Euclidean spaces. In these higher-dimensional Euclidean spaces, the self-similarity that one wants, in typical situations, can be realized in terms of standard linear self-similarity, through dilations and translations. More precisely, in these circumstances, the quotient of Rnby the decomposition can be realized topologically as an n-dimensional subset X of some RN(with N=n+ 1, for instance), in such a way that X is a smooth submanifold away from the natural singularities, and Xis self-similar around these singularities. To build such a set X, one can start with the complement of the original domain Din Rn. One would view Rn\Das an n-dimensional submanifold of RN. In place of the iteration of the basic rule for the decomposition from before, one now stacks some “basic building blocks” in RNon top of Rn\D(along the boundary of D), and then on top of the other building blocks, over and over again. These basic building blocks are given by n-dimensional smooth manifolds in RN(with boundary). They are diffeomorphic to a single “model” in Rn, which is the original domain Din Rn, minus the interiors of the mcopies of Dembedded inside D, as given by the basic “rule” that generates the decomposition. The building blocks are all diffeomorphic to each other, since they are all diffeomorphic to this same model, but they are also constructed in such a way as to be “similar” to each other. That is, they can all be given by translations and dilations of each other. Some Topics Concerning Homeomorphisms 35 This is a key difference between this construction and the original decomposition in Rn. Further, the building blocks are constructed in such a way that their ends are all similar to each other (i.e., even different ends on the same building block). Specifically, the building blocks are chosen so that when one goes to stack them on top of each other, their “ends” fit together properly, with smoothness across the interfaces. These things are not difficult to arrange. Roughly speaking, one uses the extra dimensions in RNto straighten the “ends” in this way, so that the different building blocks can be stacked properly. Typically, this would involve something like the following. One starts with the basic model in Rn, given by Dminus the interiors of the membedded copies of Din D. One then makes some translations of the membedded subdomains in D, up into the extra dimension or dimensions in RN.Up there, these subdomains can be moved or bent around, until they are similar to Ditself (i.e., being the same modulo translations and dilations). This can be done one at a time, and without changing anything near the boundary of the original domain D. In this manner, the original model region in Rnbecomes realized as an n-dimensional compact smooth submanifold (with boundary) in RN, with the ends matching up properly under similarities. To put it another way, the main “trade-off” here is that one gives up the “flatness” of the original model, as a region in Rn, to get basic building blocks in RNthat are n-dimensional curved submanifolds whose ends are similar to each other. The curving of the interiors of these building blocks compensates for the straightening of their ends. As above, one then stacks these building blocks on top of each other, one after another, to get a realization of the decomposition space by an n-dimensional subset Xof RN. (One also puts in some limiting points, at the ends of the towers of the building blocks that arise. In other words, this makes Xbe a closed subset of RN. These extra points are the singularities of X.) This subset is smooth away from the singularities, and self-similar at the singularities, because of the corresponding properties of the basic building blocks. By choosing the scale-factors associated to the ends of the basic building blocks to be less than 1, the diameters of the ends tend to 0 (and in a good way) as one stacks the building blocks on top of each other many times. This corresponds to the fact that the sets in the decomposition are supposed to be shrunk to single points in the quotient space. This is also part of the story of the “limiting points” in the previous paragraph. The limiting points are exactly the ones associated (in the end) to the 36 S. Semmes nondegenerate sets in the original decomposition in Rn, which are being shrunk to single points. The actual homeomorphic equivalence between the set Xin RNproduced through this method and the decomposition space Rn/G with which one starts is obtained using the diffeomorphic equivalence between the building blocks in RNand the original model in Rn(Dminus the interiors of the membedded copies of itself, as above). In rough terms, at the level of the topology, the same kind of construction is occurring in both places, Xand the decomposition space, and one can match them up, by matching up the individual building blocks. This is not hard to track. Instead of stacking building blocks on top of each other infinitely many times, one can stop after finitely many steps of the construction (and add in suitable plugs to fill in the holes). This gives a set which is still smooth, and diffeomorphic to Rn, and which approximates the non-smooth version that represents the decomposition space. When one makes constructions like these —either finite approximations or infinite limits— the self-similarity helps to ensure that the spaces behave geometrically about as well as they could. See [Sem3] for more information, and some slightly different versions of these basic themes. 7.2. Quotient spaces can be topologically standard, but geometrically tricky. We have seen before how decompositions of Rnmight lead to Rnagain topologically in the quotient, but do so in a manner that is still somehow nontrivial. For instance, the decomposition might arise from a nontrivial manifold factor, or lead to wild embeddings in the quotient which seem very simple (like a straight line) at the level of the decomposition. In these situations, one can still have highly nontrivial geometries from the procedures described in Subsection 7.1, even though the underlying space is topologically equivalent to Rn. As a special case, wild embeddings in the quotient can have nice metric properties in the kind of geometric realizations discussed here, while the same properties would not be possible in Rnwith the standard Euclidean metric. Here is a concrete instance of this. Let Wbe a Whitehead continuum in R3, as in Section 2. Consider the corresponding quotient space R3/W, as in Section 4. One can realize R3/W topologically as a subset of R4, where this subset is smooth away from the singular point, and has a simple self-similarity at the singular point, as in Subsection 7.1. Similarly, Some Topics Concerning Homeomorphisms 37 one can think of (R3/W)×Ras being given as a subset of R5, namely, as the product of the one in R4with R. Let us write qfor the singular point in R3/W, i.e., the point in the quotient which corresponds to W, and set L={q}×R. With respect to the embedding of (R3/W)×Rinto R5,Lhas Hausdorff dimension 1, and bounded subsets of it have finite 1-dimensional Hausdorff measure. However, the image of Linside of R4under a homeomorphism from (R3/W)×Ronto R4will be wild. As in Section 5, the image of Lin R4 under such a homeomorphism has Hausdorff dimension at least 2, with respect to the usual Euclidean metric in R4. This uses Theorem 5.2. This shows that the geometry that we have for (R3/W)×Rhas to be substantially different from the usual Euclidean geometry on R4,even though the two spaces are topologically equivalent. Specifically, even though there are homeomorphisms from (R3/W)×Ronto R4, no such homeomorphism can be Lipschitz, or even H¨older continuous of order larger than 1/2. (Recall that a mapping is Lipschitz if for each pair of points in the domain, the distance between their images is bounded by a constant times the distance between the points themselves. A mapping is H¨older continuous of order αif the distance between the images of two points is bounded by a constant times the distance between the two original points raised to the power α. For this condition, it is often natural to restrict one’s attention to pairs of points which are no more than distance 1 apart, or to points in bounded regions.) Although (R3/W)×R—with the kind of geometry described above— is quite different from R4with the usual Euclidean metric, there is a strong and nice feature that it has, in common with R4. We shall call this property “uniform local coordinates”. Since (R3/W)×Ris homeomorphic to R4, it has homeomorphic local coordinates from R4at every point. “Uniform local coordinates” asks for a stronger version of this, and is more quantitative. Specifically, around each metric ball Bin (R3/W)×R(with respect to the kind of geometry that we have), there are homeomorphic local coordinates from a standard Euclidean ball βof the same radius in R4, such that the image of βunder the coordinate mapping covers the given ball Bin (R3/W)×R, (7.1) and the modulus of continuity of the coordinate mapping and its inverse can be controlled, uniformly over all choices of metric balls Bin (R3/W)×R, and in a scale-invariant manner. (7.2) 38 S. Semmes Here “modulus of continuity” means a function ω(r) so that when two points in the domain (of a given mapping) are at distance ≤r, their images are at distance ≤ω(r). Also, rwould range through positive numbers, and ω(r) would be nonnegative and satisfy lim r→0ω(r)=0.(7.3) This last captures the continuity involved, and, in fact, gives uniform continuity. For a mapping from a compact metric space to another metric space, continuity automatically implies uniform continuity, and that implies the existence of some modulus of continuity. This is not to say that one knows much about the modulus of continuity, a priori. (One can always choose it to be monotone, for instance, but one cannot in general say how fast it tends to 0 as r→0.) Concrete examples of moduli of continuity would include ω(r)=Cr for some constant C, which corresponds to a mapping being Lipschitz with constant C,orω(r)=Cr α,α>0, which corresponds to H¨older continuity of order α. One can have much slower rates of vanishing, such as ω(r) = (log log log(1/r))−1. In our case, with the property of “uniform local coordinates”, we want to have a single modulus of continuity ω(r) which works simultaneously for all of the local coordinate mappings (and their inverses). Actually, we do not look at moduli of continuity for the mappings themselves, but renormalized versions of them. The renormalizations are given by dividing distances in the domain and range by the (common) radius of Band β. In this way, Band βare viewed as though they have radius 1, independently of what the radius was originally. This gives a kind of uniform basis for making comparisons between the behavior of the individual local coordinate mappings and their moduli of continuity. Let us return now to the special case of (R3/W)×R, with the geometry as before. In this case one can get the condition of uniform local coordinates from the existence of topological coordinates (without uniform bounds), together with the self-similarity and smoothness properties of the set. Here is an outline of the argument. (A detailed version of this, for a modestly different situation, is given in [Sem3]. Specifically, see Theorem 6.3 on p. 241 in [Sem3]. Note that the property of uniform local coordinates is called “Condition (∗∗)” in [Sem3], as in Definition 1.7 on p. 192 of [Sem3].) Let Bbe a metric ball in (R3/W)×R, for which one wants to find suitable coordinates. Assume first that Bdoes not get too close to the singular line {q}×Rin (R3/W)×R, and in fact that the radius of Some Topics Concerning Homeomorphisms 39 Bis reasonably small compared to the distance from Bto the singular line. In this case (R3/W)×Ris pretty smooth and flat in B, by construction (through the method of Subsection 7.1). This permits one to get local coordinates around Bquite easily, and with suitable uniform bounds for the moduli of continuity of the coordinate mappings and their inverses. The bounds that one gets are scale-invariant, because of the self-similarity in the geometric construction (from Subsection 7.1). In fact, one can have Lipschitz bounds in this case, as well as stronger forms of smoothness. If the ball Bis reasonably close to the singular line {q}×R, then one can reduce to the case where it is actually centered on {q}×R. That is, one could replace Bwith a ball which is centered on {q}×R, and which is not too much larger. (The radius of the new ball would be bounded by a constant times the radius of B.) This substitution does not cause trouble for the kind of bounds which are sought here. Thus we suppose that Bis centered on the line {q}×R.Wemay as well assume that the center of Bis the point (q,0). This is because (R3/W)×Rand the geometry that we have on it are invariant under translations in the Rdirection, so that one can move the center to (q,0) without difficulty, if necessary. Using the self-similarity of (R3/W)×R, one can reduce further to the case where the radius of Bis approximately 1. For that matter, one can reduce to the case where it is equal to 1, by simply increasing the radius by a bounded factor (which again does not cause problems for the uniform bounds that are being considered here). (To be honest, if one takes the geometry for R3/W to be flat outside a compact set, as in Subsection 7.1, then this reduction is not fully covered by self-similarity. That is, one should handle large scales a bit differently. This can be done, and a similar point is discussed in [Sem3] in a slightly different situation (for examples based on “Bing doubling”).) Once one makes these reductions, one gets down to the case of the single ball Bin (R3/W)×R, centered at (q,0) and with radius 1. For this single choice of scale and location, one can use the fact that (R3/W)×R is homeomorphic to R4to get suitable local coordinates. For this single ball B, there is no issue of “uniformity” in the moduli of continuity for the coordinate mappings. One simply needs amodulus of continuity. A key point, however, is that when one works backwards in the reductions just made, to go to arbitrary balls in (R3/W)×Rwhich are relatively close to the singular line {q}×R, one does get local coordinates with uniform control on the (normalized) moduli of continuity of 40 S. Semmes the coordinate mappings and their inverses. This is because of the way that the reductions cooperate with the scaling and the geometry. At any rate, this completes the outline of the argument for showing that (R3/W)×Rhas uniform local coordinates, in the sense described before, and with the kind of geometry for (R3/W)×Ras in Subsection 7.1. It is easy to see that a metric space which is bilipschitz equivalent to some Rnhas uniform local coordinates (relative to Rnrather than R4, as above). (Recall that two spaces are bilipschitz equivalent if there is a homeomorphism from one onto the other which is both Lipschitz and has Lipschitz inverse.) The required local coordinates can simply be obtained from restrictions of the global bilipschitz parameterization. The converse is not true in general, i.e., a space can have uniform local coordinates and not be bilipschitz equivalent to the corresponding Rn. An example of this is given by (R3/W )×Rwith the kind of geometry that we have been considering. One can also get much simpler examples, by taking snowflake spaces. That is, one can take Rnequipped with the metric |x−y|α, where |x−y| is the usual metric, and αis a positive real number strictly less than 1. It is not hard to check that this has the property of uniform local coordinates. It is not bilipschitz equivalent to Rn, because it has Hausdorff dimension n/α instead of Hausdorff dimension n. The previous example based on (R3/W)×Rbehaves much better than this, though, with the correct Hausdorff dimension (namely, 4) in particular. Notice that the uniform local coordinates property would imply a bilipschitz condition if the local coordinates all came from restrictions of a single global parameterization. This is because of the way that the scaling works. In general, the uniform local coordinates property allows the local coordinate mappings to change as one changes locations and scales, and this is why it allows for there to be no global bilipschitz parameterization. The case of (R3/W)×Rprovides a good example of this. Another way to think about the uniformity over all locations and scales in the uniform local coordinates property is that it is a condition which implies the existence of homeomorphic coordinates even after one “blows up” the space (in the Hausdorff or Gromov-Hausdorff senses) along any sequence of locations and scales in the space. With the uniform local coordinates property, the local coordinates could be “blown up” along with the space, with the uniform bounds for the moduli of continuity providing the equicontinuity needed to take limits of the coordinate mappings (after passing to suitable subsequences). Some Topics Concerning Homeomorphisms 41 Instead of looking at uniform local coordinates in connection with bilipschitz equivalence with Rn, one can consider quasisymmetric equivalence. Roughly speaking, a quasisymmetric mapping between two metric spaces is one that approximately preserves relative distances, in the same way that bilipschitz mappings approximately preserve actual distances. In other words, if one has three points x,y, and zin the domain of such a mapping, and if xis much closer to ythan zis, then this should also be true for their images under a quasisymmetric mapping, even if the actual distances between the points might be changed a lot. See [TukV] for more information about quasisymmetric mappings. Two metric spaces are quasisymmetrically equivalent if there is a quasisymmetric mapping from one onto the other. As with bilipschitz mappings, compositions and inverses of quasisymmetric mappings remain quasisymmetric. If a metric space admits a quasisymmetric parameterization from Rn, then it also satisfies the condition of uniform local coordinates. This is true for nearly the same reason as for bilipschitz mappings; given a quasisymmetric mapping from Rnonto the metric space, one can get suitable local coordinates for the space from restrictions of the global mapping to individual balls. There is a difference between this case and that of bilipschitz mappings, which is that one should allow some extra rescalings to compensate for the fact that distances are not approximately preserved. Specifically, a ball Bin the metric space may be covered in a nice way by the image under a quasisymmetric mapping of a ball β in Rn, but the radii of Band βneed not match up. For a bilipschitz mapping, one would be able to choose βso that it has radius which is comparable to that of B. In the quasisymmetric case, one may not have that, but if one adds an extra rescaling on Rn(depending on the choices of balls), then one can still get local coordinates with the kind of uniform control on the (normalized) moduli of continuity as in the uniform local coordinates condition. If a metric space has uniform local coordinates with respect to Rn, and if these coordinates come from restrictions and then rescalings (on Rn) of a single global parameterization, as in the preceding paragraph, then that parameterization does have to be quasisymmetric. This is an easy consequence of the definitions, and it is analogous to what happens in the bilipschitz case. If a metric space admits uniform local coordinates from some Rn,it still may not be true that it admits a quasisymmetric parameterization. This is trickier than before, and in particular one does not get examples 42 S. Semmes simply by using snowflake metrics |x−y|αon Rn. Indeed, the identity mapping on Rnis quasisymmetric as a mapping from Rnwith the standard metric to Rnwith the snowflake metric |x−y|α,0<α<1. However, there are counterexamples, going back to results of Rickman and V¨ais¨al¨a. That is, these are spaces which have uniform local coordinates (and are even somewhat nicer than that), but which do not admit quasisymmetric parameterizations. Basically, these spaces are Cartesian products, where the individual factors can behave nicely in their own right, and where the combination mixes different types of geometry. A basic example (which was the original one) is to take a product of a snowflake with a straight line. Quasisymmetric mappings try to treat different directions in a uniform manner, and in the end this does not work for parameterizations of these examples. See Lemma 4 in [Tuk], and also [V¨ai3] and [AleV]. These examples occur already in dimension 2. They do not behave well in terms of measure, though. This is a basic part of the story; compare with [AleV], [Tuk], [V¨ai3]. As usual, dimension 1 is special. There are positive results starting from more primitive conditions, and good characterizations for the existence of quasisymmetric parameterizations, in fact. See Section 4 of [TukV]. In dimension 2, there are positive results about having global quasisymmetric parameterizations for a given space, and with bounds, under additional assumptions of good behavior in terms of 2-dimensional measure, which include having Hausdorff dimension 2. Instead of “uniform local coordinates”, one can assume a priori weaker conditions about the geometry and topology. See [DaviS1], [HeiKo1], [Sem1]. For these results, it is important that the dimension be 2, and not larger, because of the way that they rely on the existence of conformal mappings. We shall say a bit more about this in Section 8. In dimension 3, there are counterexamples, even in the context of good behavior in terms of measure. These examples are based on decompositions of R3, using geometric realizations as in Subsection 7.1. This is discussed in [Sem3]. The absence of a quasisymmetric parameterization in this case is close to a result in [FreS], although the setting in [FreS] is different. Concerning the space (R3/W)×Rconsidered before, equipped with a nice geometry as from Subsection 7.1, it is not clear (to my knowledge) whether quasisymmetric parameterizations from R4exist or not. We saw earlier that bilipschitz mappings do not exist, because of the line in (R3/W)×Rwhich has to have Hausdorff dimension at least 2 after any Some Topics Concerning Homeomorphisms 43 homeomorphism from (R3/W)×Ronto R4. These considerations of Hausdorff dimension or measure do not by themselves rule out the existence of a quasisymmetric mapping, as they do for bilipschitz mappings. (Compare with [V¨ai2], for instance.) Similar remarks apply to double-suspensions of homology spheres. In particular, it is not known (to my knowledge) whether or not quasisymmetric parameterizations exist for them. 7.3. Examples that are even simpler topologically, but still nontrivial geometrically. Let us mention another class of examples, which one can also think of in terms of decompositions (although they are “trivial” in this respect). These examples are based on “Antoine’s necklaces”, which came up before, in Section 1. Antoine’s necklaces are compact subsets of R3which are homeomorphic to the usual middle-thirds Cantor set in the real line, but for which there is no global homeomorphism from R3onto itself which maps these sets into subsets of a line. In dimension 2 this does not happen, as in Chapter 13 of [Moi]. The “wildness” of these sets is manifested in a simple fundamentalgroup property. Namely, the complement of these sets in R3have nontrivial fundamental group, whereas this would not be true if there were a global homeomorphism from R3to itself which would take one of these sets to a subset of the line. This last uses the fact that these sets are totally disconnected (i.e., to have simple-connectivity of the complement in R3if the set were to lie in a line). Antoine’s necklaces are discussed in Chapter 18 of [Moi]. See also p. 71ff in [Dave2]. The basic construction for them can be described in terms of the same kind of “rules” as in Subsection 7.1. One starts with a solid torus Tin R3. Inside this torus one embeds some more tori, which are disjoint, but which form a chain that is “linked” around the hole in the original torus. (See Figure 18.1 on p. 127 of [Moi], or Figure 9.9 on p. 71 of [Dave2].) In each of these smaller tori, one can embed another collection of linking tori, in the same way as for the first solid torus. One can repeat this indefinitely. In the limit, one gets a Cantor set, which is a necklace of Antoine. Actually, we should be a little more precise here. In saying that we get a Cantor set in the limit, we are implicitly imagining that the diameters of the solid tori are going to 0 as one proceeds through the generations of the construction. This is easy to arrange, if one uses enough tori in 50 S. Semmes A conformal deformation of the standard metric is defined by 1 realvalued function of nreal variables, i.e., for the conformal factor. A general diffeomorphism in ndimensions is described by nreal-valued functions of nvariables. Thus, allowing for general changes of variables, the metrics which are conformally-equivalent to the standard metric are described by n+1 real-valued functions of nreal variables. When n=2, this is equal to n(n+1)/2, but for n>2 one has that n(n+1)/2>n+1. In fact, one knows that in dimension 3 there are numerous examples of spaces which satisfy geometric conditions analogous to ones that work in dimension 2, but which do not admit quasisymmetric parameterizations. There are also different levels of structure which occur in dimension 3, between basic geometric properties and having quasisymmetric parameterizations, and which would come together in dimension 2. Parts of this are reviewed or discussed in Section 7, especially Subsection 7.2; see [Sem3] for more information. Thus, not only does the method based on conformal and quasiconformal mappings not work in higher dimensions, but some of the basic results that one might hope to get or expect simply are not true, by examples which are pretty concrete. This is all pretty neat! One has kinds of “parallel tracks”, with geometric topology on one side, and aspects of geometry and analysis on the other. A priori, these two tracks can exist independently, even if there are ways in which each can be involved in the other. Each of these two tracks has special features in low dimensions. This concerns the existence of homeomorphisms with certain properties, for instance. Each has statements and results and machinery which make sense for the given track, and not for the other side, even if there are also some overlaps (as with applications of Riemann mappings). Each of these two tracks also starts running into trouble in higher dimensions, and at about the same time! The kinds of trouble that they encounter can be rather different a priori, even if there is again significant overlap between them. 8.1. Special coordinates that one might consider in other dimensions. Let us now briefly consider a couple of things that one might try in higher dimensions on the side of geometry and analysis, in similar veins as above. One basic approach would be to try to find and use mappings which minimize some kind of “energy”. As before, one can consider smooth metrics on smooth manifolds (like Sn), and try to get parameterizations Some Topics Concerning Homeomorphisms 51 with uniform bounds on their behavior, under modest conditions on the geometry of the spaces. (One can also try to work directly with spaces and metrics that are not smooth.) A very standard energy functional to consider would be the L2norm of the differential, as with harmonic mappings. In dimension 2, conformal mappings can be placed in this framework. One can also consider energy functionals based on Lpnorms of differentials of mappings. This is more complicated in terms of the differential equations that come up, but it can have other advantages. The choice of pas the dimension nhas some particularly nice features, for having the energy functional cooperate with the geometry (and analysis). (This is one of the ways that n=2is special; for this one can have both p=nand p= 2 at the same time!) In particular, the energy becomes invariant under conformal changes in the metric when pis equal to the dimension. In elasticity theory, one considers more elaborate energy functionals as well. For instance, these might include integral norms of the inverse of the Jacobian of the mapping, in addition to Lpnorms of the differentials. In other words, the functional can try to limit both the way that the differential becomes large and small, so that it takes into account both stretchings and compressions. In any case, although there is a lot of work concerning existence and behavior of minimizers for functionals like these, I do not really know of results in dimensions n≥3 where they can be used to obtain wellbehaved parameterizations of spaces, with bounds, under modest or general geometric conditions. This is especially true in comparison with what one can get in dimension 2, as discussed before. In dimension 3, there is another kind of special structure that one might consider. Namely, instead of metrics which are conformal deformations of the standard Euclidean metric, let us consider metrics g=gi,j for which only the diagonal entries gi,i are nonzero. In this case the diagonal entries are allowed to vary independently. For conformal deformations of the standard Euclidean metric, the offdiagonal entries are zero, and the diagonal entries are all equal. In dimension 3, the problem of making a change of variables to put a given metric into diagonal form like this is “determined”, in the same way as for conformal deformations of Euclidean metrics in dimension 2. Specifically, one can compute as follows. A general Riemannian metric is described by n(n+1)/2 real-valued functions of nvariables, which means 6 real-valued functions of 3 real variables in dimension 3. Metrics with only diagonal nonzero entries are defined by 3 real-valued functions of 52 S. Semmes 3-real variables, and changes of variables are given by 3 real-valued functions of 3 real variables as well. Thus, allowing for changes of variables, the metrics that can be reduced to diagonal metrics can be described by 6 real-valued functions of 3 real-variables, which is the same as for the total family of Riemannian metrics in this dimension. In dimensions greater than or equal to 4, this would not work, and there would again be too many Riemannian metrics in general compared to diagonal metrics and ways of reducing to them via changes of variables. Of course this is just an informal “dimension” count, and not a justification for being able to put metrics into diagonal form in dimension 3. (One should also be careful that there is not significant overlap between changes of variables and diagonal metrics, i.e., so that there was no “overcounting” for the combination of them.) However, it does turn out that one can put metrics in diagonal form (in dimension 3), at least locally. This was established in [DeTY] in the case of smooth metrics. There were earlier results in the real-analytic category. (See [DeTY] for more information.) However, this type of “normal form” does not seem to be as useful for the present type of issue as conformal parameterizations are. As in the case of conformal coordinates, part of the problem is that even if one has such a normal form, one does not a priori know anything about the behavior of the diagonal entries of the metric in this normal form. One would need methods of getting estimates without this information, and only the nature of the normal form. In the context of conformal mappings, one has extremal lengths, conformal capacities, and other conformal and quasiconformal invariants and quasi-invariants. For diagonal metrics, it is not clear what one might do. A related point is that the analysis of the partial differential equations which permits one to put smooth Riemannian metrics in dimension 3 into diagonal form is roughly “hyperbolic”, in the same way that the corresponding differential equations for conformal coordinates in dimension 2 are elliptic. See [DeTY]. This is closely connected to the kind of stability that one has for conformal mappings, and the possibilities for having estimates for them under mild or primitive geometric conditions. In a way this is all “just fair”, and nicely so. With diagonal metrics one does have something analogous to conformal coordinates in dimension 3. On the other hand, this analogue behaves differently in fundamental ways, including estimates. This is compatible with other aspects of the story as a whole, like the topological and geometric examples that one has in dimension 3 (where homeomorphisms may not exist, with the properties that one might otherwise hope for). Some Topics Concerning Homeomorphisms 53 In any event, this illustrates how analytic and geometric methods seem to behave rather differently in dimensions 3 and higher, compared to the special structure and phenomena which occur in dimension 2. This is somewhat remarkable in analogy with topological phenomena, which have similar differences between dimensions. With the topology there are both some crossings and overlaps with geometry and analysis, and much that is separate or independent. On the side of geometry and analysis, let us also note that there are some other special features in low dimensions that we have not mentioned. As a basic example, the large amount of flexibility that one has in making conformal mappings in dimension 2 leads to some possibilities in dimension 3 that are not available in higher dimensions. That is, the large freedom that one has in dimension 2 can sometimes permit one to make more limited constructions in dimension 3, e.g., by starting with submanifolds of dimension 2, and working from there (with extensions, gluings, etc.) These possibilities in dimension 3 can be much more restricted than in dimension 2, but having them at all can be significantly more than what happens in higher dimensions. Concerning variational problems, one might also keep in mind the approaches of [DaviS2], [DaviS3] (and some earlier ideas of Morel and Solimini [MoreS]). For these one does not necessarily work directly with mappings or potential parameterizations of sets, and in particular one may allow sets themselves to be variables in the minimization (rather than mappings between fixed spaces). This broader range can make it easier for the minimizations to lead to useful conclusions about geometric structure and complexity, under natural and modest conditions. In particular, one can get substantial “partial parameterizations”, as with uniform rectifiability conditions. These approaches are also nicely compatible with the trouble that one knows can occur, related to topology and homeomorphisms (and in geometrically moderate situations, as in Subsections 7.2 and 7.3, and [Sem3], [Sem4]). Finally, while we have mentioned a lot about the special phenomena that can occur in dimension 2, and what happens in higher dimensions, we should also not forget about dimension 1. This is even more special than dimension 2. This is a familiar theme in geometric topology, for the ways that one can recognize and parameterize curves. In geometry and analysis, one can look for parameterizations with bounds, and these are often constructible. 54 S. Semmes A fundamental point along these lines is the ability to make parameterizations by arclength, for curves of locally finite length. More generally, one can use parameterizations adapted to other measures (rather than length), when they are around. Arclength parameterizations provide a very robust and useful way for obtaining parameterizations in dimension 1 with good behavior and bounds. In dimension 1, simple conditions in terms of mass can often be immediately “integrated” to get well-behaved parameterizations, in ways that are not available (or do not work nearly as well) in higher dimensions, even in dimension 2. To put the matter in more concrete terms, in dimension 1 one can often make parameterizations, or approximate parameterizations, simply by ordering points in a good way. This does not work in higher dimensions. Once one has the ordering, one can regularize the geometry by parameterizing according to arclength, or some other measure (as appropriate). For another version of this, in connection with quasisymmetric mappings, see Section 4 of [TukV]. In differential-geometric language, one might say that dimension 1 is special for the way that one can make isometries between spaces, through arclength parameterizations. This no longer works in dimension 2, but one has conformal coordinates there. Neither of these are generally available in higher dimensions. In higher dimensions one has less special structure for getting the existence in general of well-behaved parameterizations, and then the kinds and ranges of geometric and topological phenomena which can exist open up in a large way. 9. Nonlinear similarity: Another class of examples A very nice and concrete situation in which issues of existence and behavior of homeomorphisms can come up is that of “nonlinear similarity”. Specifically, it is possible to have linear mappings A,Bon Rn which are conjugate to each other by homeomorphisms from Rnonto itself —i.e., B=h◦A◦h−1, where his a homeomorphism of Rnonto itself— and which are not conjugate by linear mappings! Examples of this were given in [CapS1], [CapS2]. For related matters, including other examples, conditions under which one can deduce linear equivalence, and more on the behavior of conjugating homeomorphisms when a linear equivalence does not exist, see [CapS3], [CapS4], [CapS5], [CapS+], [CapS∗], [HamP1], [HamP2], [HsiP1], [HsiP2], Some Topics Concerning Homeomorphisms 55 [KuiR], [MadR1], [MadR2], [Mio], [Rha1], [Rha2], [RotW], [Wei1], [Wei2], [Wilk]. 10. Doing pretty well with spaces which may not have nice coordinates If one has a topological or metric space (or whatever) which has nice coordinates, then that can be pretty good. However, there is a lot that one can do without having coordinates. Let us begin with some basic conditions. Let Mbe a topological space which is compact, Hausdorff, and metrizable. We shall assume that Mhas finite topological dimension, in the sense of [HurW]. In these circumstances, this is equivalent to saying that Mis homeomorphic to a subset of some Rn. (See [HurW].) For simplicity, let us assume that Mis a compact subset of some Rn. It is also convenient to ask that Mbe locally contractable. This means that for each point p∈M, and each neighborhood Uof pin M, there is a smaller neighborhood W⊆Uof pin Msuch that Wcan be contracted to a point in U. As a class of examples, finite polyhedra are locally contractable. Finite polyhedra make a nice special case to consider throughout this section, and we shall return to it several times. For another class of examples, one has cell-like quotients of topological manifolds (and of some locally contractable spaces more generally), at least when the quotient spaces have finite topological dimension. See Corollary 12B on p. 129 of [Dave2]. As a general fact about local contractability, let us note the following. Proposition 10.1. Let Mbe a compact subset of some Rn. Then Mis locally contractable if and only if there is a set V⊆Rnwhich contains M in its interior, and a continuous mapping r:V→Mwhich is a retract, i.e., r(w)=wfor all w∈V. This is a fairly standard observation. The “if” part is an easy consequence of the local contractability of Rn(through linear mappings). Specifically, to get local contractions inside of M, one makes standard linear contractions in Rn, which normally do not stay inside M, and then one applies the retraction to keep the contractions inside M. For the converse, one can begin by defining ron a discrete and reasonably-thick set of points outside M, but near M. For a point win such a set, one could choose r(w)∈Mso that it lies as close to was possible (among points in M), or is at least approximately like this. To fill 56 S. Semmes in rin the areas around these discrete points, one can make extensions first to edges, then 2-dimensional faces, and so on, up to dimension n. To make these extensions, one uses local contractability of M. It is also important that the local extensions do not go to far from the selections already made, so that r:V→Mwill be continuous in the end, and this one can get from the local contractability. The notion of “Whitney decompositions”, as in Chapter VI of [Ste], is helpful for this kind of argument. It gives a way of decomposing Rn\M into cubes with disjoint interiors, and some other useful properties. (In particular, this kind of decomposition can be helpful for keeping track of bounds, if one should wish to do so.) One can use the vertices of these cubes for the discrete set in the complement of Mmentioned above. See also [Dave2] concerning Proposition 10.1, especially p. 117ff. Let us return now to the general story. Suppose that Mis a compact subset of Rn, and that Mis locally contractable. Let r:V→Mbe a continuous retraction on M, as in Proposition 10.1. Thus Vcontains M in its interior. By replacing Vby a slightly smaller subset, if necessary, we may assume that Vis compact, and in fact that it is a finite union of dyadic cubes in Rn.(Adyadic cube in Rnis a cube which is a Cartesian product of intervals of the form [ji2−k,(ji+1)2 −k], i=1,2,... ,n, where the ji’s and kare integers.) This type of choice for Vis convenient for having nice properties in terms of homology and cohomology. In particular, Vis then a finite complex. The inclusion of Minto V, and the mapping r:V→M, induce mappings between the homology and cohomology of Mand V.If ι:M→Vdenotes the mapping coming from inclusion, then r◦ι:M→ Mis the identity mapping, and thus it induces the identity mapping on the homology and cohomology of M. Using this, one can see that the mapping from the homology of Minto the homology of Vinduced by ι is an injection (in addition to being a group homomorphism, as usual), and that the mapping from the homology of Vto the homology of M induced by ris a surjection. This follows from standard properties of homology and mappings, as in [Mas]. Similarly, rinduces a mapping from cohomology of Mto cohomology of Vwhich is injective, and ι induces a mapping from cohomology of Vinto cohomology of Mwhich is surjective. This provides a simple way in which the algebraic topology of Mcan be “bounded”, under the type of assumptions on Mthat we are making. (There are more refined things that one can also do, but we shall not worry about this here.) Local contractability, and the existence of a retraction as in Proposition 10.1, are also nice for making it clear and Some Topics Concerning Homeomorphisms 57 easy to work with continuous mappings into M. In particular, one can get continuous mappings into Mfrom continuous mappings into V, when one has a retraction r:V→M, as above. This is as opposed to standard examples like the closure of the graph of sin(1/x), x∈[−1,1]\{0}. (This set is connected but not arcwise connected.) Now let us consider the following stronger conditions on M. Definition 10.2 (Generalized k-Manifolds).Let Mbe a compact subset of Rnwhich is locally contractable. Then Mis a generalized k-manifold if for every point z∈M, the relative homology Hj(M,M\{z})is the same (up to isomorphism) as the relative homology Hj(Rk,Rk\{0}) for each j. We are implicitly working with homology defined over the integers here, and there are analogous notions with respect to other coefficient groups (like rational numbers, for instance). One may also wish to use weaker conditions than local contractability (as in [Bre2], [Wild]). There are other natural variations for this concept. If Mis a finite polyhedron, then the property of being a generalized manifold is equivalent to asking that the links of Mbe homology spheres of the right dimension (i.e., with the same homology as a standard sphere, up to isomorphism). Another class of examples comes from taking quotients of compact topological manifolds by cell-like decompositions (Sections 4 and 6), at least when the quotient space has finite topological dimension. See Corollary 1A on p. 191 of [Dave2] (and Corollary 12B on p. 129 there), and compare also with Theorem 16.33 on p. 389 of [Bre2], and [Fer]. As usual, dimensions 1 and 2 are special for generalized manifolds, which are then topological manifolds. See [Wild], Theorem 16.32 on p. 388 of [Bre2], and the introduction to [Fer]. For more on ways that generalized manifolds can arise, see [Bor2], [Bre2], [Bry+], [Bry∗], [Dave2], [Fer], [Wei2] (and the references therein). A related topic is the “recognition problem”, for determining when a topological space is a topological manifold. Some references for this include [Bry+], [Bry∗], [Can1], [Can2], [Can3], [Dave2], [Edw2], [Fer], [Wei2]. What are some properties of generalized manifolds? In what ways might they be like manifolds? A fundamental point is that Poincar´e duality (and other duality theorems for manifolds) also work for generalized manifolds. See [Bor1], [Bor2], [Bre2], [Wild] and p. 277–278 of [Spa]. This is pretty good, 58 S. Semmes since Poincar´e duality is such a fundamental aspect of manifolds. (See [BotT], [Bre1], [Mas], [MilS], [Spa], for instance.) A more involved fact is that rational Pontrjagin classes can be defined for generalized manifolds. (See the introduction to [Bry+].) For smooth manifolds, the definition of the Pontrjagin classes is classical. (See [BotT], [MilS].) More precisely, one can define Pontrjagin classes for vector bundles in general, and then apply this to the tangent bundle of a smooth manifold to get the Pontrjagin classes of a manifold. As integral cohomology classes, the Pontrjagin classes are preserved by diffeomorphisms between smooth manifolds, but not, in general, by homeomorphisms. However, a famous theorem of Novikov is that the Pontrjagin classes of smooth manifolds are preserved as rational cohomology classes by homeomorphisms in general. Further developments lead to the definition of rational characteristic classes on more general spaces. For finite polyhedra, there is an earlier treatment of rational Pontrjagin classes, which goes back to work of Thom and Rohlin and Schwarz. See Section 20 of [MilS]. More precisely, this gives a procedure by which to define rational Pontrjagin classes for finite polyhedra which are generalized manifolds, and which is invariant under piecewise-linear equivalence. (For this, the generalized-manifold condition can be given in terms of rational coefficients for the homology groups.) If one starts with a smooth manifold, then there it can be converted to a piecewise-linear manifold (unique up to equivalence) by earlier results, and the classical rational Pontrjagin classes for the smooth manifold are the same as the ones that are obtained by the procedure for polyhedral spaces. See [MilS] for more information. References [Ahl1] L. V. Ahlfors,“Complex analysis”, An introduction to the theory of analytic functions of one complex variable, Third ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978. [Ahl2] L. V. Ahlfors,“Lectures on quasiconformal mappings”,Van Nostrand Mathematical Studies 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. [Ahl3] L. V. Ahlfors,“Conformal invariants: topics in geometric function theory”, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York, 1973. Some Topics Concerning Homeomorphisms 59 [AleV] P. Alestalo and J. 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