The Revised and Uniform Fundamental Groups and Universal Covers of Geodesic Spaces

The Revised and Uniform Fundamental Groups
and Universal Covers of Geodesic Spaces

Jay Wilkins
University of Connecticut
Department of Mathematics
196 Auditorium Road, Unit 3009
Storrs, CT 06269-3009
Email: leonard.wilkins@uconn.edu
Jay Wilkins
Department of Mathematics, University of Connecticut, Storrs, CT 06029
leonard.wilkins@uconn.edu
Abstract.

Sormani and Wei proved in 2004 that a compact geodesic space has a categorical universal cover if and only if its covering/critical spectrum is finite. We add to this several equivalent conditions pertaining to the geometry and topology of the revised and uniform fundamental groups. We show that a compact geodesic space has a universal cover if and only if the following hold: 1) its revised and uniform fundamental groups are finitely presented, or, more generally, countable; 2) its revised fundamental group is discrete as a quotient of the topological fundamental group . In the process, we classify the topological singularities in , and we show that the above conditions imply closed liftings of all sufficiently small path loops to all covers of , generalizing the traditional semilocally simply connected property. A geodesic space with this new property is called semilocally -simply connected, and has a universal cover if and only if it satisfies this condition. We then introduce a topology on called the covering topology, which always makes a topological group. We establish several connections between properties of the covering topology, the existence of simply connected and universal covers, and geometries on the fundamental group.

Key words and phrases: geodesic space; critical spectrum; universal cover; revised fundamental group; uniform fundamental group; topological fundamental group.

2010 Mathematics Subject Classification: Primary 54E45; Secondary 57M10, 54H11, 20F38.

1. Introduction and Main Results

In [26], Sormani and Wei formally defined the covering spectrum of a compact geodesic space, a geometric invariant that detects one-dimensional holes of positive intrinsic diameter. They showed (Theorem 3.4, [26]) that a compact geodesic space has a universal cover if and only if its covering spectrum, , is finite. When this holds, they defined the revised fundamental group of to be the deck group of the universal cover, and they showed that it is finitely generated (Proposition 6.4, [26]).

In this paper, we extend the above results through an investigation of the geometry and topology of a slightly generalized revised fundamental group and another associated group called the uniform fundamental group. To do so, we apply the generalized covering methods developed by Berestovskii-Plaut for uniform spaces ([3]) to the more restricted but important class of geodesic spaces.

In [3], Berestovskii and Plaut defined the uniform universal covering and its deck group, the uniform fundamental group. These are generalizations of the classical universal cover and fundamental group for uniform spaces - hence, metric spaces - that are not necessarily semilocally simply connected or even locally path connected. Spaces for which the uniform universal cover exists are called coverable, and these include all geodesic spaces and, thus, Gromov-Hausdorff limits of Riemannian manifolds. The foundation for [3] is discrete homotopy theory, an analog of classical path homotopy theory that uses discrete chains and chain homotopies instead of the continuous counterparts. In [30], and with Plaut et al. in [12], the author used discrete homotopy theory to generalize the covering spectrum. When the methods of [3] are applied to a metric space , one obtains the -parameterized collection of -covers of , . These covers, in turn, determine the critical spectrum of , the set of values, , at which the equivalence class of changes as decreases to . The uniform universal cover and uniform fundamental group are inverse limits of the -covers and their deck groups, respectively (see Section 2).

With the exception of the inverse limit formulations, this construction and spectral definition parallel those of Sormani-Wei in [25] and [26]. The primary difference between the covering and critical spectra is the applicability. The Sormani-Wei construction relies on a classical method of Spanier ([28]) that requires local path connectivity of the underlying metric space , which - if is compact and connected - is equivalent to being geodesic. The Berestovskii-Plaut construction, however, can be carried out much more generally, allowing investigation of the critical spectra of more exotic and pathological metric spaces. Like the covering spectrum, the critical spectrum detects fundamental group generators, but it also detects other metric structures in the general case that do not show up in geodesic spaces (cf. [12]). Nevertheless, Plaut and the author showed in [23] that when the underlying metric space is compact geodesic, the two spectra differ only by a constant multiple, namely . Thus, the covering spectrum, appropriately rescaled, is a special case of the critical spectrum in the compact geodesic setting. In particular, this fact and Sormani-Wei’s theorem, together, show that a compact geodesic space has a universal cover if and only if its critical spectrum is finite.

Since we will be exploiting the methods of Berestovskii-Plaut and the uniform structure of the given geodesic space, our results will be presented in the language of discrete homotopy theory and the critical spectrum. The relevant technical background is given in Section 2. In this paper, a cover or covering space of will always imply a traditional, connected cover with the property that each is contained in an evenly covered neighborhood with respect to . A universal cover of will mean a traditional, categorical universal cover (not necessarily simply connected), or a cover so that, for any other cover , there is a cover such that . Except for the uniform universal cover, we will not need or use any of the recent, non-traditional generalizations of universal covers that relax the evenly covered property (cf. [6], [7], [16], [21]). When we use the uniform universal cover, it will always be explicitly referenced as such, so no confusion should result.

The fundamental observation that makes our results possible is that we can characterize local topology of a compact geodesic space, , in terms of how path loops at a base point lift not to a single -cover, , but to the -covers in the aggregate. Thus, we begin Section 3 by slightly generalizing the revised fundamental group defined by Sormani-Wei in [26]. The normal covering groups of the -covers, denoted by , intersect to form the closed lifting group, the normal subgroup representing all loops at that lift closed - that is, the lift is also a loop - to for every . The revised fundamental group, then, is , and it isomorphically injects into the uniform fundamental group of , denoted by . In fact, is isomorphic to , the first shape group of , showing that injects into , as well. When has a universal cover, agrees with the definition of Sormani-Wei, though they only define and discuss this group in that particular case. Our approach shows that and are well-defined whether has a universal cover or not. Indeed, it is a specific property of that determines when has a universal cover (Lemma 4.1).

Two obvious cases of interest occur when is trivial and when it is the whole fundamental group. In the former case, is just the fundamental group, which, then, injects into . We prove (Lemma 3.12) that is always trivial for compact, one-dimensional geodesic spaces, and determining conditions under which is trivial in general is one of the goals of the final section (see Theorem 1.4 below). When , is its own universal cover and . We show that these conditions are actually all equivalent (Corollary 4.9).

We define to be semilocally -simply connected if and only if each has a neighborhood such that every path loop in based at lifts closed to for all (Definition 3.15). This generalization of the classical semilocally simply connected definition can be algebraically reformulated in a familiar way. If is the quotient map and is the inclusion of a set , then is semilocally -simply connected at if and only if there is a neighborhood of such that the homomorphism is trivial (Lemma 3.16). There are other classical fundamental group results that have analogous statements in the revised case. The classical functor has an analog for revised fundamental groups, and a homotopy equivalence induces a revised fundamental group isomorphism (Lemma 3.17 and Corollary 3.18).

Proposition 4.3 gives sufficient conditions for to be semilocally -simply connected, namely that be countable. We then use lifting properties to classify the two types of topological singularities that obstruct semilocal simply connectedness (Definition 4.5). A sequentially singular point is one at which there is a sequence of path loops with an associated, strictly decreasing sequence such that lifts closed to but open to . These are “Hawaiian earring-type” topological singularities. A point is degenerate if every neighborhood of contains a nontrivial path loop that lifts closed to for all ; this is a generalization of “not homotopically Hausdorff.” Our first and primary theorem is

Theorem 1.1.

If is a compact geodesic space, then the following are equivalent.

  1. has a universal cover.

  2. is finite.

  3. The revised fundamental group, , is any one of the following: i) countable; ii) finitely generated; iii) finitely presented.

  4. The uniform fundamental group, , is any one of the following: i) countable; ii) finitely generated; iii) finitely presented.

  5. has no sequentially singular points.

  6. is semilocally -simply connected.

If these hold, then the universal cover is -simply connected (i.e. is trivial), its deck and covering groups, respectively, are and , and is isomorphic to .

We have already noted that is known. The proof of Theorem 1.1 will mostly show that the other statements are equivalent to , but we include for both emphasis and reference. Moreover, the implication follows directly from Lemma 4.1 and a result of Plaut and the author in [22], which is recalled in Section 2. The implications are clear, and likewise for the parts of . The new and presently most important aspects of Theorem 1.1 are the sufficiency of for to hold, the equivalence of - , and the equivalence of - to .

The reader may want to compare Theorem 1.1 to Corollary 5.7 of [11], where Cannon and Conner showed that a locally path connected, homotopically Hausdorff Peano continuum - or, equivalently, a homotopically Hausdorff compact geodesic space - has a simply connected cover if and only if is any of the following: 1) countable, 2) finitely generated, 3) finitely presented. We recharacterize Cannon-Conner’s result via the critical spectrum, showing that has a simply connected cover if and only if is finite and is trivial (Proposition 4.10).

In Section 5, we examine the relationship between universal covers and topologies on the fundamental group. Recall that in [6] Biss defined the topological fundamental group, , to be topologized as a quotient of the pointed loop space with the compact-open topology, which is equivalent to the uniform topology when is geodesic. Unfortunately, is not always a topological group as was originally claimed in [6], even in the compact geodesic case; Fabel has shown that multiplication in the Hawaiian earring group with this topology is not continuous ([15]). However, Brazas has shown that is a quasitopological group (Lemma 1.8, [8]), meaning that the inverse operation is continuous and multiplication is continuous in each variable, i.e. left and right translations are homeomorphisms (cf. [2] for details on such groups). Nevertheless, Biss’ definition has still been effectively utilized. For instance, Fabel also showed that a path connected, locally path connected metric space has a simply connected cover if and only if is discrete ([14]).

The revised fundamental group inherits a natural quotient topology from , and we call the resulting group the topological revised fundamental group, denoting it .

Theorem 1.2.

If is compact geodesic, then is a quasitopological group, and has a universal cover if and only if is discrete.

We then introduce a new topology on that is an example of a subgroup topology as defined by Bogley and Sierdaski in their preprint [7]. Specifically, the cosets of the -covering groups, , form a basis for a topology we call the covering topology on . We denote the fundamental group with this topology by , and we show that is always a topological group. The revised fundamental group with the inherited quotient topology is denoted , and we call it the revised -fundamental group. This yields our final two theorems. Theorem 1.3 shows that is just as effective as with regard to detecting universal and simply connected covers. Theorem 1.4 gives conditions for to be trivial, and it provides a nice picture of the geometric connection between and . Namely, is in bijective correspondence with the connected components of .

Theorem 1.3.

If is a compact geodesic space, then has a universal cover (respectively, simply connected cover) if and only if (respectively, ) is discrete.

Theorem 1.4.

Let be a compact geodesic space. Then the connected component of containing is , and the following are equivalent.

  1. is trivial.

  2. is totally disconnected.

  3. is Hausdorff.

  4. is a geometric group over the semigroup , with geometry .

  5. admits a compatible, left-invariant ultrametric.

If these conditions hold, then isomorphically injects into the first shape group of , .

The first statement of Theorem 1.4 and the equivalence of and are part of a more general result on subgroup topologies proved by Bogley and Sierdaski in [7] (see Lemma 5.5). The particular application, the last statement, and the equivalence of , , and to and are the new results here.

Conditions 2, 4, and 5 above are closely related. A geometric group over an abelian, partially ordered semigroup, , is a topological group, , with a local basis at the identity, , such that the following hold for all , : 1) if and only if , and ; 2) ; 3) and . The collection is called a geometry on . This notion was first defined by Berestovskii, Plaut, and Stallman in [4], and the type of geometry that admits, if any, is strongly related to what types of metrics induce its given topology. We show that the collection , indexed over - the positive reals with their usual order but operation - is always almost a geometry on , possibly lacking only one of the required conditions. That condition is precisely part 2 of the definition, i.e. that is trivial. Moreover, geometries over correspond to ultrametrics (see Section 5), and ultrametric spaces are necessarily totally disconnected.

2. Background: Discrete Homotopy Theory

Recall that a metric space is a geodesic space if any two points in are joined by a (minimal) geodesic, or an arclength parameterized curve, , having length equal to the distance between its endpoints. This is slightly different from the Riemannian definition of a geodesic, which only requires that a curve be locally minimizing. It is well-known that when is geodesic, is a connected topological space, and is a covering map, the metric on can be lifted to a unique geodesic metric on that makes a local isometry. If is also compact, this lifted metric makes a metric covering map - a traditional covering map that is also a uniform local isometry. Thus, there is no distinction between metric coverings and general connected coverings in the compact geodesic setting.

All of the spaces we consider in Section 3 and beyond will be geodesic, and compactness will be assumed when necessary. It should be noted, however, that the Bing-Moise Theorem (Theorem 8 in [5], Theorem 4 in [19]) establishes for a compact, connected, metric space the equivalence of local connectedness, local path connectedness, and the existence of a compatible geodesic metric. Thus, every Peano continuum - or compact, connected, locally connected metric space - admits a compatible geodesic metric, and any topological result that holds for compact geodesic spaces holds for all Peano continua.

We will outline the discrete homotopy constructions of [3] as applied to metric spaces. Readers familiar with the results and methods of discrete homotopy theory may want to skip this section and simply refer back to it as needed. Further explanations, details, and proofs may be found in [3] in the context of uniform spaces, and in [30] and [12] for metric spaces.

Let be a connected metric space, and fix . An -chain in is a finite sequence such that for . A basic move on an -chain is the addition or removal of a single point with the conditions that the endpoints remain fixed and the resulting chain is still an -chain. Two -chains and are -homotopic if there is a finite sequence of -chains, - called an -homotopy - such that each differs from by a basic move. The relation “-homotopic” is an equivalence relation on -chains in , and it carries the same basic concatenation and algebraic or groupoid properties as traditional path homotopy equivalence.

For a fixed base point , is the set of all equivalence classes of -chains in beginning at , , and is the endpoint map taking to . There is a natural metric, , on that makes a regular metric covering map (cf. [12], [22], or [30]). We call and its deck group, , the -cover and -group of , respectively. The -group is naturally identified with the subset of consisting of classes of -loops at , which is a group under concatenation. If an -loop is -homotopic to the trivial loop then we say is -null. Clearly, the identity of is , and we take to be a pointed map, where will always be a shorthand notation for the identity element .

The -group is uniformly discrete and left invariant as a metric subspace of . It acts discretely by isometries on via left concatenation. That is, for and , and if , then is trivial. A discrete action is necessarily free and properly discontinuous. The resulting metric quotient is homeomorphic and uniformly locally isometric to ; the two are isometric when is geodesic. When is compact geodesic, the -groups are finitely presented (Theorem 3, [22]), with a set of generators and relations of the form .

The preceding construction is independent of the base point. If is another base point and is a fixed -chain from to , then the maps and are, respectively, an isometric covering equivalence from to and an isomorphism from to . Thus, we usually just fix a base point in and use it to determine all -covers and groups. This gives us collections and , which stratify the covering spaces and fundamental group of . Indeed, if is compact geodesic, every connected cover of is covered by for small enough .

Now, we assume that is geodesic. The natural metric on mentioned above is equal to the lifted geodesic metric from (Proposition 24, [22]). Given , there is a well-defined, surjective bonding map that simply treats a -chain as an -chain, i.e. . These maps are metric covering maps between geodesic spaces and, thus, isometries when they are injective. They also satisfy the composition relation when . The restriction of to is a homomorphism onto , which we denote by . For any , is injective if and only if is injective, and these homomorphisms satisfy the obvious analog of the previously mentioned composition relation.

A positive number is a critical value of a compact geodesic space if there is a nontrivial -loop, , at that is -null for all . Equivalently, there is a nontrivial element that is in for all . The set of all critical values of is called the critical spectrum of . It should be noted that our present definition of a critical value relies on the assumption that is compact geodesic. As we mentioned in the introduction, critical values of metric spaces can be defined more generally, and the definition given here would not suffice to capture every value that should be critical for a non-compact or non-geodesic space. See [12] for the general definition and a classification of the types of critical values. For compact geodesic , however, critical values occur only in this very specific way (Lemma 3.1.10, [30]), allowing for the present simpler definition.

The critical/covering spectrum of a compact geodesic space is closed, discrete, and bounded above by in , although may be . Thus, either is finite, or it is a strictly decreasing sequence of isolated critical values converging to . This was originally shown by Sormani-Wei for in [26], and Plaut-Wilkins in [22] constructed a different direct proof for not using the equality . We have noted that these spectra detect fundamental group generators. The standard examples illustrating this involve circles (and not by coincidence - see [22]). For instance, if is the geodesic circle of circumference , , then and is trivial for , while and for (Example 17, [12]). Thus, .

There is a natural way to transfer properties between discrete chains and continuous paths. The proofs of the statements leading up to Definition 2.2 may be found in [30].

Definition 2.1.

For a path , a strong -chain along is an -chain with the following property: there exists a partition of such that for each and for each .

Note that the reversal of a strong -chain along is a strong -chain along , and if and are strong -chains along paths and , respectively, with the initial point of equal to the terminal point of , then the concatenation is a strong -chain along .

A simple Lebesgue covering argument shows that there is a strong -chain along any path, and if is any path, then any two strong -chains along are -homotopic. Moreover, if and are paths that are fixed endpoint path homotopic, then any strong -chain along is -homotopic to any strong -chain along . These statements are not true for -chains along paths without the strong condition. Taken together, these properties induce natural, well-defined, “continuous to discrete” homomorphisms from to each -group, which are surjective when is geodesic.

Definition 2.2.

Fix a base point , and let and be the fundamental and -groups, respectively, based at . For , the -homomorphism is the map taking to the -equivalence class of strong -loops along .

If is geodesic and is an -loop at , we can join each consecutive pair of points in by a minimal geodesic, making a strong -chain along the resulting broken geodesic path loop. This is an -chording of , and the homotopy class of the resulting path loop maps to under , making surjective. It is easy to see that the following commutes, where is the identity isomorphism.

(1)

The following definition and lemma (Definition 16 and Proposition 17 in [22]) will be needed for some basic chain homotopy computations later on.

Definition 2.3.

Let be a metric space and . Given an chain in , , define . If and are chains having the same number of points, define .

Lemma 2.4.

Let be an -chain in a metric space . If is a chain with the same endpoints and same number of points as , and if , then is an -chain that is -homotopic to .

The relations and for imply that and form inverse systems indexed by with reverse order. The uniform universal cover of (the UU-cover for short) and uniform fundamental group of are the resultant inverse limits

The endpoint projection is surjective and continuous but is not typically a traditional cover; the fibers , for instance, are totally disconnected but not necessarily discrete. However, is a generalized universal cover in the following senses: 1) (universality) if is a cover then there is a unique, possibly generalized, cover such that ;  2) (unique lifting) paths and path homotopies lift uniquely into ; 3) (generalized regularity) - which admits equivalent definitions as and as the deck group of - acts prodiscretely on , and is homeomorphic to . Thus, can be interpreted as a generalized fundamental group of .

The following facts are in [3]. There is a canonical homomorphism , mapping to the endpoint of the lift of at the identity in . While is surjective if and only if is path connected, is always a dense, normal subgroup of when the latter is given the inverse limit topology, which is also the subspace topology it inherits from . The kernel of contains those elements in represented by loops that lift closed to , or, equivalently, that lift closed to for all . That is, ; we will say more about this in Section 3.

Now, if is compact and is finite, then the covers stabilize as . Precisely, and are, respectively, covering equivalences and isomorphisms for all . In this case, for each , is equivalent to the traditional cover with deck group . The universality of thus implies that it - as well as each , - is a traditional universal cover. This is precisely the idea Sormani-Wei used to prove that has a universal cover when is finite, though they did so without reference to the UU-cover or any generalized universal cover. Of course, they also proved the converse; a universal cover requires that the spectra be finite, and the preceding scenario still holds.

Remark 2.5.

We have changed notation slightly from [3]. In [3], and the uniform fundamental group are denoted, respectively, by and , but we will use and differently.

Finally, when is compact geodesic, Brodskiy, Dydak, Labuz, and Mitra showed (Corollary 6.5, [9]) that the uniform fundamental group is isomorphic to the first shape group of , . We will suppress the formal and rather technical definition of (cf. [13]), since it will play no role in this paper. Like , is an inverse limit of coarse approximations to the fundamental group. Very roughly, if is the geometric realization of the nerve of an open covering, , and if is a refinement of , then there is a bonding homomorphism . Then is the resultant inverse limit , taken over a cofinal directed set consisting of open coverings admitting subordinated partitions of unity. Like finding conditions under which is trivial in the geodesic case, determining when the fundamental group isomorphically injects into the first shape group is a broad area of interest in general topology (cf. [16], [17]).

3. The Revised Fundamental Group

Much of this section will hold for general geodesic spaces, since many of the results will not involve our compact-dependent definition of a critical value. Recall that open metric balls of radius and center are denoted . To distinguish discrete chains from continuous paths, we will exclusively use ‘path’ (resp. ‘path loop’) to denote continuous curves (resp. closed curves). A path loop, is said to be based at if . We do not discuss free homotopies in the present work, so to say that two path loops based at a point are homotopic means that they are fixed endpoint path homotopic, or homotopic rel . If groups and are isomorphic, we will denote this by .

Let be a geodesic space with base point , and let be the -covers determined by this base point. Recalling Definition 2.2, we denote the subgroup by , or when the base point needs to be emphasized. Commutative diagram (1) shows immediately that these kernels form a decreasing, nested set of normal subgroups of in the following sense: if , then . We thus define the normal subgroup

and, for reasons which will soon become clear, we call the closed lifting group of at . When is the trivial subgroup of consisting of just the identity, we denote this by .

Definition 3.1.

The revised fundamental group of at is defined to be the quotient group , and we denote the standard quotient homomorphism by .

We usually suppress the base point when it is clear, and we show below that it is, in fact, immaterial. We will routinely express elements of as left cosets, . The only scenario where confusion might occur is when we need to distinguish the subset from the corresponding element in . The context, however, should always make the usage clear.

We can connect the fundamental group, revised fundamental group, and -groups by a commutative diagram. We use a standard generalization of the First Isomorphism Theorem, namely the following: if is a homomorphism, not necessarily surjective, and if contains a normal subgroup, say , then there is a unique homomorphism such that , where is the quotient homomorphism. The map is defined by . We will use this result several times.

For any , contains . Thus, there is a unique homomorphism which takes to and is such that the following diagram commutes.

(2)

Note that , and since is surjective, is surjective, also.

Let be a path in beginning at , and let denote its unique lift to the identity . It follows from Proposition 21 in [22] that the endpoint of is the equivalence class of strong -chains along . Thus, if is a path loop then the endpoint of is , which is trivial if and only if . Hence, we have

Lemma 3.2.

Let be a geodesic space, and let be given. If is a path loop at , then is a path loop at if and only if some (equivalently, every) strong -chain along is -null. That is, lifts closed to if and only if , and is the covering group of .

It follows from the regularity of that the lifts to of a path loop based at any point in are either all closed or all open, regardless of the point in the appropriate preimage to which it is lifted. Hence, if is a path loop based at , there is no ambiguity in simply stating that lifts closed or open to without specifying the particular point in to which it is lifted.

Lemma 3.2 also gives us another useful and familiar interpretation of a critical value. Suppose is compact and is a critical value of . Then there is a nontrivial -loop, , at that is -null for all . Let be an -chording of , and note that is also a strong -loop along for all . It follows, then, that lifts open to since is -nontrivial, but closed to for all since is -null. That is, there is a path loop at that lifts open to but closed to for all . This is precisely how one characterizes elements of the covering spectrum of Sormani-Wei, and we see that it is a consequence of the more general discrete formulation.

Corollary 3.3.

Let be geodesic with base point . If is a path loop based at and lying in an open ball of radius not necessarily centered at , then lifts closed to .

Proof.

Suppose lies in . Choose a strong -loop along , say , and let be the 2-point -chain . Then is an -loop, and note that is -null if and only if is. But, letting denote the relation “-homotopic,” we have

Since for each , we can successively remove , then , and so on, reducing via -homotopy to the trivial chain . Thus, is -null.

Finally, let be a path from to , and choose a strong -chain, , along . Then is a strong -chain along that is -null. By Lemma 3.2, this means that lifts closed to , which, by uniqueness of path lifts, can only hold if lifts closed, also.∎

The author has shown that the previous corollary can be strengthened to path loops lying in balls of radius ([30]). The proof is more technical, however, and we will not need that stronger result.

We include the next definition and lemma for convenient reference, since they will be used several times. The lemma is an immediate consequence of Lemma 3.2 and Corollary 52 in [23].

Definition 3.4.

A lollipop at is a path loop of the form , where is a path from to a point and is a path loop at . If - which we call the head of the lollipop - lies in a ball of radius , then we call an -lollipop. If a path loop is homotopic to a product of -lollipops, , we call this an -lollipop factorization of .

Lemma 3.5.

Let be a geodesic space. If is in , then is homotopic to a finite product of -lollipops.

It follows from the uniqueness of path lifts to covering spaces that a lollipop lifts closed to a regular cover if and only if lifts closed. Combining this with Corollary 3.3, we obtain

Corollary 3.6.

If is a product of -lollipops in a geodesic space , then lifts closed to .

Remark 3.7.

Lemma 3.5 and the statement immediately following the proof of Corollary 3.3 are the means by which one shows that . In fact, those results, together, show that , where is the subgroup generated by -lollipops. These are precisely the covering groups of the -covers, , used by Sormani and Wei to define their covering spectrum (Definition 2.3, [26]). With this fact, it is straightforward to show that is isometricaly equivalent to , and the equality then follows immediately. See [23] for details.

Proposition 3.8.

If is geodesic, then if and only if lifts closed to for all . If is also compact, then if and only if lifts closed to all connected covers of .

Proof.

The first statement follows directly from Lemma 3.2 and the definition of . The second statement follows from the first and the fact that, in the compact case, every connected cover of is covered by for some sufficiently small .∎

Thus, is precisely equal to the kernel of , and we now see why is called the closed lifting group. It is natural to wonder why is not simply defined as . While this would be more efficient, the homomorphisms and the representation are useful by themselves (see Section 4), beyond the mere fact that . Therefore, it will be convenient to have both the algebraic and geometric interpretations of the closed lifting group.

We have carried out the above constuctions using the -covers determined by an arbitrarily chosen base point, . Since our definitions fundamentally rely upon the lifts of path loops to these covers, this raises the question of whether or not our results depend on that base point, particularly since another base point would technically induce different, albeit isometrically equivalent, covers. This equivalence and the regularity of the -covers should make it evident that the lifts of path loops are independent of the point we choose to construct the -covers. For the sake of completeness, however, we will formally clear up any questions about base points.

The key result is one we already mentioned above: if is our base point in , and is a path loop based at , then lifts closed to if and only if every lollipop lifts closed to , where is any path from to . Recall that we denote the base point of by , which we always take to be the identity in . For the next two lemmas, we adopt the following notation, but we will not need it thereafter. We denote the -covers determined by using the standard pointed notation, .

Lemma 3.9.

Let be a geodesic space with base points and . A path loop based at lifts closed to if and only if lifts closed to .

Proof.

This follows from repeated applications of the aforementioned key result regarding lollipops. Let be an isometric covering equivalence (see Section 2) satisfying , where and are the respective covers. If lifts closed to , then lifts closed to , where is any path from to . Let denote the lift of to . Then is a path loop in , and it projects to under , because of the equality . Thus, lifts closed to , which further implies that lifts closed to , where is any path from to . But this implies that lifts closed to .∎

In particular, there is no ambiguity in stating that a path loop lifts closed to without referencing the base point used to determine the -covers. We will use this result without comment going forward.

Continuing the notation of the previous lemma, let and denote the closed lifting groups at and , respectively. For , let and denote the respective kernels of and . Fix a path, , from to , and define by .

Lemma 3.10.

For a geodesic space, , the following hold.

  1. The restriction of to is an isomorphism onto .

  2. The restriction of to is an isomorphism onto .

  3. .

Proof.

We first note that is an isomorphism; this is standard in fundamental group arguments. Suppose , so that is a path loop at that lifts closed to . By Lemma 3.9, lifts closed to , implying that lifts closed to . This shows that maps into . On the other hand, if , let , so that . By Lemma 3.9, lifts closed to , and it follows that lifts closed to . Thus, maps onto , proving 1.

Next, suppose , and let be given. Since , by part 1. Since was arbitrary, this shows that , and maps