1 Introduction↩︎

There are two important notions of globally nonpositive curvature defined for metric spaces. One arises from Alexandrov’s pioneering comparison geometry and is nowadays called CAT(0), while the other goes back to Busemann and will be referred to here as BNPC (also known in the literature as a Busemann space or a convex space). A complete geodesic space \(X\) is BNPC if, for any two constant-speed geodesics \(\gamma,\eta:[0,1]\to X\), the restriction of the distance function \[t\mapsto d(\gamma(t),\eta(t))\] is convex on \([0,1]\). Although CAT(\(0\)) spaces are BNPC spaces, the class of BNPC spaces is much larger than the class of CAT(\(0\)) spaces. For example, an \(L^p\) space is CAT(\(0\)) only for \(p=2\), yet it is BNPC for any \(1<p<\infty\). More generally, CAT(\(0\)) spaces generalize complete, simply connected Riemannian manifolds of nonpositive sectional curvature, whereas BNPC spaces can be viewed as a generalization of such Finsler manifolds. See Section 2 for more background information.

The purpose of this paper is to extend recent topological results of Lytchak–Nagano [1], [2] and Lytchak–Nagano–Stadler [3] (building on earlier work of Thurston [4]) for CAT(\(0\)) spaces to the setting of BNPC spaces. This gives a positive answer to the problem raised in [3]. Although many of the original results were stated for more general CAT(\(\kappa\)) spaces, we restrict ourselves here to the most basic case of nonpositive curvature. As mentioned above, the difference between Alexandrov and Busemann curvature bounds closely reflects the difference between Riemannian and Finsler geometries. Therefore, the key to the extension is to capture the Finsler nature of BNPC spaces and deduce from it the same topological properties as in the CAT(\(0\)) case.

More specifically, the main technical contribution of this paper is the development of a strainer theory for geodesically complete BNPC spaces (more precisely, for GNPC spaces defined later). Strainers are a classical tool in Alexandrov geometry, originally introduced in the study of Alexandrov spaces with curvature bounded below [5], and are also the key ingredient in the aforementioned studies of CAT(\(0\)) spaces [1], [2], [3]. However, in contrast to the CAT(\(0\)) setting, BNPC spaces do not have a canonical angle structure for introducing strainers, reflecting their Finsler-like nature. We overcome this difficulty by separating the two roles played by angles in the CAT(\(0\)) theory: one controlling the branching of geodesics and the other controlling orthogonality. These technical details will be discussed in Section 1.6.

1.1 Main results↩︎

The main results concern topological manifolds and homology manifolds with BNPC metrics. A metric space is locally BNPC if every point has a neighborhood that is BNPC with respect to the ambient metric, which we call a BNPC neighborhood.

Theorems 1 and 2 generalize, respectively, [2] of Lytchak–Nagano in the CAT(\(0\)) case, and Theorem 3 generalizes [3] of Lytchak–Nagano–Stadler. In Theorem 1, the homotopy type of a punctured ball in a BNPC neighborhood is independent of its radius (and of its openness or closedness), thanks to the unique geodesic contraction centered at \(p\). Although Theorem 1 may look different from [2], they are essentially the same, because in the CAT(\(0\)) case any small punctured ball around \(p\) is homotopy equivalent to the space of directions at \(p\) ([6]; see also Remark 106). Note that Theorems 1 and 2 are local results, whereas Theorem 3 is a global one. For more generalizations, such as general curvature bounds, see Section 1.5.

1.2 Comments↩︎

Here are some more comments on the main theorems, which largely parallel the CAT(\(0\)) case.

First, we remark that Theorems 1 and 2 for Alexandrov spaces with curvature bounded below were proved by Wu [7] (cf.[8]).

In the situation of Theorem 1, the punctured ball around \(p\) is homotopy equivalent to any sufficiently small metric sphere around \(p\); see Theorem 5. However, for dimensions \(\ge 5\), such a metric sphere need not be homeomorphic to a sphere, as a consequence of the double suspension theorem of Cannon [9] and Edwards [10] (cf.[11]). On the other hand, for dimension \(\le 4\), every sufficiently small metric sphere around \(p\) is a topological sphere, exactly as in the CAT(\(0\)) case (Corollary 107).

Theorem 2 for Alexandrov curvature bounds was a question raised by Quinn [12]. Note that for dimension \(\ge4\), a locally BNPC homology manifold is not necessarily a topological manifold. On the other hand, for dimension \(\le3\), any locally BNPC homology manifold is a topological manifold. The dimension \(1\) and \(2\) cases are general facts holding without the BNPC condition (Theorem 18). The dimension \(3\) case was essentially observed by Thurston [4] (see Theorem 31). Our argument provides an alternative proof of this fact in Corollary 105.

Theorem 3 provides a definitive answer to a question of Gromov [13], asking whether there exists a BNPC topological manifold different from Euclidean space. The same statement as Theorem 3 is known to hold in dimensions \(\le 3\) and to fail in dimensions \(\ge 5\). The dimension \(1\) and \(2\) cases follow from the contractibility of BNPC spaces. The dimension \(3\) case is a combination of the results of Brown [14] and Rolfsen [15] (Theorem 31). A counterexample in dimension \(\ge 5\) was constructed by Davis–Januszkiewicz [16] (cf.[17]) in the CAT(\(0\)) setting. Our theorem thus provides an affirmative solution to the only remaining case, extending the result of Lytchak–Nagano–Stadler [3] in the CAT(\(0\)) case.

For additional results related to the main theorems, see Sections 7.1 and 7.2.

1.3 General results↩︎

In our main theorems, the (locally) BNPC spaces under consideration are locally geodesically complete, i.e., any geodesic is locally extendable (Proposition 28). The original results of Lytchak–Nagano and Lytchak–Nagano–Stadler were applications of the theory of GCBA spaces, developed in [1], [2], [3]. A GCBA space is a separable, locally compact, locally geodesically complete, locally CAT(\(\kappa\)) space. For recent results on GCBA spaces, see [18], [19], [20], [21], [22], [23], [24], [25].

Similarly, in this paper we develop the theory of GNPC spaces. A GNPC space is a separable, locally compact, locally geodesically complete, locally BNPC space (the separability is not necessary for our purposes). Every GCBA space of nonpositive curvature is, in fact, a GNPC space. Throughout this paper, all spaces under consideration are GNPC spaces. We prove the following two theorems in this general setting; compare with [1] (cf.[26], [6]).

Lytchak–Nagano [1] further revealed the geometric structure of GCBA spaces, such as the tangent cone, measure-theoretic stratification, and almost everywhere Riemannian structure. However, we focus here on the topological aspects of GNPC spaces, and only delve into as much geometric structure as needed for the proofs of the main theorems. We leave more detailed geometric aspects for future work. See Section 8 for open problems (see also Remark 137 for the latest information after the first version of this paper appeared on arXiv).

1.4 Applications↩︎

Busemann [27] also introduced the notion of G-space, which can be viewed as a qualitative generalization of Finsler manifolds. The BNPC condition was first introduced in the context of G-spaces ([28], [27]). In simple terms, a G-space is a complete, locally compact geodesic space with local unique extendability of geodesics. See Section 7.3 for background.

In this paper we only consider G-spaces with (locally) BNPC metrics. Note that a locally BNPC G-space is nothing but a complete GNPC space without branching geodesics. The nonbranching assumption for geodesics is very strong and prevents the existence of singular points of GNPC spaces. Indeed, Andreev [29] proved that any locally (resp.globally) BNPC G-space is a topological manifold (resp.homeomorphic to Euclidean space). Compare with the general situation described in Section 1.2 (see also [30]). As a corollary of Theorem 4, we obtain an alternative proof of Andreev’s result.

Furthermore, any small metric sphere in a locally BNPC G-space is a topological sphere (Theorem 116). Hence the pathological phenomena arising from the double suspension theorem mentioned in Section 1.2 do not occur.

Andreev [31] later proved a more rigid result: the tangent cone of a locally BNPC G-space is a strictly convex normed space. In other words, any GNPC space without branching geodesics admits a Finsler-like structure. This can be viewed as a model case of the geometric structure of the “regular part” of a GNPC space. Compare with the Riemannian structure of the regular part of a GCBA space: [1], [32]. See Section 8 for more details.

As an application of Theorem 1, we obtain the following topological stability theorem for locally BNPC G-spaces. This partially generalizes [2]. The injectivity radius of a G-space \(X\) is the supremum of \(r>0\) such that for any \(p\in X\) every shortest path emanating from \(p\) of length \(<r\) extends uniquely to length \(r\) (i.e., without branching).

Note that the limit space \(X\) in Theorem 7 may fail to be a G-space if geodesics branch; it may even fail to be locally BNPC if geodesics are not unique. For instance, \(L^p\) spaces (\(1<p<\infty\)) converge to \(L^1\) or \(L^\infty\) space under the Gromov–Hausdorff topology. However, as we explain in the next subsection, the limit space locally admits a convex geodesic bicombing.

1.5 Generalizations↩︎

We now discuss two possible/potential generalizations.

One is a possible generalization to spaces with convex geodesic bicombings. The notion of convex geodesic bicombing, introduced by Descombes–Lang [33], is a generalization of the BNPC condition. Similar approaches can also be found in [34] and [26]. Roughly speaking, all these approaches weaken the BNPC condition by dropping the uniqueness of geodesics and instead imposing the convexity condition only on some suitable choice of geodesics. See Section 7.4 for a precise definition. For example, all normed spaces admit convex geodesic bicombings by choosing the affine geodesics. The advantage of this treatment is that this class is closed under limiting operations, which is necessary to prove Theorem 7. We claim

The other is a potential generalization to general curvature bounds. The original statements of Lytchak–Nagano and Lytchak–Nagano–Stadler were established for CAT(\(\kappa\)) spaces with arbitrary \(\kappa\in\mathbb{R}\) (even a global result; see [3]). By using triangle comparison, Busemann curvature bounds can also be defined for any \(\kappa\in\mathbb{R}\) (Remark 13). While it should be possible to extend our arguments to that broader setting, we do not pursue this here and leave it as an open problem (Problem 130).

1.6 Main tool↩︎

We now explain how to prove the main theorems. Our proofs are parallel to the original proofs of Lytchak–Nagano [1], [2] and Lytchak–Nagano–Stadler [3]. The original proofs can be divided into a topological part and a geometrical part. In our generalizations, the topological part is the same as the original one, and the problem to overcome lies in the geometrical part (contrary to the title of this paper).

The main technical tool in the geometrical part of the original proofs is the notion of a strainer (and its variant called an extended strainer). Roughly speaking, a strainer at a point \(p\) is a finite collection of points around \(p\) that behaves like an almost orthonormal family of vectors in the tangent space at \(p\).

Historically, strainers were first introduced by Burago–Gromov–Perelman [5] in the study of Alexandrov spaces with curvature bounded below. Lytchak–Nagano [1], [2] later adapted them to the setting of GCBA spaces. Extended strainers were introduced by Lytchak–Nagano–Stadler [3] to handle global problems on GCBA spaces. In this paper, we define strainers and extended strainers for GNPC spaces, prove their existence, and establish their topological properties. This is the main contribution of this paper.

Although the various notions of strainers in different settings are defined abstractly using angles, their underlying constructions are essentially the same and remain meaningful (whether or not they are immediately useful) in general metric spaces. Since GNPC spaces, like Finsler manifolds, have no fixed notion of angle, we now recall the concrete construction based on metric spheres.

Let \(X\) be a metric space. We denote by \(\partial B(p,r)\) the metric sphere centered at \(p\in X\) of radius \(r>0\). Fix the first point \(p_1\in X\) and choose the second point \(p_2\in\partial B(p_1,r_1)\), where \(r_1\ll1\) depends on \(p_1\) (we discuss the dependence later). Next choose the third point \(p_3\in\partial B(p_1,r_1)\cap\partial B(p_2,r_2)\), where \(r_2\ll r_1\) depends on \(p_2\). Repeating this procedure, one can construct a sequence \(p_1,\dots,p_k,p_{k+1}\) of points in \(X\) such that \[p_{i+1}\in\bigcap_{j=1}^i\partial B(p_j,r_j),\quad r_{i+1}\ll\;r_i\] for any \(1\le i\le k\), where \(r_i\) depends on \(p_i\). (Note that the construction may stop if the intersection is empty.) The sequence \(p_1,\dots,p_k\) is called a \(k\)-strainer at \(p_{k+1}\). The distance map \[f=(d(p_1,\cdot),\dots,d(p_k,\cdot)):X\to\mathbb{R}^k\] is called a \(k\)-strainer map at \(p_{k+1}\). See Figure 1.

If \(X\) is an Alexandrov or GCBA space, the strainer \(p_1,\dots,p_k\) behaves like an almost orthonormal family of vectors at \(p_{k+1}\). As a result, the associated strainer map \(f\) has some fibration property around \(p_{k+1}\). We recall this in more detail below.

Suppose \(X\) is an Alexandrov space. Then the geodesics in \(X\) are generally not extendable (but never branch). However, if \(r_i\) is sufficiently small depending on \(p_i\), the geodesics emanating from \(p_i\) become almost extendable in the \(r_i\)-neighborhood of \(p_i\), especially at \(p_{k+1}\). Moreover, if \(r_i\ll r_j\), these almost extendable geodesics from \(p_i\) and \(p_j\) to \(p_{k+1}\) are almost orthogonal at \(p_{k+1}\) for any \(1\le i\neq j\le k\). These properties, together with a deep analysis using geometric topology, imply that \(f\) is locally a trivial fibration around \(p_{k+1}\) ([35], [36]).

Suppose \(X\) is a GCBA space. Then the geodesics in \(X\) are always extendable, but may branch. Nevertheless, if \(r_i\) is sufficiently small depending on \(p_i\), the geodesics emanating from \(p_i\) are almost nonbranching in the \(r_i\)-neighborhood of \(p_i\), especially at \(p_{k+1}\). Furthermore, if \(r_i\ll r_j\), these almost nonbranching geodesics from \(p_i\) and \(p_j\) to \(p_{k+1}\) are almost orthogonal at \(p_{k+1}\) for any \(1\le i\neq j\le k\). These properties, combined with a retraction argument using geometric topology, show that \(f\) is locally a Hurewicz fibration around \(p_{k+1}\) ([1], [2]; but not a trivial fibration because of the complexity of singularity).

In both cases, the key properties of a strainer are the almost extendability (in the Alexandrov case) or the almost nonbranching (in the GCBA case), and the almost orthogonality (in both cases) of geodesics. All these properties are established using the unique notion of angle, whose existence follows from the Riemann-type curvature bounds of Alexandrov/GCBA spaces.

Now let us turn to the case of GNPC spaces. As in the case of GCBA spaces, what we want to prove are:

1. the almost nonbranching and almost orthogonality of a strainer, and

2. the fibration property of a strainer map based on them.

Regarding (1), the main difficulty is the lack of a fixed angle notion in GNPC spaces, reflecting their Finsler-like character. To overcome this, we introduce two distinct notions of angle which are tied to the branching and orthogonality of geodesics, respectively. These angles coincide in GCBA spaces but can differ in general GNPC spaces. This will be discussed in Section 4.

Regarding (2), the challenge is the asymmetry of the almost orthogonality of a GNPC strainer, which is addressed by using a modified consecutive-approximation argument. An informal analogy is as follows. Let \(f\) and \(g\) be smooth maps from \(\mathbb{R}^n\) to \(\mathbb{R}^k\) (\(n\ge k\)), whose Jacobian matrices are \[df= \left( \begin{array}{ccc|c} 1\pm\delta & & \pm\delta & \\ & \ddots & & ?\\ \pm\delta & & 1\pm\delta & \end{array} \right) ,\quad dg= \left( \begin{array}{ccc|c} 1\pm\delta & & \pm\delta & \\ & \ddots & & ?\\ \pm C & & 1\pm\delta & \end{array} \right),\] where \(\delta\) is a small positive number and \(C\) is a uniformly bounded positive number. An Alexandrov/GCBA strainer map looks like \(f\), whereas a GNPC strainer map looks like \(g\). The first \(k\) diagonal entries of \(df\) being almost \(1\) represents the almost extendability or almost nonbranching, while the other entries being almost \(0\) correspond to the almost orthogonality. The lower triangular entries of \(dg\) are responsible for the infinitesimal deviation of our GNPC metric from the standard Euclidean metric. In either case, however, if \(\delta\) is small enough compared to \(C\), the Jacobian matrices are regular and thus, after appropriate coordinate changes, \(f\) and \(g\) are locally projection maps. An analogous argument allows us to prove the fibration property of a GNPC strainer map. This will be discussed in Section 5.

In addition to the the absence of a canonical angle structure, another difficulty overcome in Sections 4 and 5 is the possible “non-differentiability” of sufficiently small metric spheres in GNPC spaces, which is not explicitly stated but is implicitly handled in our arguments. This stems from the fact that the GNPC setting differs from the GCBA setting by allowing non-smooth normed spaces, reflecting again its Finsler-like nature. This perspective will play an important role in future studies of the geometric structure of GNPC spaces; see Remarks 80 and 137.

Furthermore, in Section 6, we introduce an extended strainer for a GNPC space. Our treatment of an extended strainer is slightly different from the original one of Lytchak–Nagano–Stadler [3]: we introduce the associated half strainer map and prove its fibration property, in parallel with that of the standard strainer maps. This half-strainer formulation, and in particular the corresponding fibration theorem, does not appear in [3] in this form, and constitutes another contribution of this paper.

Hence, Sections 4, 5, and 6 constitute the geometric core of this paper. Once the existence and fibration properties of strainers and extended strainers are established, the proofs of the main topological theorems follow the GCBA case and rely mainly on topological arguments.

Organization. In Section 2, we recall the basics of BNPC spaces, introduce GNPC spaces, and present some results from geometric topology. In Section 3, we collect several facts known for CAT(\(0\)) spaces that directly extend to BNPC spaces. In Section 4, we prove key lemmas on the almost nonbranching and almost orthogonality of geodesics in GNPC spaces. In Section 5, we develop strainers for GNPC spaces and prove Theorems 4 and 5. In Section 6, we develop extended strainers for GNPC spaces in parallel with the previous section. In Section 7, we prove Theorems 1, 2, 3, 6, 7, and 8. In Section 8, we discuss open problems.

Acknowledgments. The authors are especially grateful to Stephan Stadler for giving us the opportunity to collaborate and for his insightful suggestions. The first author would like to thank Shin-ichi Ohta and Koichi Nagano for helpful discussions, and Alexander Lytchak for his valuable comments, especially on Section 8. The second author would like to thank Wenyuan Yang for his inquiry regarding potential applications to spaces admitting convex geodesic bicombings, and Shmuel Weinberger for helpful discussions on homology manifolds. The first author was supported by JSPS KAKENHI Grant Numbers 22KJ2099 and 25K23336. The second author was supported by NSFC grant 12201102.

2 Preliminaries↩︎

In this section, we recall the definition and basic properties of BNPC spaces and introduce GNPC spaces together with their tiny balls. We also present several results from geometric topology concerning homology manifolds and Hurewicz fibrations.

2.1 Notation↩︎

We denote the distance between two points \(x\) and \(y\) by \(d(x,y)\) or \(|xy|\). A shortest path (see the next subsection) between \(x\) and \(y\) is denoted by \(xy\). The open and closed metric balls, and metric sphere centered at \(p\) of radius \(r\) are denoted by \(B(p,r)\), \(\bar B(p,r)\), and \(\partial B(p,r)\), respectively. Open and closed punctured balls mean \(B(p,r)\setminus\{p\}\) and \(\bar B(p,r)\setminus\{p\}\), respectively.

In Sections 4, 5, and 6, we will use the symbols \(\delta\), \(\varkappa(\delta)\), \(\varkappa_k(\delta)\), \(\delta_k\), and \(\varepsilon_k\) with special meanings. See Notations 34 and 56.

2.2 Geodesic spaces↩︎

A shortest path is an isometric embedding of a closed interval into a metric space. A geodesic space is a metric space such that any two points can be joined by a shortest path. Any complete, locally compact geodesic space is proper, i.e., every closed ball is compact ([37]). In other words, a distance function from a point is proper in the usual sense.

A geodesic is a curve whose restriction to each sufficiently small sub-interval is a shortest path. A geodesic space \(X\) is called locally geodesically complete if every geodesic is locally extendable, or more precisely, for any geodesic \(\gamma:[a,b]\to X\), there exist \(\varepsilon>0\) and a geodesic \[\bar\gamma:[a-\varepsilon,b+\varepsilon]\to X\] such that \(\bar\gamma_{[a,b]}=\gamma\). Similarly, \(X\) is called (globally) geodesically complete if each geodesic is infinitely extendable, i.e., one can take \(\varepsilon=\infty\). In a complete metric space, local geodesic completeness implies global geodesic completeness ([37]).

By definition, every geodesic is unit-speed parametrized. In this paper, we often consider a linear reparametrization of a geodesic, also described as a constant-speed geodesic.

2.3 BNPC spaces↩︎

Now we introduce BNPC spaces, i.e., globally nonpositively curved spaces in the sense of Busemann. While there are many textbooks about CAT(\(0\)) spaces, globally nonpositively curved spaces in the sense of Alexandrov (e.g., [38], [39], [40], [37]), the references for BNPC spaces are quite limited. We refer the reader to [41] and [42].

The following remark points out simple yet crucial consequences of the above definition.

Indeed, Condition (1 ) is equivalent to the definition of a BNPC space, thanks to the triangle inequality ([41]). The above two properties are, in a sense, closely related to the two angles we will introduce in Sections 4.1 and 4.2, respectively.

Note that if \((X,d)\) is BNPC, then its rescaling \((X,\lambda d)\) remains BNPC for any \(\lambda>0\). The following are basic properties of BNPC spaces that immediately follow from the above definition (see [41]).

• A geodesic between any two points is unique; in particular, every geodesic is actually a shortest path.

• Every open or closed ball \(B\) is convex, i.e., any shortest path between two points in \(B\) is contained in \(B\).

• Every convex subset \(C\) is contractible to an arbitrary point in \(C\) along the unique shortest paths: we call this the geodesic contraction.

In particular, any BNPC space is contractible and therefore simply connected. Conversely, the Cartan–Hadamard-type globalization theorem holds: any complete, simply connected, locally BNPC space is (globally) BNPC ([49], [50]). We say that a metric space is locally BNPC if every point has a neighborhood that is BNPC with respect to the ambient metric. In particular, the universal cover of any complete, locally BNPC space is BNPC.

2.4 BNPC vs.CAT(0)↩︎

We now compare BNPC spaces with CAT(\(0\)) spaces. Recall that the Busemann monotonicity 1 is an equivalent definition of a BNPC space and we interpret it geometrically as follows.

Let \(\triangle pxy\) be a geodesic triangle in a BNPC space, and let \(\triangle\tilde{p}\tilde{x}\tilde{y}\) be its comparison triangle, that is, a geodesic triangle in the Euclidean plane with the same side lengths. For \(0\le t\le 1\), let \(tx\) and \(ty\) denote the points on the shortest paths \(px\) and \(py\) such that \(|p,tx|=t|px|\) and \(|p,ty|=t|py|\), respectively. Similarly, let \(t\tilde{x}\) and \(t\tilde{y}\) be the corresponding points on the shortest paths \(\tilde{p}\tilde{x}\) and \(\tilde{p}\tilde{y}\), respectively. Then the Busemann monotonicity 1 says \[|tx,ty|\le|t\tilde{x},t\tilde{y}|.\]

By contrast, the CAT(\(0\)) condition says \[|tx,sy|\le|t\tilde{x},s\tilde{y}|\] for any \(0\le t\le 1\) and \(0\le s\le 1\). Hence every CAT(\(0\)) space is a BNPC space. This distinction reflects the Riemannian vs. Finsler nature of CAT(\(0\)) vs. BNPC, as illustrated in the following examples.

In general, a BNPC space is CAT(\(0\)) if and only if the angle is well-defined for every hinge ([39]). Foertsch–Lytchak–Schroeder [51] characterized CAT(\(0\)) spaces among BNPC spaces in terms of the Ptolemy inequality. Another characterization was given by the second author [52] in terms of average angles.

2.5 GNPC spaces↩︎

Following Lytchak–Nagano [1], [2], we introduce

Note that BNPC is a global condition, whereas GNPC is a local condition, except for the separability (which we do not use in this paper). Lytchak–Nagano’s GCBA space with nonpositive curvature, defined by replacing “BNPC" with”CAT(\(0\))" in Definition 15, is our GNPC space. Any open subset of a GNPC space remains GNPC, and any locally BNPC homology manifold (without boundary) is GNPC (Proposition 28; see the next subsection for homology manifolds).

Following Lytchak–Nagano [1], [2], we also introduce

For each \(p\in X\), any sufficiently small ball around \(p\) is a tiny ball. We allow arbitrarily large radius to deal with global problems like Theorem 3, but after rescaling, the actual radius becomes irrelevant.

Within a tiny ball \(B\) (and its enlargement \(10\bar B\)), all properties of a BNPC space from Section 2.3 hold. Moreover, any shortest path in \(B\) is extendable to length \(9r\) in \(10\bar B\), where \(r\) is the radius of \(B\). This follows from the same argument that local geodesic completeness implies global one in a complete space (see Section 2.2).

2.6 Homology manifolds↩︎

We say that \(X\) is a homology \(n\)-manifold with boundary if it is a locally compact, separable, metrizable space of finite topological dimension such that the local (integral) homology group at each point is isomorphic to that of the closed \(n\)-cell \(D^n\) at some point, i.e., \[H_\ast(X,X\setminus\{x\})\cong H_\ast(D^n,D^n\setminus\{y\})\] for any \(x\in X\) and some \(y=y(x)\in D^n\).

The boundary of a homology \(n\)-manifold is the set of points where the \(n\)-th local homology vanishes. The following theorem is due to Mitchell [53].

A homology manifold simply refers to a homology manifold without boundary (the same convention applies to topological manifolds). Any homology \(n\)-manifold has topological dimension \(n\).

The following theorem, due to Moore ([54]), is the starting point of the inductive proofs of the main theorems.

A metric space \(X\) is called an absolute neighborhood retract, abbreviated as ANR, if for any closed subset \(A\) of a metric space \(Z\), every continuous map \(A\to X\) has a continuous extension \(U\to X\) defined on some neighborhood \(U\) of \(A\) in \(Z\). For our purposes, the following characterization is more useful: a finite-dimensional, locally compact metric space is an ANR if and only if it is locally contractible. See [55] for ANRs. Every ANR homology manifold with boundary satisfies the Poincaré–Lefschetz duality (note that our homology manifolds are of finite dimension).

2.7 Hurewicz fibrations↩︎

Here we assume all maps are continuous. A surjective map between topological spaces is a Hurewicz fibration if it satisfies the homotopy lifting property with respect to all topological spaces.

We recall the following condition from [2].

The following theorems of Ungar [56] provide sufficient conditions for a map to be a Hurewicz fibration (see also [2]).

The following theorem of Raymond [57] describes the relationship between homology manifolds and Hurewicz fibrations (see also [2]).

2.8 Recognition theorems↩︎

In the proofs of the main theorems, the key is to “recognize” topological manifolds among homology manifolds (cf.Theorem 18) and fiber bundles among Hurewicz fibrations. Our proofs rely heavily on such recognition theorems listed in [2]. However, these recognition theorems are used in the topological part of the proofs, match exactly the CAT(\(0\)) case and thus omitted here. For this reason, although the use of these theorems is essential, we avoid including their statements and numerous related references here. See [2] and the references therein.

3 Facts↩︎

In this section, we collect several facts known for CAT(\(0\)) spaces whose proofs also work for BNPC spaces. For completeness (and as a warm-up), we include the proofs whenever possible. The reader familiar with Lytchak–Nagano’s theory of GCBA spaces [1], [2] and Thurston’s result [4] may skip this section.

The first proposition is about the finite-dimensionality of GNPC spaces ([1]). A metric space \(X\) is called doubling, or more specifically \(L\)-doubling, if there exists a doubling constant \(L>0\) such that every \(r\)-ball in \(X\) can be covered by at most \(L\) balls of radius \(r/2\). Clearly, if any \(r\)-ball in \(X\) contains at most \(L\) points with pairwise distance \(\ge r/2\), then \(X\) is \(L\)-doubling (we say that such points with pairwise distance \(\ge r/2\) are \(r/2\)-separated). A metric space is locally doubling if each point has a neighborhood that is doubling, where the doubling constant may depend on each point.

The second proposition provides a sufficient condition for a BNPC space to be geodesically complete ([2], [60]; see also [61] for a slightly stronger statement). In particular, any (locally) BNPC space in the main theorems is (locally) geodesically complete.

The third proposition addresses the local topology of BNPC homology manifolds ([2], [4]).

The fourth theorem is the \(3\)-dimensional version of Theorems 2 and 3. The first half is due to Thurston [4] and its second half follows from Rolfsen [15] and Brown [14]. Later we will provide an alternative proof of the first half (Corollary 105).

The last theorem is the main result of Thurston [4]. This plays an essential role in the proof of Theorem 3, as in the CAT(\(0\)) case. Later we will show that the tameness assumption is always satisfied (Theorem 109).

4 Key lemmas↩︎

In this section, we prove key lemmas needed to develop strainers for GNPC spaces in the next section. They involve two different notions of angle, so we divide this section into two subsections: in Section 4.1, we introduce the angle related to the branching of geodesics, and in Section 4.2, we introduce the angle related to the orthogonality of geodesics. These subsections are independent; neither part relies on results from the other.

Let \(\triangle pqr\) be a geodesic triangle in a tiny ball \(B\) of a GNPC space \(X\). The two angles at \(p\) we will introduce are denoted by \[\angle_p(q,r),\quad\angle qpr.\] These two angles are in a sense related to the Busemann monotonicity 1 and Busemann convexity 2 , respectively. In general they do not coincide unless \(X\) is a GCBA space. This asymmetry of angles is the central problem we must deal with.

Before delving into the details, we establish key notations that will be used extensively in this and the next two sections.

4.1 Almost nonbranching↩︎

In this subsection, we introduce the “angle of fixed scale,” which controls the branching of geodesics. The goal of this subsection is to prove Proposition 39, the almost nonbranching of geodesics.

Throughout this subsection:

• \(X\) denotes a GNPC space and \(B\) a tiny ball.

• \(p\in B\) denotes an arbitrary fixed point.

• \(x,y\in B\setminus\{p\}\) denote variable points (near \(p\)).

• \(\tilde{\angle}_p(x,y)\) denotes the comparison angle at \(p\) of the Euclidean triangle with side-lengths \(|px|\), \(|py|\), and \(|xy|\).

• For \(\lambda\ge0\), \(\lambda x\) denotes a point on an extension of the shortest path from \(p\) to \(x\) such that \(|p,\lambda x|=\lambda|px|\).

In other words, \(\lambda x\) is the reparametrization of an extension of the shortest path \(px\) by the constant speed \(|px|\). This notation allows us to regard \(p\) as if it were the origin of a vector space. Note that, however, \(\lambda x\) is not necessarily unique for \(\lambda>1\) because of the branching of geodesics.

Let \(p,x,y\in B\) be as above. The Busemann monotonicity 1 literally says that the comparison angle \(\tilde{\angle}_p(tx,ty)\) is nondecreasing in \(t\ge 0\) (see Section 2.4). Therefore we can define the following angle.

This angle controls the branching of geodesics emanating from \(p\). In the GCBA case, the independence of this angle on parametrizations implies that the tangent cone (i.e., the Gromov–Hausdorff blow-up) at \(p\) is a Euclidean cone ([1]). Since the geodesics from the vertex of the Euclidean cone are nonbranching, one sees that the geodesics from \(p\) are almost nonbranching near \(p\) ([1]). In the GNPC case, we must restrict this argument to some almost fixed scale. The following is a uniform version of the convergence in Definition 36 with respect to some almost fixed scale.

The above lemma immediately implies the following property, which we call the almost nonbranching of geodesics.

4.2 Almost orthogonality↩︎

In this subsection, we introduce the “angle viewed from a fixed point,” controlling the orthogonality of geodesics. The goal of this subsection is to prove Proposition 47, the almost orthogonality of geodesics.

Throughout this subsection, we will use the following notation, some of which are different from those in the previous subsection:

• \(X\) denotes a GNPC space and \(B\) a tiny ball.

• \(p\in B\) and \(q\in B\setminus\{p\}\) denote arbitrary fixed points.

• \(x\in B\setminus\{q\}\) denotes a variable point (near \(q\)).

• \(\tilde{\angle} pqx\) denotes the comparison angle at \(q\) of the Euclidean triangle with side-lengths \(|pq|\), \(|px|\), and \(|qx|\).

• \(x(t)\) denotes the shortest path from \(q\) to \(x\) and its extension beyond \(x\), parametrized by arclength.

Unlike the GCBA case, the comparison angle \(\tilde{\angle} pqx(t)\) is not necessarily nondecreasing in \(t\). Therefore the existence of the above limit is a little bit nontrivial. However, the “almost monotonicity” coming from the Busemann convexity 2 guarantees the existence of the limit.

The above angle is used to control the orthogonality of geodesics around \(q\). To see this, we first prove a uniform version of the convergence in Definition 40. The proof is similar to that of Lemma 38.

Next we show a basic property of this angle. In the GCBA case, the following is a special case of the triangle inequality for angles. In the GNPC case, this can be viewed as an infinitesimal version of the Busemann convexity 2 . Note that we do not use the fixed point \(q\) here.

Combining the above three lemmas, we obtain the following key property. In the GCBA case, this is a combination of the triangle inequality and the upper semicontinuity of angles ([1]).

Finally, we reach the following property, which we call the almost orthogonality of geodesics.

5 Strainers↩︎

In this section, we develop strainers for GNPC spaces, which play central roles in the proofs of the main theorems. The definition and existence are based on the key lemmas proved in the previous section, specifically their consequences, Propositions 39 and 47. Once we establish the existence of strainers, we will not need to refer back to those lemmas when proving further properties.

The organization of this section is as follows. In Section 5.1, we define strainers and show their existence. In Section 5.2, we prove the fibration property of strainers. In Section 5.3, as consequences, we prove Theorems 4 and 5.

5.1 Definition and existence↩︎

In this subsection, we define strainers for GNPC spaces and show their existence (cf.[1]).

Throughout this subsection, \(X\) denotes a GNPC space, \(B\) a tiny ball, and \(\delta\) a small positive number (see Notation 34 and Remark 35). The definition of a strainer is given in terms of the angle \[\angle pqx\] viewed from \(p\) at \(q\) toward \(x\), introduced in Section 4.2. We again emphasize that this angle is essentially the directional derivative of the distance function from \(p\), i.e., \[\cos\angle pqx=-\left.\frac{d}{dt}\right|_{t=0}d(p,x(t)),\] where \(x(t)\) is the unit-speed shortest path from \(q\) to \(x\) (see Remark 43). Therefore, the reader should consider the following definition of a strainer as controlling the derivative of the distance function rather than the angles, unlike in the GCBA case. Note that the other angle introduced in Section 4.1 is not used for the definition of a strainer, but will be used later to show the existence of a strainer.

We first define a \(1\)-strainer, which controls the branching of geodesics.

The point of the above definition is that the inequality holds for the same \(\bar p\) independent of \(y\in U\) (otherwise, one may choose \(\bar p=\bar p(y)\) as a point on an extension of the shortest path \(py\) beyond \(y\)). Roughly speaking, this means that there is one direction at \(x\) that uniformly increases the distance from \(p\) around \(x\). If the shortest path \(px\) branches significantly at \(x\), there is no such direction. This is the same as in the GCBA case.

Based on the above definition, we next define a \(k\)-strainer (\(k\ge 2\)) that takes into account the almost orthogonality. Our definition is by induction.

The point of the above definition is that only \(\angle p_iyp_k\) and \(\angle p_iy\bar p_k\) are almost \(\pi/2\), and we make no assumptions on \(\angle p_kyp_i\) and \(\angle p_ky\bar p_i\). Thus, when we move from \(y\) to \(p_k\) or \(\bar p_k\), the distance from \(p_i\) remains almost unchanged, but there is no control on the distance from \(p_k\) when we move to \(p_i\) or \(\bar p_i\). This asymmetry is the crucial difference from the GCBA case, and is the Finsler nature of GNPC spaces we should deal with carefully. However, it turns out that the above asymmetric condition is enough to derive the same topological properties of a strainer as in the GCBA case.

Note that the definition of a strainer is an open condition. For later use, we also introduce (a lower bound for) the straining radius, which gives a quantitative estimate on the openness of a strainer.

Clearly, the straining radius is positive. It also has the following uniformity. This simple fact plays an important role in the proof of the local uniform contractibility of the fibers of a strainer map (Theorem 58).

Now we prove the existence of a strainer. First we show the existence of a \(1\)-strainer. This follows from the key lemmas in Section 4.1, specifically Proposition 39. Compare with [1].

Next we show the existence of a \(k\)-strainer, or more precisely, how to extend a strainer. This follows from the previous proposition and the key lemmas in Section 4.2, specifically Proposition 47. Compare with [1].

5.2 Fibration property↩︎

In this subsection, we establish the fibration property of strainers, which is the key ingredient in the proofs of the main theorems. This proof relies only on the definition of a strainer, so we no longer need to refer back to the key lemmas in Section 4.

Henceforth, in addition to \(\delta\) and \(\varkappa(\delta)\) from Notation 34, we often use the following notation.

The main object of this subsection is the following strainer map.

The following two theorems are the main results of this subsection; compare with [1] and [2], respectively. Let \(X\) be a GNPC space, and let \(\delta_k\) be a positive constant depending only on \(k\), as in Notation 56.

See Definition 19 for local uniform contractibility of fibers. As we will see later, Theorem 59 is a combination of Theorem 58, Proposition 55, and Theorems 20 and 21. Note that both of the above statements are local.

The key technique used in the proof of Theorem 58 is the so-called consecutive approximation, originating in the fundamental theory of Alexandrov spaces from Burago–Gromov–Perelman [5] and Perelman [35]. In some sense, what we develop here can be seen as the asymmetric version of it. We modify the standard Euclidean metric of the target space of a strainer map so that the asymmetry of the almost orthogonality can be ignored.

We remark that this type of asymmetry appeared in the technical proof of the stability theorem for Alexandrov spaces by Perelman [35], and more relevantly, in Pogorelov’s approach [65] to Busemann G-spaces with some regularity.

To prove the openness, we introduce a metric version of an open map.

In particular, for any \(0<s<r\), we have \(f(B(x,\varepsilon^{-1}s))\supset B(f(x),s)\). Therefore every \(\varepsilon\)-open map is a (topological) open map.

The following is a criterion for the \(\varepsilon\)-openness (cf.[35], [1]; see [66] for a more general statement). We say that a metric space \(Y\) is locally geodesic if each point has a neighborhood \(U\) that is geodesic in \(Y\), i.e., any two points in \(U\) are connected by a shortest path in \(Y\).

For completeness, we give the straightforward proof.

We now prove the openness of a strainer map, which is the first half of Theorem 58. Compare the statement below with [1].

We next prove the local uniform contractibility of the fibers of a strainer map, the second half of Theorem 58. Since a GNPC space is locally contractible, it suffices to construct a local uniform retraction to the fiber of a strainer map. The construction is a refinement of the previous proof of the openness. Here, we do require the almost nonbranching property, in addition to the almost orthogonality (cf.Remark 66). We also need the notion of straining radius, Definition 51, to guarantee the uniformity of the retraction. Compare the following with [1], and see also Remarks 70 and 71.

Before the proof, we introduce the following notation. We denote the (unit-speed) geodesic contraction centered at \(p\in B\) by \[\Phi_p(x,t)\] for \(x\in B\) and \(t\ge 0\), that is, \(\Phi_p(x,t)\) is the unique unit-speed shortest path from \(x\) to \(p\) for \(0\le t\le |px|\) and then stops at \(p\) for \(t\ge |px|\).

Recall that every metric ball in a tiny ball of a GNPC space is contractible via the geodesic contraction. By composing this contraction with the retraction to the fiber of a strainer map, as constructed above, we obtain the following corollary.

We are now ready to finish the proof of Theorem 58.

Finally, we derive Theorem 59 from Theorem 58. The following is a global version of Theorem 59, which follows directly from Theorems 58 and 20.

We now prove Theorem 59, using Theorem 21. The idea of the proof is to complement the given strainer map by adding one more distance function, coming from Proposition 55, to control the local contractibility of fibers (cf.[24]).

The next result follows directly from the proof above.

5.3 Consequences↩︎

In this subsection, we apply the properties of strainer maps to prove Theorems 4 and 5.

Theorem 5 follows from the special case \(k=0\) of Corollary 75. Here, by convention, any constant map to \(\mathbb{R}^0\) is a \((0,\delta)\)-strainer map for any \(\delta>0\). Hence the proof of Corollary 75 is valid for \(k=0\) (specifically, by Proposition 53 and Corollary 73).

Next, we prove Theorem 4. The following notion plays a key role.

The next lemma follows immediately from the finite dimensionality of a tiny ball (Proposition 23) and the openness of a strainer map (Proposition 65). This tells us that any strainer map will eventually become non-complementable.

We now prove another main result for a strainer map.

We now prove Theorem 4. Let \(X\) be a GNPC space and \(p\in X\). For \(\delta>0\), the local \(\delta\)-strainer number at \(p\) is the supremum of \(k\) for which each neighborhood of \(p\) contains a \((k,\delta)\)-strained point (i.e., a point with a \((k,\delta)\)-strainer). By Lemma 77, this local number is finite for \(\delta<\delta_p\), where \(\delta_p\) depends on \(p\).

6 Extended strainers↩︎

In this section, we modify the definition of a strainer to introduce an extended strainer (cf.[3]). Roughly speaking, we add a base point \(p_0\) to a strainer \(p_1,\dots,p_k\) such that \(p_1,\dots,p_k\) are contained in a metric sphere centered at \(p_0\), which may be arbitrarily large. This modification is necessary for dealing with a global problem like Theorem 3.

The contents of this section will be used only in Section 7.2, so one may skip this section until then. However, since this section is organized in parallel with the previous section, some readers may prefer to read it immediately after reading Section 5.

The organization of this section is as follows. In Section 6.1, we define extended strainers and show their existence. In Section 6.2, we prove the fibration property of extended strainers. In Section 6.3, as consequences, we prove some propositions which will be used in the proof of Theorem 3.

6.1 Definition and existence↩︎

In this subsection, we define extended strainers and show their existence. Throughout this subsection, \(X\) denotes a GNPC space, \(B\) a tiny ball, and \(\delta\) a small positive number (see Notation 34). Let \(k\ge 1\).

The difference of the above definition from the standard \((k+1)\)-strainer is that \(p_0\) is not assumed to be a \(1\)-strainer at \(x\): there is no opposite strainer \(\bar p_0\) for \(p_0\). In particular, there is no restriction on the distance between \(p_0\) and \(x\), as we will see below (recall that for the existence of a \(1\)-strainer, we have to consider a small neighborhood of a given point; see Proposition 53). The problem of the asymmetry of almost orthogonality appears here again for \(p_0\) and \(p_i\).

Similar to before, we introduce

Clearly, we have

We show the existence of an extended strainer. For convenience, we say that \(p\in B\) is an extended \((0,\delta)\)-strainer for any \(x\in B\setminus\{p\}\) and \(\delta>0\) (there is no condition, and thus it exists). Compare the following with [3].

6.2 Fibration property↩︎

In this subsection, we establish the fibration property of extended strainers, which will be used to prove Theorem 3. The main object of this subsection is the following half strainer map. See also Remark 86 below.

The following two theorems are the main results of this subsection. These are counterparts to Theorems 58 and 59 for standard strainer maps, respectively, basically saying that the same results hold for half strainer maps. As before, let \(X\) be a GNPC space and \(\delta_k\) a positive constant depending only on \(k\) (see Notation 56).

The underlying idea in the proofs is straightforward: we use the retraction onto \(\bar B(p_0,R)\) along the shortest paths to \(p_0\) so that the desired properties of \(f\) still hold when restricted to \(\bar B(p_0,R)\). However, unlike the GCBA case [3], this retraction is not “almost orthogonal” to the strainer \(p_1,\dots,p_k\), due to the asymmetry issue. More precisely, if we move \(x\) toward \(p_0\), the distances from \(p_1,\dots,p_k\) may change significantly. Instead, by Definition 81(2), \(p_1,\dots,p_k\) are almost orthogonal to this retraction, in the sense that moving \(x\) toward \(p_1,\dots,p_k\) or its opposite strainer \(\bar p_1,\dots,\bar p_k\) leaves the distance from \(p_0\) almost unchanged. This is enough to prove the desired properties of the half strainer map \(f|_{\bar B(p_0,R)}\).

We first prove Theorem 87 along the same lines as Theorem 58. The following is a counterpart to Proposition 65. Compare with [3] (see also Remark 91 below).

The following is a counterpart to Proposition 67. Compare with [3].

By the Busemann convexity 2 , we see that \(B(x,r)\cap\bar B(p_0,R)\) is convex, and hence contractible. Composing this contraction with the retraction to the fiber of a half strainer map constructed above yields a counterpart to Corollary 72.

Now the proof of Theorem 87 is exactly the same as that of Theorem 58.

As before, the following counterpart to Corollary 73 is a direct consequence of Theorems 20, 58, and 87.

Now we can prove Theorem 88 by complementing an extended strainer in the same way as Theorem 59.

The following is a direct corollary of the above proof. This is exactly the same as Corollary 75, so we omit the proof.

6.3 Consequences↩︎

In this subsection, using the properties of half strainer maps, we prove two propositions which will be used in the proof of Theorem 3. These correspond to [3].

The first is the contractibility of a “hemisphere” in the fiber of a half strainer map, which corresponds to [3]. The proof is somewhat different from the original which used a limiting argument with the property of the tangent cone of a GCBA space. Here we give in some sense a more direct proof.

As the above proof shows, the “punctured half ball” \(\Pi_+\cap B(x,r)\setminus\{x\}\) in the fiber of a half strainer map is homotopy equivalent to the boundary hemisphere \(\Pi_+\cap\partial B(x,r)\) (by using \(\theta\) from Corollary 99, which is actually a deformation retraction). Therefore we obtain the following corollary.

The second is the homotopy negligibleness of the “boundary” of the fiber of a half strainer map, which corresponds to [3]. In fact, we do not need this because its consequence, Proposition 110, is now deduced more directly from Corollaries 99 and 102 (cf.Remark 111). Nevertheless, for comparison we include the statement and proof.

Recall that a closed subset \(Z\) of a topological space \(Y\) is homotopy negligible (or a Z-set) if, for any open subset \(U\) of \(Y\), the inclusion \[U\setminus Z\hookrightarrow U\] is a homotopy equivalence. If \(Y\) is an ANR, this is equivalent to the following local condition: for any \(z\in Z\), there exists a neighborhood basis \(\mathcal{U}_z\) of contractible neighborhoods \(U_z\) with contractible complements \(U_z\setminus Z\) ([68]).

Note that our terminology and notation are slightly different from the original ones of [3]: see Remarks 86 and 93. In the original terminology, \(\hat{\Pi}\) is a fiber of an extended strainer map \((f,d(p_0,\cdot))\).

7 Proofs↩︎

In this section, we give the proofs of the main results, i.e., Theorems 1, 2, 3, 6, 7, and 8.

The organization of this section is as follows. In Section 7.1, we prove local results on BNPC manifolds, including Theorems 1 and 2. In Section 7.2, we prove global results on BNPC manifolds, including Theorem 3. The contents of these two subsections are almost the same as those in [2] and [3], respectively. When the argument is exactly the same, we will omit it and refer the reader to the original papers. In Section 7.3, we give applications to G-spaces, i.e., Theorems 6 and 7. In Section 7.4, we discuss the generalization to spaces with convex geodesic bicombings, i.e., Theorem 8.

7.1 Local results↩︎

In this subsection, we prove local results on BNPC manifolds, including Theorems 1 and 2. Recall that any locally BNPC homology manifold is a GNPC space (Proposition 28).

The proofs of Theorems 1 and 2 are based on the following technical result, which is a BNPC counterpart to [2] of Lytchak–Nagano. Recall that a \((k,\delta)\)-strainer map \(f\) at \(x\in X\) is complementable at \(x\) if there exists \(p\in X\) such that \((f,d(p,\cdot))\) is a \((k+1,\delta)\)-strainer map at \(x\) (Definition 76). For convenience, we treat the constant map to \(\mathbb{R}^0\) as a \((0,\delta)\)-strainer map for any \(\delta>0\).

Theorem 2 immediately follows from the case \(k=n\).

The \(k=n=3\) case gives an alternative proof of the first half of Theorem 31, originally by Thurston (cf.the first half of [2]).

Next we prove Theorem 1.

The following is a BNPC counterpart to the second half of [2].

As mentioned in Section 1.2, the above result does not hold for dimension \(\ge 5\), due to the double suspension theorem ([9], [10], cf.[11]). A counterexample is given by the Euclidean cone over the spherical suspension of a CAT(\(1\)) homology sphere. In general we have the following local structure around a singular point. The proof is exactly the same as [2]. The \(k=5\) case is attributed to Steve Ferry.

7.2 Global results↩︎

In this subsection, we prove global results on BNPC manifolds, including Theorem 3. The main result is the following BNPC counterpart to [3] of Lytchak–Nagano–Stadler.

Together with Theorem 32 of Thurston, this immediately yields Theorem 3.

The proof of Theorem 109 is along the same lines with the original proof of [3], which is based only on [3]. Recall that we have already proved a BNPC counterpart to [3]: see Proposition 100 (and Remark 101). Hence it remains to prove a BNPC counterpart to [3].

The following is a BNPC counterpart to [3]. The setting is the same as Proposition 103, except for the assumption that \(X\) is a homology manifold. Again, note that our terminology and notation are slightly different from the original ones: see Remarks 86 and 93.

Using the properties of half strainer maps, we provide a little more direct proof than the original one of [3] (see Remark 111 below).

Now we can prove Theorem 109 in exactly the same way as the original proof in [3].

The following corollaries are proved in exactly the same way as [3], only using the geodesic contraction with topological arguments.

Here the ideal boundary of a BNPC space is the set of geodesic rays emanating from one base point, equipped with the compact-open topology. This definition does not depend on the choice of a base point (see [71]).

7.3 Applications↩︎

In this subsection, we give two applications on G-spaces, i.e., Theorems 6 and 7.

Let us first define a G-space, introduced by Busemann [27]. The following definition may not be standard, but is compatible with our terminology and is equivalent to the original one.

For the original axiomatic definition, properties, and related problems, we refer to [74], [75], and [76]. Note that a locally BNPC G-space is nothing but a complete GNPC space without branching geodesics.

We briefly review problems on G-spaces. There are two long-standing problems asked by Busemann himself [27]:

1. Is any G-space finite-dimensional?

2. Is any finite-dimensional G-space a topological manifold? (see also [27].)

Regarding the finite-dimensionality conjecture (1), Berestovskiǐ [77] proved it under the assumption of a lower Alexandrov curvature bound or the assumption that any small metric ball is convex. The latter condition is weaker than an upper Alexandrov curvature bound and a local Busemann curvature bound (see [75] for further generalization). Compare with Proposition 23.

Regarding the manifold conjecture (2), the dimension \(\le 2\) case was proved by Busemann [27], the dimension \(3\) and \(4\) cases were proved by Krakus [78] and Thurston [60], respectively. The dimension \(\ge 5\) case still remains open. In general, it is only known that any finite-dimensional G-space is a homogeneous ANR homology manifold ([60], [75]). Here (topological) homogeneity means that any point can be moved to any other point by a self-homeomorphism of the space.

The manifold conjecture was also proved under several curvature bounds. See Berestovskiǐ’s results [79], [32] for Alexandrov curvature bounds and Andreev’s results [29], [31] for a Busemann curvature bound. In these cases, more rigid geometric structures (Riemannian or Finsler-like structures) were obtained, which can be viewed as the special cases of more general results (see Section 8 and references therein). Pogorelov [65] also proved both conjectures by assuming some regularity of distance function and obtained a Finsler structure.

Here we provide an alternative proof of Andreev’s first result [29] on the manifold conjecture (Theorem 6). In fact, due to the homogeneity of a G-space mentioned above, it is sufficient to find one manifold point for proving the manifold conjecture. With additional assumptions of curvature bounds, this is relatively easy (compared to the general case), as a result of strainer arguments.

In fact, we can prove the following stronger result. Compare with Corollary 107 and the subsequent paragraph. See also Andreev’s second result [31].

Thanks to the unique extension property of geodesics, we immediately obtain the following corollary. Compare with Corollary 113. See also [16] for examples in dimension \(\ge 5\).

Next we prove Theorem 7. For the Gromov–Hausdorff convergence, we refer to [37]. We also use some notions and results listed in [2].

We first recall the following globalization, which is actually the final step of the proof of the Cartan–Hadamard theorem (see Section 2.3). For the convenience of the reader, we include the proof (cf.[41]).

Next we prove the homotopical stability of small metric spheres in GNPC spaces with respect to the Gromov–Hausdorff convergence. Compare with [2]. Strictly speaking, we need the notion of convex geodesic bicombing here, but we defer it to the next subsection.

Now we prove Theorem 7. The proof is almost the same as [2]. Recall that a G-space has injectivity radius \(\ge r\) if every shortest path of length \(<r\) extends uniquely to length \(r\). In particular, any two points at distance \(<r\) are joined by a unique shortest path.

As a corollary, we obtain the following finiteness theorem. For \(\varepsilon>0\), the \(\varepsilon\)-packing number of a metric space \(X\) is the maximal number of \(\varepsilon\)-separated points in \(X\), i.e., points with pairwise distance \(\ge\varepsilon\).

7.4 Generalization↩︎

In this subsection, we discuss the generalization to spaces with convex geodesic bicombings introduced by Descombes–Lang [33]. In particular, we provide more details on Theorem 8.

Spaces with convex geodesic bicombings are a natural generalization of BNPC spaces. They have recently attracted the attention of geometric group theorists, because every hyperbolic group admits a proper and cocompact action by isometries on a space with a convex geodesic bicombing ([89]; more precisely, on an injective metric space), while it is a long-standing problem whether the same holds for a CAT(\(0\)) space. From our viewpoint, these spaces naturally arise as Gromov–Hausdorff limits of BNPC spaces, as already used in the previous section.

Let \(X\) be a geodesic space. A geodesic bicombing on \(X\) is a map \[\sigma:X\times X\times[0,1]\to X\] that assigns linearly reparametrized shortest paths between two points of \(X\), i.e., for any \(x,y\in X\), \(\sigma_{xy}(t):=\sigma(x,y,t)\) is a constant-speed shortest path from \(x\) to \(y\).

In addition, we always assume the consistency and reversibility of a bicombing. A geodesic bicombing \(\sigma\) on \(X\) is consistent if for any \(x,y\in X\), setting \(p:=\sigma_{xy}(s)\) and \(q:=\sigma_{xy}(t)\), where \(0\le s\le t\le 1\), we have \[\sigma_{pq}(\lambda)=\sigma_{xy}((1-\lambda)s+\lambda t)\] for all \(0\le\lambda\le1\). It is reversible if for any \(x,y\in X\), we have \[\sigma_{xy}(t)=\sigma_{yx}(1-t)\] for all \(0\le t\le1\).

Clearly, if a geodesic bicombing is convex, then it is continuous. Any BNPC space admits a unique consistent, reversible, convex geodesic bicombing. Conversely, a (complete) metric space with a convex geodesic bicombing is a BNPC space if it is uniquely geodesic, i.e., a shortest path between any two points is unique.

Any proper injective metric space of finite topological dimension admits a unique consistent, reversible, convex geodesic bicombing ([33]). A combinatorial condition for a simplicial complex equipped with some polyhedral metric to admit a convex geodesic bicombing can be found in [90]. Several theorems known for BNPC spaces are extended to spaces with convex geodesic bicombings ([63], [91], [92]).

To make the terminology compatible with ours, let us call \(\sigma_{xy}\) the \(\sigma\)-shortest path between \(x\) and \(y\) (to be more precise, its unit-speed reparametrization). Then Definition 125 is the same as Definition 9 with “shortest path” replaced by “\(\sigma\)-shortest path,” except for the completeness assumption.

For a (consistent, reversible) geodesic bicombing \(\sigma\), a \(\sigma\)-geodesic is a curve whose restriction to each sufficiently small interval is a \(\sigma\)-shortest path. Note that if \(\sigma\) is convex, then any \(\sigma\)-geodesic is a \(\sigma\)-shortest path; compare with Section 2.3. We say that a geodesic bicombing \(\sigma\) is locally extendable (resp.(globally) extendable) if any \(\sigma\)-geodesic is extendable to a \(\sigma\)-geodesic locally (resp.infinitely). These conditions are equivalent if \(X\) is complete; compare with Section 2.2 (cf.[37]).

With this terminology, instead of a GNPC space, we introduce

Then we have

This is because our arguments rely only on the contraction along geodesics and the convexity of distance function along geodesics, both of which are available for \(\sigma\)-geodesics even in the weaker setting. The uniqueness of shortest paths is never used essentially in our arguments. Even if there is a place where the uniqueness of shortest paths appears to be used, it can be replaced with the consistency of \(\sigma\)-shortest paths: for instance, see the proof of Proposition 28. The reversibility is also implicitly used, for example, in the proof of Lemma 45. Lemma 118 also holds for a continuous, consistent, reversible geodesic bicombing \(\sigma\) defined globally on \(X\) that is locally convex.

On the other hand, the non-uniqueness of shortest paths typically results in the non-existence of a unique projection to a “\(\sigma\)-convex subset” (compare [41]). However, we have never used such a projection.

The advantage of dropping the uniqueness of shortest paths is that this weaker notion is preserved under the Gromov–Hausdorff convergence, which was essential in the previous subsection. Compare the following with [1].

Applying the above proposition to tiny balls, we obtain

8 Problems↩︎

We conclude this paper by raising some remaining problems. Most of them are related to the geometric structure of GNPC spaces, which we intentionally did not explore in depth here. See also Remark 137 for the latest information.

The first is the generalization of upper curvature bounds mentioned in Section 1.5. By modifying the BNPC condition in terms of comparison angles, one can define Busemann spaces with curvature \(\le\kappa\) for arbitrary \(\kappa\in\mathbb{R}\) (see Remark 13).

The next problem is the one mentioned in Remark 27, which is true for GCBA spaces. We remark that the original proof in [1] is based on a strainer argument.

The next three problems are all related to the tangent cones of GNPC spaces, which also generalize previous results for GCBA spaces by Lytchak–Nagano [1]. Before discussing the problems, we briefly review the original results in [1] (cf.[93], [94], [32]). For analogous results for Alexandrov spaces with curvature bounded below, see [95], [96], [94], [97], [79]. We also refer to [98] for tangent cones in more general settings.

Let \(X\) be a GCBA space and \(p\in X\). The tangent cone \(T_pX\) of \(X\) at \(p\) is defined as the Gromov–Hausdorff blow-up of \(X\) at \(p\), which in turn is isometric to the Euclidean cone over the space of directions \(\Sigma_pX\) of \(X\) at \(p\). \(T_pX\) is a locally compact, geodesically complete CAT(\(0\)) space and \(\Sigma_pX\) is a compact, geodesically complete CAT(\(1\)) space.

We say that \(p\) is a \(k\)-regular point if \(T_pX\) is isometric to \(\mathbb{R}^k\) (with Euclidean metric). The set \(R^k\) of \(k\)-regular points in \(X\) is dense and has full measure with respect to the \(k\)-dimensional Hausdorff measure in the \(k\)-dimensional part \(X^k\) of \(X\). Here, the \(k\)-dimensional part \(X^k\) is the set of points in \(X\) with \(\dim T_pX=k\) (in the sense of the Hausdorff or topological dimension).

Furthermore, there exists a Lipschitz manifold \(M^k\subset X^k\), which is open in \(X\) and contains \(R^k\), such that \(M^k\) admits a Riemannian metric that is continuous on \(R^k\) and is locally compatible with the original distance.

We want to develop GNPC analogs of the above results. For simplicity, we ignore the dimensional inhomogeneity of a GNPC space and formulate very rough problems. Note that these are not rigorous conjectures.

The starting point would be the structure of the tangent cone (cf.[98]).

Considering that the tangent space of a Finsler manifold is a strictly convex normed space, a regular point of a GNPC space should be defined as a point at which the tangent cone is isometric to a strictly convex normed space. Note that, unlike a Riemannian metric modeled on the standard Euclidean metric, a Finsler norm generally varies pointwise. However, if a Finsler manifold satisfies the BNPC condition, then the Finsler metric belongs to a special class called a Berwald metric, as shown by Ivanov–Lytchak [45]. One of the properties of a Berwald metric is that the tangent norms are all mutually isometric. From our viewpoint that GNPC spaces are a generalization of Finsler manifolds with nonpositive flag curvature, one might ask the following.

Related problems and partial counterexamples can be found in [45]. For example, one can glue two different strictly convex normed spaces at one point to obtain a GNPC space such that the tangent cones at “regular points" are not isometric. However, the authors do not know more complicated counterexamples to the above problem.

Note that, unlike Riemannian manifolds, Finsler manifolds do not generally have a “canonical” measure, but for (reversible) Berwald spaces the Hausdorff measure can be regarded as a canonical measure (see [46]). Therefore it would be natural to consider the Hausdorff measure for our GNPC spaces.

Furthermore, based on the notion of a regular point, one could ask the following. Compare with Theorem 78 and Remark 79.

To attack the above problems, it would be reasonable to first consider the special case of locally BNPC G-spaces, i.e., (complete) GNPC spaces without branching geodesics, which could be viewed as the “regular part” of general GNPC spaces. Andreev [31] proved that the tangent cone of such a space is indeed a strictly convex normed space.

Another problem we would like to mention here is the diffeomorphism version of Theorem 3 in the cocompact setting. As mentioned in [99], this is open even in the CAT(\(0\)) case. See also [100] and [101] for related results.

Finally, we discuss Busemann spaces with lower curvature bounds. Recall that Theorems 1 and 2 were originally proved by Wu [7] for Alexandrov spaces with curvature bounded below. Reversing the inequality for comparison angles in Remark 13, one can also define Busemann spaces with curvature \(\ge\kappa\), where \(\kappa\in\mathbb{R}\) (cf.[52]; however, note that there is no equivalent definition in terms of the concavity of the distance function, similar to Definition 9, even for nonnegative curvature). Clearly, every Alexandrov space is a Busemann space with the same lower curvature bound. Therefore, it is natural to ask

Note that Wu’s proof [7] (cf.[8]) relies heavily on the structure theorem for Alexandrov spaces by Perelman [35], [36], which seems very hard to generalize to Busemann spaces. We point out that Lytchak–Nagano’s proof [2] can also be applied to Alexandrov spaces, by using the gradient flow of the distance function (see [102]) instead of the geodesic contraction we frequently used. However, to the best of our knowledge, it is not even known whether a Busemann space with a lower curvature bound is locally contractible. We remark that there is work of Kell [103] on Busemann spaces with nonnegative curvature.

References↩︎