Closed subsets of a CAT(0) 2-complex are intrinsically CAT(0)
CCLOSED SUBSETS OF A CAT(0) 2-COMPLEX AREINTRINSICALLY CAT(0)
RUSSELL RICKS
Abstract.
Let κ ≤
0, and let X be a locally-finite CAT( κ ) polyhedral 2-complex X , each face with constant curvature κ . Let E be a closed, rectifiably-connected subset of X with trivial first singular homology. We show that E ,under the induced path metric, is a complete CAT( κ ) space. CAT( κ ) spaces are a well-studied generalization of nonpositive curvature fromRiemannian manifolds to metric spaces. Many combinatorial constructions, suchas M κ -polyhedral complexes, involve gluing together pieces of well-understoodCAT( κ ) spaces to form new ones. We extend the class of known examples by tak-ing closed subsets of 2-dimensional polyhedral CAT( κ ) spaces and considering theinduced path metric; under proper assumptions, these subspaces are also CAT( κ ).In [3], we showed that any closed, simply-connected, rectifiably-connected subsetof the plane (with a constant curvature κ ≤ κ ) spacewhen endowed with the induced path metric. In this paper, we extend this resultto similar subspaces of a CAT( κ ) polyhedral 2-complex. In particular, we provethe following result. Theorem A (Theorem 6.1) . Let E be a closed, rectifiably-connected subspace of alocally-finite CAT( κ ) M κ -polyhedral -complex X , where κ ≤ . If H ( E ) = 0 then E , under the induced path metric, is a complete CAT( κ ) space. (Throughout this paper, we assume κ ≤ κ ) 2-complex (see Example 1.4 below, for instance), and there isno general 1-Lipschitz retraction from the full 2-complex to a closed subset, even ifthe subset is contractible.The paper proceeds as follows: We first generalize some aspects of the JordanCurve Theorem to the context of CAT( κ ) 2-complexes (Theorem 2.8). This allowsus to make assertions about the behavior of curves in Y at the points where theyare locally geodesic (Lemma 3.4). Finally, we reduce the theorem to the case where X is planar, and use a characterization of angles proved in [3, Theorem B].The author would like to thank Eric Swenson for suggesting the problem.1. Preliminaries
Write M κ for the plane, equipped with the metric of constant curvature κ .(Throughout this paper, we assume κ ≤ Date : September 4, 2019. a r X i v : . [ m a t h . M G ] A ug RUSSELL RICKS
Let X be a geodesic space, and let ∠ ( κ ) p ( q, r ) be the angle at p in the comparisontriangle (cid:52) ( p, q, r ) in M κ for (cid:52) ( p, q, r ). Suppose σ : [0 , → X and τ : [0 , → X areconstant-speed geodesic segments emanating from the point p ∈ X , with σ (1) = q and τ (1) = r . The Alexandrov angle between σ and τ is defined as ∠ p ( σ, τ ) = lim (cid:15) → + sup For a geodesic metric space X , and κ ≤ 0, we say that X isCAT( κ ) if ∠ p ( q, r ) ≤ ∠ ( κ ) p ( q, r ) for every triple of distinct points p, q, r ∈ X .Every CAT( κ ) space (with κ ≤ 0) is a CAT(0) space, and every CAT(0) spaceis uniquely geodesic. For more on CAT( κ ) spaces, we refer the reader to [1]. Definition 1.2. For distinct points x and y in a uniquely geodesc metric space Z ,write [ x, y ] Z for the geodesic in Z between x and y .Throughout this paper, we will assume X is a locally-finite CAT( κ ) M κ -polyhedral2-complex. The technical hypotheses that X be locally finite and that each face of X has curvature κ ensure that every point p ∈ X has a conical neighborhood —thatis, an open neighborhood isometric to a convex open set in the κ -cone over the linkof p , and this isometry maps p to the cone point [1, Theorem I.7.39].We will often identify a simple closed curve with its image throughout the paper. Definition 1.3. Let γ be a simple closed curve in X . Define the interior of γ tobe Int γ = { x ∈ X (cid:114) γ : [ γ ] (cid:54) = 0 ∈ H ( X (cid:114) { x } ) } . Note that γ ∪ Int γ is the intersection of the images of all singular 2-chains withboundary γ . In particular, γ ∪ Int γ is compact.We remark that the interior of a simple closed curve can be somewhat subtle, asthe following example illustrates. Example 1.4. Let X be the CAT(0) Euclidean 2-complex formed by gluing threeflat half-planes H , H , H = { ( x, y ) : x ≥ } , along the boundary edge of each. Let γ be a simple closed curve in X that follows the following pattern (see Figure 1):Trace arcs from (0 , − 1) to (0 , 1) in H , then from (0 , 1) to (0 , − 2) in H , then from(0 , − 2) to (0 , 3) in H , then from (0 , 3) to (0 , − 3) in H , then from (0 , − 3) to (0 , H , then from (0 , 2) to (0 , − 1) in H . The interior of γ is homeomorphic to apunctured torus (not a disk). It is also not open in X .Notice that replacing H by π in the definition of Int γ defines a strictly largerset in this example—the point (0 , ∈ X , for instance, lies in Int π γ but not inInt H γ . 2. The Curve Theorem The interior of a simple closed curve in X is not open in general, but it is whenrestricted to the faces (i.e., open 2-cells) of X . LOSED SUBSETS OF A CAT(0) 2-COMPLEX ARE INTRINSICALLY CAT(0) 3 γ Int γ Figure 1. The interior of the simple closed curve γ of Example 1.4is not homeomorphic to a disk, nor is it open. And though γ isnullhomologous in γ ∪ Int γ , it is not nullhomotopic (in particular, γ does not bound a disk in γ ∪ Int γ ). Lemma 2.1. Let γ be a simple closed curve in X . Then the intersection of Int γ with each face of X is open in X .Proof. Suppose x ∈ Int γ lies on a face X f of X . Find δ > U := B X ( x, δ ) ⊂ X f (cid:114) γ . Let y ∈ U ; since U is homeorphic to the closed disk, we have adeformation retraction of U (cid:114) { y } to ∂U . Thus X (cid:114) U is a deformation retract of X (cid:114) { y } by the Pasting Lemma. Hence the inclusion X (cid:114) U (cid:44) → X (cid:114) { y } is a homotopyequivalence. So we have ismorphisms H ( X (cid:114) { x } ) → H ( X (cid:114) U ) → H ( X (cid:114) { y } ).Thus [ γ ] (cid:54) = 0 ∈ H ( X (cid:114) { x } ) gives us [ γ ] (cid:54) = 0 ∈ H ( X (cid:114) { y } ), hence y ∈ Int γ .Therefore U ⊂ Int γ , which proves X f ∩ Int γ is open in X . (cid:3) For completeness of exposition, we now prove two simple lemmas. Lemma 2.2. Let D be the closed unit disk in the plane, and let x ∈ ∂D . Forevery open neighborhood U of x , there is an open neighborhood V ⊂ U of x , a point z ∈ V ∩ Int D , and a deformation retraction D (cid:114) { z } → ∂D such that the image of D (cid:114) V lies in ∂D (cid:114) V .Proof. Let U be an open neighborhood of x . Then there is some δ > V := B ( x, δ ) ⊂ U . Pick z ∈ V , and let f be the radial projection of D (cid:114) { z } onto ∂D . By convexity of V , f − ( V ) ⊂ V . Thus f is a deformation retraction such that f ( D (cid:114) V ) ⊂ ∂D (cid:114) V . (cid:3) Lemma 2.3. Let α : [0 , → B be a topological embedding into a space B suchthat α ([0 , (cid:114) ( α (0) ∪ α (1)) is open in B . Let A = α ([0 , . Assume there is acontinuous map σ : S → B of the unit circle to B , and let C = σ − ( A ) . If σ | C isa bijection onto A then [ σ ] (cid:54) = 0 ∈ H ( B ) . RUSSELL RICKS Proof. Let f : B → C/∂C be given by f | A = ( σ | C ) − and f ( b ) = ∂C for b ∈ B (cid:114) A .Let h : C/∂C → S be a homeomorphism. Then h ◦ f ◦ σ is a degree ± S → S , hence ( h ◦ f ) ∗ [ σ ] (cid:54) = 0. Therefore [ σ ] (cid:54) = 0. (cid:3) We now extend one important feature of the Jordan Curve Theorem to X : Theinterior of a simple closed curve γ in X accumulates on each point of γ . We firstprove this result for points of γ on the faces of X . Lemma 2.4. Let γ be a simple closed curve in X , and let x be a point on γ thatlies on a face of X . Then Int γ accumulates on x .Proof. Let r > B X ( x, r ) lies in a face of X . Let δ ∈ (0 , r ) be given;we show there is some point of Int γ in B X ( x, δ ). Note we may assume δ is smallenough that γ (cid:42) B X ( x, δ ). Fix δ (cid:48) with δ < δ (cid:48) < r .Let p and q be the endpoints of the maximal arc α of γ in B X ( x, δ ) on which x lies.By choice of r , the geodesics [ x, p ] X and [ x, q ] X uniquely extend to geodesics [ x, p (cid:48) ] X and [ x, q (cid:48) ] X , respectively, with p (cid:48) , q (cid:48) ∈ ∂B X ( x, δ (cid:48) ). Note that (up to reparametriza-tion) there are two arcs in ∂B X ( x, δ (cid:48) ) from q (cid:48) to p (cid:48) ; let β be one and β the other.Let η and η be the concatenated paths η i = α ∗ [ q, q (cid:48) ] X ∗ β i ∗ [ p (cid:48) , p ] X (see Figure2). Thus η and η are simple closed curves in B X ( x, r ). r δ δ x q q p p αβ Figure 2. The curve η = α ∗ [ q, q (cid:48) ] X ∗ β ∗ [ p (cid:48) , p ] X .For i = 1 , 2, let U i be the bounded component of B X ( x, r ) (cid:114) η i guaranteed bythe Jordan Curve Theorem. Note that U and U are disjoint, and by Schoenflies,each U i is homeomorphic to the closed unit disk. By Lemma 2.2, we may find anondegenerate subarc α (cid:48) of α containing x , points z i ∈ U i ∩ B X ( x, δ ), and deforma-tion retractions of U i (cid:114) { z i } onto η i mapping ( γ (cid:114) α (cid:48) ) ∩ B X ( x, r ) to η i (cid:114) α (cid:48) . Pastingthese deformation retractions together with the geodesic retraction (projection) of X onto B X ( x, δ (cid:48) ), we have a deformation retraction f : X (cid:114) { z , z } → η ∪ η suchthat γ ∩ f − ( α (cid:48) ) = α (cid:48) . Thus H ( X (cid:114) { z , z } ) f ∗ (cid:47) (cid:47) H ( η ∪ η )is an isomorphism that maps [ γ ] (cid:55)→ f ∗ [ γ ] (cid:54) = 0 by Lemma 2.3. Hence [ γ ] (cid:54) = 0 ∈ H ( X (cid:114) { z , z } ). By Mayer-Vietoris (recall that X is contractible) we have an LOSED SUBSETS OF A CAT(0) 2-COMPLEX ARE INTRINSICALLY CAT(0) 5 isomorphism H ( X (cid:114) { z , z } ) (cid:47) (cid:47) H ( X (cid:114) { z } ) ⊕ H ( X (cid:114) { z } )induced by inclusions, and thus [ γ ] (cid:54) = 0 in at least one of H ( X (cid:114) { z } ) and H ( X (cid:114) { z } ). Therefore, either z or z is an element of Int γ . (cid:3) We now drop the hypothesis that x lie on a face of X . Lemma 2.5. Let γ be a simple closed curve in X . Then Int γ accumulates onevery point of γ . In fact, the points of Int γ that lie on faces of X so accumulate.Proof. Let x ∈ γ , and let δ > γ in B X ( x, δ ). If some y ∈ γ ∩ B X ( x, δ ) lies on a face of X , Lemma 2.4 gives usa point of Int γ in B X ( x, δ ). So suppose γ ∩ B X ( x, δ ) lies in the 1-skeleton of X .Then there is a y ∈ γ ∩ B X ( x, δ ) that lies on an edge of X , and a δ (cid:48) ∈ (0 , d ( x, y ))such that B X ( y, δ (cid:48) ) contains no vertices of X but γ ∩ B X ( y, δ (cid:48) ) is a geodesic segmentalong the edge.For each face X f of X that touches the edge, choose a point y f ∈ X f suchthat δ (cid:48) < d ( y f , y ) < δ (cid:48) . Let P be the set of all points y f . Since B X ( y, δ (cid:48) ) (cid:114) P deformation retracts onto ∂B X ( y, δ (cid:48) ) ∪ ( γ ∩ B X ( y, δ (cid:48) )), by Lemma 2.3 we have [ γ ] (cid:54) =0 ∈ H ( X (cid:114) P ). Now P is discrete and closed by construction, and X is contractibleand locally contractible, so the inclusions X (cid:114) P → X (cid:114) { p } induce an isomorphism H ∗ ( X (cid:114) P ) ∼ = (cid:47) (cid:47) (cid:76) p ∈ P H ∗ ( X (cid:114) { p } ) . Thus one of the points y f must lie in Int γ . (cid:3) Our next goal is to prove a replacement property for openness of the interiorInt γ of a simple closed curve γ in X , to use when not on a face of X . We proveInt γ has the geodesic extension property, in a local sense. Lemma 2.6. Let γ be a simple closed curve in X . Then Int γ locally extendsgeodesics; that is, every geodesic [ p, q ] X ⊂ Int γ extends to a geodesic [ p, q (cid:48) ] X ⊂ Int γ with [ p, q ] X ⊂ [ p, q (cid:48) ] X .Proof. Let [ p, q ] X ⊂ Int γ be a geodesic in X . Find (cid:15) > D := B X ( q, (cid:15) )is a conical neighborhood of q , and γ ⊂ X (cid:114) D . Now D (cid:114) { q } deformation retractsonto ∂D , and X deformation retracts onto D via geodesic projection, so pastingtogether gives a deformation retraction f : X (cid:114) { q } → ∂D .Let C = { y ∈ ∂D : ∠ q ( p, y ) < π } . Since X is locally finite, the topologicalboundary P = ∂C in ∂D is finite, so ∂D (cid:114) P is a disjoint union of metric graphs.By the link condition on X [1, p. 206], the component C of ∂D (cid:114) P is contractible,and there is a lower bound on the length of circles in ∂D ; hence there exists a finiteset Q ⊂ ∂D (cid:114) C such that ∂D (cid:114) Q is a disjoint union of trees. The long exactsequence . . . (cid:47) (cid:47) H k ( ∂D (cid:114) Q ) (cid:47) (cid:47) H k ( ∂D ) ϕ k (cid:47) (cid:47) H k ( ∂D, ∂D (cid:114) Q ) (cid:47) (cid:47) . . . then has H k ( ∂D (cid:114) Q ) = 0 for k > 0, so in particular the map H ( ∂D ) ϕ (cid:47) (cid:47) H ( ∂D, ∂D (cid:114) Q ) RUSSELL RICKS is injective. Let { V y } y ∈ Q be a collection of pairwise-disjoint open sets in ∂D suchthat each y ∈ V y . Then we have a commutative diagram H ( (cid:96) y ∈ Q V y , ( (cid:96) y ∈ Q V y ) (cid:114) Q ) (cid:47) (cid:47) (cid:15) (cid:15) (cid:76) y ∈ Q H ( V y , V y (cid:114) { y } ) (cid:15) (cid:15) H ( ∂D, ∂D (cid:114) Q ) Ψ (cid:47) (cid:47) (cid:76) y ∈ Q H ( ∂D, ∂D (cid:114) { y } ) , where the vertical maps are induced by inclusion, the top map is the canonicalisomorphism, and the bottom map is induced by the inclusions ∂D (cid:114) Q → ∂D (cid:114) { y } .By excision, the vertical maps are isomorphisms, so Ψ is an isomorphism. Let Φbe the composition H ( ∂D ) ϕ (cid:47) (cid:47) H ( ∂D, ∂D (cid:114) Q ) Ψ (cid:47) (cid:47) (cid:76) y ∈ Q H ( ∂D, ∂D (cid:114) { y } ) . Since the maps f ∗ and Φ are both injective, and [ γ ] (cid:54) = 0 ∈ H ( ∂D ), there must besome q (cid:48) ∈ Q such that ( φ ◦ f ∗ )([ γ ]) (cid:54) = 0, where φ : H ( ∂D ) (cid:47) (cid:47) H ( ∂D, ∂D (cid:114) { q (cid:48) } )is the associated component function of Φ. (Hence by naturality of the long exactsequence, φ is the map in the long exact sequence of the pair ( ∂D, ∂D (cid:114) { q (cid:48) } ).)Since q (cid:48) ∈ ∂D (cid:114) C , we have ∠ q ( p, q (cid:48) ) = π , and therefore [ q, q (cid:48) ] X extends thegeodesic segment [ p, q ] X to [ p, q (cid:48) ] X . Thus it remains only to show [ q, q (cid:48) ] X ⊂ Int γ .Define W ⊂ D and U ⊂ ∂D as follows. If q (cid:48) lies on a face of X , then let F be theopen face containing q (cid:48) , and let W = F ∩ D and U = F ∩ ∂D . On the other hand,if q (cid:48) lies on an edge of X , then let G = [ q, q (cid:48) ] X , let F be the union of all open facescontaining q (cid:48) in their closure, and let W = ( F ∪ G ) ∩ D and U = ( F ∪ G ) ∩ ∂D . Nowlet x ∈ [ q, q (cid:48) ] X (cid:114) { q, q (cid:48) } be arbitrary. By construction, W is a conical neighborhoodof q (cid:48) in X , so we have a deformation retraction g : X (cid:114) { x } → ∂W . Also, ∂W (cid:114) { q (cid:48) } deformation retracts onto ∂W (cid:114) U , which deformation retracts onto { q } , being acone over q . Thus ∂W (cid:114) { q (cid:48) } is contractible, so the long exact sequence gives usan isomorphism H ( ∂W ) θ (cid:47) (cid:47) H ( ∂W, ∂W (cid:114) { q (cid:48) } ) . Let ψ be the map that makes the following diagram commute, where the other twomaps are the isomorphisms (by excision) induced by inclusion H ( U, U (cid:114) { q (cid:48) } ) (cid:117) (cid:117) (cid:41) (cid:41) H ( ∂W, ∂W (cid:114) { q (cid:48) } ) ψ (cid:47) (cid:47) H ( ∂D, ∂D (cid:114) { q (cid:48) } ) . Thus we have isomorphisms H ( X (cid:114) { x } ) g ∗ (cid:47) (cid:47) H ( ∂W ) θ (cid:15) (cid:15) H ( ∂W, ∂W (cid:114) { q (cid:48) } ) ψ (cid:47) (cid:47) H ( ∂D, ∂D (cid:114) { q (cid:48) } ) . LOSED SUBSETS OF A CAT(0) 2-COMPLEX ARE INTRINSICALLY CAT(0) 7 Let X (cid:114) D i (cid:47) (cid:47) X (cid:114) { q } and X (cid:114) D j (cid:47) (cid:47) X (cid:114) { x } be inclusion. Then the square H ( X (cid:114) D ) j ∗ (cid:47) (cid:47) i ∗ (cid:15) (cid:15) H ( X (cid:114) { x } ) g ∗ (cid:47) (cid:47) H ( ∂W ) θ (cid:15) (cid:15) H ( ∂W, ∂W (cid:114) { q (cid:48) } ) ψ (cid:15) (cid:15) H ( X (cid:114) { q } ) f ∗ (cid:47) (cid:47) H ( ∂D ) φ (cid:47) (cid:47) H ( ∂D, ∂D (cid:114) { q (cid:48) } ) , commutes, and thus q ∈ Int γ implies x ∈ Int γ for all x ∈ [ q, q (cid:48) ] X (cid:114) { q, q (cid:48) } . Bycompactness of γ ∪ Int γ , we obtain q (cid:48) ∈ Int γ , and the theorem is proved. (cid:3) Thus we can extend another feature of the Jordan Curve Theorem to X . Corollary 2.7. Let γ be a simple closed curve in X . Then Int γ extends geodesicsto γ —that is, every geodesic [ p, q ] X ⊂ Int γ extends to a geodesic [ p, q (cid:48) ] X ⊂ γ ∪ Int γ with [ p, q ] X ⊂ [ p, q (cid:48) ] X , q (cid:48) ∈ γ , and [ p, q (cid:48) ] X (cid:114) { q (cid:48) } ⊂ Int γ .Proof. Let a (cid:48) = sup { d ( p, q (cid:48) ) : q (cid:48) ∈ X such that [ p, q ] X ⊂ [ p, q (cid:48) ] X ⊂ γ ∪ Int γ } . Bycompactness of γ ∪ Int γ , there is some q (cid:48) ∈ γ ∪ Int γ such that d X ( p, q (cid:48) ) = a (cid:48) and[ p, q ] X ⊂ [ p, q (cid:48) ] X ⊂ γ ∪ Int γ . By Lemma 2.6, q (cid:48) / ∈ Int γ ; hence q (cid:48) ∈ γ .Now, it is conceivable that q (cid:48) is not the only point of γ on [ p, q (cid:48) ] X . If so, bycompactness of γ we may find the closest point q (cid:48)(cid:48) to p on [ p, q (cid:48) ] X ∩ γ . Then theconclusion of the lemma holds with q (cid:48)(cid:48) in place of q (cid:48) . (cid:3) Combining Lemma 2.5 and Corollary 2.7, we obtain the following theorem. Theorem 2.8 (“The Curve Theorem”) . Let γ be a simple closed curve in X . Then Int γ accumulates on every point of γ . Moreover, Int γ extends geodesics to γ . Detecting intrinsic geodesics Let E be a fixed closed, rectifiably-connected subspace of X with the inducedsubspace metric (also written d ), such that H ( E ) = 0. Let Y be the space E ,endowed with the induced path metric, d Y . Lemma 3.1. Let Z be a complete, rectifiably-connected metric space. Then Z , withthe induced path metric, is complete and geodesic.Proof. This is proved in Lemma 2.1 and Corollary 2.2 of [3]. (cid:3) Of course this implies Y is complete and geodesic. Lemma 3.2. If the geodesic [ x, y ] X in X between x and y satisfies [ x, y ] X ⊂ E ,then [ x, y ] X is the unique geodesic in Y between x and y .Proof. This result is immediate from the definitions. (cid:3) Lemma 3.3. Let γ be a simple closed curve in E , and α be a subarc of γ that isgeodesic in Y . If x, y ∈ α such that [ x, y ] X lies in γ ∪ Int γ , then [ x, y ] X ⊂ α .Proof. Since H ( E ) = 0, we know Int γ ⊂ E . Apply Lemma 3.2. (cid:3) RUSSELL RICKS Lemma 3.4. Let C be a closed convex subspace of X and γ be a simple closedcurve in E . Define dist C : X → R by dist C ( p ) = d ( p, C ) . Let p be a point on γ where dist C attains its maximum on γ but is not locally constant on γ . Then γ isnot locally geodesic in Y at the point p .Proof. Suppose, by way of contradiction, that V is a conical neighborhood of p in X such that γ ∩ V is geodesic. By Lemma 2.5, there is a sequence p k → p in X suchthat each p k lies on a face of X and p k ∈ Int γ . For each p k , let r k ∈ C be the closestpoint to p k in C , and let x (cid:48) k , y (cid:48) k be points of X (cid:114) { p k } such that the Alexandrov anglein X satisfies ∠ Xp k ( x (cid:48) k , r k ) = π = ∠ Xp k ( y (cid:48) k , r k ) and ∠ Xp k ( x (cid:48) k , y (cid:48) k ) = π . By Lemma 2.1,we may assume x (cid:48) k , y (cid:48) k are near enough to p k that the geodesic [ x (cid:48) k , y (cid:48) k ] X in X satisfies[ x (cid:48) k , y (cid:48) k ] X ⊂ Int γ . By Corollary 2.7, we can extend [ x (cid:48) k , y (cid:48) k ] X to a geodesic [ x k , y k ] X in X such that [ x k , y k ] X ⊂ γ ∪ Int γ and x k , y k ∈ γ . Taking the limit of the geodesics[ x k , y k ] X in X as k → ∞ , we obtain a (possibly degenerate) geodesic [ x, y ] X in X ;by compactness, [ x, y ] X ⊂ γ ∪ Int γ and x, y ∈ γ . Since each p k ∈ [ x k , y k ] X , we alsosee that p ∈ [ x, y ] X .We consider three cases for x and y . Case 1: x = p = y . Then eventually x k , y k ∈ V ∩ γ , contradicting Lemma 3.3. Case 2: x (cid:54) = p and y (cid:54) = p . Then γ locallyfollows the geodesic [ x, y ] X in X . But now p cannot be a local maximum for dist C without forcing dist C to be locally constant at p because dist C is convex in theCAT(0) space X . So Cases 1 and 2 are impossible.Finally, Case 3: Exactly one of x, y equals p . We may assume x (cid:54) = p and y = p .Choose w, q ∈ [ x, p ] X (cid:114) { x, p } such that the geodesic [ w, p ] X in X extends thegeodesic [ w, q ] X in X . By Lemma 2.5, there is a sequence q k → q in X such thateach q k lies on a face of X and q k ∈ Int γ . Note that [ w, q k ] X (cid:114) { w } ⊂ Int γ forall sufficiently large k ∈ N because V is a conical neighborhood of p in X . ByCorollary 2.7, we may extend each geodesic [ w, q k ] X in X to a geodesic [ w, z k ] X in X such that z k ∈ γ and [ w, z k ] X (cid:114) { w, z k } ⊂ Int γ . Passing to a subsequenceif necessary, we may assume ( z k ) converges in X to some point z ∈ γ . Notice p ∈ [ x, z ] X by simpleness of γ . Since [ x, z ] X ⊂ γ ∪ Int γ by compactness, if p (cid:54) = z then γ is locally geodesic in X at p , and we obtain a contradiction as in Case 2above; on the other hand, if p = z then eventually z k , w ∈ V ∩ γ , contradictingLemma 3.3 as in Case 1 above. Thus in every case, we obtain a contradiction. Thestatement of the lemma follows. (cid:3) Corollary 3.5. Y is uniquely geodesic.Proof. Suppose, by way of contradiction, that σ and τ are distinct geodesics in Y from x ∈ Y to y ∈ Y . We may assume σ and τ intersect only at x and y .Put C = [ x, y ] X and γ = σ ∪ τ . By Lemma 3.4, one of σ and τ is not locallygeodesic at some point (away from x and y ), contradicting our hypothesis on σ and τ . Therefore, Y is uniquely geodesic. (cid:3) Bootstrapping the planar case In the proof of [3, Theorem B], the essential construction is that of two unique(though not necessarily distinct) geodesic segments we call “limit segments,” basedat a given vertex, p , of a simple geodesic triangle T , with two important features:(1) The limit segments extend through T ∪ Int T to the opposite side of T .(2) The angle between the limit segments equals the Alexandrov angle of thetriangle at p (this is the content of [3, Theorem B]). LOSED SUBSETS OF A CAT(0) 2-COMPLEX ARE INTRINSICALLY CAT(0) 9 We will follow the same general outline, creating limit segments and proving thatthe angle between them equals the Alexandrov angle of the triangle at the vertex.The first part of the proof is to reduce to the case where [3, Theorem B] applies.We begin with some terminology. Definition 4.1. A geodesic triangle T in Y is called simple if it is a simple closedcurve in Y (or, equivalently, in X ). Definition 4.2. Let p and q be distinct points in Y , and let σ : [0 , → Y be aconstant-speed geodesic with σ (0) = p and σ (1) = q . Let (cid:15) > B X ( p, (cid:15) ) is a conical neighborhood of p in X . For t ∈ (0 , R (cid:15),tq be thegeodesic from p through σ ( t ) in X of length (cid:15) . If the geodesics R (cid:15),tq limit uniformlyonto a geodesic R (cid:15)q in X , we call R (cid:15)q the limit segment of σ at p (of length (cid:15) ).Since we will be talking about angles in both X and Y , we will distinguishbetween them by placing the space as a superscript, as follows. Definition 4.3. Let Z be a metric space. We will write ∠ Zp ( q, r ) for the Alexandrovangle ∠ p ( q, r ) in Z and ∠ ( κ ) ,Zp ( q, r ) for the M κ -comparison angle ∠ ( κ ) p ( q, r ) in Z .We restate [3, Theorem B] in the current context. Theorem 4.4 (Theorem B of [3]) . Assume X = M κ and E is simply connected.Let T be a simple geodesic triangle in Y with vertices p , q , and r . For all sufficientlysmall (cid:15) > , both limit segments R (cid:15)q and R (cid:15)r of T exist at p , both R (cid:15)q and R (cid:15)r liecompletely in T ∪ Int T , and ∠ Yp ( q, r ) = ∠ Xp ( R (cid:15)q , R (cid:15)r ) . We would like to use this theorem to prove the general case, but first we have toproperly transfer the setting. Lemma 4.5. Let C be a closed, convex subspace of X . Let A be a path-connectedmetric subspace of E , and let B be the union of all geodesics [ x, y ] Y in Y betweenall points x, y ∈ A , taken as a metric subspace of E . If A ⊂ C , then B ⊂ C .Proof. First suppose, by way of contradiction, that A ⊂ C but there is a point z ∈ B (cid:114) C . By definition of B , we know z lies on a geodesic [ x, y ] Y in Y for some x, y ∈ A . We may assume that [ x, y ] Y ∩ C = { x, y } . Since A is path connected, wemay find a path in A ⊂ C ∩ E joining x and y ; we may of course assume this pathis injective. Concatenating this path with the geodesic [ x, y ] Y , we obtain a simpleclosed curve in E . By Lemma 3.4, [ x, y ] Y must stay in C or it cannot be a geodesicin Y ; this contradicts our choice of [ x, y ] Y . Therefore, we must have B ⊂ C . (cid:3) Lemma 4.6. Let D be a closed, convex subspace of X which isometrically embedsin M κ . Let T be a simple geodesic triangle in Y with vertices p, q, r ∈ D . Then T ∪ Int T ⊂ D ∩ E and is compact, simply connected, and rectifiably connected as asubspace of D , and convex as a subspace of Y .Proof. Let C be the convex hull of { p, q, r } in X ; note that C is closed. ByLemma 3.4, T must lie in C . Now C , as a convex subspace of a CAT(0) space, iscontractible. Therefore, there is a singular 2-chain in C with boundary T ; hence T ∪ Int T , being the intersection of all singular 2-chains with boundary T , must liein C . Thus T ∪ Int T ⊂ C ⊂ D . Let ϕ : D → M κ be an isometric embedding. Notethat ϕ ( C ) is the convex hull of { ϕ ( p ) , ϕ ( q ) , ϕ ( r ) } in M κ , and ϕ (Int T ) = Int ϕ ( T ) is the interior of ϕ ( T ) guaranteed by the Jordan Curve Theorem in M κ . In particular, T ∪ Int T , as a metric subspace of X , is topologically a closed disk by Schoenflies.Let F = T ∪ Int T as a metric subspace of X . Since H ( E ) = 0, we knowInt T ⊂ E and therefore F ⊂ E . Let Z = F as a metric subspace of Y . Let Z be the union of all geodesics [ x, y ] Y in Y between all points x, y ∈ Z , with Z taken as a metric subspace of Y . Let F = Z as a metric subspace of X . Since F is topologically a closed disk, F is path connected. Thus F ⊂ C by Lemma 4.5.But now F ⊂ F ⊂ C ⊂ D . Since C is the convex hull of the vertices of T in a disk of M κ , no geodesic [ x, y ] Y in Y can stay in C and yet leave F withoutviolating the hypothesis that the sides of T are geodesics in Y . Thus F = F , i.e. Z is convex. This implies F is rectifiably connected, and the lemma is proved. (cid:3) The next lemma gives some surprising teeth to Lemma 4.6. Lemma 4.7. Let p ∈ Y , and let V be a conical neighborhood of p in X . Everysimple geodesic triangle in Y with one vertex p and other two vertices q, r ∈ V liesinside a closed, convex subspace C of X that isometrically embeds in M κ .Proof. Let q, r be points of E ∩ V such that the geodesic triangle T in Y withvertices p, q, r is simple. Let C be the convex hull of { p, q, r } in X ; note that C is closed and C ⊂ V . By Lemma 3.4, T must lie in C . Now if ∠ Xp ( q, r ) = π ,then C is the geodesic [ q, r ] X in X ; however, this means T ⊂ C cannot be simple,contradicting our hypothesis. Thus ∠ Xp ( q, r ) < π .Let ρ be radial projection in X (cid:114) { p } to the link of p in X , and let G be thegeodesic in the link between ρ ( q ) and ρ ( r ). Let D = ( ρ − ( G ) ∩ V ) ∪ { p } . Since V is a conical neighborhood about p , it follows that D is convex. Thus C ⊂ D ; since D isometrically embeds in M κ , so does C . (cid:3) Corollary 4.8. Let p ∈ Y , and let V be a conical neighborhood of p in X . Let T be a simple geodesic triangle in Y with one vertex p and other two vertices q, r ∈ V .There exists (cid:15) > such that both limit segments R (cid:15)q and R (cid:15)r of T exist at p , both R (cid:15)q and R (cid:15)r lie completely in T ∪ Int T , and ∠ Yp ( q, r ) = ∠ Xp ( R (cid:15)q , R (cid:15)r ) . Proof. Apply Lemma 4.7, Lemma 4.6, and Theorem 4.4. (cid:3) We began this section by mentioning two important features that held in theplanar case, which we wanted to extend to the general case. We can now proveexistence of limit segments in the general case, along with Feature 2. Theorem 4.9. Let T be a simple geodesic triangle in Y with vertices p , q , and r .For all (cid:15) > small enough, both limit segments R (cid:15)q and R (cid:15)r of T exist at p , and ∠ Yp ( q, r ) = ∠ Xp ( R (cid:15)q , R (cid:15)r ) . Proof. Let V be a conical neighborhood about p in X . Since the existence of limitsegments of T at p is local, and Y uniquely geodesic, we may assume q, r ∈ V . Thetheorem follows from Corollary 4.8. (cid:3) Extending limit segments We now turn to proving Feature 1 of the limit segments in the general case. Webegin with a result about variations of geodesics in Y . LOSED SUBSETS OF A CAT(0) 2-COMPLEX ARE INTRINSICALLY CAT(0) 11 Lemma 5.1. Let p k → p and q k → q in Y . For each k ∈ N let σ k : [0 , → Y be theconstant-speed parametrization of the geodesic in Y from p k to q k . Then the maps σ k converge uniformly in X to the constant-speed parametrization σ : [0 , → Y ofthe geodesic in Y from p to q .Proof. Let ι : Y → E be the setwise identity map, ι ( y ) = y ∈ E for all y ∈ Y . Foreach k ∈ N let σ Ek = ι ◦ σ k , and note that σ Ek : [0 , → E is d Y ( p k , q k )-Lipschitz.Since p k → p and q k → q in Y , we know d Y ( p k , q k ) → d Y ( p, q ). Since E is proper, bythe Arzel`a-Ascoli Theorem the maps σ Ek have a uniform limit σ E : [0 , → E , whichis d Y ( p, q )-Lipschitz. By uniqueness of geodesics in Y , we must have σ E = ι ◦ σ . (cid:3) Remark. If the uniform convergence in Lemma 5.1 was in Y instead of in X , thenwe could say geodesics in Y vary continuously in their endpoints. Corollary 5.2. Let ι : Y → E be the setwise identity map, ι ( y ) = y ∈ E for all y ∈ Y . Let τ , τ : [0 , → Y be two continuous paths in Y , and for each s ∈ [0 , let σ s : [0 , → Y be constant-speed parametrization of the (possibly degenerate)geodesic [ τ ( s ) , τ ( s )] Y in Y from τ ( s ) to τ ( s ) . Then the map f : [0 , × [0 , → E defined by setting f ( s, t ) = ι ( σ s ( t )) for all s, t ∈ [0 , is continuous in X . Thus a variation of geodesics in Y is a homotopy in E ⊂ X (see Figure 3). τ τ σ σ s σ Figure 3. A variation of geodesics in Y gives a homotopy in E (as a subset of X ), by Corollary 5.2.This homotopy allows us to compare interiors of two related triangles (a similarstatement for generic simple closed curves in Y would be false). Lemma 5.3. Let T be a simple geodesic triangle in Y with vertices p, q, r . Let (cid:98) T be a simple geodesic triangle in Y with vertices p, q (cid:48) , r (cid:48) , where q (cid:48) ∈ [ p, q ] Y and r (cid:48) ∈ [ p, r ] Y . There is some (cid:15) > such that Int T ∩ B X ( p, (cid:15) ) = Int (cid:98) T ∩ B X ( p, (cid:15) ) .Proof. Let P ⊂ Y be the broken-geodesic path P = [ q (cid:48) , q ] Y ∪ [ q, r ] Y ∪ [ r, r (cid:48) ] Y in Y , and Q ⊂ Y the quadrilateral Q = P ∪ [ r (cid:48) , q (cid:48) ] Y . Let G be the geodesic[ q, r ] X between q and r in X . By Lemma 3.4, p globally maximizes dist G (thedistance in X to G ) among points of T . Hence by compactness, there is some (cid:15) > G ( x ) ≤ dist G ( p ) − (cid:15) for all x ∈ P . Let C ⊂ X be the closed,convex subspace C = { x ∈ X : d ( x, G ) ≤ (cid:15) } . Then P ⊂ C , and by Lemma 4.5, Q ⊂ C . Corollary 5.2 guarantees a nulhomotopy of Q in E . Applying Lemma 4.5again, we see that this nulhomotopy lies completely in C . Therefore, T and (cid:98) T arehomotopic in E by a homotopy that is constant on X (cid:114) C ⊃ B X ( p, (cid:15) ). In particular,Int T ∩ B X ( p, (cid:15) ) = Int (cid:98) T ∩ B X ( p, (cid:15) ). (cid:3) Lemma 5.4. Let T be a simple geodesic triangle in Y with vertices p, q, r . Forsmall enough (cid:15) > , both limit segments R (cid:15)q and R (cid:15)r of T at p lie completely in T ∪ Int T .Proof. Let V be a conical neighborhood of p in X . Choose q (cid:48) ∈ [ p, q ] Y (cid:114) { p } and r (cid:48) ∈ [ p, r ] Y (cid:114) { p } such that both q (cid:48) , r (cid:48) ∈ V . Since p / ∈ [ q, r ] Y , by uniquenessof geodesics p / ∈ [ q (cid:48) , r (cid:48) ] Y ; thus we may assume the geodesic triangle (cid:98) T in Y withvertices p, q (cid:48) , r (cid:48) is simple. By Corollary 4.8, there exists (cid:15) > (cid:98) R (cid:15)q and (cid:98) R (cid:15)r of (cid:98) T exist at p , and both (cid:98) R (cid:15)q and (cid:98) R (cid:15)r lie completely in (cid:98) T ∪ Int (cid:98) T .By Lemma 5.3, we may assume Int T ∩ B X ( p, (cid:15) ) = Int (cid:98) T ∩ B X ( p, (cid:15) ); clearly we mayalso assume T ∩ B X ( p, (cid:15) ) = (cid:98) T ∩ B X ( p, (cid:15) ). Since the limit segments R (cid:15)q and R (cid:15)r of T at p coincide with the limit segments (cid:98) R (cid:15)q and (cid:98) R (cid:15)r of (cid:98) T at p , we have proved thelemma. (cid:3) Lemma 5.5. Let T be a simple geodesic triangle in Y with vertices p, q, r . Forsmall enough (cid:15) > , the limit segment R (cid:15)q of T at p can be extended to a geodesicin X from p to q (cid:48) ∈ [ q, r ] Y ⊂ T , with [ p, q (cid:48) ] X ⊂ T ∪ Int T . Similarly, the limitsegment R (cid:15)r can be extended to a geodesic in X from p to r (cid:48) ∈ [ q, r ] Y ⊂ T , with [ p, r (cid:48) ] X ⊂ T ∪ Int T .Proof. Choose (cid:15) > R (cid:15)q and R (cid:15)r of T at p lie completely in T ∪ Int T (possible by Lemma 5.4). If R (cid:15)q enters Int T , then byLemma 2.7 we can extend R (cid:15)q to a geodesic [ p, q (cid:48) ] X in X that lies completely in T ∪ Int T with q (cid:48) ∈ T ; by Lemma 3.3, q (cid:48) ∈ [ q, r ] Y , as desired. So assume R (cid:15)q ⊂ T .Assume first that R (cid:15)q ⊂ [ p, q ] Y . We may assume R (cid:15)q and R (cid:15)r lie in a conicalneighborhood V of p in X . Let w be the midpoint of R (cid:15)q ; by Lemma 2.5, wemay find be a sequence of points w k ∈ Int T such that w k → w in X and each w k lies on a face of X . For each k ∈ N , let [ p k , w k ] X be the maximal subarc ofthe geodesic [ p, w k ] X in X between p and w k such that [ p k , w k ] X (cid:114) { p k } ⊂ Int T .By Lemma 2.7, we can extend [ p k , w k ] X to a geodesic [ p k , x k ] X in X such that x k ∈ T and [ p k , x k ] X (cid:114) { p k , x k } ⊂ Int T . Because p and w lie in the conicalneighborhood V , for all sufficiently large k we find [ p, x k ] X (cid:114) { p } lies on a face of X , and thus p k ∈ T . Since T is simple and [ p, q ] Y follows R (cid:15)q , which is a geodesic in X with one endpoint p and midpoint w , we see that eventually every p k ∈ [ p, r ] Y (possibly p k = p ) and p k → p . Since w k ∈ Int T lies on [ p k , x k ] X , by Lemma 3.3we must have x k ∈ [ p, q ] Y or x k ∈ [ q, r ] Y . In the first case, we have a new simplegeodesic triangle in Y with vertices p, p k , x k formed by concatenating the geodesics[ p, p k ] Y ⊂ [ p, r ] Y ⊂ T , [ p k , x k ] X ⊂ E , and [ x k , p ] Y ⊂ [ q, p ] Y ⊂ T . But this simplegeodesic triangle has all its vertices along the (closed, convex) geodesic [ p, x k ] X in X ; by Lemma 3.4 the whole triangle must lie in [ p, x k ] X , making it degenerate andtherefore not simple. Thus only the second case is possible: x k ∈ [ q, r ] Y . Takingthe limit of geodesics [ p k , x k ] X in X as k → ∞ , we obtain a geodesic [ p, x ] X in X ; by compactness, x ∈ [ q, r ] Y and [ p, x ] X ⊂ T ∪ Int T . Thus we have proved thelemma in this case, too.Finally, assume that R (cid:15)q is not a subset of [ p, q ] Y . Since R (cid:15)q ⊂ T , we must have R (cid:15)q ⊂ [ p, r ] Y . But then R (cid:15)r = R (cid:15)q ⊂ [ p, r ] Y . So, swapping the roles of q and r in theprevious paragraph, we can extend R (cid:15)r = R (cid:15)q in T ∪ Int T to [ q, r ] Y , as desired.The proof for the limit segment R (cid:15)r is completely similar. (cid:3) LOSED SUBSETS OF A CAT(0) 2-COMPLEX ARE INTRINSICALLY CAT(0) 13 Curvature We finally arrive at the main result of the paper. Theorem 6.1. Let E be a closed, rectifiably-connected subspace of a locally-finite CAT( κ ) M κ -polyhedral -complex X , where κ ≤ . If H ( E ) = 0 then E , underthe induced path metric, is a complete CAT( κ ) space.Proof. Let Y be the space E , with the induced path metric. We want to showthat ∠ Yp ( q, r ) ≤ ∠ ( κ ) ,Yp ( q, r ) for every triple of distinct points p, q, r ∈ E . So let T be a geodesic triangle in Y with distinct vertices p, q, r ∈ E . If T is not simplein a neighborhood of p in X , then by uniqueness of geodesics, either p lies on thegeodesic [ q, r ] Y in Y or both geodesics [ p, q ] Y and [ p, r ] Y in Y must coincide for somedistance from p . In the first case, T is degenerate with ∠ Yp ( q, r ) = ∠ ( κ ) ,Yp ( q, r ) = π .In the second, ∠ Yp ( q, r ) = 0 ≤ ∠ ( κ ) ,Yp ( q, r ). Thus we may assume T is simple in aneighborhood U of p .Let [ q, q (cid:48) ] Y be the maximal subarc (possibly degenerate) of the geodesic [ q, r ] Y in Y such that [ q, q (cid:48) ] Y coincides with the geodesic [ q, p ] Y ⊂ T in Y from q to p . The geodesic triangle in Y with vertices p, q, q (cid:48) is therefore degenerate with ∠ Yp ( q, q (cid:48) ) = ∠ ( κ ) ,Yp ( q, q (cid:48) ) = 0. By Alexandrov’s Lemma [1, Lemma II.4.10], itsuffices to prove the theorem when q = q (cid:48) . Similarly, we may assume the geodesics[ r, q ] Y and [ r, p ] Y in Y do not coincide near r . Thus we may assume T is simple.By Theorem 4.9 and Lemma 5.5, we have a geodesic [ p, q (cid:48)(cid:48) ] X in X from p to q (cid:48)(cid:48) ∈ [ q, r ] Y such that [ p, q (cid:48)(cid:48) ] X ⊂ T ∪ Int T . The geodesic triangle in Y with vertices p, q, q (cid:48)(cid:48) is either simple, in which case the equation in Theorem 4.9 applies, or bothedges of the triangle from p coincide for some positive distance out from p ; in eithercase, ∠ Yp ( q, q (cid:48)(cid:48) ) = 0 ≤ ∠ ( κ ) ,Yp ( q, q (cid:48)(cid:48) ). By Alexandrov’s Lemma [1, Lemma II.4.10],it suffices to consider the case q = q (cid:48)(cid:48) . A similar argument holds for r in place of q .Thus we may assume both geodesics [ p, q ] Y and [ p, r ] Y in Y lie completely in X .Now d ( p, q (cid:48) ) = d Y ( p, q (cid:48) ) and d ( p, r (cid:48) ) = d Y ( p, r (cid:48) ). And since d ( q (cid:48) , r (cid:48) ) ≤ d Y ( q (cid:48) , r (cid:48) ),we see directly that ∠ Yp ( q (cid:48) , r (cid:48) ) = ∠ Xp ( q (cid:48) , r (cid:48) ) ≤ ∠ ( κ ) ,Xp ( q (cid:48) , r (cid:48) ) ≤ ∠ ( κ ) ,Yp ( q (cid:48) , r (cid:48) ) . (Recall ∠ Xp ( q (cid:48) , r (cid:48) ) ≤ ∠ ( κ ) ,Xp ( q (cid:48) , r (cid:48) ) by the CAT( κ ) inequality.) Thus Y is CAT( κ ).Completeness comes from Lemma 3.1. (cid:3) References 1. Martin R. Bridson and Andr´e Haefliger, Metric spaces of non-positive curvature , Springer-Verlag, Berlin, 1999.2. Alexander Lytchak and Stefan Wenger, Isoperimetric characterization of upper curvaturebounds , Acta Math. (2018), no. 1, 159–202, https://doi.org/10.4310/ACTA.2018.v221.n1.a5 .3. Russell M. Ricks, Planar CAT( κ ) subspaces , (preprint), https://arxiv.org/abs/1001.2299https://arxiv.org/abs/1001.2299