The fundamental group of a visual boundary versus the fundamental group at infinity

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1 The fundamental group of a visual boundary versus the fundamental group at infinity Greg Conner and Hanspeter Fischer June 2001 There is a natural homomorphism from the fundamental group of the boundary of any nonpositively curved geodesic space to its fundamental group at infinity. We will show that this homomorphism is an isomorphism in case the boundary admits a universal covering space, and that it is injective in case the boundary is one-dimensional. 1. Definitions A metric space is called proper if all of its closed metric balls are compact. A geodesic space is a metric space in which any two points lie in a geodesic, i.e. a subset that is isometric to an interval of the real line in its usual metric. A proper geodesic space X is said to be non-positively curved if any two points on the sides of a geodesic triangle in X are no further apart than their corresponding points on a reference triangle in Euclidean 2-space. The boundary of a non-positively curved geodesic space X, denoted by bdy X, is defined to be the set of all geodesic rays emanating from a fixed base point x 0 endowed with the compact-open topology. This definition is independent of the choice of x 0 [1, Proposition II.8.8]. Examples of such boundaries include the Sierpinski gasket and the one-dimensional Menger space [2]. While bdy X has a well-defined fundamental group π 1 bdy X, ω based at a geodesic ray ω :[0, X with ω0 = x 0, there is also the notion of a fundamental group at infinity of X based at ω, denoted by π1 X, ω. It is defined to be the limit of the inverse system whose terms are the fundamental groups of complements of compact subsets of X and whose bonds are induced by inclusion. Since the sequence of closed metric balls Bk ={x X dx, x 0 k} is cofinal in the system of compact subsets of X, weget π 1 X, ω = lim π 1 X \ B1,ω2 i 2 π 1 X \ B2,ω3 i 3 π 1 X \ B3,ω4 i 4,

2 36 π 1 of a visual boundary versus π 1 at infinity where i k is defined to be the the composition of the inclusion induced homomorphism incl k# : π 1 X \ Bk,ωk +1 π 1 X \ Bk 1,ωk + 1 and the isomorphism s k : π 1 X \ Bk 1,ωk +1 π 1 X \ Bk 1,ωk, which slides the base point from ωk +1toωk alongω. For the remainder of this note let us fix a non-positively curved geodesic space X with base point x 0 and a geodesic ray ω emanating from x 0. We shall be interested in the relationship between π 1 bdy X, ω andπ1 X, ω. 2. The natural homomorphism Lemma 1. There is a natural homomorphism ϕ : π 1 bdy X, ω π 1 X, ω. Proof. Denoting by [x 0,x] the unique geodesic in X from x 0 to x, we define a geodesic retraction map r k : Sk Sk 1 by x [x 0,x] Sk 1. Similarly we define r k : X \ Bk 1 Sk 1 by x [x 0,x] Sk 1. This allows us to write bdy X = lim S1 r 2 S2 r 3 S3 r 4, where we now interpret a geodesic ray γ :[0, X with γ0 = x 0 as the sequence γ1,γ2,. Notice that the diagram π 1 X \ B1,ω2 i 2 π1 X \ B2,ω3 i 3 r 2# incl # r 3# r 2# r 3# π 1 S1,ω1 π1 S2,ω2 commutes. Hence its top row is pro-equivalent to its bottom row. Therefore the limit of the top inverse sequence, which defines π1 X, ω, agrees with that of the bottom one. We obtain π1 X, ω = lim π 1 S1,ω1 r 2# π 1 S2,ω2 r 3# π 1 S3,ω3 r 4#. 1 The inverse limit projections q k : bdy X Sk clearly induce homomorphisms q k# : π 1 bdy X, ω π 1 Sk,ωk that commute with the homomorphisms r k#. We therefore get an induced homomorphism ϕ : π 1 bdy X, ω π1 X, ω defined by [α] [α 1 ], [α 2 ],, where for a map α :S 1, bdy X, ω we put α k = q k α. /

3 Greg Conner and Hanspeter Fischer Coincidence with the first shape group The typical examples of non-positively curved geodesic spaces have the structure of certain locally finite simplicial complexes whose simplices are isometric to simplices in a complete simply connected Riemannian manifold of some constant sectional curvature. In such spaces, metric spheres are ANRs. It is therefore not very restrictive to make the following General Assumption. Each Sk,ωk has the homotopy type of a pointed ANR. Consequently, the projection bdy X, ω q k S1,ω1 r 2 S2,ω2 r 3 S3,ω3 r 4 induces an HPol -expansion in the sense of shape theory [8]. It follows from 1 that the fundamental group at infinity π 1 X, ω coincides with the first shape group of bdy X, ω, which we will denote by ˇπ 1 bdy X, ω. We record Lemma 2. π 1 X, ω =ˇπ 1bdy X, ω. / 4. Boundaries with universal covers Theorem 1. If bdy X admits a universal covering space, then the natural homomorphism ϕ : π 1 bdy X, ω π1 X, ω is an isomorhism. We recall the definition of the pointed Čech system of a pointed compact metric space Z, z from [6] and [8]: Consider the collection C of finite open covers U of Z which contain exactly one element vu U with z vu. Then C is naturally directed by refinement. Denote by NU,vU a geometric realization of the pointed nerve of U, i.e. of the abstract simplicial complex { U, } with distinguished vertex vu. For every U, V C such that V refines U, choose a pointed simplicial map p UV :NV,vV NU,vU with the property that the vertex corresponding to an element V Vgets mapped to a vertex corresponding to an element U U with V U. Any assignment on the vertices which is induced by the refinement property will extend linearly. Then p UV is unique up to pointed homotopy and we denote its pointed homotopy class by [p UV ]. For each U C choose a pointed map p U :Z, z NU,vU such that p 1 U StU, NU U for all U U, where StU, NU denotes the open star of the vertex of NU which corresponds to U. For example, define p U based on a partition of unity subordinated to U. Again,

4 38 π 1 of a visual boundary versus π 1 at infinity such a map p U is unique up to pointed homotopy and we denote its pointed homotopy class by [p U ]. Then [p UV p V ]=[p U ], and Z, z [p U ] NU,vU, [p UV ], C isan HPol -expansion, so that ˇπ 1 Z, z = lim π 1 NU,vU,p UV #, C. 2 A proof of the following lemma can be found in Section 2 of [3]: Lemma 3. Let V C. Suppose every element of V is connected and every loop which lies in the union of any two elements of V contracts in Z, then the homomorphism p V# : π 1 Z, z π 1 NV,vV is an isomorphism. To prove Theorem 1, we let Z, z =bdyx, ω. In view of 2 and Lemma 2, it suffices now to show that for every element U Cthere is an element V Csuch that V refines U and p V# : π 1 bdy X, ω π 1 NV,vV is an isomorphism. Since by assumption, bdy X is a connected, locally path connected, semi-locally simply connected, compact metric space, every U Ccan easily be refined by an element V Cthat satisfies the requirements of Lemma One Dimensional Boundaries Theorem 2. If bdy X is one-dimensional, then the natural homomorphism ϕ : π 1 bdy X, ω π1 X, ω is injective. Proof. Suppose P k,p k is any sequence of pointed compact metric spaces having the homotopy type of pointed ANRs, and f k 1,k :P k,p k P k 1,p k 1 are continuous maps such that bdy X, ω = lim P 1,p 1 f 1,2 P 2,p 2 f 2,3 P 3,p 3 f 3,4. Then the projections f k :bdyx, ω P k,p k induce a canonical homomorphism ψ : π 1 bdy X, ω G = lim π 1 P 1,p 1 f 1,2# π 1 P 2,p 2 f 2,3# π 1 P 3,p 3 f 3,4#, defined by ψ[α] = [f 1 α], [f 2 α], [f 3 α],. Since bdy X, ω f k P 1,p 1 f 1,2 P 2,p 2 f 2,3 P 3,p 3 f 3,4 is another HPol expansion, there is an isomorphism i : ˇπ 1 bdy X, ω G such that i ϕ = ψ. The assertion of the theorem will follow if we choose the sequence P k,p k,f k 1,k such that ψ : π 1 bdy X, ω G is injective. This can be done using any one of the following three theorems. /

5 Greg Conner and Hanspeter Fischer 39 Theorem. [4] Let Z be the one-dimensional Menger space, obtained by intersecting the standard nested sequence P k of three-dimensional handlebodies. Fix a point z Z. Then the canonical homomorphism ψ : π 1 Z, z lim π 1 P 1,z incl # π 1 P 2,z incl # π 1 P 3,z incl # is injective. Remark. This theorem suffices to finish the proof of Theorem 2, since every onedimensional compact metric space Y embeds in the one-dimensional Menger space Z so that the induced homomorphism on fundamental groups π 1 Y,y π 1 Z, z is injective: π 1 Y,y incl # π1 Z, z ˇπ 1 Y,y incl # ˇπ1 Z, z f 2 f 3 f 4 Theorem. [7] Let Z be the limit of an inverse sequence P 1 P2 P3 of one-dimensional compact polyhedra and z =p n Z. Then the canonical homomorphism ψ : π 1 Z, z lim π 1 P 1,p 1 f 2# π 1 P 2,p 2 f 3# π 1 P 3,p 3 f 4# is injective. Remark. This theorem suffices to finish the proof of Theorem 2, because every one-dimensional compact metric space is the limit of an inverse sequence of onedimensional compact polyhedra. Theorem. [5] Let Z be a one-dimensional, compact, connected metric space, and z Z. ThenZ can be embedded in three-dimensional Euclidean space such that there exists a sequence P 1 P 2 P 3 of handlebodies with P k = Z and such that the canonical homomorphism is injective. ψ : π 1 Z, z lim π 1 P 1,z incl # π 1 P 2,z incl # π 1 P 3,z incl #

6 40 π 1 of a visual boundary versus π 1 at infinity References 1. M. Bridson and A. Haefliger, Metric spaces of non-positive curvature. Grundlehren der math. Wissenschaften 319, Springer Verlag N. Benakli, Polyèdres Hyperboliques, Passaga du Local au Global. Thèse, Univ. de Paris Sud J. Cannon, Geometric Group Theory. Pre-Print. 4. M. Curtis and M. Fort, Singular homology of one-dimensional spaces. Ann. Math J. Cannon and G. Conner, On the fundamental groups of one-dimensional spaces. Pre-print. 6. J. Dydak and J. Segal, Shape Theory: An Introduction. Lecture Notes in Mathematics 688, Springer Verlag K. Eda and K.Kawamura, The fundamental groups of 1-dimensional spaces. Topology and its Applications S. Mardešić and J. Segal, Shape theory: the inverse limit approach. North-Holland Mathematical Library 26, North-Holland Publishing Company Department of Mathematics Brigham Young University Provo, Utah U.S.A. conner@math.byu.edu Department of Mathematical Sciences Ball State University Muncie, IN U.S.A. fischer@math.bsu.edu