Soul Theorem for 4-dimensional Topologically Regular Open Nonnegatively Curved Alexandrov Spaces

In this paper, we study the topology of topologically regular 4-dimensional open non-negatively curved Alexandrov spaces. These spaces occur naturally as the blow-up limits of compact Riemannian manifolds with lower curvature bound. These manifolds have also been studied by Yamaguchi in his preprint [Yam2002]. Our main tools are gradient flows of semi-concave functions and critical point theory for distance functions, which have been used to study the 3-dimensional collapsing theory in the paper [CaoG2010]. The results of this paper will be used in our future studies of collapsing 4-manifolds, which will be discussed elsewhere.


Introduction
The topology of noncompact manifold with a complete metric of nonnegative sectional curvatures was studied by Gromoll-Meyer in [GM69] and Cheeger-Gromoll in [CG72].
Theorem 0.1 (Soul Theorem, [CG72]). Let M n be an n-dimensional noncompact manifold with a complete metric of nonnegative sectional curvature. Then there exists a compact totally geodesic embedded submanifold S ⊂ M with nonnegative sectional curvature, such that M n is diffeomorphic to the normal bundle ν(S) of S in M n .
If in addition to the assumption above, there exists p ∈ M n such that the sectional curvatures at p are all positive, then M n is diffeomorphic to R n . This is called Cheeger-Gromoll soul conjecture. It was proved by Perelman in [Per94].
For Alexandrov spaces, (see [BGP92] and [BBI01] for basics of Alexandrov spaces) Perelman proved a similar result: Theorem 0.2 (Soul Theorem for Alexandrov space, [Per91]). Let X n be an n-dimensional non-compact Alexandrov space with nonnegative curvature, then there exits a closed totally convex subset S ⊂ X n , such that S is a deformation retraction of X n . This paper has been accepted for publication in Proc AMS.
Similar to the soul conjecture, Cao-Dai-Mei [CDM07,CDM09] proved that if in addition to the conditions of Theorem 0.2, one assumes that X n has positive curvature in a metric ball, then X n is contractible. Unlike the manifold case, Theorem 0.2 is the best topological result one can expect, in the sense that in general X n is not homeomorphic to the normal bundle of S. By Perelman's stability theorem, if Alexandrov space is the limit of sequence of Riemannian manifolds with lower curvature bound, then it's a topological manifold. In fact Kapovitch showed in [Kap02] Theorem 0.3 ( [Kap02]). If X n is the limit of a sequence of n-dimensional Riemannian manifold with the same lower curvature bound k, then Σ p X n is homeomorphic to (n − 1)-sphere S n−1 for any p ∈ X n . Moreover all the iterated space of directions are homeomorphic to spheres.
It's still unknown whether an Alexandrov space, which satisfies the conclusion of Theorem 0.3, can be realized as a limit of Riemannian manifolds with the same dimension and same lower curvature bound. In this paper, we consider the class of 4-dimensional topologically regular open nonnegatively curved Alexandrov spaces, in the sense that all space of directions are spheres. These Alexandrov spaces occurs naturally as the blow-up limits of compact Riemannian manifolds with lower curvature bound, thus play an important role in the study of collapsing under a lower curvature bound. We will prove the following Theorem 0.4 (Main Theorem). Let X 4 be as above, S be a soul of X 4 , then X 4 is homeomorphic a open disk bundle over S: (1) If dim S = 0, then X 4 is homeomorphic to R 4 ; (2) If dim S = 1, then X 4 is homeomorphic to R 3 bundle over S 1 ; (3) If dim S = 2, then X 4 is homeomorphic to R 2 bundle over S, where S = S 2 , RP 2 , T 2 or K 2 . (4) If dim S = 3, then X 4 is homeomorphic to line bundle over S, where S = S 3 /Γ, T 3 /Γ, (S 2 × S 1 )/Γ, and Γ some subgroup of isometric group of S acting freely on S.
This theorem will be used to study the collapsing 4-manifolds, which will discussed elsewhere. Theorem 0.4 has also been studied in the preprint [Yam02] Chap 15,16 in a traditional way. Our main tools are the gradient flow of semi-concave functions and Perelman's version of Fibration theorem, which have been used extensively in [CaoG10] to study the 3-dimensional collapsing manifolds under a lower curvature bound.

Construction of Soul
In this section we recall Cheeger-Gromoll's construction of soul for X n , where X n is an n-dimensional non-negatively curved open complete Alexandrov space. (c.f. [CG72], [Per91]).
Fix p ∈ X n , the Busemann function can be defined by (1.1) where B(p, t) is the ball centered at p with radius t. Denote the superlevel set b −1 ([a, +∞)) by Ω a . We have , [Per91], [Wu79]). Let X n , Ω a be as above, then the following hold (1) The Busemann function b is concave and bounded above.
(2) Ω a is compact and totally convex for all a ≤ a 0 := max x∈X n b(x).
We call S = Ω(k) a soul of the type (s, m), if the dimension of the soul is s and the dimension of Ω(0) is m.
We call a geodesic γ : (−∞, +∞) → X n a line in a metric space X n , if d(γ(t), γ(s)) = |t − s| for ∀t, s ∈ R. The splitting theorem reduces our discussion of 4-dimension to 3-dimension when X 4 admits a line.
. Let X n be open non-negatively curved Alexandrov space and assume that X n admits a line, then X n splits isometrically as In order to handle the non-smooth metric Alexandrov space, Perelman's Stability Theorem and his version of Fibration Theorem are extensively used in this paper. Let's recall these results.
be a sequence of n-dimensional Alexandrov spaces with curv ≥ −1 converging to an Alexandrov space with same dimension: lim α→∞ X n α = X n . Then X n α is homeomorphic to X n for α large.
The stability theorem for pointed spaces can be stated in a similarly way. The fibration theorem states that Theorem 1.4 (Fibration Theorem [Per91,Per93]). Let X n be an ndimensional Alexandrov space, U a domain in X n , f : U → R k be an admissible function, having no critical point and proper on U , then it's restriction to this domain is a locally trivial fiber bundle.
We refer to [Per91] for the definitions of admissible functions and regular map.
Theorem 2.1 ([SY00]). Let X 3 be an open complete 3-manifold with a possibly singular metric of non-negative curvature. Suppose that X 3 is oriented and S s is a soul of X 3 . Then the following is true.
(1) When dim(S s ) = 1, then the soul of X 3 is isometric to a circle.
Moreover, its universal coverX 3 is isometric toX 2 × R, whereX 2 is homeomorphic to R 2 ; (2) When dim(S s ) = 2, then the soul of X 3 is homeomorphic to S 2 /Γ or T 2 /Γ. Furthermore, X 3 is isometric to one of four spaces: (3) When dim(S s ) = 0, then the soul of X 3 is a single point and X 3 is homeomorphic to R 3 .
Throughout this paper S n denotes the standard n-sphere, T n denotes the n-dimensional torus, K 2 denotes the Klein bottle and D 3 denotes the standard 3-ball.

Proof of the main theorem
A key observation by K. Grove is that the distance function to the soul has no critical point in X n − S s when X n is a smooth Riemannian manifold. However this is no longer true for Alexandrov spaces even for topologically regular one. For example let L be the closed half strip {(x, y) ∈ R 2 |x ≥ 0, 0 ≤ y ≤ 1}, then the double of L which denoted by dbl(L) is a 2-dimensional Alexandrov space with non-negative curvature, which is homeomorphic to R 2 and has point soul (0, 1 2 ), it's clear that the distance function has critical points (0, 0) and (0, 1). However for Alexandrov space, [CaoG10] derived a modified result similar to Grove's observation.  For simplicity we assume that a 0 = 0 by adding a constant if needed. Let A = Ω(0). Using Proposition 3.1, we see that f (x) = d(x, A) has no critical value within (ε, +∞). It follows by Perelman's Fibration Theorem, that X n ∼ = B X n (Ω(0), ε) for ε > 0, where B X (A, r) denoted the set of points with distance ≤ r to set A in metric X.
We will divide the proof of our Main Theorem into the flowing cases.
Proof. (cf. the proof of Theorem 2.21 in [CaoG10]) In this case, S = Ω(0) and has dimension 3. Since X 4 is topologically regular, S is a topological manifold and non-negatively curved, hence by Hamilton's classification of 3-dimensional manifolds with non-negatively Ricci curvature, S is homeomorphic to S 3 /Γ, T 3 /Γ or S 2 × S 1 /Γ, where Γ is a subgroup of the isometry group of S 3 , T 3 , S 2 × S 1 . For p ∈ S, we know Σ p (S) is homeomorphic to S 2 , which divide Σ p (X 4 ) ∼ = S 3 into two parts, denoted by A ± . Since S is totally convex, Σ p (S) is convex in Σ p (X 4 ), therefore r Σp(S) | A ± have a unique maximum point ξ ± in A ± . Denote the maximum values by ± , i.e. r Σp(S) (ξ ± ) = ± . Since Ω(0) = S is the set of maximum points for Busemann function, by the first variation theorem, we know ± ≥ π/2. On the other hand if γ : [0, ] → Σ p (X 4 ) is a shortest geodesic connecting ξ − to ξ + , and let t 0 ∈ [0, ] satisfying γ(t 0 ) ∈ Σ p (S). By triangle inequality we know Note that curv(Σ p (X 4 )) ≥ 1 implies diam(Σ p (X 4 )) ≤ π, hence the inequalities in (3.1) are equalities, in particular − = + = π/2 and Σ p (X 4 ) is the spherical suspension over Σ p (S), i.e. T p (X 4 ) splits isometrically as T p (S) × R 1 . Hence we have a normal line bundle over S. By passing to the double cover, we can assume that this line bundle is trivial, therefore S separates X 4 into two parts and X 4 has two ends. Now it's easy to see X 4 admits a line, by splitting theorem, X 4 is isometric to S × R 1 .
By the lower semi continuity of the norm of gradient of λ-concave function ∇r ∂Ω(0) (cf. [Petr07] Corollary 1.3.5), r ∂Ω(0) has no critical point in for ε δ, see Figure 2. By the proof of §3.3, we know B X 4 (∂Ω(0), ε) is homeomorphic to a D 2 bundle over ∂Ω(0) for ε δ small enough, where the homeomorphism follows from the facts that r ∂Ω(0) is concave in the interior of Ω(0) and that there is a normal line bundle over the interior of Ω(0), see §3.1.
Since the bundle (3.4) and it's sub-bundle (3.5) are both trivial bundles, one can extend the gradient flow of r Ω −10δ (0) on B X 4 (Ω −δ (0) \ Ω −2δ (0), ε) to a gradient-like flow on B X 4 (Ω(0) \ Ω −2δ (0), ε), which give the homeomorphism (see Figure 3): It follows from the proof of §3.1 that B X 4 (Ω −2δ (0), ε) is homeomorphic to the normal line bundle over Ω −2δ (0), hence combine with Theorem 2.1, our main theorem holds in these two cases. Proof. In this case, S is 2-dimensional surface with non-negatively curvature. Thus S is homeomorphic to S 2 , T 2 , RP 2 or K 2 , by Splitting Theorem, if S = T 2 or S = K 2 the universal cover X 4 splits isometrically as N 2 × R 2 , thus the theorem follows from the fact that N 2 is homeomorphic to R 2 . Now we consider the cases where S = S 2 or RP 2 .
Let {p i } N i=1 be the set of extremal points on S, where the extremal points in Alexandrov surface is defined to be the points satisfying diam(Σ p i (S)) ≤ π/2. Note that Σ p i (S) = S 1 is a convex subset of Σ p i (X 4 ) = S 3 , hence for ε > 0, we have B Tp i (X 4 ) (T p i (S), ε) = T p i (S) × D 2 . Then by Stability Theorem, we know there exits δ > 0 such that B X 4 (p i , δ) ∩ B X 4 (S, ε) ∼ = D 4 and for ε δ, we have disc bundle (3.6) where π i is the bundle projection map. In particular π −1 i (∂B S (p i , δ)) = D 2 × S 1 For p ∈ S \ (∪ N i=1 B S (p i , δ/10)), by our assumption Σ p (S) > π/2, thus there exits δ > 0 and a admissible map F p = (f 1 , f 2 ) : B S (p, δ ) → R 2 which is regular B S (p, δ ), by the lower semi-continuity of gradient of semi-concave functions, we know F p is regular in B X 4 (p, δ ), for some δ > 0 satisfying δ ≤ δ ε. Thus we have a fiber bundle: Let f (x) = d(x, S). Since G p = (f 1 , f 2 , f ) is regular in the domain A X 4 (S, ε/100, ε) ∩ B X 4 (p, δ ), for ε δ, where A X (S, ε 1 , ε 2 ) denoted the annular region, i.e. all points have distance to S between ε 1 and ε 2 . It follows from the Fibration Theorem that we have a fiber bundle: where I is a open interval. Thus ∂N = S 1 , by generalized Margulis Lemma (cf. [FY92]), N ∼ = D 2 . Now we can glue the D 2 bundle together, (this part is similar to Yamaguchi's construction in [Yam02] . By construction we have a D 2 bundle over S( δ 2 ): Hence π −1 (∂B S (p i , δ 2 )) = D 2 × S 1 . Consider the gradient flow of r p i (·) = d(p i , ·) on A S (p i , δ/2, δ), again by the lower semi-continuity of |∇r p i |, for ε δ small enough, r p i is regular in B X 4 (A S (p i , δ/2, δ), ε) hence provide a homeomorphism φ between F i := π −1 (∂B S (p i , δ 2 )) and G i := π −1 (∂B S (p i , δ)). Clearly ∂B S (p i , δ/2) isotopic to ∂B S (p i , δ) in B S (p i , 2δ), hence when restricted to the boundaries ∂F i = S 1 × S 1 and ∂G i = S 1 × S 1 , φ is isotopic to the identity, therefore we can glue the D 2 bundle together.
Proof. If the soul is S 1 , then the universal cover X 4 admits a line by the totally convexity of soul, so X 4 = N 3 × R, where N 3 is homeomorphic to R 3 by Theorem 2.1, thus Theorem 0.4 holds in this case.
Proof of the Claim: Let {p i } N i=1 be the set of extremal points in ∂Ω(0). By the Stability Theorem, B X 4 (p i , ε) ∼ = D 4 . Let γ i be the boundary curve connection p i to p i+1 with the understanding that p N +1 = p 1 . One can assume γ i is short enough such that d p i has no critical point in B X 4 (γ −δ i , ε), where δ ε, γ −δ i is the sub-curve of γ i defined by {x ∈ γ i | d(x, p i ) ≥ δ and d(x, p i+1 ) ≥ δ}. Hence by Fibration Theorem it's a locally trivial fiber bundle, Since B X 4 (p i , ε) ∼ = D 4 , we have S X 4 (p i , δ) ∩ B X 4 (γ −δ i , ε) ∼ = D 3 , hence N 3 ∼ = D 3 . This finishes the proof.
Proof. Let p = S be a soul and {α n } ∞ n=1 be a sequence of number such that lim n→∞ α n = ∞. It's clear that lim n→∞ (α n X 4 , p) = (T p (X 4 ), O). Then it follows by Stability Theorem that B αnX 4 (p, ε) ∼ = B TpX 4 (O, ε) ∼ = R 4 . Since d p (x) has no critical point in X 4 \ p, we conclude that X 4 ∼ = R 4 by Perelman's Fibration theorem.
This completes the proof of the Main Theorem.