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2 Infinite connected sum

We give more details of the definition of infinite connected sum. Let G be as in the Introduction. Let {v1, v2, ···} be the set of the vertices in G. If the connected, smooth n-manifold (n ≥ 2) Xvi associated to the vertex vi is orientable, we choose k an orientation of it. For each pair (i, j) with i ≤ j, let {eij|k = 1, 2, ···, mij} be the set of edges connecting vi to vj (of course, if there is no edge connecting vi to k vj, mij = 0, and in this case this set is empty). To each edge eij we associate a n n pair of smooth embeddings f k : R → Xv and g k : R → Xv from the standard eij i eij j n n R . We fix an orientation of the standard R . If Xv is oriented, we let f k (j ≥ i, i eij k = 1, 2, ···, mij) be orientation-preserving, and let g l (p ≤ i, l = 1, 2, ···, mpi) epi be orientation-reversing. We assume that the images of all these embeddings are disjoint from each other. Let Dn be the closed unit ball with center the origin in the standard Rn. For each i, let

mij 1 n mpi 1 n k l Yi := Xvi \ (∪j≥i ∪ =1 fe ( D ) ∪ ∪p≤i ∪ =1 ge ( D )), k ij 3 l pi 3 and let Y be the infinite disjoint union F Yi. We define an equivalence relation ∼ in Y by setting

f k (tu) ∼ g k ((1 − t)u) eij eij 1 2 Sn−1 for all (i, j) with i ≤ j, k = 1, 2, ···, mij, t ∈ ( 3 , 3 ) and u ∈ . Let X be the quotient space Y/ ∼. We call X the connected sum of Xvi according to the graph G via {f k ,g k }, and denote it by ♯GXv (f k ,g k ). eij eij i eij eij n k k n n Let M be a connected n-manifold, and ϕ = ⊔i=1ϕi : ⊔i=1D → M and k k n n ϕ˜ = ⊔i=1ϕ˜i : ⊔i=1D → M be two embeddings from the disjoint union of k copies of the standard n-disk. As in [BJ, Definition (10.1)] we say ϕ andϕ ˜ are compatibly n oriented if either M is not orientable, or, for each i (1 ≤ i ≤ k), ϕi andϕ ˜i are both orientation preserving or both orientation reversing (relative to fixed orientations of Dn and M n). The following result is well-known, see for example Theorem 3.2 in Chapter 8 of [H]. k k n n k k n Proposition 2.1. Let ϕ = ⊔i=1ϕi : ⊔i=1D → M and ϕ˜ = ⊔i=1ϕ˜i : ⊔i=1D → M n be two (smooth) embeddings from the disjoint union of k (< ∞) copies of the standard n-disk to a connected, smooth n-manifold M n (n ≥ 2). Suppose that ϕ and ϕ˜ are compatibly oriented. Then there is a diffeotopy H of M n, which is fixed outside of a compact subset of M n such that H(·, 1) ◦ ϕ =ϕ ˜.

2 (For definition of diffeotopy (or ambient isotopy), see [BJ, Definition (9.3)] and p.178 of [H].) Proof We follow closely the proof of Theorem 3.2 in Chapter 8 of [H]. We do induction on k. The k = 1 case is due to Cerf and Palais (for expositions see Chapters 9 and 10 in [BJ], Chapter III in [K] and Theorem 3.1 in Chapter 8 of [H]). Suppose the result is true for k = j. Now we consider the case k = j + 1. By assumption there exists a diffeotopy H of M n, which is fixed outside of a compact n je n j n subset of M such that He(·, 1) ◦ ϕ| ⊔i=1 D =ϕ ˜| ⊔i=1 D . (In particular, it follows n n j n n j n that He(·, 1)(ϕj+1(D )) ⊂ M \ ∪i=1ϕ˜i(D ).) Since n ≥ 2, M \ ∪i=1ϕ˜i(D ) is connected. We apply the k = 1 case to the two embeddings

n n j n He(·, 1) ◦ ϕj+1, ϕ˜j+1 : D → M \ ∪i=1ϕ˜i(D ), ˆ n j n and get a diffeotopy H of M \ ∪i=1ϕ˜i(D ) which is fixed outside of a compact n j n ˆ ˆ subset of M \ ∪i=1ϕ˜i(D ) such that H(·, 1) ◦ He(·, 1) ◦ ϕj+1 =ϕ ˜j+1. Clearly H n j n ˆ extends to a diffeotopy of M which leaves ∪i=1ϕ˜i(D ) fixed. Then Ht := Ht ◦ Het is the desired diffeotopy. ✷

Theorem 2.2. The infinite connected sum ♯GXv (f k ,g k ) is a connected, smooth i eij eij manifold, and oriented if all Xvi are oriented. Its diffeomorphism type (oriented if relevant) does not depend on the choice of embeddings f k and g k . eij eij Proof The first claim can be shown as in pp. 103-104 in [BJ] and pp. 90-91 k in [K]. Now we show the second claim. Suppose that to each edge eij we associate n n another pair of smooth embeddings f˜ k : R → Xv andg ˜ k : R → Xv from the eij i eij j n standard R . If Xv is oriented, we let f˜ k (j ≥ i, k =1, 2, ···, mij) be orientation- i eij preserving, and letg ˜ l (p ≤ i, l =1, 2, ···, mpi) be orientation-reversing. We assume epi that the images of all these embeddings f˜ k andg ˜ l are disjoint from each other. eij epi ˜ We define Yi and Y as before using fek andg ˜ek . We also introduce an equivalence e e ij ij ˜ relation in Y as before using fek andg ˜ek , and still denote it by ∼. Finally we let e ij ij ˜ ♯GXvi (fek , g˜ek ) := Y/ ∼. For each i let ij ij e

mij mpi n ϕi := ⊔j≥i ⊔ f k ⊔ ⊔p≤i ⊔ g l : ⊔R → Xv k=1 eij l=1 epi i and mij mpi n ϕ˜i := ⊔j≥i ⊔ f˜ k ⊔ ⊔p≤i ⊔ g˜ l : ⊔R → Xv . k=1 eij l=1 epi i Since G is locally finite, for each i, the above ⊔Rn is a finite disjoint union; we consider the finite disjoint union ⊔Dn contained in it. For each i, we can apply n n Proposition 2.1 to ϕi| ⊔ D andϕ ˜i| ⊔ D , and get a diffeotopy Hi(·, t) of Xvi , which is fixed outside a compact subset of Xvi , such that

n n ϕ˜i| ⊔ D = Hi(·, 1) ◦ ϕi| ⊔ D . (2.1)

3 By equation (2.1) we can define a map F : Y = ⊔Yi → Ye = ⊔Yei via

F (x)= Hi(x, 1) when x ∈ Yi for some i.

Clearly F is a diffeomorphism. Note that by equation (2.1) again F is compatible with the equivalence relations in Y and in Ye. So F induces a diffeomorphism

F : ♯GXv (f k ,g k ) → ♯GXv (f˜ k , g˜ k ). i eij eij i eij eij ✷

Remark In general, if each Xvi is orientable, the diffeomorphism type of the infinite connected sum ♯GXvi may depend on the choice of the orientations of Xvi . But it is easy to see that if each Xvi is orientable and admits an orientation-reversing dif- feomorphism, then the (unoriented) diffeomorphism type of ♯GXvi does not depend on the choice of the orientations of Xvi .

3 Proof of Theorem 1.2

Let X be a smooth 4-manifold which is homeomorphic to the standard R4. Assume that X admits a complete Riemannian metric with uniformly positive isotropic cur- vature and with bounded geometry. Clearly X contains no essential incompressible space form. By Theorem 1.1, X is diffeomorphic to an infinite connected sum of S4, RP4, S3 × S1, and /or S3×S1 according to a locally finite graph G. Since the e fundamental group of X is trivial, the graph G must be a tree, and the smooth manifold Xv associated to any vertex v in G must be diffeomorphic to the standard S4. We know that any topological manifold homeomorphic to R4 has exactly one (topological) end (for definition see for example, [DK]). It follows that the tree G has exactly one topological end also. But for a locally finite graph, there is a natural bijection between its topological ends and its graph-theoretical ends, cf. [DK] and the references therein. So the tree G has only one graph-theoretical end. (There should be a more direct argument for this fact.) Now we choose a ray γ in G, which is essentially unique. Let w0,w1,w2, ··· be the set of vertices along the ray γ. For each i, there are only finite vertices which can be connected to wi via a sequence of edges not contained in the ray γ. For each i, we do connected sum of 4 all S ’s associated to these finite vertices (including wi), the result is diffeomorphic 4 to the S associated to the vertex wi via a diffeomorphism not affecting the part 4 4 of this S where its connected sum with the two S ’s associated to wi−1 and wi+1 occurs. Then we see that X is diffeomorphic to an infinite connected sum of S4’s according to the ray [0, +∞) with a vertex wi at i (i =0, 1, 2, ···) using some gluing maps. We know that the infinite connected sum of S4’s (all with the standard ori- entation) according to the ray [0, +∞) using certain special gluing maps actually produces the standard R4. (Represent the standard R4 as the union of the unit

4 4-ball and the closed subspaces Ai bounded by the two 3-spheres with radius i and i + 1 (and each with center the origin), i =1, 2, ···. Note that each Ai may be seen as S4 with two open 4-balls removed.) We also know that S4 admits an orientation- reversing diffeomorphism. So by Theorem 2.2 and the Remark following it, X is diffeomorphic to the standard R4. ✷

Remark It is interesting to see whether or not the condition ‘with bounded geom- etry’ in Theorem 1.2 can be removed.

Acknowledgements This note was conceived during the workshop on the geometry of submanifolds and curvature flows, April 25-29, 2016 in Zhejiang Uni- versity. I would like to thank all the organizers and participants of this workshop, in particular, Prof. Zizhou Tang for his kind invitation, Prof. Hongwei Xu for his hospitality, Prof. Bing-Long Chen for his questions on my talk in this workshop, and Prof. Weiping Zhang for his encouragements.

References

[BBM] L. Bessi`eres, G. Besson and S. Maillot, Ricci flow on open 3-manifolds and positive scalar curvature, Geometry and Topology 15 (2011), 927-975. [BJ] Th. Br¨ocker and K. J¨anich, Introduction to differential topology, Cam- bridge University Press 1982. [CZ] B.-L. Chen, X.-P. Zhu, Ricci flow with surgery on four-manifolds with positive isotropic curvature, J. Diff. Geom. 74 (2006), 177-264. [DK] R. Diestel, D. K¨uhn, Graph-theoretical versus topological ends of graphs, J. Combin. Theory Ser. B 87 (2003), no.1, 197-206. [H] M. Hirsch, Differential topology, GTM 33, Springer-Verlag 1994. [Hu1] H. Huang, Ricci flow on open 4-manifolds with positive isotropic curva- ture, J. Geom. Anal. 23 (2013), no.3, 1213-1235. [Hu2] H. Huang, Four-orbifolds with positive isotropic curvature, Comm. Anal. Geom. 23 (2015), no.5, 951-991. [K] A. Kosinski, Differential manifolds, Academic Press 1993.

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, P.R. China E-mail address: [email protected]

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