Open 3-manifolds which are connected sums of closed ones Laurent Bessières, Gérard Besson, Sylvain Maillot To cite this version: Laurent Bessières, Gérard Besson, Sylvain Maillot. Open 3-manifolds which are connected sums of closed ones. 2020. hal-02462552 HAL Id: hal-02462552 https://hal.archives-ouvertes.fr/hal-02462552 Preprint submitted on 31 Jan 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Open 3-manifolds which are connected sums of closed ones Laurent Bessieres` Gerard´ Besson Sylvain Maillot January 30, 2020 Abstract We consider open, oriented 3-manifolds which are infinite connected sums of closed 3-manifolds. We introduce some topological invariants for these manifolds and obtain a classification in the case where there are only finitely many summands up to diffeomorphism. This result encompasses both the Kneser-Milnor Prime De- composition Theorem for closed 3-manifolds and the Kerekj´ art´ o-Richards´ classifi- cation theorem for open surfaces. Keywords. Topology of 3-manifolds 1 Introduction In [Sco77], P. Scott studied decompositions of open 3-manifolds as (possibly infinite) connected sums of prime manifolds which may or may not be compact. He gave exam- ples of open 3-manifolds that do not have such decompositions. By contrast, in [BBM11] the authors proved that open 3-manifolds which carry certain Riemannian metrics of pos- itive scalar curvature do admit such decompositions, and in addition all summands are compact. In order to recall the result, we introduce some terminology. L. Bessieres:` Institut de Mathematiques´ de Bordeaux, Universite´ de Bordeaux, 33405 Talence, FRANCE; e-mail: [email protected] G. Besson: Institut Fourier, Universite´ Grenoble Alpes, 100 rue des maths, 38610 Gieres,` France; e- mail: [email protected] S. Maillot: Institut Montpellierain´ Alexander Grothendieck, CNRS, Universite´ de Montpellier; e-mail: [email protected] Mathematics Subject Classification (2010): 57Mxx,57N10 1 Open 3-manifolds which are connected sums of closed ones 2 Throughout the paper, we work in the category of smooth, oriented 3-manifolds. We say that two oriented 3-manifolds M1;M2 are isomorphic if there exists an orientation- preserving diffeomorphism from M1 to M2. For convenience we say that a 3-manifold N has spherical boundary if every component of @N (if any) is a 2-sphere. Then there is a unique manifold, up to isomorphism, obtained by gluing a 3-ball to every boundary component of N. This operation is called capping-off. The capped-off manifold is denoted by Nb. Definition 1.1. Let X be a class of connected oriented 3-manifolds without boundary. An oriented 3-manifold M is said to be decomposable over X if M contains a locally finite collection of pairwise disjoint embedded 2-spheres fSig such that for every connected component C of M split along fSig, the capped-off manifold Cb is isomorphic to some member of X or to S3. The main result of [BBM11] can be formulated as follows: if M is an open 3-manifold which admits a complete Riemannian metric of bounded geometry and uniformly positive scalar curvature, then there exists a finite familly F of spherical manifolds such that M is decomposable over F [fS2 ×S1g. This begs the question of whether such manifolds, and more generally 3-manifolds which are decomposable over some finite collection of closed 3-manifolds, can be classified. In this paper we show that this is the case, using a set of invariants inspired by the work of Kerekj´ art´ o´ [Ker23] and Richards [Ric63] to classify open surfaces:3 the space of ends of M together with a colouring depending on whether a given closed prime 3-manifold appears infinitely many times in any neighbourhood of this end or not. Recall that a closed 3-manifold K is prime if it is connected, not diffeomorphic to S3, 3 and whenever K is diffeomorphic to M1#M2, one of the Mi’s is diffeomorphic to S . In this paper we shall use the word ‘prime’ only for closed manifolds. The class of oriented prime 3-manifolds will be denoted by P. Let K be a (possibly disconnected) compact oriented 3-manifold with spherical bound- ary. For each P 2 P, we denote by nP (K) the sum over all connected components L of K of the number of summands isomorphic to P in the Kneser-Milnor decomposition of the capped-off manifold Lb. Thus the Kneser-Milnor theorem implies: every closed oriented 3-manifold is decomposable over some finite collection of prime oriented 3-manifolds, and two closed, connected, oriented 3-manifolds K1;K2 are isomorphic if and only if nP (K1) = nP (K2) for every P 2 P. We now come to the definition of our invariants. Let P be an oriented prime 3- manifold. Let U be an open oriented 3-manifold.Then we define nP (U) 2 N [ f1g 3While the Kerekj´ art´ o-Richards´ theorem also applies to nonorientable surfaces, in this paper we restrict attention to orientable manifolds for simplicity. Open 3-manifolds which are connected sums of closed ones 3 as the supremum of the numbers nP (K) where K ranges over all compact submanifolds of U with spherical boundary. Let M be an open connected oriented 3-manifold. An end e of M has colour P if for every neighbourhood U of e, with compact boundary, the number nP (U) is nonzero. (Then it is actually infinite, see below.) The set of ends of M of colour P is denoted by EP (M). By Lemma 3.2 below, it is a closed subset of E(M). Note that an end may have several colours, i.e. EP (M) \ EP 0 (M) can be nonempty for 0 P; P 2 P. In fact, the closed subsets EP (M) can be quite arbitrary by our realisation theorem, Theorem 2.5 below. Let M; M 0 be open connected oriented 3-manifolds. A map φ : E(M) ! E(M 0) is 0 called colour-preserving if for every P 2 P, we have φ(EP (M)) = EP (M ). Theorem 1.2 (Main theorem). Let X be a finite collection of prime oriented 3-manifolds. Let M; M 0 be open connected oriented 3-manifolds which are decomposable over X . Then M is isomorphic to M 0 if and only if the following conditions hold: 0 (1) For every P 2 P, the numbers nP (M) and nP (M ) are equal. (2) There is a colour-preserving homeomorphism φ : E(M) ! E(M 0). Remark 1.3. 1. By Thereom 2.3 and Proposition 2.4 below, it suffices in Condi- tion (1) to assume this for all P 2 X [ fS2 × S1g, since all the other numbers vanish. 2. Example 1 below shows that the finiteness assumption on X is necessary. 3. In the case where nP (M) < 1 for all P 2 P, the manifold is determined, up to isomorphism, by E(M) and the numbers nP (M). To conclude this introduction, let us mention that the notion of infinite connected sum is also useful in large dimensions, see e.g. [Why01]. The paper is organized as follows: in Section 2, we recall, in a slight more precise form, the notion of connected sum of closed manifolds along a possibly infinite graph, discussed in [BBM11, BBMM], and explain the relationship between this notion and de- composability. We state Theorem 2.3, which allows to turn the graph into a tree, possibly at the expense of adding S2 × S1 summands. We also state Proposition 2.4 and Theo- rem 2.5, and give Example 1 mentioned above. In Section 3, we discuss the properties of special exhaustions of decomposable 3-manifolds called spherical exhaustions, which play an important role in all the subsequent proofs. Proposition 2.4 and Theorem 2.5 are proved there. Section 4 is devoted to the proof of Theorem 2.3. Section 5 is dedicated to the proof of Theorem 1.2. Some concluding remarks are gathered in Section 6. We finish Open 3-manifolds which are connected sums of closed ones 4 with an appendix reviewing the theory of spaces of ends. Acknowledgment: this work began during ANR project GTO ANR-12-BS01-0004 and completed during ANR project CCEM ANR-17-CE40-0034. 2 Connected sums along graphs In this section we reformulate our notion of decomposable manifolds in terms of perform- ing connected sums of possibly infinitely many closed oriented manifolds along some locally finite graph. This was discussed in previous articles of the authors and F. C. Mar- ques [BBM11, BBMM] (cf. [Sco77].) Here though, we need to be more precise regarding orientations, and fix some notation for future use. Let I be a subset of N and X = fXkgk2I be a family of closed connected oriented 3 3-manifolds. By convention we assume henceforth that 0 2 I and X0 = S .A coloured graph is a pair (G; f) where G is a locally finite connected graph, possibly with loops, with vertex set V (G) and f is a map from V (G) to I. Construction. For each vertex v 2 V (G), let Yv be a copy of Xf(v) with d(v) disjoint open 3-balls removed, where d(v) is the degree of v.