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O httecomputational the that , ( 2 geometrization 2 31 n ) 46 SeSection (See . , 32 aeltr eastheorem Sela’s later. came ] salse nelementary an established ] 5.1 lmnayrecursive elementary 1.1 , 5.2 st ieacomplete a give to is 1.1 and , 1.2 1.2 a eandun- remained has 2.2 . osuenor- use does 5.3 )I contrast In .) 8 o one So, . , 11 3.2 , fits if on s 46 me- 1.1 1.1 and ted za- re r- ]. c e s - ; . t 2 recursive bounds on the number of Pachner moves needed to more than the ideas of Waldhausen [51] in his classification of standardize either S3 or a Seifert-fibered space with boundary. graph manifolds. The hyperbolic case of Theorem 1.2 is new. By contrast, Later in the Bulletin article [49, Sec. 6], Thurston gives a Mijatovi´calso established a primitive recursive bound on the list of open problems and projects, including: number of Pachner moves needed to equate two hyperbolic, fiber-free, Haken 3-manifolds. However, primitive recursive 21. Develop a computer program to calculate hy- is significantly weaker than elementary recursive; the Haken perbolic structures on 3-manifolds. condition is also a significant restriction. Theorem 1.2 also Jeff Weeks’ SnapPea [53] (now SnapPy [6]), which was origi- has the advantage of combining a mixed set of methods to nally written in the 1980s, met this challenge. It is fast and handle the full generality of closed, oriented 3-manifolds. reliable in practice, it can also compute the isometries be- Remark. We leave the non-orientable versions of Theorems tween two hyperbolic 3-manifolds, and it has been supremely 1.1 and 1.2 for future work. This case includes new de- useful for a lot of research in 3-manifold topology. Snap- tails such as 3-manifolds with essential, two-sided projective Pea also supports the belief that the homeomorphism prob- planes and Klein bottles. A more thorough result would also lem follows from geometrization, given its spectacular record handle compact 3-manifolds with boundary. in practice. However, its specific algorithms are not rigor- ous. SnapPea uses ideal triangulations of cusped 3-manifolds, together with Dehn fillings to make spun triangulations of 1.1. History and discussion closed 3-manifolds; it is only conjectured that such a struc- ture always exists. SnapPea also uses non-rigorous methods As already mentioned, the geometrization conjecture has to find suitable triangulations. In particular it uses limited- often been interpreted as a classification of closed 3- precision floating point arithmetic; it has no rigorous model manifolds, and computability is one candidate standard of of necessary precision as a function of geometric complexity. what it means to classify mathematical objects. In Thurston’s (Note that SnapPy has been extended to rigorously certify an famous survey of his results in the AMS Bulletin [49, Sec. 3], answer, when SnapPea finds one.) In contrast to the SnapPea he says: data structure, and other reasons that ideal and spun triangula- tions are important, we will use triangulations with semi-ideal Riley’s work makes it clear that there is a rig- and finite tetrahedra to prove Theorem 1.1 (see Section 5.2.5). orous, but not generally practical, algorithm for To start the rigorous discussion of computable classification computing hyperbolic structures. and the homeomorphism problem, we can say that closed 3- manifolds are classified if we can: Thurston then sketches an algorithm which is similar Man- ning’s construction [29] in some ways and to our argumentsin 1. specify every closed 3-manifold by a finite data struc- other ways. This passage, and some other aspects of the Bul- ture; letin article, support the conclusion that Thurston anticipated 2. algorithmically generate a standard list of closed 3- not only the statement of Theorem 1.1, but also its proof. The manifolds without repetition; and author also discussed Theorem 1.1 in personal communication with Thurston in the late 1990s. 3. given any 3-manifold M, algorithmically identify the At first glance, an algorithm that can only find a hyperbolic standard manifold M′ such that M = M′. structure on a 3-manifold is both less general and weaker than ∼ Theorem 1.1. It is less general because a 3-manifold may For closed 3-manifolds, condition 1 is addressed by the fact also have non-hyperbolic components; it is weaker because that every 3-manifold has a unique smooth structure and a the homeomorphism problem for two hyperbolic manifolds unique PL structure. As a result, we can describe a closed M1 and M2 takes more work than just finding their hyperbolic 3-manifold as a finite simplicial complex. Unlike in higher structures. However, in the theorem that it is recursive to ge- dimensions, it is easy to check whether a simplicial complex ometrize a 3-manifold (Theorem 5.4), the hyperbolic pieces is a 3-manifold (Section 3). Conditions 2 and 3 are equivalent (Lemma 5.7) are the hardest part. The geometric structure of to an algorithm to determine whether two closed 3-manifolds the other pieces and the data to glue the pieces together are M1 and M2 are homeomorphic by the following simple argu- complicated to describe carefully, but the proof that this data ment. (Haken calls this argument the “cheapological trick” is computable only requires moves on triangulations (Corol- [52, Sec. 4].) In one direction, if both conditions 2 and 3 are lary 3.4), the principle that nested infinite loops can be com- satisfied, then Condition 3 immediately implies a homeomor- bined into a single infinite loop (Proposition 2.5), and some phism algorithm. In the other direction, given a homeomor- facts about Seifert fibrations (Lemma 5.11 and Theorem 5.12). phism algorithm, we can make can lexicographically order all In the second stage, the homeomorphism problem for hy- descriptions of all closed 3-manifolds according to condition perbolic 3-manifolds (Theorem 6.1) reduces to calculating 1, and then list only those examples that are not homeomor- isometries by Mostow rigidity, and a typical algorithm for this phic to any earlier example. This satisfies condition 2. Then is similar to one for computing a hyperbolic structure. The given a description of a closed 3-manifold M, we can search homeomorphism problem for Seifert-fibered components and the list in order to find the standard M′ ∼= M to satisfy condi- glued combinations of components (Section 6) requires little tion 3. (Haken calls this argument the “cheapological trick” 3

[52, Sec. 4]. Arguably it is not cheap after all, since it is sub- decision problem: The input consists of two simplicial com- stantially similar to actual constructions of tables of knots and plexes that are promised to be manifolds; then the yes/no de- 3-manifolds.) cision is whether they are homeomorphic. (But see Proposi- As mentioned, Manning [29] and Scott and Short [43] give tion 3.1.) partial results toward Theorem 1.1, but they use more recent A decision algorithm is a mathematical computer program, tools. In particular, Manning uses Sela’s algorithm [46] for which can be modelled by a Turing machine (or some equiva- the isomorphism problem for word-hyperbolic groups, while lent model of computation), that takes some input x A∗ and Scott and Short use the theory of biautomatic groups [8]. can do one of three things: (1) Terminate with the∈ answer Both Short-Scott and Aschenbrenner-Friedl-Wilton [4, “yes”, (2) terminate with the answer “no”, or (3) continue in Sec. 2.1] mention a particular subtlety in approaches to The- an infinite loop. Similarly, a function algorithm can terminate orem 1.1 that are based on analyzing the fundamental group and report an output y A , or it can continue in an infinite ∈ ∗ π1(M) or the fundamental groups of its components. Namely, loop. Given a multivalued function f , then a function algo- π1(M) is insensitive to the orientation of M. Worse, if rithm is only required to calculate one of the values of f (x) on input x. A complexity class or computability class is some set of de- M ∼= W1 #W2 #...#Wn cision or function problems, typically defined by the existence is a decomposition of M into prime summands, then the ori- of algorithms of some kind. For example, a decision prob- entation of each summand Wk can be chosen separately with- lem d or a function problem f is recursive (or computable or out changing π1(M). Or the summands can be lens spaces; decidable) if it is computed by an algorithm that always ter- two lens spaces can have the same fundamental group with- minates. By definition, the complexity class R is the set of out even being unoriented homeomorphic. We surmount this recursive decision problems. By abuse of notation, R can also subtlety by modelling all 3-manifolds and their components denote the set of recursive, promise decision problems; or the with triangulations that are decorated with orientations; see set of recursive function problems, with or without a promise. Section 5.2. The following proposition is elementary.

Proposition 2.1. If d is a recursive promise problem, and if the promise itself is recursive, then d is a recursive non- ACKNOWLEDGMENTS promise problem if we let d(x)= no when the promise is not satisfied. The author would like to thank David Futer, Bus Jaco, Misha Kapovich, Jason Manning, Peter Scott, and Henry The complexity class RE is the set of recursively enumer- Wilton for very useful discussions. The author would also able decision problems. These are problems with an algo- like to thank the anonymous referee for a careful reading that rithm that terminates with “yes” when the answer is yes; but led to significant revisions. if the answer is “no”, the algorithm might not terminate. The complexity class coRE is defined in the same way as RE, but with yes and no switched. We review the following standard 2. COMPUTABILITY propositions and theorems. Proposition 2.2. A non-promise decision problem d is in RE We recommend Arora-Barak [3] and the Complexity Zoo if and only if there is an algorithm that lists all solutions to [54] for modern introductions to models of computation and d(x)= yes without repetition. complexity classes. Proposition 2.2 justifies the name “recursively enumerable” for the class RE. (Note that the solutions can be listed in order 2.1. Recursive and recursively enumerable problems if and only if d R.) The proof is left as an exercise. Also, in the spirit of Proposition∈ 2.2, a decision problem d can be identified with the set of solutions to d(x)= yes; in this way Let A be a finite alphabet and let A∗ be the set of all finite words over that alphabet. A decision problem is a function we can call a set recursive, recursively enumerable, etc. Proposition 2.3. R = RE coRE. d : A∗ yes,no . ∩ →{ } The proof of Proposition 2.3 is elementary but important: A function problem is likewise a function f : A A , which RE ∗ → ∗ Given separate algorithms for both the “yes” and “no” an- can be multivalued. The input space A∗ is equivalent to many swers, we can simply run them in parallel; one of them will other types of input by some suitable encoding: Finite se- finish. The proposition and its proof reveal the importantpoint quences of strings, finite simplicial complexes, etc. that a recursive algorithm might come with no bound whatso- A decision problem or a function problem can be a promise ever on its execution time. problem, meaning that it is defined only on some subset of in- puts P A∗ which is called a promise. Whether two closed Theorem 2.4 (Turing). The halting problem is in RE but not n-manifoldsare⊆ PL homeomorphicis an example of a promise in R. In particular, RE = R. 6 4

Informally, the halting problem is the question of whether asymptotic notation as a function of the input length n = x a given algorithm with a given input terminates. Let h be the to a decision problem d(x). For example, we could ask for| a| halting decision problem, where the input x in the value h(x) polynomial-time algorithm, by definition one that runs in time is an encoding of an algorithm and its input (or, traditionally, O(nk) for some fixed k. an encoding of a Turing machine). It is easy to show that h is We have two reasons to consider a fairly generous bound in RE-complete in the following sense: Given a problem d RE, this paper. First, the recursive class R is unfathomably gener- there is a recursive function f such that d(x)= h( f (x)) for∈ any ous, so any explicit boundcan be considered a major improve- input x. Any other problem in RE with this same property can ment. Second, the computational complexity of a problem also be called RE-complete, or halting complete. or algorithm depends somewhat on the specific computational model, but certain relatively generous complexity bounds are Proposition 2.5. Let G be a graph structure on A∗. If the substantially model-independent. edge set of G is recursively enumerable, then so is the set of We consider a traditional Turing machine first. By (infor- pairs (x,y), where x and y are vertices in the same connected mal) definition, a Turing machine is a finite-state “head” with component of G. an infinite linear memory tape, and deterministic dynamical Proposition 2.5 is important for recursively enumerable in- behavior. We say that an algorithm is elementary recursive if finite searches. The interpretation of the proposition, which it runs in time is conveyed by the proof, is that nested infinite loops can be n .2 reorganized into a single infinite loop. .. O22  Proof. By Proposition 2.2, we can model a recursively enu-  k  | {z } merable set by an algorithm that lists its elements. The propo- sition states that the elements can be listed without repetition; for some constant k. We call the corresponding complexity ER ER but this is optional, since we can store all of the elements al- class . By abuse of terminology, we use to refer to both ready listed and omit duplicates. decision problems and function problems, and to numerical ER We use a recursive bijection f between the natural numbers bounds. (Note that if f (n) can be computed in , then a ER N and N , the set of finite sequences of elements of N. We running time bound of O( f (n)) is itself a subclass of .) ∗ Without reviewing rigorous definitions, we list some of the can express any element of N∗ uniquely in a finite alphabet that consists of the ten digits and the comma symbol. We many variations in the computational model that do not affect ER can then list of all of these strings first by length, and then the class in the following proposition. The proposition is ER in lexicographic order for each fixed length. We can then let not really needed in this paper except to motivate as an f (n) be the nth listed string. important complexity class. The only tacit dependence is that We can now convert the value f (n) to a finite path a random access machine is somewhat closer to both intuitive descriptions of algorithms and actual computers than a Turing (x0,...,xk) in the graph G, in such a way that every finite path is realized. If machine with a linear tape is. Proposition 2.6. Each of the following four computational f (n) = (n ,...,n ), 0 k models is the same as standard ER. then we let x0 be the n0th string in A∗. For each j > 0, we 1. A Turing machine with an n-dimensional tape, or a let x j be the n jth neighbor of x j 1. In order to find the n jth − random access tape addressable by a separate address neighbor of x j 1, we list theelementsof theedgesetof G until − tape, with an elementary recursive bound on computa- the edge (x j 1,x j) arises as the jth edge from x j 1. There is − − tion time. the technicality that x j 1 might not have an n jth neighbor if it only has finitely many− neighbors. To avoid this problem, 2. A Turing machine restricted to an elementary recursive we intersperse trivial edges of the form (x,x) infinitely many bound on computational space and unrestricted compu- times, for every string x A∗, along with the non-trivial edges tation time. of G. ∈ Since the algorithm finds every finite path in G, it finds ev- 3. A randomized Turing machine whose answers are prob- ery pair of vertices x and y in the same connected component. ably correct, with an elementary recursive bound on Thus, the set of such pairs is recursively enumerable. computation time. 4. A quantum Turing machine that can compute in quan- tum superposition, with an elementary recursive bound 2.2. Elementary recursive problems on computation time. As mentioned after Proposition 2.3, a recursive algorithm need not have any explicit upper bound on its execution time, Proof. Instead of a self-contained proof, we justify each case beyond the tautological bound that running it is a way to cal- of the proposition with specific references to Arora-Barak [3]. culate how long it runs. This motivates smaller complexity 1. This follows from Exercises 1.7 and 1.9 in Arora-Barak. classes that are defined by explicit bounds. The most common notation for a bound on the execution time of an algorithm is 2. This follows from Theorem 4.2 in Arora-Barak. 5

3. This reduces to case 4 by the proof of Corollary 10.11 Remark. The field of real algebraic numbers together with re- in Arora-Barak. liable equality testing is implemented in Sage [41].

4. This reduces to case 2 by the proof of Theorem 10.23 Theorem 2.8 (Tarski-Seidenberg [44, 47]). It is recursive to in Arora-Barak. determine whether there is a solution to a finite list of poly- nomial equalities and inequalities with coefficients in Qˆ in Remark. An elementary recursive bound is also a major im- finitely many variables; or to find a solution. provement over another bound that is popular in logic and computer science: primitive recursive. An algorithm is prim- Actually, Tarski and Seidenberg proved the stronger result itive recursive if it runs in time O(n[k]b) for some fixed k and that it is recursive to decide any assertion over R expressed b, where the kth operation a[k]b is defined inductively as fol- with polynomial relations and first-order quantifiers. lows:

a[1]b = a + b a[2]b = ab a[3]b = ab 3. TRIANGULATIONS OF MANIFOLDS AND MOVES a[k + 1]b = a[k](a[k]( (a[k]a) )). ··· ··· b | {z } In this section, we will analyze the form of the input to For example, the operation a[4]b, which is called tetration, is Theorem 1.1. We will show that given simplicial complexes defined as a tower of exponentials of height b. The primitive as input to the homeomorphism problem, we can first confirm recursive complexity class is denoted PR. We can organize that they are 3-manifolds. (It is also easy to confirm that input strings actually represent simplicial complexes, in some con- PR into a complexity hierarchy by defining Ek to be the set of functions computable in time O(n[k + 1]b) for some fixed venient data type.) Thus Proposition 2.1 applies: we can view the homeomorphism problem as a non-promise problem. Ac- b. Then E2 = P, E3 = ER, E4 consists of complexity bounds which are bounded towers of tetrations, etc. tually, Proposition 3.1 below is overkill for this purpose, since it is much harder in dimension n = 4 than in dimension n = 3. We then discuss moves between triangulations of a mani- 2.3. Computable numbers fold, mainly to establish Corollary 3.4. In light of Proposi- tion 2.3, Corollary 3.4 is an easy half of Theorem 1.1, one A computable real number r R is a real number with a that holds in any dimension n. computable sequence of bounding∈ rational intervals. In other Proposition 3.1. If Θ is a finite simplicial complex of dimen- words, there is an algorithm that generates rational numbers sion n 4, then it is recursive to determine whether it is a a ,b Q such that x [a ,b ] and b a 0. Many stan- n n n n n n closed≤ PL n-manifold, and whether or not it is orientable. dard algorithms∈ from numerical∈ analysis,− including→ field oper- ations, integration of continuous functions, Newton’s method, Proof. The proof is partly by induction on dimension n. The etc., have the property that if the input consists of computable result is trivial if n = 0, where we need only check that Θ is numbers, then so does the output. One main limitation of a single point. Otherwise, we first check that Θ is connected, computable real numbers is that inequality tests such as a > b and we must check that the link Λ of every vertex is both a or a = b are only recursively enumerable, not recursive. In closed (n 1)-manifold and a PL n-sphere. The former con- other6 words, if a = b, then there is an algorithm to eventually − 6 dition is the inductive step. The latter condition requires an confirm this fact and say which one is greater; but there is no algorithm to recognize an (n 1)-sphere. If Λ is a closed 1- terminating algorithm that always confirms that a = b. manifold, then it is immediately− a 1-sphere, i.e., a circle. If Λ We can avoid this shortcoming of the field of computable is a closed 2-manifold, then we can compute its Euler charac- numbers by passing to a smaller subfield where equality is teristic. If Λ is a closed 3-manifold, then Theorem 1.1 implies also recursive. In particular, we will use Qˆ = R Q¯ , the real ∩ that it is recursive to determine if Λ is a 3-sphere, although algebraic closure of the rational numbers Q, which has this this result was obtained without geometrization by Rubinstein property. and Thompson (Theorem 8.7)[40, 48]. Theorem 2.7. There is an encoding of the elements of Qˆ such We can check that Θ is orientable (and orient it) algorith- that field operations, order relations, and conversion to com- mically by computing its simplicial homology. putable real numbers are all recursive. The stellar and bistellar subdivision theorems establish that One encoding of a real algebraic number x that can be used every two triangulations of a compact n-manifold, in particu- to prove Theorem 2.7 is to describe it by a minimal polyno- lar a compact 3-manifold, are connected by a finite sequence mial together with an isolating interval x [a,b] with rational of explicit moves. See Lickorish [28] for a modern treatment ∈ endpoints to distinguish x from its Galois conjugates. Note and a historical review. that the isolating interval may be made arbitrarily small since algebraic numbers are computable, for instance by Newton’s Theorem 3.2 (Alexander-Newman). If two finite simplicial method. Note also that a computable encoding of elements complexes Θ1 and Θ2 are PL equivalent, then they are con- of Qˆ yields a computable encoding of elements of Q¯ C the nected by a sequence of stellar subdivision moves and their field of all algebraic numbers. ⊆ inverses. 6

Briefly, a stellar move in a simplicial complex Θ consists each candidate for Θ2, there are only finitely many combina- of replacing the star st(∆) of some simplex ∆ in Θ with a cone torial choices for a function from the simplices of Θ2 to the over the subcomplex of simplices in st(∆) that do not contain simplices of Θ1. For each such choice, we can first check ∆. Equivalently, the apex v of this cone is placed in the interior that the simplices of Θ2 that land in a k-simplex ∆ Θ1 sup- of ∆, and all simplices that contain ∆ are subdivided to support port a simplicial cycle that represents the fundamental∈ class the new vertex v. in Hk(∆,∂∆). We solve for each such cycle for all ∆ (where each must be unique if Θ2 indeed subdivides Θ1). Then the Theorem 3.3 (Pachner). If Θ1 and Θ2 are two triangulations constraint that each simplex of Θ must be positively oriented of a compact, PL manifold M, then they are connected by bis- 2 in Θ1 yields we obtain algebraic inequalities for the positions tellar moves. of all vertices. We can then apply Theorem 2.8 to see if there A bistellar move of a triangulation of an n-manifold M is a solution for those positions. consists of a stellation followed an inverse stellation at the We conclude this section with the following theorem which same vertex. Equivalently, two triangulations of M differ by combines results of P.S. Novikov, Boone, Adian, Rabin, a bistellar move when there is a minimal cobordism between Markov, and S.P. Novikov [39]. them consisting of a single (n + 1)-simplex. In particular, a shellable triangulation of M I yields a sequence of bistellar Theorem 3.6 (NBARMN). The isomorphism problem for moves. × finitely presented groups, the PL homeomorphism problem for Lickorish points out that Newman conjectured and par- 4-manifolds, and the recognition of Sn among PL n-manifolds tially proved Theorem 3.3 in an earlier paper, before he and for each n 5 are all halting-complete. Alexander separately proved Theorem 3.2. Bistellar moves ≥ It is not known whether either topologicalor PL recognition are also called Pachner moves, although arguably they should 4 be called Newman-Pachner moves. of S is recursive. Theorem 3.3 also holds for ideal or semi-ideal triangula- Remark. The homeomorphismproblem for PL n-manifolds in tions of a compact 3-manifold with torus boundary compo- Theorem 3.6, or even recognition of Sn, needs to be handled nents. (In other words, it holds for a 3-dimensional pseudo- with some care, for several reasons. First, because recogniz- manifold with singular points with torus links, which are the ing whether the input is a PL n-manifold is (by Theorem 3.6!) ideal vertices.) an uncomputable promise when n 6. Second, because there ≥ Theorems 3.2 and 3.3 each have the following corollary. are closed manifolds that are homeomorphicbut not PL home- omorphic [23]. Third, because there are simplicial complexes Corollary 3.4. The PL homeomorphism problem for compact that are not PL n-manifolds at all, but that are homeomorphic RE PL n-manifolds is in . to Sn, for each n 5[7]. The proof of Theorem 3.6 dispenses ≥ Remark. There is a proof of Corollary 3.4 that works directly with all of these concerns as follows. Given an input x to the halting problem h(x) and an integer n 4, there is an algo- from the definition of PL equivalence without using Theo- ≥ rem 3.2 or 3.3, nor even Proposition 2.5. For each n, choose a rithm that constructs an n-manifold M(x) such that: n n linear embedding of an n-simplex ∆ R . Then in general a 1. M(x) is manifestly a closed PL manifold. geometric refinement is a simplicial complex⊆ Θ with a homeo- morphism onto ∆ which is affine-linear on each simplex of Θ. 2. M(x) is PL homeomorphic to Sn when n 5, or to a 2 2 ≥ Likewise a refinement of a simplicial complex Θ is another connected sum of copies of S S when n = 4, if and 1 × simplicial complex Θ2 with a homeomorphism f : Θ2 Θ1, only if M(x) is simply connected. such that f yields a geometric refinement of each simplex→ of 3. M(x) is simply connected if and only if h(x)= yes. Θ1. By definition, two complexes Θ1 and Θ2 are PL equiva- n lent if they share a refinement Θ3. Now, we can let each ∆ Remark. By contrast with Theorem 3.6, the PL homeomor- have rational vertices (i.e., vertices in Qn), and after that we phism problem for simply connected n-manifolds with n 5 can perturb any geometric refinement so that its vertices are is recursive [36]. ≥ all rational. The set of rational mutual refinements of two fi- nite complexes Θ1 and Θ2 is recursive by direct verification. (In other words, given a simplicial complex Θ3, and given ra- 4. SOME NOTATION tional target positions for its vertices in both Θ1 and Θ2, we can algorithmically check whether this data yields a mutual We summarize some notation for specific topological refinement.) Therefore the question of whether there exists a spaces, beyond the most standard notation that Sn is an n- mutual refinement is directly recursively enumerable. sphere, Dn is an n-disk, Pn is real projective n-space, and I = D1 is an interval. Proposition 3.5. If Θ1 is a finite simplicial complex with n1 simplices (of arbitrary dimension) and n n , then it is re- We let X ⋉Y denote a fiber bundle with base X and fiber 2 1 ˜ cursive to produce a complete list of geometric≥ subdivisions Y. Although the notation X Y is reasonably standard for a × ⋉ Θ of Θ with n simplices. twisted bundle, we prefer to write X Y, for two reasons. 2 1 2 First, because the notation specifies which side is the fiber; ⋉ ⋊ Proof. There are only finitely many simplicial complexes Θ2 we can write X Y ∼= Y X. Second, because a fiber bundle with n2 simplices, and they can be generated recursively. For is analogous to a semidirect product in group theory. 7

We review Seifert’s description of oriented Seifert-fibered Theorem 5.1 (Kneser-Milnor [24, 34]). Every closed, ori- spaces [45]. If F is a compact surface which may or may not ented 3-manifold (other than S3) is a connected sum of prime, be orientable, then there is a unique, canonically oriented I- closed, oriented 3-manifolds (none of which are S3). The sum- bundle F ⋉ I. If F is orientable, then this I-bundle is simply mands are unique up to oriented homeomorphism. F I; in this case we assume an orientation for the base F 3 and× the fiber I. We consider the double F ⋉S1 of F ⋉I, which We will adopt the convenience that a 3-sphere S counts as 1 a prime 3-manifold, notwithstanding that Theorem 5.1 would again when F is orientable is just F S . 3 If p , p ,..., p are points of F, then× we can apply a Dehn be easier to state if S were instead interpreted as the “unit” in 1 2 n the terminology of unique factorization. surgery with slope bk/ak to a solid torus neighborhood of the 1 fiber over pk in F ⋉ S , where ak is a positive integer and bk Theorem 5.2 (Jaco-Shalen-Johansson [19, 20]). A closed, is a relatively prime integer of either sign. The resulting ori- oriented, prime 3-manifold has a minimal collection of in- ented 3-manifold N is thus construction from its Seifert data, compressible tori, unique up to isotopy and possibly empty, namely the multiset with the property that the complementary regions are either Seifert-fibered or atoroidal. F,(a1,b1),(a2,b2),...,(an,bn) . { } The decomposition in Theorem 5.2 is called the JSJ decom- In general we interpret F as an orbifold. If a 2, then we k position. We can call the tori JSJ tori, and the complementary interpret p F as an orbifold point of order a ,≥ and the circle k k regions JSJ components. We will use M to denote a general over it is an∈exceptional fiber. By Seifert’s classification, the closed, oriented 3-manifold; then W to denote a prime sum- integers a with a 2 together with the residues b Z/a k k k k mand of M; then N to denote a JSJ component of W. are all topological invariants≥ of the fibration of N. If∈F and therefore N has boundary, then this is a complete set of invari- Theorem 5.3 (Thurston-Hamilton-Perelman). Suppose that ants. If N is closed, then the Euler number N is an oriented, prime, atoroidal 3-manifold which is either b closed or has torus boundary components. Then N is either e(N)= b + ∑ k Seifert-fibered, or it is closed and has a unique hyperbolic ak k structure, or its interior N∗ has a unique, complete hyperbolic is the only additional necessary invariant. Thus there is a structure with torus cusps. canonicalized version of the Seifert data in the form As everyone knows, Theorem 5.3 was conjectured and partly proven by Thurston [49], then fully proven by Perel- F,b,(a1,b1),(a2,b2),...,(an,bn) , { } man using the Ricci flow program of Hamilton [35]. (Note where b represents (1,b) and otherwise ak 2 and 1 bk < that Theorem 5.3 implicitly includes the Poincar´econjecture ≥ ≤ ak. If F is non-compact, then b is irrelevant and we omit it in in the Seifert-fibered case.) the canonical form. Remark. Mixing the JSJ decomposition with hyperbolization With the notation of fiber bundles and Seifert-fibered is a less pure approach than Thurston’s decomposition into spaces, we name these specific manifolds: geometric components, but we find it convenient for Theo- 1. We use S1 S1 to denote the standard 2-torus, and T to rem 1.1. We could recognize spherical and Euclidean com- denote an arbitrary× 2-torus, i.e., T = S1 S1. ponents with the same methods as hyperbolic components ∼ × (Lemma 5.7), while several of the other Thurston geometries 2. K2 = S1 ⋉ S1 is the 2-dimensional Klein bottle. induce canonical Seifert fibrations. In fact, every Seifert- fibered 3-manifold or component is geometric. Conversely, 3. L(m,n) is the lens space defined by the Seifert data every geometric 3-manifold or component is hyperbolic un- S2,0,(n,m) . { } less it is Seifert-fibered or a Sol manifold. 4. R(m,n) denotes the prism space defined by the Seifert data P2,0,(m,n) . { } 5.2. Statement of computational geometrization

5. GEOMETRIZATION IS RECURSIVE Theorem 5.4. If M is a triangulated 3-manifold, then it is re- cursive to compute a decorated triangulation which is adapted The goal of this section is to prove Theorem 5.4, which to its geometric decomposition. says that the geometric decomposition of a 3-manifold M is Before proving Theorem 5.4, we need to state it more pre- computable. cisely. When Θ is a decorated, adapted triangulation of M it means that: 5.1. Statement of geometrization 1. M has a distinguished (but possibly empty) collection of disjoint, separating 2-spheres, each triangulated with We begin with three results that, together, are one formu- 4 triangles in Θ, that separates it into prime summands lation of the geometrization theorem for closed, oriented 3- W . Each W is closed; it inherits its triangulation from manifolds. Θ{ and} its holes are plugged with fresh tetrahedra. 8

2. The triangulation of each W supports a distinguished 5.2.3. Orientations (but possibly empty) collection of disjoint thickened tori T I and restricts to a shelled triangulation of each × To be precise, we can decorate each tetrahedron by ordering one. These thickened tori separate W into JSJ compo- its vertices, where two orderings are equivalent if they differ nents N . { } by an even permutation. 3. The tetrahedra at all stages are consistently oriented, to express an orientation of each summandW and each JSJ 5.2.4. Seifert-fibered components component N that is consistent with the orientation of M. We begin with preliminaries on cellulations and barycentric subdivisions that we will also need in Section 5.2.5. 4. If N is Seifert-fibered with base F, then we make a tri- A cellulation of a topological space X is a CW complex Λ angulation which is adapted to Seifert’s description of with a homeomorphism to X. The complex Λ is regular if Λ N by Dehn surgery on F S1 when F is orientable, or is locally finite; and if the attaching map of each k-cell is an × 1 embedding in the (k 1)-skeleton, so that each closed k-cell Dehn surgery on a twisted bundle F ⋉S when F is non- − orientable. This includes the case where N = W = M is of Λ is embedded in X. a 3-sphere. We will use the following standard proposition to model regular CW complexes using triangulations. 5. If N is hyperbolic, then it is marked as the barycentric Proposition 5.5 ([26, Sec. 10.3.5]). Every regular CW com- subdivision of a regular, adapted cellulation Λ. Λ comes plex Λ has a barycentric subdivision Θ which is a simplicial from a geometric triangulation Λ∗ of N∗ in which each complex, and the spaces of Θ and Λ are homeomorphic. tetrahedron has at most one ideal vertex. If ∆ Λ∗ has See Figure 1 for an example. an ideal vertex, then it is truncated to a triangular∈ prism in Λ; if not, then it is kept in Λ. Each tetrahedron in Λ∗ is also decorated with its dihedral angles.

We proceed to explain each stage of the definition.

5.2.1. The prime decomposition Figure 1. A barycentric subdivision of part of a cellulation of a sur- Note that we take a triangulation to be a simplicial complex face. structure rather than a generalized triangulation. In geomet- ric topology, for instance in the SnapPea census, a general- If N is a Seifert-fibered component, then as described in ized triangulation is sometimes defined to be a CW-complex Section 4, it has a base orbifold F with one circle for each whose cells are simplices and whose attaching maps take sim- boundary torus of N. The fibration has canonical Seifert data plices to simplices. In particular, a simplex in a generalized F,b,(a1,b1),(a2,b2),...,(an,bn) , triangulation need not have distinct vertices and two simplices { } may have the same vertices. We can form a connected sum with b omitted when F or N has boundary. The data indicates of two triangulated 3-manifolds by removing a single tetrahe- surgery with slope bk/ak at the fiber over some pk F and (if ∈ dron from each one and gluing the sphere boundaries. it exists) surgery with slope b at p0 F. ∈ We choose a triangulation ΘF of F such that each pk (in- cluding p0, if it exists) lies in the interior of a triangle, and such that all of these triangles are disjoint. We can lift ΘF to a 5.2.2. Shelled triangulations cellulation Λ such that the solid torus ∆ S1 over each trian- × gle ∆ in ΘF is tiled by two vertical triangular prisms. We take If X is a closed n-manifold with two triangulations Θ0 and the barycentric subdivision of ΛF to obtain a triangulation of Θ1, then a shelled triangulation of X I is a simplicial com- F I or F ⋉ I. If a triangle ∆ Λ contains some pk, we re- × × 1 ∈ plex Θ0 1 whose (n + 1)-simplices are numbered. Taken in move the solid torus ∆ S (which is now triangulated with , × order, the (n + 1)-simplices connect the triangulation Θ0 of 72 tetrahedra) and glue it back using Dehn surgery. The glu- 1 X 0 to the triangulation Θ1 of X 1 via a sequence of ing involves a homeomorphism of the boundary ∂(∆ S ), bistellar×{ } moves. Note that this combinatorial×{ } restriction on which we implement with a shelled triangulation of a× thick- Θ0 1 implies Θ0 1 is PL homeomorphic to X I. In other ened torus, as in Section 5.2.2. The result is a triangulation , , × words, if we build Θ0,1 from a sequence of bistellar moves, of N, which we decorate with information about how it was and if X 0 and X 1 are disjoint in the result, then Θ0,1 constructed, so that the canonical Seifert data is part of the is a triangulation×{ } of X×{I.} decoration. × 9

5.2.5. Hyperbolic components V and thus at most one ideal vertex. We can let Θ = Λ3 be a second barycentric subdivision, which is then a simplicial If the component N is hyperbolic, then we choose a geo- complex and still has the property that each tetrahedron has at most one ideal vertex. metric triangulation Θ∗ of N∗, meaning one whose tetrahedra lift to geometric tetrahedra in the universal cover H3. More precisely, if Nˆ is the compactification of N given by collaps- Given a semi-ideal triangulation of N∗, we can truncate the ing each torus boundary component to a point, we assume a cusps so that each semi-ideal tetrahedron becomes a triangu- continuous map lar prism, as in Figure 2. A barycentric subdivision of this cellulation is then the desired adapted triangulation.

f : Θ∗ Nˆ → such that the image f (∆) of each combinatorial tetrahedron ∆ lifts to a geometric tetrahedron in the standard compactifica- tion H3 of hyperbolic space. If none of the vertices of f (∆) are at infinity, then f (∆) is finite; if they are all at infinity, then f (∆) is ideal; and if some are at infinity, then f (∆) is semi- ideal. We will assume that all of the simplices of our Θ∗ are either finite or semi-ideal with one ideal vertex. If N = N∗ is closed, then all tetrahedra in Θ∗ must be finite; if N has boundary components and thus N∗ has cusps, then some of the tetrahedra must be semi-ideal. Before proceeding further, we contrast this with some other models that have also been studied as geometric triangula- Figure 2. A tetrahedron truncated at one vertex. tions. In some treatments f is not a homeomorphism but only a homotopy equivalence. In the case we can still ask for the restriction of f to each tetrahedron ∆ to be affine-linear in the Klein model of H3. However, f (∆) may be degenerate, 5.3. Proof of Theorem 5.4 meaning that it has zero volume, or it may be flipped over, meaning that it has negative signed volume relative to the ori- Lemma 5.7. It is recursive to find a geometric triangulation 3 entation of Θ∗ and the standard orientation of H . In another of a hyperbolic 3-manifold N which is either closed or has variation, which is often used when N is closed, the inverse torus boundary components, using either of two descriptions map g : Nˆ Θ∗ is defined, and a finite set of closed geodesic of each dihedral angle 0 < α < π of each tetrahedron: curves in N→ˆ collapse to ideal points; but the inverse image of 1. Each imaginary exponential exp i is specified as an any open tetrahedron in Θ∗ is still a geometric tetrahedron in ( α) Nˆ . Suchastructureisa spun triangulation, because a geodesic element of Q¯ . circle C N is approached by cusps of ideal tetrahedra that 2. wind helically⊆ around it. In particular SnapPea uses spun tri- Each angle α is given as a computable real number. angulations. Hence, it is in RE to determine if N is hyperbolic. Ideal geometric triangulations are especially desirable in computational hyperbolicgeometry because they are rigid and Although the second case of Lemma 5.7 immediately fol- algebraically the simplest. However, it is only a conjecture lows from the first one, we will give a separate proof of each that every suitable hyperbolic manifold has an ideal, possibly case. Moreover, even the weaker second case of Lemma 5.7 spun geometric triangulation. Such a structure does always is sufficient to prove Theorem 1.1. exist with degenerate or flipped-over tetrahedra, but these are Remark. Manning [29, Thm. 5.2] also proves Lemma 5.7, but less desirable. We will use finite and semi-ideal tetrahedra in as a corollary of a harder result. His results show (without order to avoid this impasse. The following proposition is then geometrization) that it is recursive to decide whether N is hy- standard: perbolic, when there is an algorithm for the word problem for

Proposition 5.6. If N∗ is a complete hyperbolic manifold π1(N). He also uses a single polyhedral fundamental domain which is either cusped or closed, then it has a geometric trian- to describe the geometry of N. Although this differs from a gulation with finite and semi-ideal tetrahedra (none of which hyperbolic triangulation, which is what we use, the two mod- are spun, degenerate, or flipped over). Also, every semi-ideal els are somewhat interchangeable for our purposes. tetrahedron has only one ideal vertex. Proof of case 1 of Lemma 5.7. Suppose that Θ∗ is a geometric Proof. We can choosea point p N∗ and consider the Voronoi triangulation of N∗. We can model each tetrahedron ∆ Θ∗ tiling of its orbit in H3. Each Voronoi∈ cell is a fundamental (non-uniquely) by choosing four vertices in the Poincar´eu∈ p- domain and yields a cellulation Λ1 of Nˆ . Λ1 is not in general per half-space model, including one on the boundary if ∆ is regular, but it has a barycentric subdivision Λ2 which is regu- semi-ideal. (Note that the ideal vertices of Θ∗ are marked in lar. Moreover, each simplex of Λ2 has at most one vertex of advance.) There is an algebraic formula for each finite edge 10 length ℓ and each dihedral angle α of ∆, if these are repre- converge from an approximate solution x0 to an exact solu- sented by their exponential values exp(ℓ) and exp(iα). The tion x∞. Neuberger [37] points out that an ODE analogue of main matching condition for Θ∗ to be geometric is that if two Newton’s method, which is called the continuous Newton’s tetrahedra share a finite edge, then the edge lengths agree; method, simplifies the Newton-Kantorovich result. and the total dihedral angle around each edge equals 2π. The Theorem 5.8 (Newton-Kantorovich-Neuberger[37, Thm. 2]). first condition is immediately an algebraic condition, although n Let Bε (x0) R be the open ball of radius ε > 0 around x0 note that if an edge is semi-ideal, then it has infinite length and n ⊂ ∈ its length equation is vacuous. R , and let The second condition is almost an algebraic condition since n f : Bε (x0) R the product of the exponentiated angles must be 1; this shows → that the total angle is a multiple of 2π, although not which be aC2-smooth function with non-singular matrix derivative one. However, this can be remedied with additional algebraic Df. Suppose that inequalities, recalling that we are allowed real algebraic equa- 1 tions for the real and imaginary parts cos(α) and sin(α) of (D f (x))− f (x0) < ε (1) each complex variable exp(iα). Suppose that every edge of || || n Θ has at most n incident tetrahedra. Then we can make a for all x Bε (x0), where is the Euclidean norm on R . ∗ ∈ || · || finite covering of the unit circle S1 C by rational rectangles Then there is a unique x∞ Bε (x0) such that f (x∞)= 0. Also, ∈ such that each one has an angular⊆ extent of less than 2π/n. given a solution x∞ such that D f (x∞) is non-singular,equation We can then loop over choices for which rectangle contains (1) eventually holds as x0 x∞, moreover with ε 0. → → each exponentiated angle exp iα . If eachangle is confinedto ( ) Although we will not reprove Theorem 5.8, we can discuss such a rectangle, we can know whether the sum of the angles where the theorem and its proofcome from. Newton’s method around an edge is specifically 2π and not some other multiple to find a root of a univariate function f : (a,b) R begins at of 2π. → an approximate root x0 (a,b) and applies the iteration After forming algebraic equations for all of the geometric ∈ data, the equations have a solution in terms of real algebraic f (xn) numbers when they have a solution at all. For any fixed trian- xn+1 = xn , − f (xn) gulation Θ, we can thus use Theorem 2.7 (not Theorem 2.8; ′ see the remark after the proof) to search for a solution and which in favorable cases converges to a solution x∞ of the eventually find it, if it exists. We must also search over trian- equation f (x)= 0. If f is multivariate as in Theorem 5.8, gulations using Theorem 3.2 or Theorem 3.3. Since the result then this has the well-known matrix generalization is a nested infinite search (over triangulations and then candi- 1 date geometric structures), we can apply Proposition 2.5. xn+1 = xn (D f (xn))− f (x). − Finally in the continuous version, we let x 0 x and define Remark. Although an infinite search for a solution to alge- ( )= 0 the ODE braic gluing equations is preposterous in practice, it is good enough for an algorithm in RE. Alternatively, for each tri- 1 x′(t)= (D f (x(t)))− f (x). angulation, we can apply the more difficult Theorem 2.8 to − determine in R if there is a solution. Then in favorable cases the limit Remark. If we allowed geometric triangulations with fully x∞ = lim x(t) ideal edges, then it would not be enough for the sum of the t ∞ → angles around such an edge e to be 2π. Since e goes to it- self under hyperbolic translation as well as rotation, gluing is again a solution to f (x)= 0. together the tetrahedra that contain e could create a non-trivial Remark. Although Neuberger’spaper on the continuous New- translational holonomy. The two conditions together, that the ton’s method is more recent than Thurston’s work, Kan- total angle is 2π and the translational holonomy vanishes, are torovich’s earlier, more complicated formula also suffices for known as a Neumann-Zagier gluing relation [38]. Lemma 5.7 and Theorem 1.1. Remark. Instead of calculating lengths and angles using po- If the equation f (x)= 0 has a non-singular Jacobian D f in sitions of vertices in hyperbolic geometry, we can also relate a neighborhood of a solution, as in Theorem 5.8, then the sys- them directly using formulas from hyperbolic and spherical tem of equations is also called transverse or first-order rigid. trigonometry. We will need a generalization of this concept. Given a smooth function The separate proof of the second case of Lemma 5.7 works directly with computable numbers, in effect using numerical U Rn f : U Rm, analysis to calculate better and better approximate solutions. ⊆ → In this approach, we need a criterion to know that an approx- where n and m need not be equal, if D f has constant rank imate solution is close to an exact one. Given a smooth mul- kr, then the image f (U) is a manifold and f is a submersion tivariate equation f (x)= 0, the Newton-Kantorovich theorem onto its image. In this case the equation f (x)= 0 is first-order 1 [21] establishes a sufficient criterion for Newton’s method to rigid except for the directions parallel to the manifold f − (0). 11

By the implicit function theorem, we can discard some set of Proof. We can search through triangulations until we find one n rk coordinates in the domain and project to some set of that is a barycentric subdivision of a cellulation by triangular k −coordinates in the target to achieve unconditional first-order prisms. It is then easy to check whether the prisms fit together rigidity that satisfies the hypotheses of Theorem 5.8. following the rules in Section 5.2. To establish first-order rigidity in our case, we will need a corollary of the Calabi-Weil rigidity theorem. Finally, a torus T that has matching Seifert-fibered structure on both sides is not needed and is not a JSJ torus. It is easy Theorem 5.9 (Calabi-Weil [22, Sec. 8.10]). If N is a closed, to see this case in the proof of Theorem 5.2. The more subtle hyperbolic 3-manifold, then the induced representation of its possibility is that one or both sides might have more than one fundamental group, Seifert fibration. Fortunately this is rare for Seifert-fibered 3 manifolds with boundary. It is addressed by the following ρ : π1(N) Isom(H ), → result. is first-order rigid except for conjugacy. (I.e., it is infinites- Theorem 5.12 (Waldhausen [16, Thm. VI.17 & Lem. VI.19]). imally rigid at the level of the first derivative.) The same is Let N be an oriented 3-manifold with non-empty boundary ∂N true if N is cusped, among representations that are parabolic and which has at least one Seifert fibration. Then the fibration at the torus cusps. is uniquely determined up to isotopy by its restriction to ∂N, Corollary 5.10 (Stated by Izmestiev [15, Sec. 1.3]). If Θ is and is outright unique except the following cases: a geometric triangulation of a closed or cusped hyperbolic 3- 1. If N is a solid torus D2 S1, then every fibration of ∂N manifold N∗, then it is first-order rigid except for motion of extends to a fibration of× N over a disk D2 with at most the non-ideal vertices. one exceptional fiber. Since we could not find a proof of Corollary 5.10 in the 2. If N is a thickened torus S1 S1 I, then every fibration literature, we provide one in Section 5.4. is a trivial circle bundle over× an× annulus. There is such 1 Proof of case 2 of Lemma 5.7. We fix the model of each tetra- a fibration for every rational slope in a single torus S 1 × hedron in the upper half space model so that it has exactly six S . degrees of freedom, or five if one of the vertices is ideal. Af- 3. If N is a twisted I-bundle K2 ⋉ I over a Klein bottle K2, ter ordering the vertices v1,v2,v3,v4, we can put vertex v1 at then it has two non-isotopic fibrations. One fibration is (0,0,1); we can put vertex v2 directly below it (or at (0,0,0), over a Mobius¨ strip with Seifert data S1 ⋉ I , and one allowing it to be the ideal vertex); and we can put v3 at a posi- is over a disk D2 with Seifert data D{2,(2,1)},(2,1) . tion of the form (a,0,b). We approximate the positions of the { } vertices approximately with rational numbers. We can then Proof of Theorem 5.4. We search over triangulations Θ of M approximate the lengths and angles of each tetrahedron in the using stellar or bistellar moves, and decorations of them, same form, as well as the first and second derivatives of the to find an adapted triangulation as described in Section 5.2. lengths and angles as a function of the main variables, the A suitable decoration consists of distinguished spheres and separate positions of the vertices in the ideal models of the thickened tori, and a reverse barycentric subdivision in each tetrahedra. JSJ component N to make triangular prisms in the Seifert- Suppose that there are n non-ideal vertices. By the implicit fibered case and a combination of once-truncated and ordi- function theorem, any exact solution to the gluing equations nary tetrahedra in the hyperbolic case. Within this search, we for the tetrahedra can be perturbed so that some 3n of the co- search for geometric data to describe the hyperbolic structure ordinates are exactly rational. Also by the implicit function of each N which is not Seifert-fibered. Since these are nested, theorem, some 3n of the angle conditions are implied by the infinite searches, we combine them using the RE search algo- other angle conditions and can be omitted. Finally, the fixed rithm of Proposition 2.5. By the geometrization theorem, we coordinates and omitted angle conditions can be chosen so will eventually find a Θ that fits the description of Section 5.2. that the remaining system of constraints, which we can write We examine the JSJ components to verify that all of the abstractly as f (x)= 0, has a non-singular Jacobian D f . spheres and tori are essential and that no two are parallel. We Moreover, the mapping f is real analytic with an explicit veto Θ if it includes a Seifert-fibered solid torus (by checking formula. Thus, given an approximate solution x0 which is that the base orbifold is a disk with at most one exceptional within ε of a true solution and ε is small enough, we can ma- fiber). If all JSJ components are either hyperbolic or Seifert- 1 jorize (D f (x))− on the ball Bε (x0) using Taylor series, to fibered, and if none are solid tori, then every sphere and torus || || confirm equation (1). is essential and no two spheres are parallel. Two distinct tori are parallel if and only if the component between them is a As Lemma 5.7 addresses the hyperbolic case of Theo- thickened torus; we veto this as well. We also need to veto a rem 5.4, we turn to a lemma and a proposition that address Seifert-fibered component homeomorphic to P3 P3, which is the Seifert-fibered case. # the only Seifert-fibered space that is not a prime 3-manifold. Lemma 5.11. It is recursive to find an adapted triangulation This space has Seifert data P2,0 . of a Seifert-fibered manifold N which is either closed or has Finally, we need to check{ that all} of the tori are JSJ tori. We torus boundary components. Hence, it is in RE to determine veto Θ if there is a torus T that has Seifert-fibered components if it is Seifert-fibered. on both sides that: 12

1. restrict to the same fibration of T ; or triangulation, then we make Θ by truncating the ideal vertices of Θ∗. We then want to make a cocycle α from γ. To do this, 2. could be refibered to restrict to the same fibration of T . ˜ H3 we first choose a specific isometry N∗ ∼= . Then we choose Case 1 is easy to recognize. By Theorem 5.12 and the com- an orthonormal tangent frame at each vertex of Θ. Given an edge e Θ, we let α(e) be the element of G = Isom+(H3) ments after, in case 2 we only have to consider two types of ∈ Seifert-fibered components, which can be recognized explic- that takes the tailv ˜ ofalifte ˜ totheheadw ˜, and takes the lifted itly from any of their fibrations: frame ofv ˜ to the lifted frame ofw ˜. If N∗ is cusped, then we require that each truncation edge in N is assigned a parabolic 1 1 2a. N ∼= S S I, or element that fixes the corresponding ideal vertex in N∗. × × H1 N 2 In this setting, Theorem 5.9 says that ( ;g)= 0 in the ⋉ 1 2b. N ∼= K I. closed case and H (N,∂N;g,p)= 0 in the cusped case, where Case 2a is only possible if N is glued to itself to make W a g is the parabolic Lie subalgebra of g. The theorem is typi- torus bundle over a circle, S1 ⋉ (S1 S1), because we do not cally proved using de Rham cohomology rather than cellular allow parallel tori. The torus is needed× if and only if W is a cohomology, but these models of cohomology are isomorphic Sol manifold. We can verify this case by confirming that the as usual. More explicitly, every 1-cocycle α′ = δβ, where β holonomy matrix in SL(2,Z) has distinct, real eigenvalues. is an g-valued 0-cochain on the vertices of Θ. 2 ⋉ Finally, suppose that γ′ is a first-order deformation of the In case 2b, N ∼= K I only has one torus boundary com- ponent T , so its refibration does not affect any other torus. In hyperbolic structure γ of Θ∗ that satisfies the first derivative this case the fibration of N may have Seifert data S1 ⋉ I or of the gluing equations. Then we can lift γ′ to a cocycle α′ D2,(2,1),(2,1) . The resulting binary choice may{ occur} on (non-uniquely) in the same way that γ lifts to α. Then Theo- one{ or both sides} of T, and we veto Θ if the Seifert fibrations rem 5.9 provides β, and β descends to a first-order motion of extend across T for any of these choices. the vertices of Θ∗ that induces the deformation γ′.

5.4. Proof of Corollary 5.10 6. HOMEOMORPHISM IS RECURSIVE

The idea of the proof is that we can convert a first-order The goal of this section is to prove Theorem 1.1 in this sec- tion, postponing only the proof of Theorem 6.1 below until deformation of a triangulation of N∗ to a deformation of a representation of ρ, in much the same way thatwe can convert Section 7. a triangulation to ρ in the first place. Proof. In general, if Γ is a discrete group (such as the fun- 6.1. Connected sums damental group of a topological space) and G is a Lie group, then we can describe a first-order deformation of a homomor- If M1 and M2 are two closed, oriented 3-manifolds given by phism ρ : Γ G as a homomorphism triangulations, then by Theorem 5.4, we know the direct sum → decompositions of each one into prime 3-manifolds. These (ρ,ρ′) : Γ G ⋉ g. → summands can be freely permuted and can only be matched in finitely many ways. If we search over the ways to match Here g is the Lie algebra of G viewed as a group under addi- them, we then reduce the oriented homeomorphism problem tion, while G ⋉ g is the semidirect product in which the non- ? ? normal subgroup G acts on the normal subgroup g by conju- M1 ∼= M2 to the oriented homeomorphism problem W1 ∼= W2 gation. Also, (ρ,ρ′) should reduce to ρ under the quotient for prime summands W1 and W2. To review, each summand Wk map inherits an orientation from its parent Mk; in the reverse direc- tion, there is no ambiguity in forming an oriented connected π : G ⋉ g G. sum. → Note that G ⋉ g is also the total space of the tangent bundle TG. In other words, the extension ρ′ is a choice of a tan- 6.2. One JSJ component gent vector ρ (g) T G for every g Γ, such that the pairs ′ ∈ ρ(g) ∈ (ρ(g),ρ′(g)) together make a group homomorphism. We switch to the other end of geometric decompositions to Suppose that Γ = Γ1(X) is the fundamentalgroupof a based analyze a single pair of JSJ components N1 W1 M1 and CW complex X. Then we can model ρ (non-uniquely) as a ⊆ ⊆ N2 W2 M2. We are interested not only in the isomorphism non-commutative cellular cocycle α C1(X;G). Given ρ, we ⊆ ⊆ ∈ problem, but also in the effect of the mapping class group of can likewise model the extension ρ′ (also non-uniquely) as a 1 a component N on the boundary ∂N. commutative cocycle α′ C (X;g), where here g is a coeffi- cient system twisted by α∈. Theorem 6.1. Suppose that N is an oriented, hyperbolic JSJ Now let X = N, where N has a cellulation Θ that comes summand such that N∗ is either closed or cusped. Then the from a closed or cusped hyperbolic structure on N∗ and a geo- mapping class group of N is its isometry group. It is a finite metric triangulation Θ∗. If N is cusped and Θ∗ is a semi-ideal group and its computation is recursive. If N1 and N2 are two 13 such manifolds, then they are homeomorphicif and only if they 2. A Seifert-fibered space with base P2 and at most one are isometric, and recognizing this condition is recursive. exceptional fiber is either a lens space L(4,n), a prism space R(m,n),orP3 #P3. Again, we will prove Theorem 6.1 in Section 7. Note that if N is hyperbolic and has torus boundary components, then 3. The space with Seifert data each such component inherits a Euclidean structure from the 2 hyperbolic structure on N∗. S ,b,(2,1),(2,1),(a1,b1) Suppose instead that N is Seifert-fibered (and, as usual, ori- { } ented). Then in the direct sense the automorphism problem is a prism space R(m,n). only matters for Theorem 1.1 when the JSJ graph is non-trivial 4. The twisted bundle K2 ⋉S1 which is the double of K2 ⋉I and thus N has boundary. However, we will learn the relevant has the double of its two fibrations, namely the Seifert automorphism properties from an associated closed Seifert- data K2,0 and the Seifert data fibered space. { } Lemma 6.2. Let N be a closed, oriented 3-manifold which is S2,0,(2,1),(2,1),(2,1),(2,1) . { } decorated with a Seifert fibration with Seifert data

F,b,(a1,b1),(a2,b2),...,(an,bn) . Theorem 6.3 comes with simple formulas for which lens { } space or prism space is obtained, which we omit. In particular, Then: the answer is recursive and (as we will later want) elementary recursive. 1. The exceptional fibers of N are freely permutable by au- tomorphisms of the Seifert fibration, provided that the permutation preserves the orbifold number ak 2 and ≥ 6.3. The JSJ graph the residue bk Z/ak of each exceptional fiber. ∈ 2. Any finite set of regular fibers is freely permutable. If W is a prime 3-manifold, then its JSJ decomposition is modelled by a labelled graph Γ, whose vertices represent JSJ 3. If the base F is orientable, then N has a homeomor- components and whose edges represent connecting tori. Each phism that inverts all fibers together, but they cannot be vertex is labelled by the homeomorphism type of its compo- inverted separately. nent, which is either Seifert-fibered or hyperbolic. In addition, each edge is decorated with gluing data and peripheral data 4. If the base F is non-orientable, then given two disjoint which will be described precisely in the proof of Theorem 1.1 finite sets A B F, N has a homeomorphism that in- , below. verts the fibers⊆ over A in place and fixes the fibers over B. Remark. This graph structure inspired the term graph man- ifold for a prime 3-manifold whose JSJ components are all Seifert-fibered [51]. This terminology is standard but ironic, Proof. Cases 1, 2, and 4 can all be established by isotopies of since geometrization shows that the same graph concept is im- F that move points that correspond to the distinguished fibers. portant for all prime 3-manifolds. In case 4, using the hypothesis that F is non-orientable, we can move a point p A around an orientation-reversing loop The labelled graph Γ is an invariant of W , which at first in F that stays away∈ from B and from the rest of A. glance may seem like a complete invariant, provided that the Meanwhile in case 3, the fibration itself is orientable, which homeomorphism problem for each JSJ component is recur- means that an orientation of any one fiber induces a canonical sive. However, it is not that simple, because we have to know orientation of all fibers. On the other hand, Seifert’s construc- the allowed permutations of the torus boundary components tion of the fibration via vertical Dehn surgery on F S1 is of a JSJ component N, and the allowed homeomorphisms of invariant with respect to inverting the S1 factor and simulta-× each torus boundarycomponent. Finally, we need to deal with 2 1 1 neously applying an orientation-reversing homeomorphism to the special case that N is either K ⋉ I or S S I and thus × × F. has more than one Seifert fibration. We also need the counterpart to Theorem 5.12 for closed Proof of Theorem 1.1. (Proof using case 1 of Lemma 5.7.) As Seifert-fibered spaces. explained in Section 6.1, it suffices to solve the homeomor- ? Theorem 6.3 (Waldhausen [16, Thm. VI.17]). If N is a phism problem W1 ∼= W2 for prime 3-manifolds W1 and W2. closed, oriented Seifert-fibered 3-manifold, then its Seifert fi- The proof is divided into three steps. In steps 1 and 2, we let bration is unique up to homeomorphism except in the follow- W be a prime 3-manifold and let Γ be its JSJ graph. It is recur- ing cases: sive to calculate Γ and the isomorphism types of its vertices. Step 1: We address the cases in which a JSJ component of 1. A Seifert-fibered space with base S2 and at most two W has more than one Seifert fibration. If any component is exceptional fibers is either a lens space L(m,n),S2 S1, a K2 ⋉ I, then its two fibrations (described in Theorem 5.12) or S3 × are inequivalent; we choose one of them and use it for every 14 occurrence of K2 I in W. If a JSJ component N is a thick- can use the algorithm in step 1 to determine if they are home- ened torus, then as× in the proof of Theorem 5.4, W is a Sol omorphic. manifold and a torus bundle over a circle, S1 ⋉ (S1 S1). In Otherwise we can assume that W and W have non-trivial × 1 2 this case the homeomorphism type of W is given by a pair of JSJ graphs Γ1 and Γ2, and that each JSJ torus has a canoni- conjugacy classes in SL(2,Z), one for each orientation of the cal decoration as described in step 2. To determine if W1 and base circle. Recall that the conjugacy classes in SL(2,Z) can W2 are homeomorphic, we search over graph isomorphisms be classified with the aid of the isomorphism f : Γ1 Γ2. For every pair of T1 W1 and T2 W2 that are matched→ by f , we search over mapping⊆ classes⊆ that pre- PSl(2,Z) = C2 C3. serve the canonical decorations of T and T . In the innermost ∼ ∗ 1 2 part of the search, we want to calculate whether the home- If g SL(2,Z) has non-zero trace (which it does if W is Sol), omorphisms of the JSJ tori extends to each matched pair of then∈ its conjugacyclass is given by the sign of its trace and its JSJ components N1 W1 and N2 W2. If N1 and N2 are hy- reduced cyclic word in C2 C3. perbolic, then we can⊆ use Theorem⊆ 6.1 to determine if the Step 2: We suppose that∗ W is not a Sol torus bundle over homeomorphism ∂N1 ∼= ∂N2 extends. If they are both Seifert- a circle. If T is a JSJ torus in W and one side of T is a hy- fibered, then we can use Lemma 6.2 to determine whether perbolic component N, then T inherits a Euclidean structure the corresponding closed Seifert-fibered manifolds Nˆ1 and Nˆ2 which we can normalize to have area 1. This Euclidean struc- have a homeomorphism that extends the given homeomor- ture can be described by a quadratic form Q on the first ho- phism ∂N1 ∼= ∂N2. Note that we can employ Lemma 6.2 mology H1(T )= H1(T ;Z), where Q(c) is the square of the because any relevant homeomorphism N1 ∼= N2 preserves the minimum length of c H1(T ). Moreover, the coefficients of ∈ fibration at the boundary, and is thus isotopic to a fibration- Q are real algebraic numbers computable from the geometry preserving homeomorphism by Theorem 5.12. of N. On the other hand, if N is Seifert-fibered, then the in- duced fibration of T selects a line in H1(T ), bywhichwemean Proof of Theorem 1.1. (Proof using case 2 of Lemma 5.7.) If a rank-one subgroup L H1(T ) with a torsion-free quotient the geometric data of each hyperbolic JSJ component N of a ⊆ H1(T )/L. summandW is described with computable real numbers rather Since T has two sides, it is then decorated by a pair of than real algebraic numbers, then the induced Euclidean struc- quadratic forms on H1(T ), or a quadratic form and a line, or ture on a JSJ torus T ∂N is only given by a convergent se- a pair of lines. In the third case when both sides of T are quence of approximations.⊆ Thus, it is not possible to defini- Seifert-fibered, the fibrations must be mismatched at T , so the tively calculate the isometries of T or the shortest cycles, as two lines L1,L2 H1(T ) must be distinct. Hence they consti- expressed with the quadratic form Q(c). However, all non- ⊆ tute a rational line basis in the sense that isometries and all non-zero classes in H1(T ) that are not short- est are eventually revealed. This yields an algorithm in coRE H1(T ;Q) = (L1 Q) (L2 Q). ? ⊗ ⊕ ⊗ for the homeomorphism problem M1 ∼= M2, which is enough We also obtain a rational line basis in the second case, when to show that the problem is recursive per the discussion at the one side is Seifert-fibered and produces a line L1 = L, and the beginning of Section 7. other side is hyperbolic and produces a quadratic form Q. In this case, there exist a finite set of pairs of homology classes c H1(T ) L1 that minimize Q(c). If there is only one such 7. PROOFS OF THEOREM 6.1 ± ∈ \ pair, we let L2 be the line generated by c. If there is more ± then one, we let L2 be the line generated by the first such pair In this section we will give several proofs of Theorem 6.1. in the clockwise direction from L1. Recall that Corollary 3.4 says that the existence of a PL home- Note that each possible decoration of T induced by the ge- RE omorphism N1 ∼= N2 is in ; it is also easy to check whether ometry on both of its sides has a finite stabilizer in the ori- it preserves orientation. So, by Proposition 2.3, it suffices to Z ented mapping class group SL(H1(T )) ∼= SL(2, ) of T . Ifwe show that homeomorphism is in coRE, although only one of order the two sides of T , then the stabilizer usually has two the proofs will make use of this directly. By a similar argu- elements; in rare cases it is a cyclic group of order 4 or 6. ment, finding elements in the mapping class group of a single If N is a Seifert-fibered component of W , then each of its N is in RE; the remaining task is an algorithm to show that the torus boundary components is decorated by a rational line ba- list is complete. sis. We can thus make a closed Seifert-fibered space Nˆ by Recall that if N has boundary, then its interior N∗ is cusped collapsing a circle fibration of each component T ∂N that ⊆ and has a semi-ideal triangulation Θ∗. In this case, Θ is a cel- represents the opposite line in H1(T ), the one that does not lulation in which semi-ideal tetrahedra are once truncated. We come from N itself. Each torus component of ∂N becomes a want to geometrize the truncation that produces Θ. We con- distinguished fiber in Nˆ which may be either regular or excep- sider a horospheric truncation which is almost but not quite tional. unique, with the following three properties: Step 3: Suppose that W1 and W2 are two prime, closed, ori- ented 3-manifolds. If they do not have any JSJ tori, then they 1. The horosphere sections lie entirely within the semi- are both Seifert-fibered, and we can use Theorem 6.3 to tell ideal tetrahedra of Θ∗, and therefore do not intersect if they are the same. If they are both Sol torus bundles, we each other. 15

2. For some common integer n, every horospheric torus We can obtain an upper bound ℓ on the diameter of all of N by n has area 2− . adding bounds on the diameters of the cells in Θ. Then, we let D be a convex fundamental domain for N; it has the same 3. We do not use the smallest value of n that satisfies con- volume and diameter at most 2ℓ. Thus we obtain an estimate ditions 1 and 2. similar to (2), but more robust: For convenience, we let N = N and Θ = Θ if N is closed. ∗ ∗ Vol(D) c′ Some of the proofs make use of the following lemma. r > > . 2Vol(B(p,ℓ + 1)) 2Vol(B(p,ℓ + 1)) Lemma 7.1. It is recursive to obtain a lower bound in the Remark. Without an explicit bound on least-volume closed or injectivity radius of N and Θ. cusped hyperbolic manifold, the third proof has the unusual First proof. Suppose first that N is closed. For each vertex feature of non-constructively proving that an algorithm exists, v Θ, let Uv be the open star of Θ containing p. Then the i.e., without fully stating the algorithm. Meyerhoff [30] estab- ∈ collection Uv is a finite open coverof N. It follows just from lished the first lower bound { } topology that there is some radius ε such that every ball of 2 radius ε is contained in some U . For an explicit calculation, Vol(N) v ≥ 54 let Θ be a barycentric subdivision of Θ, and for each v Θ, ′ ∈ let Xv be the closed star of v Θ ; then the sets Xv are a closed in the closed case. In the same paper, he and Jørgensen estab- ∈ ′ cover. We can calculate or bound the distance from Xv to N lished \ Uw for some w Θ with Xv Uw. The minimumofallof these distances is thus∈ a lower bound⊆ for the injectivity radius. √3 √3 ε Vol(N∗) = Vol(N) ≥ 4 ⇒ ≥ 8 Second proof. In general we use the notation B(p,r) for a hy- perbolic ball of radius r centered at p. in the cusped case. The exact minimum values are now known Let r be the exact injectivity radius of N, andlet p be a point [9]. on a closed geodesic of N of length 2r. Then p ∆ for some ∈ First proof of Theorem 6.1. This proof is similar to one given cell ∆ Θ, andwe canlet ℓ be an upper bound of the diameter by Scott and Short [43]. We assume geometric triangulations of ∆. Then∈ in the universal cover Θ1∗ and Θ2∗ of N1∗ and N2∗. 3 If N1∗ and N2∗ are homeomorphic and therefore isometric, N˜ N˜ ∗ = H , ⊆ ∼ then we can intersect the tetrahedra of Θ1∗ and Θ2∗ to make a tiling of N N by various convex cells with 8 or fewer sides; we obtain that at least 1/2r lifts of ∆ intersect B(p,1), and 1 ∼= 2 we can then take a barycentric subdivision to make tetrahedra. thus at least this many copies of ∆ are contained in B(p,ℓ+1). We thus obtain a mutual refinement Θ of Θ and Θ . Ifwe Thus 3 1 2 can bound the complexity of Θ3, then we can find it with a 1 Vol(B(p,ℓ + 1)) finite search or show that it does not exist, rather than using , 2r ≤ Vol(∆) stellar or bistellar moves in both the up and down directions. Let ∆1 Θ1∗ and ∆2 Θ2∗ be two tetrahedra in the separate ∈ ∈ ˜ hence triangulations. In the universal cover N1∗, they can only inter- sect in a single cell with at most 8 sides. In N∗ itself they can Vol(∆) 1 r > . (2) intersect many times; however, only as often as different lifts 2Vol(B(p,ℓ + 1)) of ∆1 intersect one fixed lift of ∆2. If ∆1 and/or ∆2 are semi- ideal, then their lifts intersect if and only if their truncations We can calculate an upper bound of this form, if necessary do. There is a recursive volume bound on the number of pos- using a lower bound for the numerator and an upper bound sible intersections by the same argument as the second proof for the denominator, for every cell in Θ, since we do not know of Lemma 7.1. the position of the shortest geodesic loop in advance. Having bounded the necessary complexity of a mutual re- Third proof. This proof is a variation of the second proof us- finement Θ3, we can now search over separate refinements Θ3 ing the entire diameter and volume of N. Jørgensen and of Θ1 and Θ4 of Θ2 using Proposition 3.5, and look for an orientating-preserving simplicial isomorphism Θ3 = Θ4. The Thurston provedthat the set of possible volumes of N∗ is well- ∼ ordered. In particular, there is one of least volume, so there is same method can be used to calculate the mapping class group some constant c > 0 such that of a single N. Second proof. Suppose that X and X are two compact met- Vol(N ) > c. 1 2 ∗ ric spaces, and suppose that for each ε > 0 we have a way S S X X Our construction of the geometry of N spares more than half to make finite ε-nets 1 and 2 for 1 and 2, and calculate or approximate all distances within S1 and within S2. If X1 of the volume of N∗, so and X2 are isometric, then there is a function f : S1 S2 that → Vol(N ) c changes distances by at most 2ε. On the other hand, if there is Vol(N) > ∗ > = c . 2 2 ′ such a function for every ε, then X1 and X2 must be isometric. 16

+ 3 In our case, we let Xk = Nk, where we make sure to use obtain atlases of charts for N1 and N2 using the Isom (H ) n the same truncation area 2− to geometrize N1 and N2 given pseudogroup. In fact, everything is constructed in the sub- the geometries of N1∗ and N2∗. We calculate a common lower group and sub-pseudogroupwith real algebraic matrix entries. bound δ on the injectivity radius. If there is an isometry between N1 and N2, then their at- We can choose some convenient coordinates inside each lases combine into a larger atlas. There are only finitely many cell ∆ Θk. We then have the ability to calculate geodesic possible patterns of intersection between the balls of N1 and ∈ segments in Nk that are made of geodesic segments in the sep- the balls of N2. For each such pattern, we obtain a finite sys- arate tetrahedra. If ∆ is truncated, then the geodesic segment tem of algebraic equalities and inequalities, which says first might hug the truncation boundary for part of its length, but that the intersection pattern is what is promised, and second it still has a finite description. Without more work, we don’t that the gluing maps between the atlases are consistent. The- know which of these geodesics are shortest geodesics. How- orem 2.8 then says that it is recursive to determine whether ever, if a geodesic is shorter than δ, then it is shortest. Taking this system of equations has a solution. Since we work in + 3 δ ε 0, we can make ε-nets of both N1 and N2 and look the group Isom (H ), we are looking only for orientation- for≫ approximate→ isometries between these ε-nets; it suffices to preserving isometries. check distances below the fixed value δ. More explicitly, we can use the covering by open stars Sv in the first proofof Lemma 7.1. Thereisa δ such that if d(x,y) < 8. HOMEOMORPHISM IS IN ER δ, then x and y and even the connecting short geodesic are all in some open star. We will use the basic fact that a finite composition of ER This algorithm does not by itself ever prove that N1 and N2 functions is in ER. In other words, if an algorithm has a are isometric, only that they aren’t. Thus it shows that the bounded number of stages that expand its data by an expo- homeomorphism problem is in coRE. This is good enough by nential amount or otherwise by an ER amount, then it is still Proposition 2.3 and Corollary 3.4. in ER. The algorithm also does not by itself determine whether the isometry is orientation-preserving. However, this is very lit- tle extra work. Given ε δ and given ε-nets S1 N1 and 8.1. The outer proof ≪ ⊆ S2 N2, we can let p1, p2, p3, p4 be 4 points in S1 that lie ⊆ in a ball of radius δ/2 and that make an approximately regu- In this section we will prove Theorem 1.2. The proof is a lar tetrahedron ∆. If f : S1 S2 is an approximate isometry, → combination of the proof of Theorem 1.1 together with several then we can check whether f flips over ∆. If no orientation- computational improvements. We summarize these computa- preserving isometry exists, then when ε is small enough, ei- tional improvements in this section by stated some supporting ther f will cease to exist or ∆ will be flipped over. theorems which we will prove ourselves (or prove by citation) We can use similar methodsto find the mapping class group with two main supporting tools. The first tool is normal sur- of a single N, since by Mostow rigidity it is also the isometry face theory, which we can use to find essential spheres and group of N. We assume that N has boundary, which is techni- tori and Seifert fibrations. Note that Jaco, Letscher, and Ru- cally short of the full generality of Theorem 6.1, but enoughto binstein [17] sketched a similar approach. The second tool is prove Theorem 1.1. Just as with the method to check whether an ER version of Theorem 2.8 [10], which we useto boundthe and approximate f preserves orientation, we can when ε is complexity of a geometric triangulation of a hyperbolic man- small enough compute the effect of f on H1(∂N), which de- ifold, and the complexity of recognizing small Seifert-fibered termines which isometry is close to f (if any). spaces.

Third proof. In this proof, we restrict attention to case 1 of Theorem 8.1. It is in ER to find and triangulate the prime Qˆ summands W of a closed, oriented 3-manifold M, to find Lemma 5.7 and thus work over the ring of real alge- { } braic numbers. We assume real algebraic coordinates for and triangulate the JSJ components N within each prime { } H3 and for its isometry group Isom+(H3); for example we summandW, to find their JSJ graph Γ, and to recognize which can take H3 to be the set of positive, unit timelike vectors components N are Seifert-fibered and find their fibrations. in 3 + 1-dimensional Minkowski geometry, and we can take We will prove most of Theorem 8.1 in Section 8.2 using Isom+(H3)= ISO(3,1). We again assume that N and N are 1 2 normal surface theory. Note that Jaco, Letscher, and Rubin- made from N and N using a common truncation area 2 n. 1∗ 2∗ − stein [17] sketched ideas that are similar to our proof. We assume geometric triangulations Θ and Θ of N and 1∗ 2∗ 1∗ Theorem 8.1 also has one lingering case which is more dif- N with real algebraic descriptions. Using these triangula- 2∗ ficult. Recall that a Seifert-fibered space is small if it is non- tions, we can find finite, open coverings of N and N by met- 1 2 Haken (and therefore closed). ric balls B(p,ε), where each point p has a real algebraic po- sition and the common radius is (a) also real algebraic, and Theorem 8.2. Recognizing small Seifert-fibered spaces is in (b) less than half of the injectivity radius of N1 and N2. Then ER. we can give each ball the same algebraic coordinates as H3, and we can also calculate the relative position of every pair We will prove Theorem 8.2 in Section 8.4 using a com- of balls as some element in Isom+(H3). In other words, we bination of normal surface theory and algebraic methods. 17

Note that Li [27] shows that recognizing small Seifert-fibered Lemma 8.4 (Haken). The number of elementary disks in a spaces with infinite π1 is recursive, and his algorithm should fundamental surface in M is boundedabove by an exponential be elementary recursive. However, we will use a different ap- in t. (Thus it is elementary recursive.) proach for this part of the theorem. Li also addresses the finite We can represent a normal surface S by listing all triangles π1 case in two different ways. Without assuming geometriza- and quadrilaterals in order in each tetrahedron ∆ Θ. It is tion (which was still open at the time), he cites work of Rubin- ∈ stein and Rannard-Rubinstein on small Seifert-fibered spaces. then easy to separate S into connected components and calcu- late the topology of each component. This is exponentially in- He also outlines a simplified argument for the finite π1 caes that depends on geometrization; we give a detailed argument efficient compared to algorithms such as Agol-Hass-Thurston which is in a similar spirit. [2], but it has no effect on whether the resulting algorithm is in ER. Theorem 8.3. If a compact, oriented 3-manifold N has a We define a complete set of essential 2-spheres in a 3- closed or cusped hyperbolic structure, then it is in ER to manifold M is a collection C such that cutting M along each find a geometric triangulation and specify its geometric data 2-sphere in M and capping off the resulting boundary compo- with algebraic numbers. The isomorphism and automorphism nents produces irreducible 3-manifolds. Likewise a complete problems are also both in ER. set of essential disks is a collection C of properly embedded disks which are not boundary parallel, such that the compres- We will prove Theorem 8.3 in Section 8.5 using both sion of every disk in C renders M boundary-incompressible. Mostow rigidity and methods from algebraic geometry. Theorem 8.5 (Jaco-Tollefson). Let M be a compact, oriented, Proof of Theorem 1.2. We consider each stage of the proof triangulated 3-manifold. Then: of Theorem 1.1 in turn. The proof begins with a geometric 1. M has a collection of disjoint, fundamental surfaces recognition of a single closed, oriented 3-manifold M in The- which form a complete set of essential 2-spheres [18, orem 5.4. This is not elementary recursive as described, but Thm. 5.2]. we can replace it with Theorem 8.1 to find the direct sum and JSJ decomposition. We can then apply Theorem 8.3, which 2. If M has no essential 2-spheres, then it has a collection is an ER version of Lemma 5.7, to calculate the hyperbolic of disjoint, fundamental surfaces which form a complete structure of each hyperbolic summand N. This includes a de- set of essential disks [18, Thm. 6.2]. scription of the Euclidean structure of each torus component 3. of ∂N. If M has no essential 2-spheres or disks, and it has an essential torus, then it has one which is fundamental Finally given two closed, oriented 3-manifolds M , M , 1 2 [18, Cor. 6.8]. we first decompose them into summands. For each bijection ? 4. If M has no essential 2-spheres or disks, and it has an among the summands, we want to calculate W1 ∼= W2 for each pair of matching summands. This is a calculation with JSJ essential annulus, then it has one which is fundamental graphs which is done in Section 6.3 to complete the proof [18, Cor. 6.8]. of Theorem 1.1, and this part is already elementary recur- sive. Actually Jaco and Tollefson show that each type of sur- face described in Theorem 8.5 is a vertex surface, which is a special case of a fundamental surface. Also case 1 of The- 8.2. Normal surfaces orem 8.5 is stated for closed manifolds, but the proof is the same for manifolds with boundary. Finally, given Lemma 8.4, Let M be a compact 3-manifold with triangulation Θ. Re- the surfaces produces by Theorem 8.5 all have an elementary call that a normal surface S M intersects each tetrahedron recursive bound on their size. ∆ Θ in 7 types of elementary⊆ disks, namely 4 types of tri- We will use the following variation of Theorem 8.5. ∈ angles and 3 types of quadrilaterals. The surface S = Sv is Theorem 8.6 (Hass-K. [14]). If M is a closed, oriented 3- 7t given by a vector v Z 0 that lists the number of each type manifold with a triangulation Θ, then it has a collection of ∈ ≥ of elementary disk. If v is such a vector, then Sv is embed- disjoint normal surfaces which form a complete set of essen- ded (and uniquely defined) provided that it only uses at most tial spheres and tori, such that the total number of elementary one type of quadrilateral in each tetrahedron. After specifying disks is bounded above by an exponential in t. which type of quadrilateral is allowed in each tetrahedron, the normal surface equations then have a polytopal cone Briefly, Theorem 8.6 uses a generalization of the normal surface equations which we call the disjoint normal surface 5t 7t C Z 0 Z 0 equations. (They are similar to the crushed triangulation tech- ⊆ ≥ ⊆ ≥ nique defined by Casson [17].) They are equations for a nor- of solutions. We define a fundamental surface Sv is one whose mal surface S which is disjoint from a fixed normal surface vector v C is not the sum of two other solutions in C. If Sv is R M. To prove Theorem 8.6, we find each surface S or T ∈ ⊆ non-orientable, then S2v is its orientable double cover and we sequentially as a fundamental surface, relative to the union of call it fundamental as well. previous surfaces. 18

Theorem 8.7 (Rubinstein [40], Thompson [48]). Recognizing To see that this is an exhaustive list, we recall that if Q the 3-sphere S3 is in ER. is Seifert-fibered with boundary and is atoroidal, then with one exception its orbifold base is planar, and the total num- The proof of Theorem 8.7 uses a variant known as almost ber of boundary circles plus exceptional fibers is at most normal surfaces that are allowed one exceptional intersection 2 ⋉ three. The only exception is that Q ∼= K I has a M¨obius with a tetrahedron that is either an octagon, or a triangle anda strip base; but it also has its other fibration with Seifert data quadrilateral with a connecting annulus. The original papers D2,(2,1),(2,1) . only claim a recursive algorithm, but the algorithm is based { We claim that} we can recognize each possibility for Q by a on normal surface theory. In fact, the proof also uses dis- bounded number of applications of Theorem 8.5; the recogni- joint normal surface equations. Schleimer [42] also refines the tion algorithm is therefore in ER. Indeed, each case reduces to Rubinstein-Thompson algorithm to show that 3-sphere recog- earlier cases when a putative essential surface is found. The NP nition is in the complexity class , which is a much better most subtle case is case 5, where we can check whether a can- ER bound than just . didate torus in Q is essential by checking that it is not com- pressible (case 1) and does not bound a thickened torus (case Proof of Theorem 8.1. As a first step, we check whether M ∼= S3 using Theorem 8.7. If not, we search over collections C 3). Note that if Q is a solid torus (case 1) or a thickened torus of normal surfaces in M with a suitable elementary recursive that is not glued to itself (case 3), or if Q has an essential torus complexity bound in order to find a set of surfaces that meets (case 5), then the collection C in W should be rejected. the conclusion of case 1 of Theorem 8.5. To test whether In case 6, we use Theorem 8.2 to determine if W is small a given collection C is one that we want, we first calculate Seifert-fibered, and if so, its homeomorphism type. whethereach surfacein it is a 2-sphere. Thenwe can cut along If Q is a thickened torus which is glued to itself, then W is a all of the spheres (and retriangulate) and cap them to make a torus bundle over a circle, and we can find its monodromyma- trix A SL(2,Z) with a homology calculation. We can solve multiset of summands of M. For each non-separating sphere, ∈ we create a separate S2 S1 summand. What remains is a the conjugacy problem in SL(2,Z) using the usual trick that × PSl(2,Z) = C3 C2, and thus determine the homeomorphism putative prime factorization W , but we must check whether ∼ ∗ the summands are irreducible{ and} not S3. We can use Theo- type of W . (Note that W may be either Sol, Nil, or Euclidean.) rem 8.7 to check that no summand W is S3. If not, then we Otherwise we can piece together the JSJ decomposition N of W from the (often non-unique) atoroidal decompo- can again use case 1 of Theorem 8.5 to look for an essential { } 2-sphere in W, and again use Theorem 8.7 to check whether it sition Q . In each remaining case Q has either 0 Seifert fi- { } 2 is essential. brations (if it is hyperbolic), or 2 fibrations (if it is K ⋉ I), For each summand W, we similarly search for a collection or 1 fibration (in all other Seifert-fibered cases). Using the C that meets the conclusion of Theorem 8.6. We can check recognition of Q, we can calculate its Seifert data and express the fibration of each boundary torus T ∂Q by the corre- that each surface in C is a torus. We can then cut W along ⊆ sponding line L H1(T ). We can then piece together adja- C to make a putative decomposition of W into atoroidal com- ⊆ ponents Q . We have the following possibilities for each cent fibered components to make JSJ components, when the component{ Q} : fibrations match. The Seifert data produced in this manner is not necessarily canonical, but canonicalizing it is straightfor- 1. Q has an essential disk, which necessarily cuts it into a ward. ball. In this case, Q = S1 D2 is a solid torus. ∼ × 2. Q has no essential disk, but it has a separating essential annulus that cuts it into two solid tori. In this case Q 8.3. Algebraic algorithms fibers over a disk with two exceptional fibers. We list several complexity bounds concerning algebraic 3. Q does not have a separating essential annulus, but it numbers and solutions to algebraic equations. does have a non-separating essential annulus that cuts it into a solid torus. In this case Q fibers over an annulus Theorem 8.8 (Collins, Monk, Vorobiev-Grigoriev, W¨uthrich or a M¨obius strip with at most one exceptional fiber. [10, Thm. 4]). Suppose that a set S Rn is defined by a finite set of polynomial equalities and inequalities⊆ over Q. Then it 4. Q has a separating annulus that cuts it into two thick- is in ER to calculate a representative finite set F Qˆ n, with 1 ⊆ ened tori. In this case Q ∼= S F, where F is a pair of one point p F in each connected component of S. pants. × ∈ Theorem 8.8 is of course an ER version of Theorem 2.8. In 5. Q has an essential torus, specifically an incompressible the statement of the theorem, each element α Qˆ described torus which is not boundary-parallel. by its minimal polynomial a(x) Z[x] and an isolating∈ inter- val α [b,c] that contains exactly∈ one root of a(x). 6. Q = W is closed and has no essential torus. In this case ∈ Q is either hyperbolic or small Seifert-fibered. Lemma 8.9. If α Qˆ is a non-zero complex root of a polyno- 7. Q has boundary, and it has no essential disk, annulus, or mial a(x) Z[x], then∈ there is an ER upper bound on α and torus. In this case, Q is hyperbolic. α 1 . ∈ | | | − | 19

Proof. Let n = dega and write Proof. A function from the generators of Γ to n n matrices × n n 1 forms a representation a(x)= anx + an 1x − + + a1x + a0. − ··· ρ : Γ GL(n,C) We can assume without loss of generality that a0 = 0. If → n 6 α > ∑k ak, then us that anα is larger than the total norm of if and only if the matrix entries satisfy equations that come |all| of the other terms, so| by the| triangle inequality, a(α) = 0. 6 from the relators of the presentation of Γ. We also want This establishes ∑k ak as an upper boundon α . For the lower g I. To this end, we assume another matrix of variables 1 | | ρ( ) = bound, we can observe that β = α− is a root of the polyno- Y and6 impose the condition mial n n 1 n 1 tr(Y(ρ(g) I)) = 1. b(x)= a0x + a1x − + + an 1x + an = x a(x− ). − ··· − We can thus repeat the argument. By hypothesis ρ and Y exist, and Theorem 8.8 then pro- duces an algebraic, elementary recursive solution. By Theo- If x1,...,xn are algebraic numbers, then we say that an rem 8.10, the matrix entries are generated by an integer prim- algebraic number z is an integral primitive element if each itive element z, and by Lemma 8.11, we can replace z by a xk = fk(z) for some integer polynomial fk Z[x]. It is a result residue α Z/p for some prime p computable in ER. We of Galois that every finite set of algebraic numbers∈ has a prim- thus get a modular∈ representation itive element; we are interested in a computationally bounded ρp : Γ GL(n,Z/p) version. → Theorem 8.10 (Koiran [25, Thm. 4]). If x1,...,xn are alge- and a matrix Yp over Z/p again with the same properties. braic numbers, then they have an integer primitive element z Since Yp exists, ρp(g) cannot be the identity. which can be computed in ER, and such that polynomials fk Remark. Assuming the Generalized Riemann Hypothesis, with fk(z)= xk can also be computed in ER. Koiran’s work implies Theorem 8.12 with a much better Koiran states the result in the form xk = fk(z)/ak, where the bound, namely that log(p) can be bounded by a polynomial denominator ak is an elementary recursive integer. However, in n, the length of the presentation of Γ, and the length of the it is not difficult to modify z to eliminate these denominators. word w. (Proof: Let f Z[x] be the minimal polynomial of z and let ∈ z z1 = . 8.4. The small Seifert-fibered case f (0)∏k ak

Then both 1/ak and fk(z) are expressible as integer polynomi- In this section we will prove Theorem 8.2. als in z1. Therefore, so is their product.) Let N be a closed, oriented 3-manifold which has been rec- Lemma 8.11. If h Z[x] is an integer polynomial, then there ognized as irreducible and atoroidal by part of the algorithm is a prime p that can∈ be computed in ER such that h(x) has a in the proof of Theorem 8.1. We want to distinguish between root in Z/p. the case that N is small Seifert-fibered and the case that N is hyperbolic; and in the former case, find its homeomorphism Proof. If d = degh, then h attains the values 1 at most 2d ± type. By Theorem 8.7, we also may as well assume that times. Therefore there is an integer a with a d such that N == S3. We divide the proof into two cases, according to h(a) is not 1, and we can let p be a prime| divisor|≤ of h(a). 6 ∼ ± whether π1(N) is finite or infinite.

Proposition 8.13. It is in ER to determine if π1(N) is finite Using the results so far in this section, we obtain an ele- and compute its oriented homeomorphism type. mentary recursive version of Mal’cev’s theorem, which says that finitely generated residually linear groups are residually Proof. We first calculate whether N is a lens space. We cal- finite. Our computational version requires a finitely present culate H1(N) by applying the Smith normal form algorithm to its chain complex. If H (N) is infinite, then N is not small Theorem 8.12. Let Γ be a finitely presented group, let g 1 ∈ Seifert-fibered. Otherwise the cardinality of H1(N) is elemen- Γ 1 be a non-trivial element given by a word w in the gen- tary recursive. We can calculate whether N is a lens space erators\{ } of Γ, and suppose that there is a representation by checking whether H1(N) = Z/m is cyclic and calculat- ∼ 3 ρ : Γ GL(n,C) ing whether its abelian universal cover N˜ is isomorphic to S . → To determine the parameter n in the homeomorphism type of that distinguishes g from the identity. Then Γ admits a finite N ∼= L(m,n), we can calculate the Reidemeister torsion of the representation twisted homology of N over the ring Z[ζ], where ζ is an mth root of unity. This is a determinant calculation which is a pri- ρp : Γ GL(n,Z/p) → ori elementary recursive. Note that this torsion determines the that distinguishes g from the identity, where p is a prime num- oriented homeomorphism type of N. ber. Moreover, we can find such a p and ρp in ER given the If N is spherical but not a lens space, then it has a finite presentation of Γ, the word w, and the integer n. covering space of order at most 12 which is a lens space. We 20 thus obtain an elementary recursive bound on the cardinality of π1(N). Using a presentation of π1(N) obtained from the tri- angulation of N, we can search exhaustively among surjective homomorphisms φ : π1(N) Γ, where Γ is a finite candi- → date for π1(N). For each such surjective homomorphism, we can build the corresponding covering space N˜ and calculate Θ ˜ 3 3 whether N ∼= S . If this happens, then we know that N ∼= S /Γ as unoriented 3-manifolds. Finally in the spherical case, we want to pass from the un- Φ oriented to the oriented homeomorphism type of N when N is spherical but not lens. (Note that every such N is chi- ral.) As a first warm-up, recall that we can distinguish the Figure 3. Isotopy of a triangulation Θ of F˜ (initially in blue) so that lens space L(4,1) from its reverse L(4, 1)= L(4,3) by com- in its new position (in red), its edges are parallel to a π1(N)-invariant puting its Reidemeister torsion. As a− second warm-up, we triangulation Φ. consider the simplest prism space R(1,2) whose fundamental group Γ = π1(R(1,2)) is the quaternionic 8-element group. We can build R(1,2) as a coset space inside SU(2): and ∆ Θ is any triangle that has p as a vertex, then the gluing maps between∈ charts glue the circle over p in such a way that Γ SU(2) R(1,2) = SU(2)/Γ. it shortens by a factor of a and becomes a singular fiber of N. ⊆ ∼ The group Γ has four cyclic subgroups of order 4 which Proposition 8.14. It is in ER to determine if N is small Seifert- are not conjugate in Γ itself, but which are conjugate in fibered with infinite π1(N), and if so, compute its oriented SU(2). Matching this calculation to the Seifert data, R(1,2) homeomorphism type. has three double covers which are all-oriented homeomorphic Proof. If π1(N) is infinite and N is small Seifert-fibered, then to L(4,1). We can thus calculate the orientation of N by cal- the base F of N is a 2-sphere with orbifold points of order culating whether any double cover is L(4,1) or L(4,3). a1 a2 a3 with In general if N is spherical but not lens, then the quotient ≥ ≥ 1 1 1 group π1(N)/Z(π1(N)) is either a dihedral group, which is + + 1. the prism case; or the isometry group of a Platonic solid: a1 a2 a3 ≤ tetrahedral, octahedral, or icosahedral. In the case that N ∼= In the equality case F is Euclidean and N is either Euclidean or R(m,n) and m is odd, as well as in the Platonic cases, π1(N) Nil; otherwise F is hyperbolic and the geometry of N is either has a unique subgroup isomorphic to the quaternionic group. H2 R or Isom^(H2). The orbifold fundamental group π (F) Thus we can form the corresponding covering space N˜ = 1 ∼ is a× von Dyck group D(a ,a ,a ) which is the orientation- R(1,2) and calculate its orientation as in the second warmup. 1 2 3 preserving subgroup of index two in the corresponding trian- Mean while if N = R(m,n) and m is even, then the center ∼ gle group ∆(a ,a ,a ) in either Isom(E2) or Isom(H2), and Z(π (N)) = Z/(2m) has a unique subgroup of order 4. Thus 1 2 3 1 ∼ π (N) is a central extension of D(a ,a ,a ) by Z. π (N) has a canonically chosen cyclic subgroup of order 4, 1 1 2 3 1 We first consider the case in which F is Euclidean, which and we can again form N˜ and calculate whether it is L(4,1) or implies that (a ,a ,a ) is either (3,3,3), (4,4,2), or (6,3,2). L(4,3). 1 2 3 In this case there is a homomorphism To prove Proposition 8.14, we will use a different combi- φ : π1(N) G, natorial model of Seifert-fibered spaces than the one in Sec- → tion 5.2.4. If N is Seifert-fibered with base F, then we can where the target group is respectively the dihedral group D3, consider a triangulation Θ of F with the orbifold points placed D4, or D6, such that the corresponding regular cover N˜ is a at the vertices. For each triangle ∆ Θ, we make a solid torus circle bundle over a torus. Given a putative choice for G and ∆ S1 which we interpret as a chart∈ for a circle bundle with φ, we can construct the regular cover N˜ and apply the large structure× group S1. Then we can construct N with an atlas of Seifert-fibered case of Theorem 8.1 to recognize it. If N˜ is in- charts of this type. (It is an atlas with closed charts rather than deed a circle bundle overa torus, then we know that N must be open charts, but this is valid in context.) When two triangles small Seifert-fibered with a Euclidean base, and the remaining ∆1 and ∆2 intersect in an edge, we glue the charts together question is to confirm that G and φ were correctly chosen and with a transition map thus compute the specific Seifert data of N. Either N˜ is S1 S1 S1 (so that N itself is Euclidean), or it 1 R Z is a circle bundle× over×S1 S1 with a non-trivial Euler number f12 : ∆1 ∆2 S ∼= / . ∩ → and thus has Nil geometry.× We thus obtain an explicit form 3 2 We can assume that each transition map f12 is affine-linear if of π1(N˜ ) which is either Z or a central extension of Z by Z. lifted to R, so that the value of f12 lies in Q/Z, and its slop is At the same time, since the recognition of the structure of N˜ also in Q. Morevoer, it is not hard to convert the Seifert data is based on normal surface theory, it thus yields a retriangula- for N into these transition functions, for any triangulation of tion of N˜ in ER from the one triangulation induced by the in- F. Finally, note that if p F is an orbifold point of order a put triangulation of N to one that reveals the Seifert structure. ∈ 21

Therefore we obtain an explicit (and elementary recursive) de- 8.5. The hyperbolic case scription of π1(N) as an extension of the finite group G by π1(N). We can thus match π1(N) to the corresponding model To prove Theorem 8.3, we will need a quick mutual corol- Seifert-fibered space to determine the unoriented homeomor- lary of Theorem 8.8 and the proof of Proposition 3.5. phism type of N. Finally we have to calculate the oriented homeomorphism type. In the Euclidean case, the recognition Corollary 8.15. If Θ1 is a finite simplicial complex with n1 3 of N˜ gives us an orientation of π1(N˜ ) = Z . Similarly in the simplices (of arbitrary dimension) and n2 n1, then it is in ∼ 2 ≥ Nil case, when π1(N˜ ) is an extension of Z by Z, the orienta- ER to produce a complete list of geometric subdivisions Θ2 of tion of N˜ still gives an orientation of both the center Z and the Θ1 with n2 simplices. quotient Z2, up to switching both orientations. This fixes an orientation of N itself. Proof of Theorem 8.3. Let Θ be the input triangulation of N as a compact manifold, and let Θ∗ be the result of adding a cone The argument when the base F is hyperbolic is similar to to each component of ∂N to make a semi-ideal combinatorial the Euclidean case but more complicated. In this case, π1(N) triangulation of N∗. The manifold N∗ also has a hyperbolic has a non-trivial homomorphism to Isom(H2), which in turn structure which we interpret as a separate manifold. We re- embeds in SL(2,C), so Theorem 8.12 tells us that π1(N) has name the hyperbolic version X and assume a homeomorphism a non-trivial finite quotient G which we can find in ER even if f : N∗ X. we do not know the linear representation that explains that it → exists.

We construct the finite cover N˜ of N corresponding to the 1 1 quotient map φ : π1(N) G, and we apply part of Theo- 2 5 5 2 rem 8.1 to determine if N→˜ is large Seifert-fibered, and if so f calculate its fibration and its base F˜. As before, we can first learn from this whether N is indeed small Seifert-fibered. Sec- ond, as before this part of Theorem 8.1 gives us an ER retri- 3 4 3 4 ˜ angulation from the initial triangulation of N as a covering Θ∗ space of N, to a triangulation that reflects its Seifert fibration. In particular, we obtain a triangulation Θ of F˜ together with an atlas of charts to describe N˜ . Third, we can choose an ori- 1 1 entation of F˜ and an orientation of the circle fibers so that the 2 5 5 2 two orientations together are consistent with the orientation of g N˜ inherited from N. Fourth, using the orbifold point orders of F˜ and the cardinality of G, we obtain an ER upper bound on a1, a2, and a3. 3 4 3 4 Θ∗ Λ For each candidate for (a1,a2,a3), the given representation 3 of D(a1,a2,a3) in Isom(H ) is rigid. It is still rigid even as a representation of π1(N), because any nearby representation 1 1 must still annihilate the kernel Z(π1(N)). Passing to the cov- 2 5 5 2 1 ering space F˜, we obtain a preferred hyperbolic structure on g− F˜ and we can realize Θ as a geometric triangulation. (In two dimensions, every triangulation of a hyperbolic surface is ge- ometric.) Now Theorem 8.8, combined with the fact that the 3 4 3 4 retriangulation of N˜ is in ER, gives us an ER upper bound on Ψ Φ the lengths of the edges of Θ. At the same time, we get a second (possibly generalized) triangulation Φ of F˜ by tiling it Figure 4. We straighten f to g, then subdivide the image, and finally by lifts of the triangular fundamental domain of the triangle subdivide the domain to make g a simplicial map. In the proof the group ∆(a1,a2,a3); this triangulation is both geometric and image is barycentrically subdivided, but any refinement which is a π1(N)-invariant. As illustrated in Figure 3, we can isotop Θ triangulation suffices. so that its edges run parallel to Φ. (Since this isotopy cannot introduce crossings, and since two edges of Θ might follow We fix the vertices of N∗ in the map f , and straighten all some of the same edges of Φ as illustrated, they cannot land of the tetrahedra, to make a map g that represents Θ∗ as a exactly onto these edges.) This isotopy shows that Φ and Θ self-intersecting geometric triangulation of X. Since g is ho- have a mutual refinement that can be found in ER. Thus we motopic to f (or properly homotopic of N∗ is not compact), can search over retriangulations of F˜ in ER until we find one it has (proper) degree 1. The simplices of g may be flat or that is π1(N)-invariant. We can then use this to compute the flipped over, but g has degree 1 since it is homotopic to f . Seifert structure on N, moreover preserving the orientation in- The self-intersections of g(Θ∗) yield a cellulation Λ of X with formation inherited from the fibration of N˜ . convex cells. Thus Λ has a barycentric subdivision Φ which 22

1 is a geometric triangulation of X. Also let Ψ = g− (Φ). Then can calculate an ER bound for its data complexity, using the Ψ is a refinement of the triangulation Θ∗, and g is now a sim- complexity bounds in the statement of Theorem 8.8. plicial map from Ψ to Φ. See Figure 4. (The figure uses a In particular, the existence of the map g gives us ER bounds simplicial refinement of the self-intersections which is sim- on the parametersused in the third proofof Lemma 7.1. Using pler than barycentric subdivision; this is not important for the Lemma 8.9, the existence of g yields an ER upper bound on proof.) the diameter ℓ of X ′ and then a lower bound on its injectivity Now suppose that we do not know the hyperbolic structure radius r. of N, only that it must have one because it prime, atoroidal, We can now follow the first proof of Theorem 6.1. If and acylindrical. If we are given Ψ as a combinatorial refine- ∆1,∆2 Θ are two tetrahedra, then the intersection com- ∈ ∗ ment of the triangulation Θ∗, then we can search for Φ as a plexity of g(∆1) and g(∆2) is no worse than that of g(∆1′ ) ER simplicial quotient of Ψ, such that we can solve the hyper- and g(∆2′ ), and is bounded by an function of ℓ and r. bolic gluing equations for Φ to recognize it as a geometric This yields an ER bound on the complexity of the refinements triangulation of a hyperbolic manifold X. We obtain a candi- Φ and Ψ. Recall that Ψ is a refinement of Θ∗, which is a date map g : N∗ X. If g has degree 1, and there is also a slightly modified version of the input description of N. Hav- degree 1 map h :→X N , then Mostow rigidity tells us that ing bounded the complexity of Ψ, we can search for it using → ∗ g and h are both homotopy equivalences and that X and N∗ Corollary 8.15 and solve for Φ and its geometry. We can also are homeomorphic. (Note that there can be a degree 1 map in discard g if it does not have degree 1. one direction between two hyperbolic 3-manifolds that is not Thus far, the algorithm finds an ER collection of candidate a homotopy equivalence [5], even though this cannot happen maps g : X ∗ N of degree 1, where N varies as well as g. At in the case of hyperbolic surfaces.) We can search for h by the least one of→ these maps is a homotopy equivalence. Instead same method of simplicial subdivision that we used to find of finding an inverse h, We can repeat the algorithm to look g. This establishes an algorithm to calculate the hyperbolic for degree one maps among the target manifolds N . This { } structure of N∗. induces a transitive relation among these manifolds. If N is We claim that a modified version of this algorithm is in ER. chosen at the top of this relation, then the associated map g : We first consider ER candidates for the map g. To do this, X ∗ N must be a homotopy equivalence. we make a non-commutative cocycle α C1(N;G) as in the To→ solve the isomorphism problem, we find geometric tri- ∈ proof of Corollary 5.10, where angulations Φ1 and Φ2 of the manifolds N1∗ and N2∗. We can again follow the first proof of Theorem 6.1, except now with G + H3 ER = Isom ( ) ∼= ISO(3,1), an bound on the complexity of Φ1 and Φ2, and we can again use Corollary 8.15. The same argument applies for the and with the extra restriction that α is parabolic on each com- calculation of the isometry group of a single N∗. ponent of ∂N. These cocycle equations are algebraic, so The- orem 8.8 guarantees a representative set of solutions. By Mostow or Calabi-Weil rigidity, one of components of the so- 9. OPEN PROBLEMS lution space yields a discrete homomorphism

+ 3 Theorem 1.2, together with the fact that ER is a fairly gen- ρ : π1(N) Isom (H ) → erous complexity class, suggests the following conjectures. that describes the hyperbolic geometry of X. If we assign Conjecture 9.1. If M is a closed, Riemannian 3-manifold, some point p H3 to one of the vertices of Θ, then in the ∈ ER then Ricci flow with surgery on M can be accurately simulated closed case, its orbit under α is in and can be extended on in ER. each simplex of Θ to the map g. In the cusped case, there are also ideal vertices whose position on the sphere at infinity can In other words, we conjecture that Perelman’s proof of ge- be calculated from α as well. ometrization can be placed in ER. If N has boundary, then we also want a truncated version of Θ∗ which is larger than the original Θ, and slightly different Conjecture 9.2. Every closed, hyperbolic manifold N has a from the horospheric truncation description in Section 7. If finite-sheeted Haken covering which is computable in ER. that ∆ Θ is semi-ideal, then let p be its ideal vertex, let F ∗ In other words, we conjecture that the statement of the vir- be the∈ hyperplane containing the face of ∆ opposite to p, and tual Haken conjecture, now the theorem of Agol et al [1], can let F be the hypersphere at distance log(2) from F which is ′ be placed in ER. Maybe the known proof can be as well. on the same side as p. Then we truncate ∆ with F′ to make ∆ ; or if ∆ Θ is a non-ideal tetrahedron, we let ∆ = ∆. ′ ∈ ∗ ′ Conjecture 9.3. Any two triangulations of a closed 3- We let X ′ X be the union of all ∆′. (In the closed case, we ER ⊆ manifold M have a mutual refinement computable in . obtain X ′ = X.) X ′ can have a complicated shape because the truncations are usually mismatched, but we can calculate the Conjecture 9.3 does not follow from our proof of Theo- positions of its vertices, and it is easy to confirm that it has at rem 1.2, because the algorithm in Theorem 8.3 only estab- least half of the volume of X. lishes a simplicial homotopy equivalence and then relies on Our algorithm does not know which cocycle α gives us a Mostow rigidity. However, the rest of the proof of Theo- desired g and we do not compute this directly. Instead, we rem 1.2 uses a bounded number of normal surface dissections, 23 which does establish an ER mutual refinement according to ER is the union of an alternating, nested sequence of time and the arguments of Mijatovi´c[31, 32]. Also, Conjecture 9.2 space complexity classes, as follows: and the Haken case of Conjecture 9.3 would together imply the hyperbolic case of Conjecture 9.3, which would then im- P PSPACE EXP EXPSPACE EEXP . ply the full conjecture. Mijatovi´c[33] also established that ⊆ ⊆ ⊆ ⊆ ⊆ ··· any two triangulations of a fiber-free Haken 3-manifold have a primitive recursive mutual refinement. Here P is the set of decision problemsthat can be solved in de- Cases 3 and 4 of Proposition 2.6 are expected to be false for terministic polynomial time; PSPACE is solvability in poly- typical bounds on complexity that are better than ER. Thus, in nomial space with unrestricted (but deterministic) computa- discussing further improvements to Theorem 1.2, we should tion time; EXP is deterministic time exp(poly(n)); etc. The consider qualitative complexity classes, such as the famous author does not know where a careful version of our proof of NP, rather than just bounds on execution time. For one thing, Theorem 1.2 would land in this hierarchy.

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