Collapsing 3-manifolds
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1 Collapsing 3-manifolds L. Bessières, G. Besson, M. Boileau, S. Maillot, J. Porti 4th William Rowan Hamilton Geometry and Topology Workshop HMI, TCD August 29, 2008 Collapsing 3-manifolds p.1/15
2 Main Theorem (Perelman) (Mn, 3 g n ) n seq. of closed, irreducible and orientable 3-mflds s.t. π 1 (Mn) 3 1 g n volume collapses g n has local curvature bounds (LCB) Then Mn 3 is a graph manifold for n large. Graph: union of Seifert fibered Collapsing 3-manifolds p.2/15
3 Main Theorem (Perelman) (M 3 n, g n ) n seq. of closed, irreducible and orientable 3-mflds s.t. π 1 (M 3 n) 1 g n volume collapses g n has local curvature bounds (LCB) Graph: union of Seifert fibered Then M 3 n is a graph manifold for n large. Those metrics appear at the end of Perelman s Ricci flow Shioya-Yamaguchi: Alexandrov spaces (without π 1 1 nor LCB) Collapsing 3-manifolds p.2/15
4 Main Theorem (Perelman) (M 3 n, g n ) n seq. of closed, irreducible and orientable 3-mflds s.t. π 1 (M 3 n) 1 g n volume collapses g n has local curvature bounds (LCB) Graph: union of Seifert fibered Then M 3 n is a graph manifold for n large. Those metrics appear at the end of Perelman s Ricci flow Shioya-Yamaguchi: Alexandrov spaces (without π 1 1 nor LCB) Kleiner-Lott and Morgan-Tian (without π 1 1) Collapsing 3-manifolds p.2/15
5 Main Theorem (Perelman) (M 3 n, g n ) n seq. of closed, irreducible and orientable 3-mflds s.t. π 1 (M 3 n) 1 g n volume collapses g n has local curvature bounds (LCB) Graph: union of Seifert fibered Then M 3 n is a graph manifold for n large. Those metrics appear at the end of Perelman s Ricci flow Shioya-Yamaguchi: Alexandrov spaces (without π 1 1 nor LCB) Kleiner-Lott and Morgan-Tian (without π 1 1) BBBMP: Proof relies on Thurston s geometrization for Haken mnflds. This doesn t prove Poincaré, but required for hyperbolization. Part of a larger project. Collapsing 3-manifolds p.2/15
6 Volume collapse (M 3 n, g n ) n volume collapses: ε n 0 s.t. for all x M 3 n there exists 0 < ρ n (x) < 1 satisfying: sec 1 ρ n (x) 2 on B(x, ρ n(x)) and vol(b(x, ρ n (x))) ρ n (x) 3 ε n Collapsing 3-manifolds p.3/15
7 Volume collapse (M 3 n, g n ) n volume collapses: ε n 0 s.t. for all x M 3 n there exists 0 < ρ n (x) < 1 satisfying: sec 1 ρ n (x) 2 on B(x, ρ n(x)) and vol(b(x, ρ n (x))) ρ n (x) 3 ε n Scale invariant inequalities: Normalize by an homotethy to have ρ n (x) = 1, get: sec 1 and vol(b(x, 1)) ε n. When sec C, a bound on the volume is equivalent to a bound on the injectivity radius Collapsing 3-manifolds p.3/15
8 Volume collapse (M 3 n, g n ) n volume collapses: ε n 0 s.t. for all x M 3 n there exists 0 < ρ n (x) < 1 satisfying: sec 1 ρ n (x) 2 on B(x, ρ n(x)) and vol(b(x, ρ n (x))) ρ n (x) 3 ε n Scale invariant inequalities: Normalize by an homotethy to have ρ n (x) = 1, get: sec 1 and vol(b(x, 1)) ε n. When sec C, a bound on the volume is equivalent to a bound on the injectivity radius ρ n (x)= collapsing scale Collapsing 3-manifolds p.3/15
9 Local curvature bounds Dfn. A sequence (Mn, 3 g n ) has local curvature bounds if: δ > 0 r = r(δ) > 0, K 0 = K 0 (δ) and K 1 = K 1 (δ) such that: If 0 < r < r(δ), vol(b(x,r)) r > δ and sec 1 3 r on B(x, r), then 2 Rm(x) < K 0 r 2 and Rm(x) < K 1 r 3 The constants r(δ), K 0 (δ) and K 1 (δ) are independent of n. Collapsing 3-manifolds p.4/15
10 Local curvature bounds Dfn. A sequence (Mn, 3 g n ) has local curvature bounds if: δ > 0 r = r(δ) > 0, K 0 = K 0 (δ) and K 1 = K 1 (δ) such that: If 0 < r < r(δ), vol(b(x,r)) r > δ and sec 1 3 r on B(x, r), then 2 Rm(x) < K 0 r 2 and Rm(x) < K 1 r 3 The constants r(δ), K 0 (δ) and K 1 (δ) are independent of n. Scale invariant inequalities. When r 0, vol(b(x,r)) r 4π and r 2 Thus such an r always exist, but r ρ n (x) = collapsing scale r x ρ n (x) Collapsing 3-manifolds p.4/15
11 Scheme of the proof (M 3 n, g n ) irr, π vol. collapses + Loc.Curv.Bds M 3 n graph Collapsing 3-manifolds p.5/15
12 Scheme of the proof (M 3 n, g n ) irr, π vol. collapses + Loc.Curv.Bds M 3 n graph 1 Find local models: metric balls B(x, ν x ) that are Volume collapsed: sec 1 ν and vol(b(x,ν x)) x 2 ν < ε x 3 n 0 Homeomorphic to B 3, S 1 D 2, S 2 I, T 2 I or K 2 I. Collapsing 3-manifolds p.5/15
13 Scheme of the proof (M 3 n, g n ) irr, π vol. collapses + Loc.Curv.Bds M 3 n graph 1 Find local models: metric balls B(x, ν x ) that are Volume collapsed: sec 1 ν and vol(b(x,ν x)) x 2 ν < ε x 3 n 0 Homeomorphic to B 3, S 1 D 2, S 2 I, T 2 I or K 2 I. 2 For one of the local models V = B(x, ν x ) the image of π 1 (V) π(m 3 n) is nontrivial. Hence M 3 n \ V is Haken, and geometric, by Thurston. Collapsing 3-manifolds p.5/15
14 Scheme of the proof (M 3 n, g n ) irr, π vol. collapses + Loc.Curv.Bds M 3 n graph 1 Find local models: metric balls B(x, ν x ) that are Volume collapsed: sec 1 ν and vol(b(x,ν x)) x 2 ν < ε x 3 n 0 Homeomorphic to B 3, S 1 D 2, S 2 I, T 2 I or K 2 I. 2 For one of the local models V = B(x, ν x ) the image of π 1 (V) π(m 3 n) is nontrivial. Hence M 3 n \ V is Haken, and geometric, by Thurston. 3 Construct a covering of dim 2 of M 3 n \ V by local models B(x, ν x ). (dim 2 means that every point belongs to at most three open sets). and dim 0 on (M 3 n \ V). Hence the simplicial volume M 3 n \ V = 0. Collapsing 3-manifolds p.5/15
15 Scheme of the proof 1 Find local models: metric balls B(x, ν x ) that are Volume collapsed: sec 1 ν and vol(b(x,ν x)) x 2 ν < ε x 3 n 0 Homeomorphic to B 3, S 1 D 2, S 2 I, T 2 I or K 2 I. 2 For one of the local models V = B(x, ν x ) the image of π 1 (V) π(m 3 n) is nontrivial. Hence M 3 n \ V is Haken, and geometric, by Thurston. 3 Construct a covering of dim 2 of M 3 n \ V by local models B(x, ν x ). (dim 2 means that every point belongs to at most three open sets). and dim 0 on (M 3 n \ V). Hence the simplicial volume M 3 n \ V = 0. Thus M 3 n \ V is graph, and so is M 3 n. Collapsing 3-manifolds p.5/15
16 Local models Lemma D > 0 n 0 = n 0 (D) such that for n > n 0 : either (M 3 n, g n ) is 1 D -close to X3 compact with sec 0 or x M 3 n ν x > 0 with: B(x, ν x ) 1 D -close to N ν x (S) X 3, X 3 noncompact with sec 0 and soul S, diam(s) ν x D vol(b(x,ν x)) ν 3 x < 1 D and sec B(x,ν x ) 1 ν 2 x Collapsing 3-manifolds p.6/15
17 Local models Lemma D > 0 n 0 = n 0 (D) such that for n > n 0 : either (M 3 n, g n ) is 1 D -close to X3 compact with sec 0 or x M 3 n ν x > 0 with: B(x, ν x ) 1 D -close to N ν x (S) X 3, X 3 noncompact with sec 0 and soul S, diam(s) ν x D vol(b(x,ν x)) ν 3 x < 1 D and sec B(x,ν x ) 1 ν 2 x Cheeger-Gromoll: X 3 = normal bundle of its soul S =, S 1, S 2, T 2, K 2. Hence B(x, ν x ) = B 3, S 1 D 2, S 2 I, T 2 I, K 2 I. S x ν x Collapsing 3-manifolds p.6/15
18 Local models Lemma D > 0 n 0 = n 0 (D) such that for n > n 0 : either (M 3 n, g n ) is 1 D -close to X3 compact with sec 0 or x M 3 n ν x > 0 with: B(x, ν x ) 1 D -close to N ν x (S) X 3, X 3 noncompact with sec 0 and soul S, diam(s) ν x D vol(b(x,ν x)) ν 3 x < 1 D and sec B(x,ν x ) 1 ν 2 x Proof by contradiction. Assume there is D 0 and x n M 3 n a seq. of counterexamples Set λ n = inf{r vol(b(x,r) r 3 1 D 0 } Claim: Up to subsequence 1 λ n B(x n, ρ n (x n )) X 3 Riemannian with sec 0 Collapsing 3-manifolds p.6/15
19 Blow up x n M 3 n seq. of counterexamples. Let λ n = inf{r vol(b(x,r) r 3 1 D 0 }. Claim: 1 λ n B(x n, ρ n (x n )) X 3 Riemannian with sec 0. Collapsing 3-manifolds p.7/15
20 Blow up x n M 3 n seq. of counterexamples. Let λ n = inf{r vol(b(x,r) r 3 1 D 0 }. Claim: 1 λ n B(x n, ρ n (x n )) X 3 Riemannian with sec 0. Proof By continuity, vol(b(x n,λ n )) λ = 1 vol(b(x,r)) 3 n D 0. (lim r 0 r = 4π 3 3 ). We knew sec 1 ρ n (x n ) on B(x 2 n, ρ n (x n )) and vol(b(x n,ρ n (x n ))) ρ n (x n ) ε 3 n. Since ε n 0 ρ n(x n ) λ n Collapsing 3-manifolds p.7/15
21 Blow up x n M 3 n seq. of counterexamples. Let λ n = inf{r vol(b(x,r) r 3 1 D 0 }. Claim: 1 λ n B(x n, ρ n (x n )) X 3 Riemannian with sec 0. Proof By continuity, vol(b(x n,λ n )) λ = 1 vol(b(x,r)) 3 n D 0. (lim r 0 r = 4π 3 3 ). We knew sec 1 ρ n (x n ) on B(x 2 n, ρ n (x n )) and vol(b(x n,ρ n (x n ))) ρ n (x n ) ε 3 n. Since ε n 0 ρ n(x n ) λ n Hence 1 λ n B(x n, ρ n (x n )) has curvature λ2 n ρ n (x n ) 0 2 and HG-converges to a complete metric space X 3 Collapsing 3-manifolds p.7/15
22 Blow up x n M 3 n seq. of counterexamples. Let λ n = inf{r vol(b(x,r) r 3 1 D 0 }. Claim: 1 λ n B(x n, ρ n (x n )) X 3 Riemannian with sec 0. Proof By continuity, vol(b(x n,λ n )) λ = 1 vol(b(x,r)) 3 n D 0. (lim r 0 r = 4π 3 3 ). We knew sec 1 ρ n (x n ) on B(x 2 n, ρ n (x n )) and vol(b(x n,ρ n (x n ))) ρ n (x n ) ε 3 n. Since ε n 0 ρ n(x n ) λ n Hence 1 λ n B(x n, ρ n (x n )) has curvature λ2 n ρ n (x n ) 0 2 and HG-converges to a complete metric space X 3 Local Curvature Bounds: (LCB apply by the choice of λ n ) Rm, Rm < C unif. on balls of bounded radius in 1 λ n B(x n, ρ n (x n )) X 3 is Riemannian and the convergence is C Collapsing 3-manifolds p.7/15
23 Back to local models We have shown: Lemma D > 0 n 0 = n 0 (D) such that for n > n 0 : either (M 3 n, g n ) is 1 D -close X3 compact with sec 0 or x M 3 n ν x > 0 with: B(x, ν x ) 1 D -close to N ν x (S) X 3, X 3 noncompact with sec 0 and soul S, diam(s) ν x D vol(b(x,ν x)) ν 3 x < 1 D and sec B(x,ν x ) 1 ν 2 x B(x, ν x ) = B 3, S 1 D 2, S 2 I, T 2 I, K 2 I. Collapsing 3-manifolds p.8/15
24 Back to local models Lemma D > 0 n 0 = n 0 (D) such that for n > n 0 : either (M 3 n, g n ) is 1 D -close X3 compact with sec 0 or x M 3 n ν x > 0 with: B(x, ν x ) 1 D -close to N ν x (S) X 3, X 3 noncompact with sec 0 and soul S, diam(s) ν x D vol(b(x,ν x)) ν 3 x < 1 D and sec B(x,ν x ) 1 ν 2 x B(x, ν x ) = B 3, S 1 D 2, S 2 I, T 2 I, K 2 I. Compact mfls with curv 0 are understood (geometric) We assume only non compact models B(x, ν x ) occur: S x ν x Collapsing 3-manifolds p.8/15
25 A local model has nontrivial π 1 M 3 n covered by local models B(x, ν x ) 1 D N ν x (S) X 3 noncompact, Claim If D suff. large, then there exists a local model V = B(x, ν x ) s.t. Im(π 1 (V) π 1 (M 3 n)) {1} With the claim M 3 n \ V is Haken. Collapsing 3-manifolds p.9/15
26 A local model has nontrivial π 1 M 3 n covered by local models B(x, ν x ) 1 D N ν x (S) X 3 noncompact, Claim If D suff. large, then there exists a local model V = B(x, ν x ) s.t. Im(π 1 (V) π 1 (M 3 n)) {1} With the claim M 3 n \ V is Haken. Proof By contradiction. Assume Im(π 1 (B(x, ν x )) π 1 (M 3 n)) = {1}, x M 3 n. Find a covering of M 3 n by B(x, ν x ) and shrink it to have dim 2. Theorem (Gómez-Larrañaga, González-Acuña) N 3 irr., or., closed, π 1 (N 3 ) 1. Then N 3 doesn t admit a covering {U i } i I of dim 2 such that Im(π 1 (U i ) π 1 (N 3 )) = {1} i I. Collapsing 3-manifolds p.9/15
27 Construction of the covering DEF: triv(x) = sup r π 1 (B(x, r)) π 1 (M 3 n) is trivial and B(x, r) B(x, r ) with sec B(x,r ) 1 (r ) 2 and vol(b(x,r )) (r ) 3 1 D triv(x) ν x (We assume π 1 (B(x, ν x )) π 1 (M 3 n) trivial) Collapsing 3-manifolds p.10/15
28 Construction of the covering DEF: triv(x) = sup r π 1 (B(x, r)) π 1 (M 3 n) is trivial and B(x, r) B(x, r ) with sec B(x,r ) 1 (r ) 2 and vol(b(x,r )) (r ) 3 1 D triv(x) ν x (We assume π 1 (B(x, ν x )) π 1 (M 3 n) trivial) r(x) = min{1, 1 11 triv(x)} B(x, r(x)) B(y, r(y)) 3 4 r(x) r(y) 4 3. y r y x triv(y) triv(x) r x B(x, r(x)) B(y, r(y)) triv(x) triv(y) Collapsing 3-manifolds p.10/15
29 Construction of the covering DEF: triv(x) = sup r π 1 (B(x, r)) π 1 (M 3 n) is trivial and B(x, r) B(x, r ) with sec B(x,r ) 1 (r ) 2 and vol(b(x,r )) (r ) 3 1 D triv(x) ν x (We assume π 1 (B(x, ν x )) π 1 (M 3 n) trivial) r(x) = min{1, 1 11 triv(x)} B(x, r(x)) B(y, r(y)) 3 4 r(x) r(y) 4 3. Chose B(x 1, 1 4 r 1), B(x 2, 1 4 r 2),... pairwise disjoint and maximal. By maximality B(x 1, 2 3 r 1),...,B(x q, 2 3 r q) cover M 3 n. 1 4 r(y) y 1 4 r i x i d(y, x i ) 1 4 r(y) r i 2 3 r i Collapsing 3-manifolds p.10/15
30 Construction of the covering DEF: triv(x) = sup r π 1 (B(x, r)) π 1 (M 3 n) is trivial and B(x, r) B(x, r ) with sec B(x,r ) 1 (r ) 2 and vol(b(x,r )) (r ) 3 1 D triv(x) ν x (We assume π 1 (B(x, ν x )) π 1 (M 3 n) trivial) r(x) = min{1, 1 11 triv(x)} B(x, r(x)) B(y, r(y)) 3 4 r(x) r(y) 4 3. Chose B(x 1, 1 4 r 1), B(x 2, 1 4 r 2),... pairwise disjoint and maximal. By maximality B(x 1, 2 3 r 1),...,B(x q, 2 3 r q) cover M 3 n. 1 4 r(y) 1 4 r i y x i d(y, x i ) 1 4 r(y) r i 2 3 r i Want to shrink this covering to have dim 2 (for D large enough). Collapsing 3-manifolds p.10/15
31 Schrinking the covering to dim 2 {B(x 1, r 1 ),...,B(x q, r q )} covering of M 3 n with nerve K Collapsing 3-manifolds p.11/15
32 Schrinking the covering to dim 2 {B(x 1, r 1 ),...,B(x q, r q )} covering of M 3 n with nerve K Collapsing 3-manifolds p.11/15
33 Schrinking the covering to dim 2 {B(x 1, r 1 ),...,B(x q, r q )} covering of M 3 n with nerve K Collapsing 3-manifolds p.11/15
34 Schrinking the covering to dim 2 {B(x 1, r 1 ),...,B(x q, r q )} covering of M 3 n with nerve K Collapsing 3-manifolds p.11/15
35 Schrinking the covering to dim 2 {B(x 1, r 1 ),...,B(x q, r q )} covering of M 3 n with nerve K f = 1 P φi (φ 1,...,φ q ) : M 3 n K characteristic map where φ i : B(x i, r i ) [0, 1] test function φ i B(xi, 2 3 r i) = 1, φ i B(xi,r i ) = 0, φ i 4 r i 1 φ i r i r i B(x i,r i ) Collapsing 3-manifolds p.11/15
36 Schrinking the covering to dim 2 {B(x 1, r 1 ),...,B(x q, r q )} covering of M 3 n with nerve K f = 1 P φi (φ 1,...,φ q ) : M 3 n K characteristic map where φ i : B(x i, r i ) [0, 1] test function φ i B(xi, 2 3 r i) = 1, φ i B(xi,r i ) = 0, φ i 4 r i f C r i on B(x i, r i ) for uniform C, because M 3 n = B(x, 2 3 r i) Collapsing 3-manifolds p.11/15
37 Schrinking the covering to dim 2 {B(x 1, r 1 ),...,B(x q, r q )} covering of M 3 n with nerve K f = 1 P φi (φ 1,...,φ q ) : M 3 n K characteristic map where φ i : B(x i, r i ) [0, 1] test function φ i B(xi, 2 3 r i) = 1, φ i B(xi,r i ) = 0, φ i 4 r i f C r i on B(x i, r i ) for uniform C, because Mn 3 = B(x, 2 3 r i) Strategy: Retract f to f (2) : Mn 3 K (2) and take V i = (f (2) ) 1 (open star of the i-th vertex of K (2) ). Collapsing 3-manifolds p.11/15
38 Retract f to the 2-skeletton {B(x 1, r 1 ),...,B(x q, r q )} covering of M 3 n with nerve K f : M 3 n K characteristic map f C r i on B(x i, r i ) Want to retract f to f (2) : M 3 n K (2) Collapsing 3-manifolds p.12/15
39 Retract f to the 2-skeletton {B(x 1, r 1 ),...,B(x q, r q )} covering of M 3 n with nerve K f : M 3 n K characteristic map f C r i on B(x i, r i ) Want to retract f to f (2) : M 3 n K (2) Assume we have f : M 3 n K (3) (dim(m 3 n) = 3). Claim: For D large enough, f does not fill any 3 simplex 3 Compose with radial projection 3 \ { } 3, to get f (2) :M 3 n K (2). Collapsing 3-manifolds p.12/15
40 Retract f to the 2-skeletton {B(x 1, r 1 ),...,B(x q, r q )} covering of M 3 n with nerve K f : M 3 n K characteristic map f C r i on B(x i, r i ) Want to retract f to f (2) : M 3 n K (2) Assume we have f : M 3 n K (3) (dim(m 3 n) = 3). Claim: For D large enough, f does not fill any 3 simplex 3 vol(im(f) 3 )) vol(f(b(x i, r i ))) f 3 vol(b(x i, r i )) ( ) 3 C ri 3 D C3 D <vol( 3 ) r i Compose with radial projection 3 \ { } 3, to get f (2) :M 3 n K (2). Collapsing 3-manifolds p.12/15
41 A Haken manifold M 3 n \ V V = B(x, ν x ) such that im(π 1 (V) π 1 (B(x, ν x ))) {1} V = S 1 D 2, T 2 I or K I. (and V = B 3, S 2 I) Hence M 3 n \ V is a Haken manifold with = T 2 or T 2 T 2. Want to show that the simplicial volume of M 3 n \ V vanishes. Collapsing 3-manifolds p.13/15
42 Simplicial volume (Gromov s norm) Def The simplicial volume of an orientable compact manifold N n is { N n = inf λσ [ } λ σ σ] = [N n, N n ] H n (N n, N n ;R) Collapsing 3-manifolds p.14/15
43 Simplicial volume (Gromov s norm) Def The simplicial volume of an orientable compact manifold N n is { N n = inf λσ [ } λ σ σ] = [N n, N n ] H n (N n, N n ;R) f : (M n, M n ) (N n, N n ) then M n deg(f) N n S n = 0 N 3 = 0 if N 3 Seifert fibered N 3 = v 3 vol(n 3 ) if N 3 hyperbolic (Gromov-Thurston). Collapsing 3-manifolds p.14/15
44 Simplicial volume (Gromov s norm) Def The simplicial volume of an orientable compact manifold N n is { N n = inf λσ [ } λ σ σ] = [N n, N n ] H n (N n, N n ;R) f : (M n, M n ) (N n, N n ) then M n deg(f) N n S n = 0 N 3 = 0 if N 3 Seifert fibered N 3 = v 3 vol(n 3 ) if N 3 hyperbolic (Gromov-Thurston). If N 3 has a geometric decomposition with hyperbolic pieces N 3 1,..., N 3 k and some Seifert pieces then N3 = v 3 vol(n 3 i ), v (Gromov-Soma) Corollary: If N 3 has a geometric decomposition and N 3 = 0 then all pieces are Seifert fibered. Collapsing 3-manifolds p.14/15
45 Covering of M 3 n \ V V = B(x, ν x ) such that im(π 1 (V) π 1 (B(x, ν x ))) {1} V = S 1 D 2, T 2 I or K I. (and V = B 3, S 2 I) Hence M 3 n \ V is a Haken manifold with = T 2 or T 2 T 2. Collapsing 3-manifolds p.15/15
46 Covering of M 3 n \ V V = B(x, ν x ) such that im(π 1 (V) π 1 (B(x, ν x ))) {1} V = S 1 D 2, T 2 I or K I. (and V = B 3, S 2 I) Hence M 3 n \ V is a Haken manifold with = T 2 or T 2 T 2. As before, find a covering U 1,...,U q of M 3 n \ V such that: dim of the covering is 2 (and dim 0 on (M 3 n \ V)) im(π 1 (U i ) π 1 (M 3 n \ V)) is virtually abelian. Collapsing 3-manifolds p.15/15
47 Covering of M 3 n \ V V = B(x, ν x ) such that im(π 1 (V) π 1 (B(x, ν x ))) {1} V = S 1 D 2, T 2 I or K I. (and V = B 3, S 2 I) Hence M 3 n \ V is a Haken manifold with = T 2 or T 2 T 2. As before, find a covering U 1,...,U q of M 3 n \ V such that: dim of the covering is 2 (and dim 0 on (M 3 n \ V)) im(π 1 (U i ) π 1 (M 3 n \ V)) is virtually abelian. Theorem (Gromov) If N 3 has a covering {U i } of dim 2 (dim 0 at N 3 ) and im(π 1 U i π 1 N 3 ) virtualy abelian, then N 3 = 0 Thus M 3 n \ V is graph, and so is M 3 n. Collapsing 3-manifolds p.15/15
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