On open 3-manifolds School on Geometric Evolution Problems G. Besson Universit´eGrenoble Alpes Centro De Giorgi June, 23rd 2014 INSTITUT i f FOURIER Outline Surfaces Topological classification by B. Ker´ekj´art´o(1923) and I. Richards (1963). Involves the structure at infinity of non compact surfaces. The question Post-Perelman Question : what could be a good statement for the classification (geometrization) of open 3-manifolds ? Topological classification by B. Ker´ekj´art´o(1923) and I. Richards (1963). Involves the structure at infinity of non compact surfaces. The question Post-Perelman Question : what could be a good statement for the classification (geometrization) of open 3-manifolds ? Surfaces Involves the structure at infinity of non compact surfaces. The question Post-Perelman Question : what could be a good statement for the classification (geometrization) of open 3-manifolds ? Surfaces Topological classification by B. Ker´ekj´art´o(1923) and I. Richards (1963). The question Post-Perelman Question : what could be a good statement for the classification (geometrization) of open 3-manifolds ? Surfaces Topological classification by B. Ker´ekj´art´o(1923) and I. Richards (1963). Involves the structure at infinity of non compact surfaces. I let X a simply connected closed 3-manifolds, X n {∗} is contractible. 3 I The only contractible 3-manifolds is R . 3 3 I One-point compactification of R is S . In 1935 he realizes the mistake and constructed the first contractible 3-manifold not homeomorphic to R3, the so-called Whitehead manifold. Whitehead's "proof " of the Poincar´econjecture J. H. C. Whitehead suggested in 1934 a proof based on the following scheme : 3 I The only contractible 3-manifolds is R . 3 3 I One-point compactification of R is S . In 1935 he realizes the mistake and constructed the first contractible 3-manifold not homeomorphic to R3, the so-called Whitehead manifold. Whitehead's "proof " of the Poincar´econjecture J. H. C. Whitehead suggested in 1934 a proof based on the following scheme : I let X a simply connected closed 3-manifolds, X n {∗} is contractible. 3 3 I One-point compactification of R is S . In 1935 he realizes the mistake and constructed the first contractible 3-manifold not homeomorphic to R3, the so-called Whitehead manifold. Whitehead's "proof " of the Poincar´econjecture J. H. C. Whitehead suggested in 1934 a proof based on the following scheme : I let X a simply connected closed 3-manifolds, X n {∗} is contractible. 3 I The only contractible 3-manifolds is R . In 1935 he realizes the mistake and constructed the first contractible 3-manifold not homeomorphic to R3, the so-called Whitehead manifold. Whitehead's "proof " of the Poincar´econjecture J. H. C. Whitehead suggested in 1934 a proof based on the following scheme : I let X a simply connected closed 3-manifolds, X n {∗} is contractible. 3 I The only contractible 3-manifolds is R . 3 3 I One-point compactification of R is S . , the so-called Whitehead manifold. Whitehead's "proof " of the Poincar´econjecture J. H. C. Whitehead suggested in 1934 a proof based on the following scheme : I let X a simply connected closed 3-manifolds, X n {∗} is contractible. 3 I The only contractible 3-manifolds is R . 3 3 I One-point compactification of R is S . In 1935 he realizes the mistake and constructed the first contractible 3-manifold not homeomorphic to R3 Whitehead's "proof " of the Poincar´econjecture J. H. C. Whitehead suggested in 1934 a proof based on the following scheme : I let X a simply connected closed 3-manifolds, X n {∗} is contractible. 3 I The only contractible 3-manifolds is R . 3 3 I One-point compactification of R is S . In 1935 he realizes the mistake and constructed the first contractible 3-manifold not homeomorphic to R3, the so-called Whitehead manifold. Outline 3 I T1 is unknotted in S and Ti is a null-homotopic knot in Ti−1, for i > 1. On the picture Ti+2 ⊂ Ti+1 ⊂ Ti ⊂ Ti−1. Whitehead manifolds Take T1 ⊃ T2 ⊃ T3 ⊃ ::: solid tori. On the picture Ti+2 ⊂ Ti+1 ⊂ Ti ⊂ Ti−1. Whitehead manifolds Take T1 ⊃ T2 ⊃ T3 ⊃ ::: solid tori. 3 I T1 is unknotted in S and Ti is a null-homotopic knot in Ti−1, for i > 1. On the picture Ti+2 ⊂ Ti+1 ⊂ Ti ⊂ Ti−1. Whitehead manifolds Take T1 ⊃ T2 ⊃ T3 ⊃ ::: solid tori. 3 I T1 is unknotted in S and Ti is a null-homotopic knot in Ti−1, for i > 1. Whitehead manifolds Take T1 ⊃ T2 ⊃ T3 ⊃ ::: solid tori. 3 I T1 is unknotted in S and Ti is a null-homotopic knot in Ti−1, for i > 1. On the picture Ti+2 ⊂ Ti+1 ⊂ Ti ⊂ Ti−1. 3 3 I X = S n W ⊂ S is a (the) whitehead manifold (genus one). Theorem (J.H.C. Whitehead) X is contractible and not homeomorphic to R3. Whitehead manifolds I W = \Ti is the Whitehead continuum. Theorem (J.H.C. Whitehead) X is contractible and not homeomorphic to R3. Whitehead manifolds I W = \Ti is the Whitehead continuum. 3 3 I X = S n W ⊂ S is a (the) whitehead manifold (genus one). Whitehead manifolds I W = \Ti is the Whitehead continuum. 3 3 I X = S n W ⊂ S is a (the) whitehead manifold (genus one). Theorem (J.H.C. Whitehead) X is contractible and not homeomorphic to R3. Hausdorff dimension > 1. Whitehead continuum Whitehead continuum Hausdorff dimension > 1. 3 I Si = S n Ti is diffeomorphic to a solid torus. I X is a limit of solid tori. Whitehead manifolds : remarks 3 Ti is knotted in Ti−1 but not in S , hence I X is a limit of solid tori. Whitehead manifolds : remarks 3 Ti is knotted in Ti−1 but not in S , hence 3 I Si = S n Ti is diffeomorphic to a solid torus. Whitehead manifolds : remarks 3 Ti is knotted in Ti−1 but not in S , hence 3 I Si = S n Ti is diffeomorphic to a solid torus. I X is a limit of solid tori. Whitehead link The idea is that the core of Ti and the meridian of Ti−1 form the Whitehead link. Whitehead link The idea is that the core of Ti and the meridian of Ti−1 form the Whitehead link. ; role of the two curves could be exchanged. Whitehead link Whitehead link is symmetric Whitehead link Whitehead link is symmetric ; role of the two curves could be exchanged. Whitehead link Whitehead link is symmetric ; role of the two curves could be exchanged. 3 I Define Si = S n Ti . I Si is a solid torus, I Si ⊂ Si+1 knotted in the same way, S I symmetry ) X = Si . I A more general construction. Whitehead manifolds Whitehead link is symmetric ; consequences. I Si is a solid torus, I Si ⊂ Si+1 knotted in the same way, S I symmetry ) X = Si . I A more general construction. Whitehead manifolds Whitehead link is symmetric ; consequences. 3 I Define Si = S n Ti . I Si ⊂ Si+1 knotted in the same way, S I symmetry ) X = Si . I A more general construction. Whitehead manifolds Whitehead link is symmetric ; consequences. 3 I Define Si = S n Ti . I Si is a solid torus, S I symmetry ) X = Si . I A more general construction. Whitehead manifolds Whitehead link is symmetric ; consequences. 3 I Define Si = S n Ti . I Si is a solid torus, I Si ⊂ Si+1 knotted in the same way, I A more general construction. Whitehead manifolds Whitehead link is symmetric ; consequences. 3 I Define Si = S n Ti . I Si is a solid torus, I Si ⊂ Si+1 knotted in the same way, S I symmetry ) X = Si . Whitehead manifolds Whitehead link is symmetric ; consequences. 3 I Define Si = S n Ti . I Si is a solid torus, I Si ⊂ Si+1 knotted in the same way, S I symmetry ) X = Si . I A more general construction. I Si is null homotopic in Si+1 i.e. it is homotopic to a point. k k I For any k-sphere S in X , 9i such that S ⊂ Si , I 8k; πk (X ) = f1g ; X is contractible. Non-homeomorphy to R3 (less easy) I X is not simply connected at infinity. Definition X topological space is simply connected at infinity if 8C compact, 9D compact, D ⊃ C s.t. any loop in X n D is null homotopic in X n C. Whitehead manifolds Contractibility (easy) k k I For any k-sphere S in X , 9i such that S ⊂ Si , I 8k; πk (X ) = f1g ; X is contractible. Non-homeomorphy to R3 (less easy) I X is not simply connected at infinity. Definition X topological space is simply connected at infinity if 8C compact, 9D compact, D ⊃ C s.t. any loop in X n D is null homotopic in X n C. Whitehead manifolds Contractibility (easy) I Si is null homotopic in Si+1 i.e. it is homotopic to a point. I 8k; πk (X ) = f1g ; X is contractible. Non-homeomorphy to R3 (less easy) I X is not simply connected at infinity. Definition X topological space is simply connected at infinity if 8C compact, 9D compact, D ⊃ C s.t. any loop in X n D is null homotopic in X n C. Whitehead manifolds Contractibility (easy) I Si is null homotopic in Si+1 i.e. it is homotopic to a point. k k I For any k-sphere S in X , 9i such that S ⊂ Si , Non-homeomorphy to R3 (less easy) I X is not simply connected at infinity.