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An Evolution of The Topological Spherical Space Form Problem An Evolution of The Topological Spherical Space Form Problem Dennis Dreesen, Paul Igodt*, Nansen Petrosyan K.U.Leuven Campus Kortrijk Warsaw, July 7, 2009 Dennis Dreesen, Paul

  1. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Group actions Properly discontinuous actions Definition G is said to act properly discontinuously on X if for any compact subset C ⊆ X , # { g ∈ G | C ∩ gC � = ∅} < ∞ . Any action of a finite group is properly discontinuous. In all of the previous examples actions are properly discontinuous. Ex. 3. Q , as a subgroup of R , acts freely but not properly discontinuously on R . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  2. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Group actions Properly discontinuous actions Definition G is said to act properly discontinuously on X if for any compact subset C ⊆ X , # { g ∈ G | C ∩ gC � = ∅} < ∞ . Any action of a finite group is properly discontinuous. In all of the previous examples actions are properly discontinuous. Ex. 3. Q , as a subgroup of R , acts freely but not properly discontinuously on R . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  3. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Group actions Universal covering space A connected topological space is called simply connected if its fundamental group is trivial. Let M be a connected manifold. The universal covering space is the unique connected simply connected manifold M together with the covering map p : � � M → M . Ex. 4. R is the universal cover of S 1 and p : R → S 1 , x �→ e xi . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  4. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Group actions Universal covering space A connected topological space is called simply connected if its fundamental group is trivial. Let M be a connected manifold. The universal covering space is the unique connected simply connected manifold M together with the covering map p : � � M → M . Ex. 4. R is the universal cover of S 1 and p : R → S 1 , x �→ e xi . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  5. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Group actions Universal covering space A connected topological space is called simply connected if its fundamental group is trivial. Let M be a connected manifold. The universal covering space is the unique connected simply connected manifold M together with the covering map p : � � M → M . Ex. 4. R is the universal cover of S 1 and p : R → S 1 , x �→ e xi . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  6. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Group actions Universal covering space A connected topological space is called simply connected if its fundamental group is trivial. Let M be a connected manifold. The universal covering space is the unique connected simply connected manifold M together with the covering map p : � � M → M . Ex. 4. R is the universal cover of S 1 and p : R → S 1 , x �→ e xi . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  7. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Group actions Universal covering space A connected topological space is called simply connected if its fundamental group is trivial. Let M be a connected manifold. The universal covering space is the unique connected simply connected manifold M together with the covering map p : � � M → M . Ex. 4. R is the universal cover of S 1 and p : R → S 1 , x �→ e xi . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  8. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Group actions The fundamental group and the universal cover Definition Let G act on a space X . The quotient of the action is defined to be the space X / G = { ¯ x | x ∈ X , ¯ x = ¯ y iff ∃ g ∈ G , gx = y } . Fundamental characterization Let M be a connected manifold. Then the fundamental group π of M acts freely and properly discontinuously on the universal cover � M and � M /π ∼ = M . Ex. 5. Let T 2 = S 1 × S 1 . Then π 1 ( T 2 ) = Z 2 acts freely and properly discontinuously on � T 2 ∼ = R 2 . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  9. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Group actions The fundamental group and the universal cover Definition Let G act on a space X . The quotient of the action is defined to be the space X / G = { ¯ x | x ∈ X , ¯ x = ¯ y iff ∃ g ∈ G , gx = y } . Fundamental characterization Let M be a connected manifold. Then the fundamental group π of M acts freely and properly discontinuously on the universal cover � M and � M /π ∼ = M . Ex. 5. Let T 2 = S 1 × S 1 . Then π 1 ( T 2 ) = Z 2 acts freely and properly discontinuously on � T 2 ∼ = R 2 . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  10. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Group actions The fundamental group and the universal cover Definition Let G act on a space X . The quotient of the action is defined to be the space X / G = { ¯ x | x ∈ X , ¯ x = ¯ y iff ∃ g ∈ G , gx = y } . Fundamental characterization Let M be a connected manifold. Then the fundamental group π of M acts freely and properly discontinuously on the universal cover � M and � M /π ∼ = M . Ex. 5. Let T 2 = S 1 × S 1 . Then π 1 ( T 2 ) = Z 2 acts freely and properly discontinuously on � T 2 ∼ = R 2 . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  11. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Group actions The fundamental group and the universal cover Definition Let G act on a space X . The quotient of the action is defined to be the space X / G = { ¯ x | x ∈ X , ¯ x = ¯ y iff ∃ g ∈ G , gx = y } . Fundamental characterization Let M be a connected manifold. Then the fundamental group π of M acts freely and properly discontinuously on the universal cover � M and � M /π ∼ = M . Ex. 5. Let T 2 = S 1 × S 1 . Then π 1 ( T 2 ) = Z 2 acts freely and properly discontinuously on � T 2 ∼ = R 2 . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  12. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Group actions The fundamental group and the universal cover Definition Let G act on a space X . The quotient of the action is defined to be the space X / G = { ¯ x | x ∈ X , ¯ x = ¯ y iff ∃ g ∈ G , gx = y } . Fundamental characterization Let M be a connected manifold. Then the fundamental group π of M acts freely and properly discontinuously on the universal cover � M and � M /π ∼ = M . Ex. 5. Let T 2 = S 1 × S 1 . Then π 1 ( T 2 ) = Z 2 acts freely and properly discontinuously on � T 2 ∼ = R 2 . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  13. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Group actions The fundamental group and the universal cover Definition Let G act on a space X . The quotient of the action is defined to be the space X / G = { ¯ x | x ∈ X , ¯ x = ¯ y iff ∃ g ∈ G , gx = y } . Fundamental characterization Let M be a connected manifold. Then the fundamental group π of M acts freely and properly discontinuously on the universal cover � M and � M /π ∼ = M . Ex. 5. Let T 2 = S 1 × S 1 . Then π 1 ( T 2 ) = Z 2 acts freely and properly discontinuously on � T 2 ∼ = R 2 . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  14. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Group actions The three main questions (1) The topological spherical space form problem When does a finite group act freely on a sphere S n ? (2) The topological Euclidean space form problem When does a countable group act freely and properly discontinuously on some Euclidean space R k ? (3) The hybrid problem What countable groups act freely and properly discontinuously on some S n × R k ? Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  15. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Solution Outline The Topological Spherical Space Form Problem 1 Group actions Solution The Topological Euclidean Space Form Problem 2 Historical background Group cohomology Cohomological dimension Solution Free and Proper Group Actions on S n × R k 3 Current results Talelli’s conjecture Groups with jump cohomology General conjecture for solvable groups Isometric actions Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  16. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Solution Theorems of Smith and Artin-Tate Theorem (R.G. Smith, 1938) If a finite group G acts freely on S n , then every abelian subgroup of G is cyclic. forward Theorem (Artin-Tate, 1956) A finite group has all abelian subgroups cyclic if and only if its cohomology is periodic. For instance, does the dihedral group D 6 = � x , y | x 2 = y 3 = ( xy ) 2 = e � act freely on a sphere? Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  17. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Solution Theorems of Smith and Artin-Tate Theorem (R.G. Smith, 1938) If a finite group G acts freely on S n , then every abelian subgroup of G is cyclic. forward Theorem (Artin-Tate, 1956) A finite group has all abelian subgroups cyclic if and only if its cohomology is periodic. For instance, does the dihedral group D 6 = � x , y | x 2 = y 3 = ( xy ) 2 = e � act freely on a sphere? Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  18. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Solution Theorems of Smith and Artin-Tate Theorem (R.G. Smith, 1938) If a finite group G acts freely on S n , then every abelian subgroup of G is cyclic. forward Theorem (Artin-Tate, 1956) A finite group has all abelian subgroups cyclic if and only if its cohomology is periodic. For instance, does the dihedral group D 6 = � x , y | x 2 = y 3 = ( xy ) 2 = e � act freely on a sphere? Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  19. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Solution Milnor’s condition Theorem (Milnor, 1957) Let T : S n → S n be a map such that T ◦ T = Id without fixed points. Then for every f : S n → S n of odd degree there exists a point x ∈ S n such that Tf ( x ) = fT ( x ) . Corollary If a finite group G acts freely on S n , then any element of order 2 must be in the center Z ( G ) . Proof. Let t ∈ G be of order 2 and let g ∈ G . Then there exists x ∈ S n so that tg ( x ) = gt ( x ) . This implies tg = gt . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  20. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Solution Milnor’s condition Theorem (Milnor, 1957) Let T : S n → S n be a map such that T ◦ T = Id without fixed points. Then for every f : S n → S n of odd degree there exists a point x ∈ S n such that Tf ( x ) = fT ( x ) . Corollary If a finite group G acts freely on S n , then any element of order 2 must be in the center Z ( G ) . Proof. Let t ∈ G be of order 2 and let g ∈ G . Then there exists x ∈ S n so that tg ( x ) = gt ( x ) . This implies tg = gt . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  21. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Solution Milnor’s condition Theorem (Milnor, 1957) Let T : S n → S n be a map such that T ◦ T = Id without fixed points. Then for every f : S n → S n of odd degree there exists a point x ∈ S n such that Tf ( x ) = fT ( x ) . Corollary If a finite group G acts freely on S n , then any element of order 2 must be in the center Z ( G ) . Proof. Let t ∈ G be of order 2 and let g ∈ G . Then there exists x ∈ S n so that tg ( x ) = gt ( x ) . This implies tg = gt . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  22. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Solution Milnor’s condition Theorem (Milnor, 1957) Let T : S n → S n be a map such that T ◦ T = Id without fixed points. Then for every f : S n → S n of odd degree there exists a point x ∈ S n such that Tf ( x ) = fT ( x ) . Corollary If a finite group G acts freely on S n , then any element of order 2 must be in the center Z ( G ) . Proof. Let t ∈ G be of order 2 and let g ∈ G . Then there exists x ∈ S n so that tg ( x ) = gt ( x ) . This implies tg = gt . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  23. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Solution Milnor’s condition Theorem (Milnor, 1957) Let T : S n → S n be a map such that T ◦ T = Id without fixed points. Then for every f : S n → S n of odd degree there exists a point x ∈ S n such that Tf ( x ) = fT ( x ) . Corollary If a finite group G acts freely on S n , then any element of order 2 must be in the center Z ( G ) . Proof. Let t ∈ G be of order 2 and let g ∈ G . Then there exists x ∈ S n so that tg ( x ) = gt ( x ) . This implies tg = gt . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  24. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Solution Milnor’s condition Theorem (Milnor, 1957) Let T : S n → S n be a map such that T ◦ T = Id without fixed points. Then for every f : S n → S n of odd degree there exists a point x ∈ S n such that Tf ( x ) = fT ( x ) . Corollary If a finite group G acts freely on S n , then any element of order 2 must be in the center Z ( G ) . Proof. Let t ∈ G be of order 2 and let g ∈ G . Then there exists x ∈ S n so that tg ( x ) = gt ( x ) . This implies tg = gt . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  25. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Solution Classification Fact: Milnor’s condition together with periodicity is also sufficient for a group to act freely on some S n . Let n be a positive integer. A group G is said to satisfy the n -condition if every subgroup of order n is cyclic. Theorem (Madsen-Thomas-Wall, 1978) A finite group G acts freely on some sphere S n if and only if G satisfies p 2 - and 2 p -conditions for all primes p . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  26. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Solution Classification Fact: Milnor’s condition together with periodicity is also sufficient for a group to act freely on some S n . Let n be a positive integer. A group G is said to satisfy the n -condition if every subgroup of order n is cyclic. Theorem (Madsen-Thomas-Wall, 1978) A finite group G acts freely on some sphere S n if and only if G satisfies p 2 - and 2 p -conditions for all primes p . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  27. An Evolution of The Topological Spherical Space Form Problem The Topological Spherical Space Form Problem Solution Classification Fact: Milnor’s condition together with periodicity is also sufficient for a group to act freely on some S n . Let n be a positive integer. A group G is said to satisfy the n -condition if every subgroup of order n is cyclic. Theorem (Madsen-Thomas-Wall, 1978) A finite group G acts freely on some sphere S n if and only if G satisfies p 2 - and 2 p -conditions for all primes p . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  28. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Historical background Outline The Topological Spherical Space Form Problem 1 Group actions Solution The Topological Euclidean Space Form Problem 2 Historical background Group cohomology Cohomological dimension Solution Free and Proper Group Actions on S n × R k 3 Current results Talelli’s conjecture Groups with jump cohomology General conjecture for solvable groups Isometric actions Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  29. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Historical background Euclidean space forms Question 2. What countable groups act freely and properly discontinuously on R k ? Euclidean Space Form Problem. When does a group act freely, properly discontinuously, and isometrically on R k ? Definition Let M be Riemannian manifold. A diffeomorphism f : M → M is said to be an isometry, if � u , v � p = � df p ( u ) , df p ( v ) � f ( p ) , for all p ∈ M and u , v ∈ T p M . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  30. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Historical background Euclidean space forms Question 2. What countable groups act freely and properly discontinuously on R k ? Euclidean Space Form Problem. When does a group act freely, properly discontinuously, and isometrically on R k ? Definition Let M be Riemannian manifold. A diffeomorphism f : M → M is said to be an isometry, if � u , v � p = � df p ( u ) , df p ( v ) � f ( p ) , for all p ∈ M and u , v ∈ T p M . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  31. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Historical background Euclidean space forms Question 2. What countable groups act freely and properly discontinuously on R k ? Euclidean Space Form Problem. When does a group act freely, properly discontinuously, and isometrically on R k ? Definition Let M be Riemannian manifold. A diffeomorphism f : M → M is said to be an isometry, if � u , v � p = � df p ( u ) , df p ( v ) � f ( p ) , for all p ∈ M and u , v ∈ T p M . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  32. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Historical background Cocompact actions Isom ( R k ) ∼ = R k ⋊ O ( k ) . Theorem (Bieberbach, 1911) Let Γ act freely, properly discontinuously, and isometrically on R k such that R k / Γ is compact. Then Γ is torsion-free, Γ ∩ R k ∼ = Z k , and Γ / (Γ ∩ R k ) is finite. Geometric Reformulation Let M be a closed connected flat Riemannian manifold of dimension k . Then M admits a normal Riemannian covering by a flat k -dimensional torus. M = R k → T k → M ⇒ π 1 ( T k ) ∼ � = Z k ⊳ π 1 ( M ) . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  33. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Historical background Cocompact actions Isom ( R k ) ∼ = R k ⋊ O ( k ) . Theorem (Bieberbach, 1911) Let Γ act freely, properly discontinuously, and isometrically on R k such that R k / Γ is compact. Then Γ is torsion-free, Γ ∩ R k ∼ = Z k , and Γ / (Γ ∩ R k ) is finite. Geometric Reformulation Let M be a closed connected flat Riemannian manifold of dimension k . Then M admits a normal Riemannian covering by a flat k -dimensional torus. M = R k → T k → M ⇒ π 1 ( T k ) ∼ � = Z k ⊳ π 1 ( M ) . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  34. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Historical background Cocompact actions Isom ( R k ) ∼ = R k ⋊ O ( k ) . Theorem (Bieberbach, 1911) Let Γ act freely, properly discontinuously, and isometrically on R k such that R k / Γ is compact. Then Γ is torsion-free, Γ ∩ R k ∼ = Z k , and Γ / (Γ ∩ R k ) is finite. Geometric Reformulation Let M be a closed connected flat Riemannian manifold of dimension k . Then M admits a normal Riemannian covering by a flat k -dimensional torus. M = R k → T k → M ⇒ π 1 ( T k ) ∼ � = Z k ⊳ π 1 ( M ) . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  35. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Historical background Cocompact actions Isom ( R k ) ∼ = R k ⋊ O ( k ) . Theorem (Bieberbach, 1911) Let Γ act freely, properly discontinuously, and isometrically on R k such that R k / Γ is compact. Then Γ is torsion-free, Γ ∩ R k ∼ = Z k , and Γ / (Γ ∩ R k ) is finite. Geometric Reformulation Let M be a closed connected flat Riemannian manifold of dimension k . Then M admits a normal Riemannian covering by a flat k -dimensional torus. M = R k → T k → M ⇒ π 1 ( T k ) ∼ � = Z k ⊳ π 1 ( M ) . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  36. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Historical background Cocompact actions Isom ( R k ) ∼ = R k ⋊ O ( k ) . Theorem (Bieberbach, 1911) Let Γ act freely, properly discontinuously, and isometrically on R k such that R k / Γ is compact. Then Γ is torsion-free, Γ ∩ R k ∼ = Z k , and Γ / (Γ ∩ R k ) is finite. Geometric Reformulation Let M be a closed connected flat Riemannian manifold of dimension k . Then M admits a normal Riemannian covering by a flat k -dimensional torus. M = R k → T k → M ⇒ π 1 ( T k ) ∼ � = Z k ⊳ π 1 ( M ) . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  37. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Historical background Cocompact actions Isom ( R k ) ∼ = R k ⋊ O ( k ) . Theorem (Bieberbach, 1911) Let Γ act freely, properly discontinuously, and isometrically on R k such that R k / Γ is compact. Then Γ is torsion-free, Γ ∩ R k ∼ = Z k , and Γ / (Γ ∩ R k ) is finite. Geometric Reformulation Let M be a closed connected flat Riemannian manifold of dimension k . Then M admits a normal Riemannian covering by a flat k -dimensional torus. M = R k → T k → M ⇒ π 1 ( T k ) ∼ � = Z k ⊳ π 1 ( M ) . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  38. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Historical background Cocompact actions Isom ( R k ) ∼ = R k ⋊ O ( k ) . Theorem (Bieberbach, 1911) Let Γ act freely, properly discontinuously, and isometrically on R k such that R k / Γ is compact. Then Γ is torsion-free, Γ ∩ R k ∼ = Z k , and Γ / (Γ ∩ R k ) is finite. Geometric Reformulation Let M be a closed connected flat Riemannian manifold of dimension k . Then M admits a normal Riemannian covering by a flat k -dimensional torus. M = R k → T k → M ⇒ π 1 ( T k ) ∼ � = Z k ⊳ π 1 ( M ) . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  39. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Historical background Solution to the Euclidean Space Form Problem Theorem Let Γ act freely, properly discontinuously, and isometrically on R k . Then Γ acts freely, properly discontinuously, and isometrically on R m with compact quotient for some m ≤ k . Therefore, Γ ∩ R m ∼ = Z m , and Γ / Z m is finite. Proof sketch. The quotient R k / Γ can be deformation retracted onto a compact totally geodesic submanifold call it M . Let m = dim ( M ) . Then π 1 ( M ) = Γ acts freely, properly discontinuously, and isometrically on � M = R m . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  40. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Historical background Solution to the Euclidean Space Form Problem Theorem Let Γ act freely, properly discontinuously, and isometrically on R k . Then Γ acts freely, properly discontinuously, and isometrically on R m with compact quotient for some m ≤ k . Therefore, Γ ∩ R m ∼ = Z m , and Γ / Z m is finite. Proof sketch. The quotient R k / Γ can be deformation retracted onto a compact totally geodesic submanifold call it M . Let m = dim ( M ) . Then π 1 ( M ) = Γ acts freely, properly discontinuously, and isometrically on � M = R m . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  41. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Historical background Solution to the Euclidean Space Form Problem Theorem Let Γ act freely, properly discontinuously, and isometrically on R k . Then Γ acts freely, properly discontinuously, and isometrically on R m with compact quotient for some m ≤ k . Therefore, Γ ∩ R m ∼ = Z m , and Γ / Z m is finite. Proof sketch. The quotient R k / Γ can be deformation retracted onto a compact totally geodesic submanifold call it M . Let m = dim ( M ) . Then π 1 ( M ) = Γ acts freely, properly discontinuously, and isometrically on � M = R m . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  42. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Historical background Solution to the Euclidean Space Form Problem Theorem Let Γ act freely, properly discontinuously, and isometrically on R k . Then Γ acts freely, properly discontinuously, and isometrically on R m with compact quotient for some m ≤ k . Therefore, Γ ∩ R m ∼ = Z m , and Γ / Z m is finite. Proof sketch. The quotient R k / Γ can be deformation retracted onto a compact totally geodesic submanifold call it M . Let m = dim ( M ) . Then π 1 ( M ) = Γ acts freely, properly discontinuously, and isometrically on � M = R m . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  43. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Historical background Solution to the Euclidean Space Form Problem Theorem Let Γ act freely, properly discontinuously, and isometrically on R k . Then Γ acts freely, properly discontinuously, and isometrically on R m with compact quotient for some m ≤ k . Therefore, Γ ∩ R m ∼ = Z m , and Γ / Z m is finite. Proof sketch. The quotient R k / Γ can be deformation retracted onto a compact totally geodesic submanifold call it M . Let m = dim ( M ) . Then π 1 ( M ) = Γ acts freely, properly discontinuously, and isometrically on � M = R m . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  44. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Historical background Solution to the Euclidean Space Form Problem Theorem Let Γ act freely, properly discontinuously, and isometrically on R k . Then Γ acts freely, properly discontinuously, and isometrically on R m with compact quotient for some m ≤ k . Therefore, Γ ∩ R m ∼ = Z m , and Γ / Z m is finite. Proof sketch. The quotient R k / Γ can be deformation retracted onto a compact totally geodesic submanifold call it M . Let m = dim ( M ) . Then π 1 ( M ) = Γ acts freely, properly discontinuously, and isometrically on � M = R m . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  45. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Historical background Solution to the Euclidean Space Form Problem Theorem Let Γ act freely, properly discontinuously, and isometrically on R k . Then Γ acts freely, properly discontinuously, and isometrically on R m with compact quotient for some m ≤ k . Therefore, Γ ∩ R m ∼ = Z m , and Γ / Z m is finite. Proof sketch. The quotient R k / Γ can be deformation retracted onto a compact totally geodesic submanifold call it M . Let m = dim ( M ) . Then π 1 ( M ) = Γ acts freely, properly discontinuously, and isometrically on � M = R m . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  46. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Group cohomology Outline The Topological Spherical Space Form Problem 1 Group actions Solution The Topological Euclidean Space Form Problem 2 Historical background Group cohomology Cohomological dimension Solution Free and Proper Group Actions on S n × R k 3 Current results Talelli’s conjecture Groups with jump cohomology General conjecture for solvable groups Isometric actions Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  47. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Group cohomology K (Γ , 1 ) -complex For any discrete Γ there exists a CW-complex X which is a K (Γ , 1 ) -space. That is � Γ if i = 1 π i ( X ) = 0 otherwise Then � X is a contractible CW-complex on which Γ acts freely and properly discontinuously. Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  48. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Group cohomology K (Γ , 1 ) -complex For any discrete Γ there exists a CW-complex X which is a K (Γ , 1 ) -space. That is � Γ if i = 1 π i ( X ) = 0 otherwise Then � X is a contractible CW-complex on which Γ acts freely and properly discontinuously. Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  49. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Group cohomology K (Γ , 1 ) -complex For any discrete Γ there exists a CW-complex X which is a K (Γ , 1 ) -space. That is � Γ if i = 1 π i ( X ) = 0 otherwise Then � X is a contractible CW-complex on which Γ acts freely and properly discontinuously. Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  50. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Group cohomology K (Γ , 1 ) -complex For any discrete Γ there exists a CW-complex X which is a K (Γ , 1 ) -space. That is � Γ if i = 1 π i ( X ) = 0 otherwise Then � X is a contractible CW-complex on which Γ acts freely and properly discontinuously. Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  51. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Group cohomology Group cohomology Definition Cohomology of a group Γ with coefficients in a Γ -module M is defined as H i (Γ , M ) = H i ( X , M ) for any i ≥ 0 , where X is a K (Γ , 1 ) -complex. Ex. 6. S 1 is a K ( Z , 1 ) -complex. H i ( Z , Z ) = H i ( S 1 , Z ) , ∀ i ≥ 0 . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  52. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Group cohomology Group cohomology Definition Cohomology of a group Γ with coefficients in a Γ -module M is defined as H i (Γ , M ) = H i ( X , M ) for any i ≥ 0 , where X is a K (Γ , 1 ) -complex. Ex. 6. S 1 is a K ( Z , 1 ) -complex. H i ( Z , Z ) = H i ( S 1 , Z ) , ∀ i ≥ 0 . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  53. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Group cohomology Group cohomology Definition Cohomology of a group Γ with coefficients in a Γ -module M is defined as H i (Γ , M ) = H i ( X , M ) for any i ≥ 0 , where X is a K (Γ , 1 ) -complex. Ex. 6. S 1 is a K ( Z , 1 ) -complex. H i ( Z , Z ) = H i ( S 1 , Z ) , ∀ i ≥ 0 . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  54. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Cohomological dimension Outline The Topological Spherical Space Form Problem 1 Group actions Solution The Topological Euclidean Space Form Problem 2 Historical background Group cohomology Cohomological dimension Solution Free and Proper Group Actions on S n × R k 3 Current results Talelli’s conjecture Groups with jump cohomology General conjecture for solvable groups Isometric actions Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  55. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Cohomological dimension Cohomological dimension Definition Cohomological dimension of Γ is defined by cd (Γ) = sup { n : H n (Γ , M ) � = 0 for some Γ -module M } . cd ( Z n ) = n . Γ ′ < Γ ⇒ cd (Γ ′ ) ≤ cd (Γ) . ⇐ H ∗ (Γ ′ , M ) ∼ = H ∗ (Γ , Coind Γ Γ ′ M ) . If cd (Γ) < ∞ , then Γ is tor-free. ⇐ H 2 i ( Z m , Z ) = Z m for all i > 0. Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  56. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Cohomological dimension Cohomological dimension Definition Cohomological dimension of Γ is defined by cd (Γ) = sup { n : H n (Γ , M ) � = 0 for some Γ -module M } . cd ( Z n ) = n . Γ ′ < Γ ⇒ cd (Γ ′ ) ≤ cd (Γ) . ⇐ H ∗ (Γ ′ , M ) ∼ = H ∗ (Γ , Coind Γ Γ ′ M ) . If cd (Γ) < ∞ , then Γ is tor-free. ⇐ H 2 i ( Z m , Z ) = Z m for all i > 0. Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  57. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Cohomological dimension Cohomological dimension Definition Cohomological dimension of Γ is defined by cd (Γ) = sup { n : H n (Γ , M ) � = 0 for some Γ -module M } . cd ( Z n ) = n . Γ ′ < Γ ⇒ cd (Γ ′ ) ≤ cd (Γ) . ⇐ H ∗ (Γ ′ , M ) ∼ = H ∗ (Γ , Coind Γ Γ ′ M ) . If cd (Γ) < ∞ , then Γ is tor-free. ⇐ H 2 i ( Z m , Z ) = Z m for all i > 0. Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  58. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Cohomological dimension Cohomological dimension Definition Cohomological dimension of Γ is defined by cd (Γ) = sup { n : H n (Γ , M ) � = 0 for some Γ -module M } . cd ( Z n ) = n . Γ ′ < Γ ⇒ cd (Γ ′ ) ≤ cd (Γ) . ⇐ H ∗ (Γ ′ , M ) ∼ = H ∗ (Γ , Coind Γ Γ ′ M ) . If cd (Γ) < ∞ , then Γ is tor-free. ⇐ H 2 i ( Z m , Z ) = Z m for all i > 0. Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  59. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Cohomological dimension Cohomological dimension Definition Cohomological dimension of Γ is defined by cd (Γ) = sup { n : H n (Γ , M ) � = 0 for some Γ -module M } . cd ( Z n ) = n . Γ ′ < Γ ⇒ cd (Γ ′ ) ≤ cd (Γ) . ⇐ H ∗ (Γ ′ , M ) ∼ = H ∗ (Γ , Coind Γ Γ ′ M ) . If cd (Γ) < ∞ , then Γ is tor-free. ⇐ H 2 i ( Z m , Z ) = Z m for all i > 0. Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  60. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Cohomological dimension Cohomological dimension Definition Cohomological dimension of Γ is defined by cd (Γ) = sup { n : H n (Γ , M ) � = 0 for some Γ -module M } . cd ( Z n ) = n . Γ ′ < Γ ⇒ cd (Γ ′ ) ≤ cd (Γ) . ⇐ H ∗ (Γ ′ , M ) ∼ = H ∗ (Γ , Coind Γ Γ ′ M ) . If cd (Γ) < ∞ , then Γ is tor-free. ⇐ H 2 i ( Z m , Z ) = Z m for all i > 0. Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  61. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Cohomological dimension Geometric dimension Definition Geometric dimension of Γ is defined as gd (Γ) = inf { n : n = dim ( X ) where X is a K (Γ , 1 ) -complex } . If F is a free group, then gd ( F ) = 1. This is because F acts freely and properly discontinuously on its Cayley graph Y and Y / F is a K ( F , 1 ) -complex. cd (Γ) ≤ gd (Γ) . ⇐ H ∗ (Γ , M ) = H ∗ ( X , M ) . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  62. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Cohomological dimension Geometric dimension Definition Geometric dimension of Γ is defined as gd (Γ) = inf { n : n = dim ( X ) where X is a K (Γ , 1 ) -complex } . If F is a free group, then gd ( F ) = 1. This is because F acts freely and properly discontinuously on its Cayley graph Y and Y / F is a K ( F , 1 ) -complex. cd (Γ) ≤ gd (Γ) . ⇐ H ∗ (Γ , M ) = H ∗ ( X , M ) . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  63. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Cohomological dimension Geometric dimension Definition Geometric dimension of Γ is defined as gd (Γ) = inf { n : n = dim ( X ) where X is a K (Γ , 1 ) -complex } . If F is a free group, then gd ( F ) = 1. This is because F acts freely and properly discontinuously on its Cayley graph Y and Y / F is a K ( F , 1 ) -complex. cd (Γ) ≤ gd (Γ) . ⇐ H ∗ (Γ , M ) = H ∗ ( X , M ) . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  64. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Cohomological dimension Geometric dimension Definition Geometric dimension of Γ is defined as gd (Γ) = inf { n : n = dim ( X ) where X is a K (Γ , 1 ) -complex } . If F is a free group, then gd ( F ) = 1. This is because F acts freely and properly discontinuously on its Cayley graph Y and Y / F is a K ( F , 1 ) -complex. cd (Γ) ≤ gd (Γ) . ⇐ H ∗ (Γ , M ) = H ∗ ( X , M ) . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  65. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Cohomological dimension Geometric dimension Definition Geometric dimension of Γ is defined as gd (Γ) = inf { n : n = dim ( X ) where X is a K (Γ , 1 ) -complex } . If F is a free group, then gd ( F ) = 1. This is because F acts freely and properly discontinuously on its Cayley graph Y and Y / F is a K ( F , 1 ) -complex. cd (Γ) ≤ gd (Γ) . ⇐ H ∗ (Γ , M ) = H ∗ ( X , M ) . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  66. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Solution Outline The Topological Spherical Space Form Problem 1 Group actions Solution The Topological Euclidean Space Form Problem 2 Historical background Group cohomology Cohomological dimension Solution Free and Proper Group Actions on S n × R k 3 Current results Talelli’s conjecture Groups with jump cohomology General conjecture for solvable groups Isometric actions Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  67. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Solution Solution to the space form problem Question. What countable groups act freely and properly discontinuously on R k ? Theorem (Johnson, 1969) Let Γ be a countable group. Then, cd (Γ) < ∞ if and only if Γ acts freely, properly discontinuously, and smoothly on some R n . ( ⇐ ): If Γ acts freely and properly discontinuously on some R n , then R n / Γ has a structure of a K (Γ , 1 ) -complex. This shows cd (Γ) ≤ gd (Γ) ≤ dim ( R n / Γ) = n . forward Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  68. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Solution Solution to the space form problem Question. What countable groups act freely and properly discontinuously on R k ? Theorem (Johnson, 1969) Let Γ be a countable group. Then, cd (Γ) < ∞ if and only if Γ acts freely, properly discontinuously, and smoothly on some R n . ( ⇐ ): If Γ acts freely and properly discontinuously on some R n , then R n / Γ has a structure of a K (Γ , 1 ) -complex. This shows cd (Γ) ≤ gd (Γ) ≤ dim ( R n / Γ) = n . forward Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  69. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Solution Solution to the space form problem Question. What countable groups act freely and properly discontinuously on R k ? Theorem (Johnson, 1969) Let Γ be a countable group. Then, cd (Γ) < ∞ if and only if Γ acts freely, properly discontinuously, and smoothly on some R n . ( ⇐ ): If Γ acts freely and properly discontinuously on some R n , then R n / Γ has a structure of a K (Γ , 1 ) -complex. This shows cd (Γ) ≤ gd (Γ) ≤ dim ( R n / Γ) = n . forward Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  70. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Solution Solution to the space form problem Question. What countable groups act freely and properly discontinuously on R k ? Theorem (Johnson, 1969) Let Γ be a countable group. Then, cd (Γ) < ∞ if and only if Γ acts freely, properly discontinuously, and smoothly on some R n . ( ⇐ ): If Γ acts freely and properly discontinuously on some R n , then R n / Γ has a structure of a K (Γ , 1 ) -complex. This shows cd (Γ) ≤ gd (Γ) ≤ dim ( R n / Γ) = n . forward Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  71. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Solution Solution to the space form problem Question. What countable groups act freely and properly discontinuously on R k ? Theorem (Johnson, 1969) Let Γ be a countable group. Then, cd (Γ) < ∞ if and only if Γ acts freely, properly discontinuously, and smoothly on some R n . ( ⇐ ): If Γ acts freely and properly discontinuously on some R n , then R n / Γ has a structure of a K (Γ , 1 ) -complex. This shows cd (Γ) ≤ gd (Γ) ≤ dim ( R n / Γ) = n . forward Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  72. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Solution The forward direction ( ⇒ ): Since Γ is countable, it admits a finite dimensional free- Γ -CW-complex such that X / Γ is countable. By a result of Milnor, we can assume X / Γ is l.f. and simplicial. It is therefore isomorphic to a closed simplicial subcomplex of some R q . Let Y be a smooth regular nbhd of this subcomplex. Then Y is a smooth submanifold of R q with π 1 ( Y ) = Γ . Let W = � Y , then Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  73. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Solution The forward direction ( ⇒ ): Since Γ is countable, it admits a finite dimensional free- Γ -CW-complex such that X / Γ is countable. By a result of Milnor, we can assume X / Γ is l.f. and simplicial. It is therefore isomorphic to a closed simplicial subcomplex of some R q . Let Y be a smooth regular nbhd of this subcomplex. Then Y is a smooth submanifold of R q with π 1 ( Y ) = Γ . Let W = � Y , then Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  74. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Solution The forward direction ( ⇒ ): Since Γ is countable, it admits a finite dimensional free- Γ -CW-complex such that X / Γ is countable. By a result of Milnor, we can assume X / Γ is l.f. and simplicial. It is therefore isomorphic to a closed simplicial subcomplex of some R q . Let Y be a smooth regular nbhd of this subcomplex. Then Y is a smooth submanifold of R q with π 1 ( Y ) = Γ . Let W = � Y , then Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  75. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Solution The forward direction ( ⇒ ): Since Γ is countable, it admits a finite dimensional free- Γ -CW-complex such that X / Γ is countable. By a result of Milnor, we can assume X / Γ is l.f. and simplicial. It is therefore isomorphic to a closed simplicial subcomplex of some R q . Let Y be a smooth regular nbhd of this subcomplex. Then Y is a smooth submanifold of R q with π 1 ( Y ) = Γ . Let W = � Y , then Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  76. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Solution The forward direction ( ⇒ ): Since Γ is countable, it admits a finite dimensional free- Γ -CW-complex such that X / Γ is countable. By a result of Milnor, we can assume X / Γ is l.f. and simplicial. It is therefore isomorphic to a closed simplicial subcomplex of some R q . Let Y be a smooth regular nbhd of this subcomplex. Then Y is a smooth submanifold of R q with π 1 ( Y ) = Γ . Let W = � Y , then Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  77. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Solution End of proof W is a contractible finite dim manifold with a free and properly discontinuous and smooth action of Γ . Let n − 1 = dim ( W ) , then W × R is simply connected at infinity. Let D n ⊂ W × R . W × R − D n admits a boundary at infinity. By the h -cobordism theorem, W × R − D n ∼ = ∂ D n × [ 1 , ∞ ] . Hence, W × R ∼ = R n and has the desired action of Γ . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  78. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Solution End of proof W is a contractible finite dim manifold with a free and properly discontinuous and smooth action of Γ . Let n − 1 = dim ( W ) , then W × R is simply connected at infinity. Let D n ⊂ W × R . W × R − D n admits a boundary at infinity. By the h -cobordism theorem, W × R − D n ∼ = ∂ D n × [ 1 , ∞ ] . Hence, W × R ∼ = R n and has the desired action of Γ . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  79. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Solution End of proof W is a contractible finite dim manifold with a free and properly discontinuous and smooth action of Γ . Let n − 1 = dim ( W ) , then W × R is simply connected at infinity. Let D n ⊂ W × R . W × R − D n admits a boundary at infinity. By the h -cobordism theorem, W × R − D n ∼ = ∂ D n × [ 1 , ∞ ] . Hence, W × R ∼ = R n and has the desired action of Γ . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  80. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Solution End of proof W is a contractible finite dim manifold with a free and properly discontinuous and smooth action of Γ . Let n − 1 = dim ( W ) , then W × R is simply connected at infinity. Let D n ⊂ W × R . W × R − D n admits a boundary at infinity. By the h -cobordism theorem, W × R − D n ∼ = ∂ D n × [ 1 , ∞ ] . Hence, W × R ∼ = R n and has the desired action of Γ . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  81. An Evolution of The Topological Spherical Space Form Problem The Topological Euclidean Space Form Problem Solution End of proof W is a contractible finite dim manifold with a free and properly discontinuous and smooth action of Γ . Let n − 1 = dim ( W ) , then W × R is simply connected at infinity. Let D n ⊂ W × R . W × R − D n admits a boundary at infinity. By the h -cobordism theorem, W × R − D n ∼ = ∂ D n × [ 1 , ∞ ] . Hence, W × R ∼ = R n and has the desired action of Γ . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  82. An Evolution of The Topological Spherical Space Form Problem Free and Proper Group Actions on S n × R k Current results Outline The Topological Spherical Space Form Problem 1 Group actions Solution The Topological Euclidean Space Form Problem 2 Historical background Group cohomology Cohomological dimension Solution Free and Proper Group Actions on S n × R k 3 Current results Talelli’s conjecture Groups with jump cohomology General conjecture for solvable groups Isometric actions Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  83. An Evolution of The Topological Spherical Space Form Problem Free and Proper Group Actions on S n × R k Current results Periodic cohomology Question 3. When does a countable group act freely and properly discontinuously on S n × R k ? Lemma If Γ acts freely and properly discontinuously on S n × R k , then Γ has periodic cohomology after dimension k . Proof sketch. Let X = ( S n × R k ) / Γ . By the Gysin exact sequence, · · · → H i + n ( X , M ) → H i (Γ , M ) → H i + n + 1 (Γ , M ) → H i + n + 1 ( X , M ) → . . . Thus, H i (Γ , M ) ∼ = H i + n + 1 (Γ , M ) for all Γ -modules M and i > k . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

  84. An Evolution of The Topological Spherical Space Form Problem Free and Proper Group Actions on S n × R k Current results Periodic cohomology Question 3. When does a countable group act freely and properly discontinuously on S n × R k ? Lemma If Γ acts freely and properly discontinuously on S n × R k , then Γ has periodic cohomology after dimension k . Proof sketch. Let X = ( S n × R k ) / Γ . By the Gysin exact sequence, · · · → H i + n ( X , M ) → H i (Γ , M ) → H i + n + 1 (Γ , M ) → H i + n + 1 ( X , M ) → . . . Thus, H i (Γ , M ) ∼ = H i + n + 1 (Γ , M ) for all Γ -modules M and i > k . Dennis Dreesen, Paul Igodt*, Nansen Petrosyan An Evolution of The Topological Spherical Space Form Problem

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