Manifolds with complete metrics of positive scalar curvature

Transcription

1 Manifolds with complete metrics of positive scalar curvature Shmuel Weinberger Joint work with Stanley Chang and Guoliang Yu May 5 14, 2008

2 Classical background. Fact If S p is the scalar curvature at a point p in a manifold M n, then Vol M (B ɛ (p)) = Vol R n(b ɛ ) S p Cɛ n+2 +

3 Classical background. Fact If S p is the scalar curvature at a point p in a manifold M n, then Vol M (B ɛ (p)) = Vol R n(b ɛ ) S p Cɛ n+2 + Remark (Kazhdan-Warner) Suppose M is a compact n-manifold with n > 2.

4 Classical background. Fact If S p is the scalar curvature at a point p in a manifold M n, then Vol M (B ɛ (p)) = Vol R n(b ɛ ) S p Cɛ n+2 + Remark (Kazhdan-Warner) Suppose M is a compact n-manifold with n > 2. If f : M R is a function with f (p) < 0 for some p M, then there is a metric g so that f (p) is the scalar curvature of (M, g) at p.

5 Classical background. Fact If S p is the scalar curvature at a point p in a manifold M n, then Vol M (B ɛ (p)) = Vol R n(b ɛ ) S p Cɛ n+2 + Remark (Kazhdan-Warner) Suppose M is a compact n-manifold with n > 2. If f : M R is a function with f (p) < 0 for some p M, then there is a metric g so that f (p) is the scalar curvature of (M, g) at p. If there is a metric g with positive scalar curvature, then any function f : M R is the scalar curvature of some metric.

6 Classical background. Fact If S p is the scalar curvature at a point p in a manifold M n, then Vol M (B ɛ (p)) = Vol R n(b ɛ ) S p Cɛ n+2 + Remark (Gauss-Bonnet) If a surface Σ 2 has a complete metric of positive scalar curvature, then Σ 2 = S 2 or Σ 2 = RP 2.

7 Background. Theorem (tiyah-singer, due to Lichnerowicz) If M n is a compact spin manifold admitting a metric of positive scalar curvature, then Â(M), [M] = 0.

8 Background. Theorem (tiyah-singer, due to Lichnerowicz) If M n is a compact spin manifold admitting a metric of positive scalar curvature, then Â(M), [M] = 0. Example (K 3 Surface) K3 is spin. sign K3 = 16, so Â(K3), [K3] 0 K3 has no metric of positive scalar curvature.

9 Background. Theorem (tiyah-singer, due to Lichnerowicz) If M n is a compact spin manifold admitting a metric of positive scalar curvature, then Â(M), [M] = 0. Example (CP 2 ) CP 2 has a metric with positive scalar curvature, sign CP 2 = 1, CP 2 is not spin.

10 Background. Theorem (tiyah-singer, due to Lichnerowicz) If M n is a compact spin manifold admitting a metric of positive scalar curvature, then Idea of Proof: Â(M), [M] = 0.

11 Background. Theorem (tiyah-singer, due to Lichnerowicz) If M n is a compact spin manifold admitting a metric of positive scalar curvature, then Â(M), [M] = 0. Idea of Proof: Since M is spin, M has a Dirac operator /D.

12 Background. Theorem (tiyah-singer, due to Lichnerowicz) If M n is a compact spin manifold admitting a metric of positive scalar curvature, then Â(M), [M] = 0. Idea of Proof: Since M is spin, M has a Dirac operator /D. /D /D = + Scal.

13 Background. Theorem (tiyah-singer, due to Lichnerowicz) If M n is a compact spin manifold admitting a metric of positive scalar curvature, then Â(M), [M] = 0. Idea of Proof: Since M is spin, M has a Dirac operator /D. /D /D = + Scal. If Scal > 0, then + Scal > 0,

14 Background. Theorem (tiyah-singer, due to Lichnerowicz) If M n is a compact spin manifold admitting a metric of positive scalar curvature, then Â(M), [M] = 0. Idea of Proof: Since M is spin, M has a Dirac operator /D. /D /D = + Scal. If Scal > 0, then + Scal > 0, then ker /D = 0 and ker /D = 0,

15 Background. Theorem (tiyah-singer, due to Lichnerowicz) If M n is a compact spin manifold admitting a metric of positive scalar curvature, then Â(M), [M] = 0. Idea of Proof: Since M is spin, M has a Dirac operator /D. /D /D = + Scal. If Scal > 0, then + Scal > 0, then ker /D = 0 and ker /D = 0, then ind /D = ker /D ker /D = 0,

16 Background. Theorem (tiyah-singer, due to Lichnerowicz) If M n is a compact spin manifold admitting a metric of positive scalar curvature, then Idea of Proof: Â(M), [M] = 0. Since M is spin, M has a Dirac operator /D. /D /D = + Scal. If Scal > 0, then + Scal > 0, then ker /D = 0 and ker /D = 0, then ind /D = ker /D ker /D = 0, so Â(M), [M] = 0 by tiyah-singer.

17 Simply connected case. Theorem (Gromov-Lawson, Hitchin, Stolz) If M n with n > 4 and M simply connected, then M has a metric of positive scalar curvature iff

18 Simply connected case. Theorem (Gromov-Lawson, Hitchin, Stolz) If M n with n > 4 and M simply connected, then M has a metric of positive scalar curvature iff M is not spin, or

19 Simply connected case. Theorem (Gromov-Lawson, Hitchin, Stolz) If M n with n > 4 and M simply connected, then M has a metric of positive scalar curvature iff M is not spin, or M is spin, and Â(M), [M] = 0?

20 Simply connected case. Theorem (Gromov-Lawson, Hitchin, Stolz) If M n with n > 4 and M simply connected, then M has a metric of positive scalar curvature iff M is not spin, or M is spin, and ind /D = 0 KO n (pt).

21 Non-simply connected case. Question Does T n have a metric of positive scalar curvature?

22 Non-simply connected case. Question Does T n have a metric of positive scalar curvature? Theorem (Schoen-Yau for n 7, Gromov-Lawson for all n) T n has no metric of positive scalar curvature.

23 Non-simply connected case. Question Does T n have a metric of positive scalar curvature? Theorem (Schoen-Yau for n 7, Gromov-Lawson for all n) T n has no metric of positive scalar curvature. Theorem (Gromov-Lawson) K\G/Γ, a compact locally symmetric space, has no complete metric of positive scalar curvature.

24 Non-simply connected case. Question Does T n have a metric of positive scalar curvature? Theorem (Schoen-Yau for n 7, Gromov-Lawson for all n) T n has no metric of positive scalar curvature. Theorem (Gromov-Lawson) K\G/Γ, a compact locally symmetric space, has no complete metric of positive scalar curvature. dditionally, noncompact hyperbolic manifolds do not have metrics of positive scalar curvature.

25 Non-simply connected case. Question Does T n have a metric of positive scalar curvature? Theorem (Schoen-Yau for n 7, Gromov-Lawson for all n) T n has no metric of positive scalar curvature. Theorem (Gromov-Lawson) K\G/Γ, a compact locally symmetric space, has no complete metric of positive scalar curvature. dditionally, noncompact hyperbolic manifolds do not have metrics of positive scalar curvature. Proof. Following Rosenberg, same as before but using K(C π).

26 What happens for interiors of manifolds with boundary? M M M is the interior of a manifold with boundary.

27 What happens for interiors of manifolds with boundary? M M M is the interior of a manifold with boundary. The boundary has a fundamental group π 1 M.

28 What happens for interiors of manifolds with boundary? M M M is the interior of a manifold with boundary. The boundary has a fundamental group π 1 M. This suggests an obstruction in KO(Bπ 1 M, Bπ 1 M).

29 What happens for interiors of manifolds with boundary? M M M is the interior of a manifold with boundary. The boundary has a fundamental group π 1 M. This suggests an obstruction in KO(Bπ 1 M, Bπ 1 M). ssembly map for pairs into L-theory. But in the C -algebra setting only works really well if the fundamental group injects.

30 What happens for interiors of manifolds with boundary? M M M is the interior of a manifold with boundary. The boundary has a fundamental group π 1 M. This suggests an obstruction in KO(Bπ 1 M, Bπ 1 M). ssembly map for pairs into L-theory. But in the C -algebra setting only works really well if the fundamental group injects. One mystery of the Baum-Conjecture is the functoriality aspect (even in the torsion free case.) Why should there be functoriality associated to homomorphisms?

31 Proper homotopy equivalence of non-compact manifolds. Definition M is simply connected at infinity if every compact K M is contained in a larger compact C K, so that M C is simply connected.

32 Proper homotopy equivalence of non-compact manifolds. Definition M is simply connected at infinity if every compact K M is contained in a larger compact C K, so that M C is simply connected. Theorem (Browder-Livesay-Levine) M n, n > 5, is the interior of a manifold with simply connected boundary iff M has finitely generated homology and M is simply connected at infinity.

33 The case of lattices. Theorem (Block-W) K\G/Γ has a complete metric of positive scalar curvature iff Q-rk(Γ) > 2.

34 The case of lattices. Theorem (Block-W) K\G/Γ has a complete metric of positive scalar curvature iff Q-rk(Γ) > 2. Theorem (Chang) K\G/Γ never has a complete metric of positive scalar curvature in the obvious QI class. Idea of Proof:

35 The case of lattices. Theorem (Block-W) K\G/Γ has a complete metric of positive scalar curvature iff Q-rk(Γ) > 2. Theorem (Chang) K\G/Γ never has a complete metric of positive scalar curvature in the obvious QI class. Idea of Proof: Marry Novikov idea to Roe s partitioned manifold index theorem.

36 The case of lattices. Theorem (Block-W) K\G/Γ has a complete metric of positive scalar curvature iff Q-rk(Γ) > 2. Theorem (Chang) K\G/Γ never has a complete metric of positive scalar curvature in the obvious QI class. Idea of Proof: Marry Novikov idea to Roe s partitioned manifold index theorem. We will discuss it in more detail later.

37 Fundamental group at infinity. Definition (Fundamental group at infinity) K 1 K 2 K 3 K 4 K 5 K 1 K 2 K 3 M

38 Fundamental group at infinity. Definition (Fundamental group at infinity) K 1 K 2 K 3 K 4 K 5 K 1 K 2 K 3 M π 1 (M) = Γ means that the pro-system π 1 (M K 1 ) π 1 (M K 2 ) π 1 (M K 3 )

39 Fundamental group at infinity. Definition (Fundamental group at infinity) K 1 K 2 K 3 K 4 K 5 K 1 K 2 K 3 M π 1 (M) = Γ means that the pro-system π 1 (M K 1 ) π 1 (M K 2 ) π 1 (M K 3 ) is pro-equivalent to the constant system Γ Γ.

40 Proper homotopy equivalence of non-compact manifolds. Theorem (Siebenmann s thesis) The obstruction to putting a boundary on a tame manifold lies in K 0 (Zπ 1 ).

41 Proper homotopy equivalence of non-compact manifolds. Theorem (Siebenmann s thesis) The obstruction to putting a boundary on a tame manifold lies in K 0 (Zπ 1 ). Takes the fear out of non-compactness when you are tame.

42 Proper homotopy equivalence of non-compact manifolds. Theorem (Siebenmann s thesis) The obstruction to putting a boundary on a tame manifold lies in K 0 (Zπ 1 ). Takes the fear out of non-compactness when you are tame. If tame at infinity, then there is a relative assembly theory, relative Novikov conjecture for L-classes and so on.

43 Proper homotopy equivalence of non-compact manifolds. Theorem (Siebenmann s thesis) The obstruction to putting a boundary on a tame manifold lies in K 0 (Zπ 1 ). Takes the fear out of non-compactness when you are tame. If tame at infinity, then there is a relative assembly theory, relative Novikov conjecture for L-classes and so on. If not, it s somewhat harder to describe the relevant assembly maps that enter the theory, but not impossible.

44 To what extent does the theory of pairs capture the issues? Question To what extent does the theory of pairs capture the issues? K 1 K 2 K 3 K 4 K 5

45 To what extent does the theory of pairs capture the issues? K 1 K 2 K 3 K 4 K 5 nswer In the findamental group tame case, pretty well

46 To what extent does the theory of pairs capture the issues? K 1 K 2 K 3 K 4 K 5 nswer In the findamental group tame case, pretty well but not in general.

47 To what extent does the theory of pairs capture the issues? K 1 K 2 K 3 K 4 K 5 nswer In the findamental group tame case, pretty well but not in general. There are lim 1 terms, 0 lim 1 H lf 1(K i ) H lf (M) lim H lf (K i ) 0,

48 To what extent does the theory of pairs capture the issues? K 1 K 2 K 3 K 4 K 5 nswer In the findamental group tame case, pretty well but not in general. There are lim 1 terms, 0 lim 1 H lf 1(K i ) H lf (M) lim H lf (K i ) 0, and other terms measured off group homology s limits.

49 Key example: Whitehead manifold. Example h 1 (S 1 D 2 ),

50 Key example: Whitehead manifold. Example h 1 (S 1 D 2 ), h 2 (S 1 D 2 ),

51 Key example: Whitehead manifold. Example h 1 (S 1 D 2 ), h 2 (S 1 D 2 ), h 3 (S 1 D 2 ),...

52 Key example: Whitehead manifold. Whitehead = S 3 \ i hi (S 1 D 2 ).

53 Key example: Whitehead manifold. Whitehead = S 3 \ hi (S 1 D 2 ). i Remark Whitehead 6 R3, but R Whitehead = = R4.

54 Key example: Whitehead manifold. Whitehead = S 3 i h i (S 1 D 2 ). Remark There are uncountably many variants of this construction.

55 Key example: Whitehead manifold. Whitehead = S 3 \ hi (S 1 D 2 ). i Question What does the moduli space of these manifolds look like?

56 Key example: Whitehead manifold. Whitehead = S 3 \ hi (S 1 D 2 ). i Question What does the moduli space of these manifolds look like? little bit like the space of Penrose tilings.

57 Key example: Whitehead manifold. Whitehead = S 3 \ i Remark No nice metric on these manifolds. hi (S 1 D 2 ).

58 Observations about the Whitehead manifold. S 1 D 2

59 Observations about the Whitehead manifold. S 1 D 2 is aspherical

60 Observations about the Whitehead manifold. S 1 D 2 is aspherical (by Papakyriakopoulos sphere theorem, because is the complement of a non-split link)

61 Observations about the Whitehead manifold. S 1 D 2 is aspherical (by Papakyriakopoulos sphere theorem, because is the complement of a non-split link) Naively construed, π1 is trivial.

62 The Whitehead manifold and positive scalar curvature. Theorem The Whitehead manifold has no complete metric of positive scalar curvature. Proof:

63 The Whitehead manifold and positive scalar curvature. Proof: S 1 D 2

64 The Whitehead manifold and positive scalar curvature. Proof: S 1 D 2

65 The Whitehead manifold and positive scalar curvature. Proof: Ā Ā Ā

66 The Whitehead manifold and positive scalar curvature. Proof: Ā Ā Ā DW = double of Whitehead manifold along

67 The Whitehead manifold and positive scalar curvature. Proof: Ā Ā Ā DW = double of Whitehead manifold along DW has a positive scalar curvature metric except at Ā

68 The Whitehead manifold and positive scalar curvature. Proof: Ā Ā Ā DW = double of Whitehead manifold along DW has a positive scalar curvature metric except at Ā DW has positive scalar curvature metric at infinity

69 The Whitehead manifold and positive scalar curvature. Proof: Ā Ā Ā DW = double of Whitehead manifold along DW has a positive scalar curvature metric except at Ā DW has positive scalar curvature metric at infinity a contradiction.

70 Digression: Roe s Partition Manifold Theorem. Theorem (Roe) If V is spin and Z positive scalar curvature at infinity, then ind = 0. Modern Philosophy Z V

71 Digression: Roe s Partition Manifold Theorem. Theorem (Roe) If V is spin and Z positive scalar curvature at infinity, then ind = 0. Modern Philosophy The partition defines a virtual vector bundle on the space at infinity. Z V

72 Digression: Roe s Partition Manifold Theorem. Theorem (Roe) If V is spin and Z positive scalar curvature at infinity, then ind = 0. Modern Philosophy The partition defines a virtual vector bundle on the space at infinity. Only the ends of the space at infinity are independent of the quasi-isometry class of the metric. Z V

73 The non-simply connected case. Ā Ā Ā If V is not simply connected, attractive to couple Roe s theorem to C (π 1 V ).

74 The non-simply connected case. Ā Ā Ā If V is not simply connected, attractive to couple Roe s theorem to C (π 1 V ). In this case, V =.

75 The non-simply connected case. Ā Ā Ā If V is not simply connected, attractive to couple Roe s theorem to C (π 1 V ). In this case, V =. But [ /D] K 2 ( ) dies on pushing forward by K 2 ( ) K 2 (pt),

76 The non-simply connected case. Ā Ā Ā If V is not simply connected, attractive to couple Roe s theorem to C (π 1 V ). In this case, V =. But [ /D] K 2 ( ) dies on pushing forward by K 2 ( ) K 2 (pt), so /D on does not obstruct positive scalar curvature.

77 The non-simply connected case. Ā Ā Ā If V is not simply connected, attractive to couple Roe s theorem to C (π 1 V ). In this case, V =. But [ /D] K 2 ( ) dies on pushing forward by K 2 ( ) K 2 (pt), so /D on does not obstruct positive scalar curvature. On the other hand, K 2 ( ) K 2 (C (Z 2 )) is injective.

78 Tilt horizontal direction to the vertical. Idea Use π 1 (DW ) rather than π 1 ( ) = Z 2. Question Do we know strong Novikov conjecture for π 1 (DW )? DW aspherical. H 2 ( ) H 2 (DW ) K 2 ( ) K 2 (DW ) K 2 (π 1 DW ) Theorem (Connes-Gromov-Mascovicci) Novikov conjecture holds for all 2-dimensional cohomology classes.

79 Taming 3-manifolds. Theorem If M 3 is of finite type and has positive scalar curvature at infinity, then M is the interior of a manifold with boundary.

80 Taming 3-manifolds. Theorem If M 3 is of finite type and has positive scalar curvature at infinity, then M is the interior of a manifold with boundary. Example S 3 cantor set = L 3 #L 3.

81 Taming 3-manifolds. Theorem If M 3 is of finite type and has positive scalar curvature at infinity, then M is the interior of a manifold with boundary. Example S 3 cantor set = L 3 #L 3. Proof: Whitehead case applies.

82 Taming 3-manifolds. Theorem If M 3 is of finite type and has positive scalar curvature at infinity, then M is the interior of a manifold with boundary. Example S 3 cantor set = L 3 #L 3. Proof: Whitehead case applies. Perelman is used

83 Taming 3-manifolds. Theorem If M 3 is of finite type and has positive scalar curvature at infinity, then M is the interior of a manifold with boundary. Example S 3 cantor set = L 3 #L 3. Proof: Whitehead case applies. Perelman is used though Hamilton is probably enough.

84 Contractible manifolds and positive scalar curvature. Theorem For all n, there is a contractible manifold M n having no complete metric of positive scalar curvature.

85 Contractible manifolds and positive scalar curvature. Theorem For all n, there is a contractible manifold M n having no complete metric of positive scalar curvature. Proof: n = 1 R.

86 Contractible manifolds and positive scalar curvature. Theorem For all n, there is a contractible manifold M n having no complete metric of positive scalar curvature. Proof: n = 1 R. n = 2 R 2.

87 Contractible manifolds and positive scalar curvature. Theorem For all n, there is a contractible manifold M n having no complete metric of positive scalar curvature. Proof: n = 3 The Whitehead manifold.

88 Contractible manifolds and positive scalar curvature. Theorem For all n, there is a contractible manifold M n having no complete metric of positive scalar curvature. Proof: n = 3 The Whitehead manifold. n = 4 The Mazur manifold. D 2 D 2 S 1 D 3 S 1

89 Contractible manifolds and positive scalar curvature. Theorem For all n, there is a contractible manifold M n having no complete metric of positive scalar curvature. Proof: n = 3 The Whitehead manifold. n = 4 The Mazur manifold. n > 4 Variations on the Mazur manifold.

90 Dimension four? Question re there are interesting 4-manifolds that have complete metrics of positive scalar curvature?

91 Dimension four? Question re there are interesting 4-manifolds that have complete metrics of positive scalar curvature? Question Is R 4 the only contractible 4-manifold with positive scalar curvature?