The Milnor 7-sphere does not admit a special generic map into R 3

Transcription

1 The Milnor 7-sphere does not admit a special generic map into R 3 Dominik Wrazidlo Institute of Mathematics for Industry (IMI) Kyushu University September 16, 2018

2 Introduction M n : connected closed smooth n-manifold, n 1

3 Introduction M n : connected closed smooth n-manifold, n 1 Definition A smooth map f : M n R p, p {1,..., n}, is called special generic if for every singular point x of f there exist local charts (x 1,..., x n ) centered at x and (y 1,..., y p ) centered at f (x) in which f takes the form (x 1,..., x n ) (y 1,..., y p ) = (x 1,..., x p 1, x 2 p + + x 2 n).

4 Introduction E. Calabi, Quasi-surjective mappings and a generalization of Morse theory, Proc. U.S.-Japan Seminar in Differential Geometry, Kyoto (1965),

5 Introduction E. Calabi, Quasi-surjective mappings and a generalization of Morse theory, Proc. U.S.-Japan Seminar in Differential Geometry, Kyoto (1965), Example (p = 1) f : M n R is a s.g.m. f is a Morse function with only minima and maxima f is a Morse function with exactly 2 critical points

6 Introduction E. Calabi, Quasi-surjective mappings and a generalization of Morse theory, Proc. U.S.-Japan Seminar in Differential Geometry, Kyoto (1965), Example (p = 1) f : M n R is a s.g.m. f is a Morse function with only minima and maxima f is a Morse function with exactly 2 critical points For n 4, M n admits a s.g.m. into R M n is homeomorphic to S n

7 Introduction E. Calabi, Quasi-surjective mappings and a generalization of Morse theory, Proc. U.S.-Japan Seminar in Differential Geometry, Kyoto (1965), Example (p = 1) f : M n R is a s.g.m. f is a Morse function with only minima and maxima f is a Morse function with exactly 2 critical points For n 4, M n admits a s.g.m. into R M n is homeomorphic to S n Example (M n = S n ) For every p {1,..., n}, S n R n+1 pr R p is a s.g.m.

8 Introduction History of special generic maps: E. Calabi (1965): (perhaps) first appearance of special generic maps in the literature O. Burlet and G. de Rham (1974): formulated notion of special generic maps of 3-manifolds into R 2, and introduced Stein factorization as an important tool for studying s.g.m. P. Porto and Y.K.S. Furuya (1990): extended notion of special generic maps to higher dimensions since 1990 s: study of topology of special generic maps by K. Sakuma, O. Saeki, Y. Hara, R. Sadykov, N. Kitazawa,...

9 Introduction M n : connected closed smooth n-manifold, n 1

10 Introduction M n : connected closed smooth n-manifold, n 1 Problem. Determine the smallest integer p > n for which there exists an immersion/embedding M n R p.

11 Introduction M n : connected closed smooth n-manifold, n 1 Problem. Determine the smallest integer p > n for which there exists an immersion/embedding M n R p. Question. Can an analogous problem be studied for certain smooth maps f : M n R p, where p n?

12 Introduction M n : connected closed smooth n-manifold, n 1 Problem. Determine the smallest integer p > n for which there exists an immersion/embedding M n R p. Question. Can an analogous problem be studied for certain smooth maps f : M n R p, where p n? cannot avoid f having singularities since M n is closed

13 Introduction M n : connected closed smooth n-manifold, n 1 Problem. Determine the smallest integer p > n for which there exists an immersion/embedding M n R p. Question. Can an analogous problem be studied for certain smooth maps f : M n R p, where p n? cannot avoid f having singularities since M n is closed need to consider all target dimensions separately

14 Introduction M n : connected closed smooth n-manifold, n 1 Problem. Determine the smallest integer p > n for which there exists an immersion/embedding M n R p. Question. Can an analogous problem be studied for certain smooth maps f : M n R p, where p n? cannot avoid f having singularities since M n is closed need to consider all target dimensions separately Problem (O. Saeki, 1993). Study the set S(M n ) = {p {1,..., n} s.g.m. M n R p }.

15 Special Generic Maps on Homotopy Spheres Theorem (O. Saeki, 1993) For an exotic n-sphere Σ n, n 7, {1, 2, n} S(Σ n ) {1, 2,..., n 4, n}.

16 Special Generic Maps on Homotopy Spheres Theorem (O. Saeki, 1993) For an exotic n-sphere Σ n, n 7, {1, 2, n} S(Σ n ) {1, 2,..., n 4, n}. Corollary For a homotopy sphere M n of dimension n 7, M n = S n S(M n ) = {1, 2,..., n}.

17 Special Generic Maps on Homotopy Spheres Theorem (O. Saeki, 1993) For an exotic n-sphere Σ n, n 7, {1, 2, n} S(Σ n ) {1, 2,..., n 4, n}. Corollary For a homotopy sphere M n of dimension n 7, M n = S n S(M n ) = {1, 2,..., n}. Problem. Find exotic n-spheres Σ n 1 and Σn 2 such that S(Σ n 1) S(Σ n 2).

18 Main Result Theorem (M. Weiss, 1993) If M n is a homotopy sphere of dimension n 7, then Gromoll filtration of M n (Morse perfection of M n ) + 1.

19 Main Result Theorem (M. Weiss, 1993) If M n is a homotopy sphere of dimension n 7, then Gromoll filtration of M n (Morse perfection of M n ) + 1. Theorem (W., 2017) Let M n denote a homotopy sphere of dimension n 7. Then, (a) Gromoll filtration of M n fold perfection of M n, (b) fold perfection of M n (Morse perfection of M n ) + 1.

20 Gromoll Filtration ϕ D n D n M n = D n ϕ D n (twisted sphere) ϕ: D n = D n or. pres.

21 Gromoll Filtration ϕ D n D n M n = D n ϕ D n (twisted sphere) ϕ: D n = D n or. pres. The Gromoll filtration of M n is the greatest integer k 1 for which the diffeomorphism ϕ can be chosen to be of the form ϕ = ϕ g : D n 1 n 1 g id D D n 1 D n 1, where g : D n 1 = D n 1 is an or. pres. diffeomorphism such that g = id D n 1 near D n 1, and g i (z 1,..., z n 1 ) = z i for i = 1,..., k 1.

22 Gromoll Filtration A Riemannian manifold X n is called globally δ-pinched if all sectional curvatures at all points of X lie in the interval (1, 1/δ].

23 Gromoll Filtration A Riemannian manifold X n is called globally δ-pinched if all sectional curvatures at all points of X lie in the interval (1, 1/δ]. Theorem (H.E. Rauch, M. Berger, W. Klingenberg) If X n is a complete simply connected Riemannian manifold that is globally 1/4-pinched, then X n is homeomorphic to the sphere S n.

24 Gromoll Filtration A Riemannian manifold X n is called globally δ-pinched if all sectional curvatures at all points of X lie in the interval (1, 1/δ]. Theorem (H.E. Rauch, M. Berger, W. Klingenberg) If X n is a complete simply connected Riemannian manifold that is globally 1/4-pinched, then X n is homeomorphic to the sphere S n. Theorem (D. Gromoll, 1966) There exists a sequence δ k 1 (k ), 1/4 = δ 1 < < δ k < < 1, such that whenever X n is a complete simply connected Riemannian manifold of dimension n 7 that is globally δ k -pinched for some k {1,..., n}, then X n has Gromoll filtration k.

25 Morse Perfection The Morse perfection of M n is the greatest integer k 1 for which there exists a smooth map such that η : S k M n R

26 Morse Perfection The Morse perfection of M n is the greatest integer k 1 for which there exists a smooth map such that η : S k M n R η restricts for every s S k to a Morse function η s : M n R, η s (x) = η(s, x), with exactly 2 critical points, and

27 Morse Perfection The Morse perfection of M n is the greatest integer k 1 for which there exists a smooth map such that η : S k M n R η restricts for every s S k to a Morse function η s : M n R, η s (x) = η(s, x), with exactly 2 critical points, and η s = η s for every s S k.

28 Fold Perfection Definition (first version) A s.g.m. f 0 : M n R p (p < n) is called standard if there is a smooth homotopy of s.g.m. f t : M n R p, t [0, 1], such that f 1 (M n ) = D p, and every fiber f 1 1 (y), y D p, is connected.

29 Fold Perfection Definition (first version) A s.g.m. f 0 : M n R p (p < n) is called standard if there is a smooth homotopy of s.g.m. f t : M n R p, t [0, 1], such that f 1 (M n ) = D p, and every fiber f 1 1 (y), y D p, is connected. Proposition If M n admits a standard s.g.m. into R p, then M n also admits a standard s.g.m. into R q for all q < p.

30 Fold Perfection Definition (first version) A s.g.m. f 0 : M n R p (p < n) is called standard if there is a smooth homotopy of s.g.m. f t : M n R p, t [0, 1], such that f 1 (M n ) = D p, and every fiber f 1 1 (y), y D p, is connected. Proposition If M n admits a standard s.g.m. into R p, then M n also admits a standard s.g.m. into R q for all q < p. Definition The fold perfection of a homotopy n-sphere M n (n 7) is the greatest integer p {1,..., n 1} for which there exists a standard s.g.m. M n R p.

31 Fold Perfection Definition (first version) A s.g.m. f 0 : M n R p (p < n) is called standard if there is a smooth homotopy of s.g.m. f t : M n R p, t [0, 1], such that f 1 (M n ) = D p, and every fiber f 1 1 (y), y D p, is connected. Proposition If M n admits a standard s.g.m. into R p, then M n also admits a standard s.g.m. into R q for all q < p. Definition The fold perfection of a homotopy n-sphere M n (n 7) is the greatest integer p {1,..., n 1} for which there exists a standard s.g.m. M n R p. Example Every homotopy n-sphere M n (n 7) has fold perfection 2.

32 Gromoll Filtration of Σ n Fold Perfection of Σ n D n 1 π n+1 k+1 D k+1 ϕ g π n k D k D n 2 π n+1 k+1 π n 1 k g = π n 1 k = π n k ϕ g = π n k

33 Fold Perfection of Σ n (Morse Perfection of Σ n ) + 1 s S p 1 D p Σ n s.s.g.m. η s 1 0 R, s 1

34 Definition of Stein Factorization The Stein factorization of a map f : X Y is the quotient W f := X / f of X by the following equivalence relation: x 1 f x 2 : y := f (x 1 ) = f (x 2 ), and x 1, x 2 belong to the same component of f 1 (y).

35 Definition of Stein Factorization The Stein factorization of a map f : X Y is the quotient W f := X / f of X by the following equivalence relation: x 1 f x 2 : y := f (x 1 ) = f (x 2 ), and x 1, x 2 belong to the same component of f 1 (y). The quotient map q f : X X / f induces a factorization of f via X f Y q f f W f = X / f.

36 Example: Height Function on the 2-Torus T 2 R f q f f W f

37 Stein Factorization of Special Generic Maps Proposition If f : M n R p (p < n) is a special generic map, then (i) W f is a smooth p-manifold with boundary W f, (ii) q f is smooth, (iii) f is an immersion, (iv) the singular locus of f is S(f ) = q 1 f ( W f ), (v) q f restricts to a diffeomorphism S(f ) = W f, and = (vi) q f induces an isomorphism q f : π 1 (M) π 1 (W f ). M n f R p q f f W f.

38 Stein Factorization and Homotopy Spheres Proposition (O. Saeki, 1993) Let f : M n R p (p < n) be a s.g.m. Then, M n is a homotopy sphere if and only if the Stein factorization W f is contractible.

39 Stein Factorization and Homotopy Spheres Proposition (O. Saeki, 1993) Let f : M n R p (p < n) be a s.g.m. Then, M n is a homotopy sphere if and only if the Stein factorization W f is contractible. Example Any contractible smooth manifold Q p can be realized as some W f!

40 Stein Factorization and Homotopy Spheres Proposition (O. Saeki, 1993) Let f : M n R p (p < n) be a s.g.m. Then, M n is a homotopy sphere if and only if the Stein factorization W f is contractible. Example Any contractible smooth manifold Q p can be realized as some W f! In fact, Σ n = (Q p D n p+1 ) admits a s.g.m. f : Σ n R p such that W f = Q.

41 Stein Factorization and Homotopy Spheres Proposition (O. Saeki, 1993) Let f : M n R p (p < n) be a s.g.m. Then, M n is a homotopy sphere if and only if the Stein factorization W f is contractible. Example Any contractible smooth manifold Q p can be realized as some W f! In fact, Σ n = (Q p D n p+1 ) admits a s.g.m. f : Σ n R p such that W f = Q. S(f ) must be a homology sphere, but if π 1 ( Q) 1, then S(f ) = W f = Q is not a homotopy sphere!

42 Stein Factorization and Homotopy Spheres Proposition (O. Saeki, 1993) Let f : M n R p (p < n) be a s.g.m. Then, M n is a homotopy sphere if and only if the Stein factorization W f is contractible. Example Any contractible smooth manifold Q p can be realized as some W f! In fact, Σ n = (Q p D n p+1 ) admits a s.g.m. f : Σ n R p such that W f = Q. S(f ) must be a homology sphere, but if π 1 ( Q) 1, then S(f ) = W f = Q is not a homotopy sphere! Definition (alternative version) A s.g.m. f : M n R p (p < n) is called standard if the Stein factorization W f is diffeomorphic to D p.

43 Special Generic Maps on Fake Real Projective Spaces Proposition (Saeki-Sakuma-W., 2018) If P n is a closed smooth n-manifold which is homotopy equivalent to RP n, then { S(P n {n}, if P n is stably parallelizable, ) =, else.

44 Special Generic Maps on Fake Real Projective Spaces Proposition (Saeki-Sakuma-W., 2018) If P n is a closed smooth n-manifold which is homotopy equivalent to RP n, then { S(P n {n}, if P n is stably parallelizable, ) =, else. Proof. Observation: The universal cover of P n is a homotopy sphere, and the covering map π : Σ n P n has degree 2.

45 Special Generic Maps on Fake Real Projective Spaces Proposition (Saeki-Sakuma-W., 2018) If P n is a closed smooth n-manifold which is homotopy equivalent to RP n, then { S(P n {n}, if P n is stably parallelizable, ) =, else. Proof. Observation: The universal cover of P n is a homotopy sphere, and the covering map π : Σ n P n has degree 2. Proof splits up into two claims: (1) S(P n ) {n} (2) n S(P n ) P n is stably parallelizable

46 Special Generic Maps on Fake Real Projective Spaces (1) S(P n ) {n}. (2) n S(P n ) P n is stably parallelizable.

47 Special Generic Maps on Fake Real Projective Spaces (1) S(P n ) {n}. Suppose there is a s.g.m. f : P n R p, p < n. (2) n S(P n ) P n is stably parallelizable.

48 Special Generic Maps on Fake Real Projective Spaces (1) S(P n ) {n}. Suppose there is a s.g.m. f : P n R p, p < n. The composition g = f π : Σ n R p is a special generic map whose Stein factorization W g is contractible. (2) n S(P n ) P n is stably parallelizable.

49 Special Generic Maps on Fake Real Projective Spaces (1) S(P n ) {n}. Suppose there is a s.g.m. f : P n R p, p < n. The composition g = f π : Σ n R p is a special generic map whose Stein factorization W g is contractible. The covering map π : Σ n P n of degree 2 induces a covering map between Stein factorizations W g W f of degree 2. (2) n S(P n ) P n is stably parallelizable.

50 Special Generic Maps on Fake Real Projective Spaces (1) S(P n ) {n}. Suppose there is a s.g.m. f : P n R p, p < n. The composition g = f π : Σ n R p is a special generic map whose Stein factorization W g is contractible. The covering map π : Σ n P n of degree 2 induces a covering map between Stein factorizations W g W f of degree 2. Contradiction: 1 = χ(w g ) = 2 χ(w f ). (2) n S(P n ) P n is stably parallelizable.

51 Special Generic Maps on Fake Real Projective Spaces (1) S(P n ) {n}. Suppose there is a s.g.m. f : P n R p, p < n. The composition g = f π : Σ n R p is a special generic map whose Stein factorization W g is contractible. The covering map π : Σ n P n of degree 2 induces a covering map between Stein factorizations W g W f of degree 2. Contradiction: 1 = χ(w g ) = 2 χ(w f ). (2) n S(P n ) P n is stably parallelizable. n odd: P n orientable claim holds by Eliashberg (1970)

52 Special Generic Maps on Fake Real Projective Spaces (1) S(P n ) {n}. Suppose there is a s.g.m. f : P n R p, p < n. The composition g = f π : Σ n R p is a special generic map whose Stein factorization W g is contractible. The covering map π : Σ n P n of degree 2 induces a covering map between Stein factorizations W g W f of degree 2. Contradiction: 1 = χ(w g ) = 2 χ(w f ). (2) n S(P n ) P n is stably parallelizable. n odd: P n orientable claim holds by Eliashberg (1970) n even: P n non-orientable P n not stably parallelizable; f : P n R n s.g.m. S(f ) stably parallelizable, χ(s(f )) even, but χ(s(f )) χ(p n ) (mod 2) by Fukuda (1985)

53 Application: Milnor Spheres Suppose n = 4k 1 for some integer k 2.

54 Application: Milnor Spheres Suppose n = 4k 1 for some integer k 2. Definition The Milnor n-sphere Σ n M is the unique homotopy n-sphere satisfying Σ n M = W n+1 for some parallelizable cobordism W n+1 of signature σ(w n+1 ) = 8.

55 Application: Milnor Spheres Suppose n = 4k 1 for some integer k 2. Definition The Milnor n-sphere Σ n M is the unique homotopy n-sphere satisfying Σ n M = W n+1 for some parallelizable cobordism W n+1 of signature σ(w n+1 ) = 8. Theorem (M. Weiss, 1993) Gromoll filtration of Σ n M = 2 = (Morse perfection of Σn M ) +1.

56 Application: Milnor Spheres Suppose n = 4k 1 for some integer k 2. Definition The Milnor n-sphere Σ n M is the unique homotopy n-sphere satisfying Σ n M = W n+1 for some parallelizable cobordism W n+1 of signature σ(w n+1 ) = 8. Theorem (M. Weiss, 1993) Gromoll filtration of Σ n M = 2 = (Morse perfection of Σn M ) +1. Corollary (W., 2017) (a) The fold perfection of Σ n M is 2. (b) Moreover, since every s.g.m. f : Σ n M R3 is standard, we have S(Σ 7 M ) = {1, 2, 7}.

57 Filtrations of Groups of Homotopy Spheres Θ n : group of homotopy n-spheres (Kervaire-Milnor, 1963)

58 Filtrations of Groups of Homotopy Spheres Θ n : group of homotopy n-spheres (Kervaire-Milnor, 1963) Problem. Compare the Gromoll filtration Γ n n 1 Γ n 1 = Θ n to the fold filtration Fn 1 n F1 n = Θ n.

59 Filtrations of Groups of Homotopy Spheres Θ n : group of homotopy n-spheres (Kervaire-Milnor, 1963) Problem. Compare the Gromoll filtration to the fold filtration Γ n n 1 Γ n 1 = Θ n F n n 1 F n 1 = Θ n. Facts: Γ n k and F n k are subgroup filtrations of Θ n.

60 Filtrations of Groups of Homotopy Spheres Θ n : group of homotopy n-spheres (Kervaire-Milnor, 1963) Problem. Compare the Gromoll filtration to the fold filtration Γ n n 1 Γ n 1 = Θ n F n n 1 F n 1 = Θ n. Facts: Γ n k and F k n are subgroup filtrations of Θ n. We have Γ n k F k n by part (a) of our main result.

61 Filtrations of Groups of Homotopy Spheres Θ n : group of homotopy n-spheres (Kervaire-Milnor, 1963) Problem. Compare the Gromoll filtration Γ n n 1 Γ n 1 = Θ n to the fold filtration Fn 1 n F1 n = Θ n. Facts: Γ n k and F n k are subgroup filtrations of Θ n. We have Γ n k F k n by part (a) of our main result. For instance, Γ and Γ (Crowley-Schick, 2013).

62 Thank you for your attention!