Embeddings of lens spaces in 2 CP 2 constructed from lens space surgeries

Transcription

1 Embeddings of lens spaces in 2 CP 2 constructed from lens space surgeries Motoo TANGE (Univ. of Tsukuba) Yuichi YAMADA (Univ. of Electro-Comm., Tokyo) 2014 Aug. 25 Knots and Low Dimensional Manifolds a Satellite Conference of Seoul ICM 2014 Busan, Republic of Korea 1 / 25

2 1 Introduction 2 Lens space surgery 3 Main Results 2 / 25

3 Introduction This work is a continuation after our [Tange-Y (JKTR, 2012)] Four-dimensional manifolds constructed by lens space surgeries along torus knots Motivation [Saito-Teragaito 2010] (2008 Arxiv) Knots yielding homeomorphic lens spaces by Dehn surgery 2010 Nov. 4-dim. topology in Hiroshima University. Sasahira: Instanton Floer homology over lens spaces Some L(p, q) can NOT be smoothly embedded in CP 2 CP 2. [Sasahira] (2010) a doubled CP 2 : a connected sum of two copies of CP 2. 3 / 25

4 1.2. Framed link describes 3-dim and 4-dim manifolds ex. C(2, 15) = T (2, 15) T (2, 3) Q. What is this manifold? 4 / 25

5 Framed link (L; n) = (K 1 K 2... K r ; n 1, n 2,..., n r ) Let K be a knot (L be a link) in S 3 = B 4, n Z (or n = p/q Q {1/0}), called framing. Definition (3-manifold; Dehn surgery) M 3 (K; n) = ( S 3 \intn(k) ) ϕn (D 2 S 1 ) where ϕ n : (D 2 S 1 ) ( S 3 \intn(k) ) is a homeomorphism s.t. ϕ n ( D 2 {pt}) = n[m] + [l] (or p[m] + q[l]) [m], [l] H 1 ( (S 3 \intn(k))) is the meridian-longitude system. Definition The core {o} S 1 in M(K; n) is called a dual knot of the surgery. 5 / 25

6 A Framed link also describes a 4-manifold Definition (4-manifold; Kirby diagram, 2-handle attach) X 4 (K; n) = B 4 ϕn (D 2 D 2 ) where ϕ n : ( D 2 ) D 2 B 4 is an embedding s.t. ϕ n ( D 2 {o}) = K, ϕ n ( D 2 {pt}) = n[m] + [l]. Fact For n Z, X 4 (K; n) = M 3 (K; n) 6 / 25

7 3-dim Example: Lens space L(p, q) p q = a 1 a a a n (a i > 1) p q a 1 a 2 a 3 = a n n r a 1 a 2 a 3 a n n 1 r For n Z, r Q L(p, q) = L(p, q ) q q or q q 1 mod p. 7 / 25

8 4-dim Example: In this talk, we need only 2-handles. We do not draw 4-handles CP 2 CP 2 S 2 S 2 S 2 S 2 A split sum describes a connected sum. Thus, 1 1 this describes a connected sum CP 2 CP 2 of two copies of CP 2 : a doubled CP 2. 8 / 25

9 Dehn-Rolfsen move and Kirby calculus Theorem (Dehn-Rolfsen moves, Kirby calculus) The 3-manifolds are homeo. M(L; n) = M(L ; n ) framed links (L; n), (L ; n ) are moved to each other by (K1) Blow-up/down (K2) Handle slide and isotopy. (K1) Blow-up/ down changes the 4-manifold ±1 is related to add/remove CP 2 or CP 2 9 / 25

10 (K2) Handle slide 2 Unchanges 3- and 4-manifold h 2 h 1 h 2 h 1 n 2 = n 2 ± 2lk(h 1, h 2 ) + n 1 10 / 25

11 Lens space surgery Study Which knots K yield lens spaces by Dehn surgery? M(K; n) = L(P, Q)? 1. Torus knots Lemma ([Moser 71]) Note that n = ±P. M(T (p, q); pq ± 1) = L(pq ± 1, p 2 ) 2. 2 cable of torus knots 3. Also Hyperbolic knots [Fintushel Stern 80,...] becomes a resarch area (as in Professor Jiajun Wang s talk on Yesterday) 11 / 25

12 Berge s original List [ 90] 12 / 25

13 According to Professor Sungmo Kan s talk on Saturday, the classification of Berge s list has been updated. Type VII and VIII knots in the following pages are now called OPT. (knots in Once Punctured Torus) 13 / 25

14 Today, we focus pairs (K, K ), K: Torus knot, K : Type VII or VIII K K 14 / 25

15 TypeVII (σ = +) and VIII (σ = ) knots : k σ (p, q) Let F + be the fiber surface of the left-handed trefoil, F that of Fig8 knot. The knot k σ (p, q) is defined as a (p, q)-curve in F σ. Lemma ([Berge 90]) M(k σ (p, q); P) = L(P, Q), where P = σa 2 + ab + b 2, Q = (a/b) 2 mod P. F + k + (2, 3) k (2, 5) P = = 19 P = = 31. F 15 / 25

16 [Saito-Teragaito] There exist some (a sequence of) pairs of knots K K and M(K; p) = L(p, q) = ±M(K ; p), for all pairs of geometric types (ex. torus-hyperbolic), except cable-cable, of knots K, K. Rem. Recently, [J. Greene] showed Resulting lens spaces of lens space surgeries are those in the Berge s list. Q. How many (different) knots can yield a same lens space? 16 / 25

17 Main Results Our purpose is, A. Complete List of pairs (K, K ) s.t. K is a torus knot, K is Type VII or VIII, and M(K; p) = L(p, q) = ±M(K ; p), B. Study the 4-manifold X 4 (K; p) X 4 (K ; p). In the case M(K; p) = M(K ; p), the 4-manifold is 1-conn. and (the intersection form is) definite, thus is homeo. to ±(CP 2 CP 2 ), by Freedman s theorem. 17 / 25

18 On A, we had six sequences of pairs indexed by an integer i. In TABLE in the next pages, a sequence of pairs of pairs of integers ((X, Y ), (Z, W )) with P means a pair of knots K = T (X, Y ) and K = k σ (Z, W ) satisfies M(K; P) = L(P, q) = ±M(K ; P) 18 / 25

19 ((K i, L i ), (M i, N i )) with P i Torus knot T (K, L) and TypeVIII knot k (M, N) [ ] [ ] [ ] [ ] K0 1 M0 0 =, =, 2 1 [ Ki+1 L i+1 ] = L 0 [ ] [ Ki L i ], N 0 [ Mi+1 N i+1 ] = [ i K L P M N ] [ Mi P = KL 1 = M 2 + MN + N 2, K 2 + (N/M) ±2 0 mod P. N i ]. 19 / 25

20 TABLE (K K but M(K; P) = L(P, q) = ±M(K ; P)) T-VII orientation-preserving two sequences ((A, B), (C, D)) with P = AB 1 = C 2 + CD + D 2 T-VII orientation-reversing a sequence ((S, T ), (U, V )) with P = ST 1 = U 2 + UV + V 2 T-VIII orientation-reversing a sequence ((E, F ), (G, H)) with P = EF + 1 = G 2 + GH + H 2 T-VIII orientation-reversing two sequences ((K, L), (M, N)) with P = KL 1 = M 2 + MN + N 2 ex. K = T (K i, L i ) and K = k (M i, N i ) of TypeVIII satisfies M(K; P) = L(P, q) = M(K ; P) 20 / 25

21 Theorem (A: TABLE is complete) Suppose that a pair of lens space surgeries along (K, K ) s.t. K is a torus knot, K is Type VII or VIII, and M(K; p) = L(p, q) = ±M(K ; p ) We may retake K as K = T (a, b) with a, b > 0, by taking (K!; p) instead of (K; p), and K also. Then the retaken pair ((K; p), (K ; p)) is in TABLE. Method. Calculus in Z[ ] works well. 21 / 25

22 Theorem (B: 4-dim. problem) For every pair of lens space surgeries along (K, K ) s.t. K is a torus knot, K is Type VII or VIII, and M 3 (K; p) = L(p, q) = ±M 3 (K ; p), the closed 4-manifold X 4 (K; p) X 4 (K ; p) is diffeomorphic to the standard 4-manifold S 2 S 2 or ± (CP 2 CP 2 ). Method. Search both dual knots by 3-dim. Dehn-Rolfsen moves, and do 4-dim. Kirby calculus with only (K2) handle slides. All diagrams are indexed. 22 / 25

23 Corollary Some Lens spaces can be smoothly embedded in CP 2 CP 2. L(S i T i 1, Si 2), L(E if i + 1, Ei 2) L(K i L i 1, Ki 2), L(k il i 1, ki 2) CP 2 CP 2. Rem. [Sasahira] and [TY] shows L(28657, 2) CP 2 CP 2, L(28657, 7921) CP 2 CP 2. If p is prime and p 1 mod 16,... L(28657, 7921) = M(T (199, 144), 28657) = M(T (89, 322), 28657). 23 / 25

24 Thank you very much! 24 / 25

25 Q. What is this manifold? ex. C(2, 15) = T (2, 15) T (2, 3) Answer. S 3 as 3-dim. CP 2 CP 2 as 4-dim. It also shows L(29, 4) CP 2 CP / 25