Whitney towers, gropes and Casson-Gordon style invariants of links

Transcription

1 Whitney towers, gropes and Casson-Gordon style invariants of links Min Hoon Kim POSTECH August 25, 2014

2 Part 1 : Friedl-Powell invariant

3 Let L be a 2-component link in S 3 with lk(l) = 1 and H be the Hopf link. X L = S 3 ν(l), X H = S 1 S 1 I.

4 Let L be a 2-component link in S 3 with lk(l) = 1 and H be the Hopf link. X L = S 3 ν(l), X H = S 1 S 1 I. Let Y X L be annuli neighborhood of meridians of L.

5 Let L be a 2-component link in S 3 with lk(l) = 1 and H be the Hopf link. X L = S 3 ν(l), X H = S 1 S 1 I. Let Y X L be annuli neighborhood of meridians of L. Y Y X L Note that there is a homeomorphism i: (X L, Y ) (X L, X L Y ).

6 For a prime p, ϕ: π 1 (X L ) Z/p i Z/p j, meridians (1, 0), (0, 1). (X ϕ L, Y ϕ ) (X L, Y ) : the p i+j -fold cover induced from ϕ.

7 For a prime p, ϕ: π 1 (X L ) Z/p i Z/p j, meridians (1, 0), (0, 1). (X ϕ L, Y ϕ ) (X L, Y ) : the p i+j -fold cover induced from ϕ. There is a linking form λ L : th 1 (X ϕ L, Y ϕ ) th 1 (X ϕ L, Y ϕ ) Q/Z. The adjoint of λ L is given by th 1 (X ϕ L, Y ϕ ) i PD th 2 (X ϕ L, Y ϕ ) β UCT Hom(tH 1 (X ϕ L, Y ϕ ), Q/Z) (Here, for an abelian group G, tg denotes the torsion subgroup of G.)

8 Lemma (Friedl-Powell) Suppose that L is concordant to H and W is the exterior of concordance. Then, there is a metabolizer P = P for the linking form λ L given by P = Ker(tH 1 (X ϕ L, Y ϕ ) th 1 (W ϕ, Y ϕ )).

9 Lemma (Friedl-Powell) Suppose that L is concordant to H and W is the exterior of concordance. Then, there is a metabolizer P = P for the linking form λ L given by Remark Recall that P = Ker(tH 1 (X ϕ L, Y ϕ ) th 1 (W ϕ, Y ϕ )). P = {x th 1 (X ϕ L, Y ϕ ) λ L (x, y) = 0 y P }.

10 Using X L = XH which respects meridians and longitudes, form For example, M H = S 1 S 1 S 1. M L = X L X H.

11 Using X L = XH which respects meridians and longitudes, form M L = X L X H. For example, M H = S 1 S 1 S 1. From now, ϕ: π 1 (M L ) Z/p i Z/p j, meridians (1, 0), (0, 1).

12 Using X L = XH which respects meridians and longitudes, form M L = X L X H. For example, M H = S 1 S 1 S 1. From now, ϕ: π 1 (M L ) Z/p i Z/p j, meridians (1, 0), (0, 1). Choose homomorphisms φ: H 1 (M ϕ L ) H 1(M ϕ L )/ torsion = Z 3 χ: H 1 (M ϕ L ) Z/qk for some prime q with (p, q) = 1.

13 Using X L = XH which respects meridians and longitudes, form M L = X L X H. For example, M H = S 1 S 1 S 1. From now, ϕ: π 1 (M L ) Z/p i Z/p j, meridians (1, 0), (0, 1). Choose homomorphisms φ: H 1 (M ϕ L ) H 1(M ϕ L )/ torsion = Z 3 χ: H 1 (M ϕ L ) Z/qk for some prime q with (p, q) = 1. So, we can regard χ φ: M ϕ L K(Z/qk Z 3, 1).

14 From Atiyah-Hirzebruch spectral sequence, χ φ 0 φ: M ϕ L T 3 K(Z/q k Z 3, 1) represents a torsion element in Ω 3 (Z/q k Z 3 ). Choose s N and a 4-manifold W such that W = s(m ϕ L T 3 ), φ and χ extend over W.

15 Using the ring homomorphsim, Z[Z/q k Z 3 ] = Z[Z/q k ][Z 3 ] Q(ξ q k)[z 3 ] C[Z 3 ] C(Z 3 ) we can define H (W ; C(Z 3 )).

16 Using the ring homomorphsim, Z[Z/q k Z 3 ] = Z[Z/q k ][Z 3 ] Q(ξ q k)[z 3 ] C[Z 3 ] C(Z 3 ) we can define H (W ; C(Z 3 )). Now, we consdier the intersection forms: λ C(Z 3 )(W ): H 2 (W ; C(Z 3 )) H 2 (W ; C(Z 3 )) C(Z 3 ) λ Q (W ): H 2 (W ; Q) H 2 (W ; Q) Q. Definition (Friedl-Powell invariant) τ(l, χ) := (λ C(Z 3 )(W ) C(Z 3 ) λ Q (W )) 1 s L0 (C(Z 3 )) Q

17 Theorem (Friedl-Powell) Suppose that L is concordant to H. Then, there is a metabolizer P th 1 (X ϕ L, Y ϕ ) for the linking form λ L with the following property:

18 Theorem (Friedl-Powell) Suppose that L is concordant to H. Then, there is a metabolizer P th 1 (X ϕ L, Y ϕ ) for the linking form λ L with the following property: For any χ: H 1 (M ϕ L ) Z/qk, if χ H1 (X ϕ L ) factors through χ H1 (X ϕ L ) : H 1 (X ϕ L ) H 1(X ϕ L, Y ϕ ) δ Z/q k and δ vanishes on P, τ(l, χ) = 0 L 0 (C(Z 3 )) Q.

19 Part 2: Casson-Gordon versus Friedl-Powell

20 Casson-Gordon Friedl-Powell K : knot L : 2-component link with lk(l) = 1

21 Casson-Gordon Friedl-Powell K : knot L : 2-component link with lk(l) = 1 M K (X L, Y )

22 Casson-Gordon Friedl-Powell K : knot L : 2-component link with lk(l) = 1 M K (X L, Y ) ϕ: H 1 (M K ) Z/p i ϕ: H 1 (X L, Y ) Z/p i Z/p j

23 Casson-Gordon Friedl-Powell K : knot L : 2-component link with lk(l) = 1 M K (X L, Y ) ϕ: H 1 (M K ) Z/p i ϕ: H 1 (X L, Y ) Z/p i Z/p j th 1 (M ϕ K ) th 1(M ϕ K ) Q/Z th 1(X ϕ L, Y ϕ ) th 1 (X ϕ L, Y ϕ ) Q/Z

24 Casson-Gordon Friedl-Powell K : knot L : 2-component link with lk(l) = 1 M K (X L, Y ) ϕ: H 1 (M K ) Z/p i ϕ: H 1 (X L, Y ) Z/p i Z/p j th 1 (M ϕ K ) th 1(M ϕ K ) Q/Z th 1(X ϕ L, Y ϕ ) th 1 (X ϕ L, Y ϕ ) Q/Z χ φ: H 1 (M ϕ K ) Z/qk Z χ φ: H 1 (M ϕ L ) Z/qk Z 3

25 Theorem (Cochran-Orr-Teichner 03) If K bounds a Whitney tower or grope of height 3.5 in D 4, then the Casson-Gordon invariant τ(k, χ) vanishes.

26 Theorem (Cochran-Orr-Teichner 03) If K bounds a Whitney tower or grope of height 3.5 in D 4, then the Casson-Gordon invariant τ(k, χ) vanishes. Our main result is the following: Theorem If L is height 3.5 Whitney tower or grope concordant to H in S 3 I, then the Friedl-Powell invariant τ(l, χ) vanishes.

27 Remark [Cha] For n > 2, there are links which are height n grope concordant to H, but not height n.5 grope concordant to H.

28 Part 3: Sketch of proof of the main theorem

29 Lemma (Cha) If two links are height (h + 2) grope or Whitney tower concordant, then their exteriors are h-solvable cobordant. So, we can use a 1.5-solvable cobordism between X L and X H. Lemma Suppose that W is a 1.5-solvable cobordism between X L and X H. Then, there is a metabolizer P = P for the linking form λ L given by P = Ker(tH 1 (X ϕ L, Y ϕ ) th 1 (W ϕ, Y ϕ )).

30 Sketch of proof Let W = W XH I X H I, 2r = dim Q H 2 (W, X L ; Q) and t = p i+j. If χ vanishes on a metabolizer P, then χ extends to W. Enough to prove λ Q (W ) = 0 (easy) and λ C(Z 3 )(W ) = 0. Euler characteristic argument gives dim C(Z 3 ) H 2 (W ; C(Z 3 )) = 2rt. From 2-lagrangians, there is L H 2 (W ; C(Z 3 )) where λ C(Z 3 ) L L 0. To prove dim C(Z 3 ) L = rt, we used an injectivity theorem of Friedl-Powell.