ETA INVARIANTS AS SLICENESS OBSTRUCTIONS AND THEIR RELATION TO CASSON-GORDON INVARIANTS

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1 ETA INVARIANTS AS SLICENESS OBSTRUCTIONS AND THEIR RELATION TO CASSON-GORDON INVARIANTS STEFAN FRIEDL Abstract. We classify the metabelian unitary representations of π 1 (M K ), where M K is the result of zero-surgery along a knot K S 3. We show that certain eta invariants associated to metabelian representations π 1 (M K ) U(k) vanish for slice knots and that even more eta invariants vanish for ribbon knots and doubly slice knots. We show that this result contains the Casson-Gordon sliceness obstruction. It turns out that eta invariants can in many cases be easily computed for satellite knots. We use this to study the relation between the eta invariant sliceness obstruction, eta-invariant ribboness obstruction, and the L 2 eta invariant sliceness obstruction recently introduced by Cochran, Orr and Teichner. In particular we give an example of a knot which has zero eta invariant and zero metabelian L 2 eta invariant sliceness obstruction but is not ribbon. 1. Introduction A knot K S n+2 is a smooth submanifold homeomorphic to S n, K is called slice if it bounds a smooth (n + 1)-disk in D n+3. Isotopy classes of knots form a semigroup under connected sum. This semigroup can be turned into a group by modding out the subsemigroup of slice knots. This group is called the knot concordance group in dimension n. It is a natural goal to attempt to understand this group and find complete invariants for when a knot represents a non-zero element in this group. Knot concordance in the high-dimensional case, i.e. the case n > 1, is well understood. Kervaire [K65] first showed that all even dimensional knots are slice. For an odd dimensional knot K Levine [L69] showed that K is slice if and only if K is algebraically slice, i.e. if the Seifert form has a subspace of half-rank on which it vanishes. This reduces the task of detecting slice knots to an algebraic problem which is well-understood (cf. [L69b]). The classical case n = 1 turned out to be much more difficult to understand. Casson and Gordon [CG78], [CG86] defined certain sliceness obstructions (cf. section 5.1) and used these to give the first examples of knots in S 3 that are algebraically slice but not geometrically slice. Many more examples have been given since then, the most subtle ones were found recently by Cochran, Orr and Teichner [COT01] (cf. section 7) and Kim [K02]. Date: February 11,

2 2 STEFAN FRIEDL We will show how to use eta invariants to detect non slice knots. Given a closed smooth three manifold M and a unitary representation α : π 1 (M) U(k), Atiyah Patodi Singer [APS75] defined η α (M) R, called the eta invariant of (M, α) which has the property that if (W 4, β : π 1 (W ) U(k)) is such that (W 4, β) = (M 3, α), then η α (M) = sign β (W ) ksign(w ) where sign β (W ) denotes the twisted signature of W. For a knot K we study the eta invariants associated to the closed manifold M K, where M K denotes the result of zero framed surgery along K S 3. Eta-invariants in the context of link theory were first studied by Levine [L94]. For a group G define the derived series inductively we study eta invariants corresponding to metabelian representations, i.e. representations which factos through π 1 (M K )/π 1 (M K ) (2). Letsche [L00] first studied metabelian eta invariants in the context of knot concordance. Denote the universal abelian cover of M K by M K, then H 1 ( M K ) has a natural Λ := Z[t, t 1 ] module structure, we therefore write H 1 (M K, Λ) for H 1 ( M K ). Since π 1 (M K )/π 1 (M K ) (1) = Z we get isomorphisms π 1 (M K )/π 1 (M K ) (2) = π1 (M K )/π 1 (M K ) (1) π 1 (M K ) (1) /π 1 (M K ) (2) = Z H1 (M K, Λ) where n Z acts on H 1 (M K, Λ) by multiplication by t n. The following proposition gives a classification of all metabelian unitary representations of π 1 (M K ), i.e. of all unitary representations of Z H 1 (M K, Λ). Proposition. [4.1] Let H be a Λ-module, then any irreducible representation Z H U(k) is conjugate to a representation of the form α (k,z,χ) : Z H U(k) (n, h) z n n χ(h) χ(th) χ(t k 1 h) where z S 1 and χ : H H/(t k 1) S 1 is a character which does not factor through H/(t l 1) for some l < k. Let K be a slice knot with slice disk D and let α (k,z,χ) be a representation of π 1 (M K ). Consider the following diagram α (z,χ) π 1 (M K ) Z H 1 (M K, Λ) Z H 1 (M K, Λ)/(t k 1) U(k) π 1 (N D ) Z H 1 (N D, Λ) Z H 1 (N D, Λ)/(t k 1) U(k) If χ vanishes on Ker{H 1 (M K, Λ)/(t k 1) H 1 (N D, Λ)/(t k 1)} then χ extends to χ N : H 1 (N D, Λ)/(t k 1) S 1 since S 1 is divisible. From theorem 3.2 it follows

3 ETA INVARIANTS AS SLICENESS OBSTRUCTIONS 3 that η(m K, α (z,χ) ) = sign α(z,χn ) (N D) sign(n D ). But sign(n D ) = 0 since H (N D ) = H (S 1 ); furthermore Letsche [L00] showed that if χ is of prime power order and z S 1 is transcendental, then χ N can be chosen to be of prime power order too and sign α(z,χn ) (N D) = 0. Denote the set of all α (z,χ) with z transcendental and χ of prime power order by P irr,met k (π 1 (M K )). Note that H 1 (M K, Λ)/(t k 1) = H 1 (L k ) where L k denotes the k fold cover of S 3 branched along K. If k is a prime power, then Casson and Gordon showed that Ker{H 1 (L k ) = H 1 (M K, Λ)/(t k 1) H 1 (N D, Λ)/(t k 1)} is a metabolizer for the linking pairing H 1 (L k ) H 1 (L k ) Q/Z. This proves the following theorem. Theorem. [4.5] Let K be a slice knot, k a prime power, then there exists P k H 1 (L k ) such that P k = Pk with respect to the linking pairing H 1(L k ) H 1 (L k ) Q/Z, such that for any irreducible representation α (z,χ) P k (π 1 (M K )) with χ(p k ) 0 we get η α(z,χ) (M K ) = 0. In section 5.1 we recall the Casson Gordon sliceness obstruction theorem which till the advent of Cochran Orr Teichner s L 2 eta invariant sliceness obstruction proved to be the most effective obstruction. In theorem 5.6 we show that an algebraically slice knot has zero Casson-Gordon sliceness obstruction if and only if it satisfies the vanishing conclusion of theorem 4.5. The eta invariant approach has several advantages over the Casson-Gordon approach. For example, it is clear that if α 1,..., α l are representations as in theorem 4.5, then their tensor product will also extend over π 1 (N D ). Furthermore there exists a simple criterion when α 1 α l P irr,met k (π 1 (M K )), which then guarantees that η α1 α l (M K ) = 0 (cf. theorem 4.7). This gives a potentially stronger sliceness obstruction than the Casson Gordon obstruction. We then turn to ribbon knots (for a definition cf. section 6.1). The only fact we use is that a ribbon knot has a slice disk D such that π 1 (M K ) π 1 (N D ) is surjective. In particular a representation of π 1 (M K ) extends over π 1 (N D ) if it vanishes on Ker{π 1 (M K ) π 1 (N D )}. This allows us to prove the following theorem. Theorem. [6.3] Let K S 3 be a ribbon knot. Then there exists P H 1 (M, Λ) such that P = P with respect to the Blanchfield pairing H 1 (M, Λ) H 1 (M, Λ) Q(t)/Z[t, t 1 ] and such that for any α (z,χ) P irr,met k (π 1 (M K )) with χ(p ) 0 we get η α (M K ) = 0. This is a much stronger obstruction theorem than the sliceness obstruction theorems and could potentially provide a way to disprove the conjecture that each slice knot is ribbon. In section 6.2 we discuss Gilmer s [G93] and Letsche s [L00] sliceness theorems. It turns out that in fact both theorems have gaps in their proofs, at this point they give ribbonness obstructions, which can be shown to be contained in theorem 6.3.

4 4 STEFAN FRIEDL We take a quick look at doubly slice knots, prove a doubly slice obstruction theorem (theorem 6.5), and point out that doubly slice knots actually satisfy the ribbon obstruction theorem. It seems that doubly slice knots have a higher chance of being ribbon than ordinary slice knots. Recently Cochran, Orr and Teichner [COT01], [COT02] defined the notion of (n) solvability for a knot, n 1 N which has the following properties. 2 (1) A slice knot is (n) solvable for all n. (2) A knot is (0.5) solvable if and only if it is algebraically slice. (3) A (1.5) solvable knot has zero Casson Gordon obstruction. Given a homomorphism ϕ : π 1 (M) G to a group G, Cheeger and Gromov defined the L 2 eta invariant η ϕ (2) (M) which satisfies a theorem similar to 3.2 if we replace the twisted signature by Atiyah s L 2 signature. Cochran, Orr and Teichner used L 2 eta invariants to find examples of knots which are (2.0) solvable, which in particular have zero Casson Gordon invariants, but which are not (2.5) solvable. Using similar ideas Kim [K02] found examples of knots which are (1.0) solvable and have zero Casson Gordon invariants, but which are not (1.5) solvable. A quick summary of this theory will be given in section 7. In section 8 we give examples that show that L 2 eta invariants are not complete invariants for a knot being (0.5) solvable and (1.5) solvable. We also show that there exists a knot which is (1.0) solvable, has zero Casson Gordon invariants and zero L 2 eta invariant of level 1, but does not satisfy the conclusion of theorem 6.3, i.e. is not ribbon. It s not known whether this knot is slice or not. Furthermore we give an example of a ribbon knot where the conclusion of theorem 4.5 does not hold for non prime power characters; this shows that the set P irr,met k (π 1 (M K )) is in a sense maximal. In all cases we use a satellite construction to get knots whose eta invariants can be computed explicitly by methods introduced by Litherland [L84]. This paper is essentially the author s thesis. I would like to express my deepest gratitude towards my advisor Jerry Levine for his patience, help and encouragement. 2. Basic knot theory and linking pairings as sliceness obstructions Throughout this paper we will always work in the smooth category. In the classical dimension the theory of knots in the smooth category is equivalent to the theory in the locally flat category. The same is not true for the notion of sliceness. Since a smooth submanifold is always locally flat smooth slice disks are locally flat, but the converse is not true, i.e. there exist knots which are topologically slice, but not smoothly slice (cf. [G86], [E95]) Basic knot theory. By a knot we understand a smooth oriented submanifold of S 3 diffeomorphic to S 1. A smooth surface F S 3 with (F ) = S 1 will be called a Seifert surface for K. Note that a Seifert surface inherits an orientation from K, in particular the map H 1 (F ) H 1 (S 3 \F ), a a + induced by pushing into the positive

5 ETA INVARIANTS AS SLICENESS OBSTRUCTIONS 5 normal direction is well-defined. The pairing H 1 (F ) H 1 (F ) Z (a, b) lk(a, b + ) is called the Seifert pairing of F. Any matrix A representing such a pairing is called a Seifert matrix for K. Seifert matrices are unique up to S-equivalence (cf. [M65, p. 393] or [L70]). In particular the Alexander polynomial K (t) := det(at A t ) Z[t, t 1 ]/{±t l } is well-defined and independent of the choice of A. Let N(K) be a solid torus neighborhood of K. A meridian µ of K is a nonseparating simple closed curve in (N(K)) that bounds a disc in N(K). The notion of meridian is well-defined up to homotopy in S 3 \ K. Alexander duality shows that there exists an isomorphism ɛ : H 1 (S 3 \ K) Z such that ɛ(µ) = 1. We ll mostly study invariants of M K, the result of zero-framed-surgery along K. This manifold has the advantage over S 3 \ K that it is a closed manifold associated to K. Note that H 1 (M K ) = H 1 (S 3 \ K) = Z Slice knots. We say that two knots K 1, K 2 are concordant if there exists a smooth submanifold V S 3 [0, 1] such that V = S 1 [0, 1] and such that (V ) = K 0 0 K 1 1. A knot K S 3 is called slice if it is concordant to the unknot. Equivalently, a knot is slice if there exists a smooth disk D D 4 such that (D) = K. We say that the Seifert pairing on H 1 (F ) is metabolic if there exists a subspace H of half-rank such that the Seifert pairing vanishes on H. This is the case if and only if K has a metabolic Seifert matrix A, i.e. if A is of the form ( ) 0 B A = C D where 0, B, C, D are square matrices. If the Seifert pairing of a knot is metabolic then we say that K is algebraically slice. We recall the following classical theorem (cf. [L97, p. 87ff]). Theorem 2.1. A slice knot is algebraically slice. For a slice disk D we define N D := D 4 \ N(D). We summarize a couple of wellknown facts about N D. Lemma 2.2. If K is a slice knot and D a slice disk, then (1) (N D ) = M K, (2) H (N D ) = H (S 1 ), (3) there exists a map H 1 (N D ) Z extending ɛ : H 1 (M K ) Z Universal abelian cover of M K and the Blanchfield pairing. Let K be a knot. Denote the infinite cyclic cover of M K corresponding to ɛ : H 1 (M K ) Z by M K. Then Z = t acts on M K, therefore H 1 ( M K ) carries a Λ := Z[t, t 1 ]-module structure. We ll henceforth write H 1 (M K, Λ) for H 1 ( M K ).

6 6 STEFAN FRIEDL In the following we ll give Λ the involution induced by t = t 1. Let S := {f Λ f(1) = 1}. The Λ-module H 1 (M K, Λ) is S-torsion since for example the Alexander polynomial K (t) lies in S and K (t) annihilates H 1 (M K, Λ). We recall the definition of the Blanchfield pairing and its main properties (cf. [B57]). Lemma 2.3. The pairing λ Bl : H 1 (M K, Λ) H 1 (M K, Λ) S 1 Λ/Λ (a, b) 1 p(t) i= (a ti c)t i where c C 2 (M K, Λ) such that (c) = p(t)b for some p(t) S, is well-defined, non-singular and hermitian over Λ. For any Λ-submodule P H 1 (M K, Λ) define P := {v H 1 (M K, Λ) λ Bl (v, w) = 0 for all w P } If P H 1 (M, Λ) is such that P = P then we say that P is a metabolizer for λ Bl. If λ Bl has a metabolizer we say that λ Bl is metabolic. Theorem 2.4. [K75] The Blanchfield pairing for a knot K is metabolic if and only if K is algebraically slice. Definition. For a Z module A denote by T A the Z torsion submodule of A and let F A := A/T A. Proposition 2.5. [L00, prop. 2.8] If K is slice and D any slice disk, then Q := Ker{H 1 (M K, Λ) F H 1 (N D, Λ)} is a metabolizer for the Blanchfield pairing. Note that Ker{H 1 (M K, Λ) H 1 (N D, Λ)} is not known to be a metabolizer Finite cyclic covers and linking pairings. Let K be a knot, k some number. Denote by M k the k-fold cover of M K corresponding to π 1 (M K ) ɛ Z Z/k, and denote by L k the k-fold cover of S 3 branched along K S 3. Note that H 1 (L k ) and H 1 (M k ) also have a Λ module structure. Lemma 2.6. There exist natural isomorphisms We ll henceforth identify these groups. H 1 (M k ) = H 1 (L k ) Z H 1 (M k ) = H 1 (M K, Λ)/(t k 1) Z H 1 (L k ) = H 1 (M K, Λ)/(t k 1) Proof. The second statement follows from the long exact sequence The other statements follow easily. H i ( M K ) tk 1 H i ( M K ) H i (M k )...

7 ETA INVARIANTS AS SLICENESS OBSTRUCTIONS 7 Proposition 2.7. [G77, p. 17] k H 1 (L k ) = K (e 2πij/k ) where 0 on the right hand side means that H 1 (L k ) is infinite. Lemma 2.8. Let k be any integer such that H 1 (L k ) is finite, then the map j=1 λ L,k = λ L : H 1 (L k ) H 1 (L k ) Q/Z (a, b) 1 a c n mod Z where c C 2 (L k ) such that (c) = na, defines a symmetric, non-singular pairing. The pairing of the lemma is called the linking pairing of H 1 (L k ) = T H 1 (M k ). For any Z-submodule P k T H 1 (M k ) define P k := {v T H 1 (M k ) λ L (v, w) = 0 for all w P k } If P k is a Λ submodule with P = P then we say that P k is a metabolizer for λ L and we say that λ L is metabolic. Note that if P k is a metabolizer for the linking pairing, then P k 2 = H 1 (L k ) Homology of prime-power covers and the linking pairing. The following corollary shows that prime power covers of manifolds behave in a more controlled way. Lemma 2.9. Let Y be a topological space such that H (Y ) = H (S 1 ) and let k = p s where p is a prime number. Then H (Y k ) = H (Y ) torsion. Furthermore gcd( T H 1 (Y k ), p) = 1. For a proof cf. [CG86, p. 184] Corollary Let k be a prime power. (1) H 1 (M k ) = Z T H 1 (M k ) T H 1 (M k ) = H 1 (L k ) = H 1 (M, Λ)/(t k 1) (2) Let D be a slice disk, denote the k-fold cover of N D by N k, then H 1 (N k ) = Z T H 1 (N k ) = Z H 1 (N, Λ)/(t k 1). (3) Q := Ker{T H 1 (M k ) T H 1 (N k )} is a metabolizer for λ L. Remark. Let K be a slice knot, D a slice disk. Then P := Ker{H 1 (M K, Λ) F H 1 (N D, Λ)} is a metabolizer for the Blanchfield pairing and Q k := Ker{T H 1 (M k ) T H 1 (N k )} is a metabolizer for λ L,k. Furthermore P k := π(p ) H 1 (M K, Λ)/(t k 1) can be shown to be a metabolizer for for λ L,k. It is not known whether P k = Q k. If yes, then this would show that all the metabolizers Q k can be lifted to a metabolizer for the Blanchfield pairing, which would resolve problems which appeared in [G93] and [L00] (cf. section 6.2).

8 8 STEFAN FRIEDL 3. Introduction to eta-invariants and first application to knots 3.1. Eta invariants. Definition. Let C be a complex, hermitian matrix, i.e. C = C t, then we define the signature sign(c) to be the number of positive eigenvalues of C minus the number of negative eigenvalues. The following is an easy exercise. Lemma 3.1. If C is hermitian then sign(p C P t ) = sign(c) for any P with det(p ) 0. If furthermore det(c) and C is of the form ( ) 0 B C = B t D where B is a square matrix, then sign(c) = 0. Let M 3 be a closed manifold and α : π 1 (M) U(k) a representation. Atiyah, Patodi, Singer [APS75] associated to (M, α) a number η(m, α) called the (reduced) eta invariant of (M, α). The main theorem to compute the eta invariant is the following (cf. [APS75]). Theorem 3.2. (Atiyah-Patodi-Singer index theorem) If there exists (W 4, β : π 1 (W ) U(k)) with (W, β) = r(m 3, α) for some n N, then η α (M) = 1 r (sign β(w ) ksign(w )) where sign β (W ) denotes the signature of the twisted intersection pairing on H α 2 (W ) Application of eta invariants to knots. For a knot K we ll study the eta invariants associated to the closed manifold M K. Eta invariants in the context of knot theory were first studied by Levine [L94] who used the eta invariant to find links which are not concordant to boundary links. Definition. For a group G the derived series is defined by G (0) := G and inductively G (i+1) := [G (i), G (i) ] for i > 0. The inclusion map S 3 \ K M K defines a homomorphism π 1 (S 3 \ K) π 1 (M K ), the kernel is generated by the longitude of K which lies in π 1 (S 3 \K) (2). In particular π 1 (S 3 \ K)/π 1 (S 3 \ K) (i) π 1 (M K )/π 1 (M K ) (i) is an isomorphism for i = 0, 1, 2. Let M O be the zero-framed surgery on the trivial knot. Then M O = S 1 S 2 which bounds S 1 D 3 which is homotopic to a 1-complex. This proves the following. Lemma 3.3. Let K be the trivial knot, then for any α : π 1 (M O ) U(k) we get η α (M K ) = 0.

9 ETA INVARIANTS AS SLICENESS OBSTRUCTIONS U(1)-representations. Let K be a knot, A a Seifert matrix, then we define the signature function σ(k) : S 1 Z of K as follows (cf. [L69, p. 242]) σ z (K) := sign(a(1 z) + A t (1 z)) It is easy to see that this is independent of the choice of A. Proposition 3.4. Let K be a knot, µ a meridian and let α : π 1 (M K ) U(1) be a representation. (1) Let z := α(µ), then η α (M K ) = σ z (K) (2) The function z σ z (K) is locally constant outside of the zeros of K (t). (3) If K is algebraically slice and z S 1 such that K (z) 0, then σ z (K) = 0. In particular if z is a prime power root of unity, then σ z (K) = 0. Proof. (1) cf. [L84]. (2) The function z σ z (A) is continuous outside of the set of z s for which σ z (A) is singular, an easy argument shows that these z s are precisely the zeros of K (t). (3) If z S 1 such that K (z) 0 then A(1 z) + A t (1 z) is non singular and the first part follows from lemma 3.1. The second part follows from K (1) = 1 and the well known that Φ p r(1) = p for a prime p. Let K be a knot, m N, then we can form mk by iterated connected sum. We K is (algebraically) torsion if mk is (algebraically) slice for some m. Levine [L69] showed that for n > 1 a knot K S 2n+1 is slice if and only if K is algebraically slice. Levine [L69b] and Matumuto [M77] showed that K is algebraically torsion if and only if η α (M K ) = 0 for all α : π 1 (M K ) U(1) of prime power order. In particular, in the case n > 1 the U(1)-eta invariant detects any non torsion knot. In the classical case n = 1 Casson and Gordon [CG86] first found an example of a non torsion knot which is algebraically slice. The goal of this thesis is to study to which degree certain non-abelian eta invariants can detect such examples. 4. Metabelian eta invariants and the main sliceness obstruction theorem A group G is called metabelian if G (2) = {e}, a representation ϕ : π 1 (M) U(k) is called metabelian if it factors through π 1 (M)/π 1 (M) (2). Metabelian eta invariants in the context of knot concordance were first studied by Letsche [L00]. Several of our results can be found similarly in his paper. For a knot K let π := π 1 (M K ), then consider 1 π (1) /π (2) π/π (2) π/π (1) 1

10 10 STEFAN FRIEDL Note that π 1 ( M K ) = π 1 (M K ) (1) where M K denotes the infinite cyclic cover of M K, hence H 1 (M K, Λ) = H 1 ( M K ) = π 1 (M K ) (1) /π 1 (M K ) (2). Since π/π (1) = H 1 (M K ) = Z the above sequence splits and we get isomorphisms π/π (2) = π/π (1) π (1) /π (2) = Z H1 (M K, Λ) where n Z acts by multiplication by t n. This shows that studying metabelian representations of π 1 (M K ) corresponds to studying representations of Z H 1 (M K, Λ) Metabelian representations of π 1 (M K ). For a group G denote by Rk irr(g) (resp. R irr,met k (G)) the set of conjugacy classes of irreducible, k-dimensional, unitary (metabelian) representations of G. Recall that for a knot K we can identify R irr,met k (π 1 (M K )) = R irr (Z H 1 (M K, Λ)) Proposition 4.1. Let H be a Λ module. Let z S 1 and χ : H H/(t k 1) S 1 a character, then α (z,χ) : Z H U(k) (n, h) z n k n χ(h) χ(th) χ(t k 1 h) defines a representation. If χ does not factor through H/(t l 1) for some l < k, then α (z,χ) is irreducible. Conversely, any irreducible representation α Rk irr (Z H) is (unitary) conjugate to α (z,χ) with z, χ as above. Proof. It is easy to check that α (z,χ) is well-defined. Now let α Rk irr (Z H). Denote by χ 1,..., χ l : H S 1 the different weights of α : 0 H U(k). Since H is an abelian group we can write C k = l i=1v χi where V χi := {v C k α(0, h)(v) = χ i (h)v for all h} is the weight space corresponding to χ i. Recall that the group structure of Z H is given by In particular for all v H therefore for A := α(1, 0) we get (n, h)(m, k) = (n + m, t m h + k) (j, 0)(0, t j h) = (j, t j h) = (0, h)(j, 0) A j α(0, t j h) = α(j, 0)α(0, t j h) = α(j, t j h) = α(0, h)α(j, 0) = α(0, h)a j This shows that α(0, t j h) = A j α(0, h)a j. Now let v V χ(h), then α(0, h)av = Aα(0, th)a 1 Av = Aα(0, th)v = Aχ(th)v = χ(th)av i.e. α(1, 0) : V χi (h) V χi (th). Since α is irreducible it follows that, after reordering, χ i (v) = χ 1 (t i v) for all i = 1,..., l. Note that A j induces isomorphisms between the

11 ETA INVARIANTS AS SLICENESS OBSTRUCTIONS 11 weight spaces V χi and that A l : V χ1 V χ1 is a unitary transformation. In particular it has an eigenvector v, hence Cv CAv... CA l 1 v spans an α-invariant subspace. Since α is irreducible it follows that l = k and that each V χi is one-dimensional. Since α is a unitary representation we can find a unitary matrix P such that P Ce i = V i, in particular, α 1 := P 1 αp has the following properties. (1) α(0 H) = diag(χ(h), χ(th),..., χ(t k 1 h)), (2) for some z 1,..., z k S z k z α(1, 0) := z k 1 0 Here we denote by diag(b 1,..., b k ) the diagonal matrix with entries b 1,..., b k. Let z := k i=1 z i and let Q := diag(d 1,..., d k ) where d i := Q 1 α 1 Q has the required properties. i 1 j=1 z j. Then α z i 1 2 := 4.2. Eta invariants as concordance invariants. We quote some definitions, initially introduced by Levine [L94]. Let G be a group, then a G-manifold is a pair (M, α) where M is a compact oriented 3-manifold α is a homomorphism α : π 1 (M i ) G defined up to inner automorphism. We call two G-manifolds (M j, α j ), j = 1, 2, homology G bordant if there exists a G-manifold (N, β) such that (N) = M 1 M 2, H (N, M j ) = 0 for j = 1, 2 and, up to inner automorphisms of G, β π 1 (M j ) = α j. We will compare eta invariants for homology G bordant G manifolds. Definition. For a Λ-torsion module H define Pk irr (Z H) to be the set of conjugacy classes of representations which are conjugate to α (z,χ) with z S 1 transcendental and χ : H/(t k 1) S 1 factoring through a group of prime power order. If W is a manifold with H 1 (W ) = Z then we define P irr,met k (π 1 (W )) := Pk irr(z H 1(W, Λ)). We need the following theorem, which is a slight reformulation of a theorem by Letsche [L00, cor. 3.10] which in turn is based on work by Levine [L94]. Theorem 4.2. Let H be a Λ-torsion module, G := Z H. If (M 1, α 1 ), (M 2, α 2 ) are homology G bordant and θ P k (G), then η θ α1 (M 1 ) = η θ α2 (M 2 ). Proof. Let α (z,χ) P irr k (Z H), i.e. z S1 transcendental and χ : H/(t k 1) Z/m S 1 where m is a prime power. The theorem follows from a result by Letsche [L00, cor. 3.10] once we show that the set P k (Z H) as defined above is the same as the set P k (Z H) defined by Letsche. This in turn follows follows immediately from the observation that (1) all the eigenvalues of α (z,χ) (1, 0) are of the form ze 2πij/k, in particular all are transcendental,

12 12 STEFAN FRIEDL (2) α (z,χ) : Z H U(k) factors through Z (Z/m) k and (Z/m) k is a group of prime power order, where Z acts on (Z/m) k by cyclic permutation, i.e. by 1 (v 1,..., v k ) := (v k, v 1,..., v k 1 ). (3) H 1 (Z H) = Z and Z H Z (Z/m) k induces an isomorphism of the first homology groups. Let K be a slice knot with slice disk D and let α (z,χ) R irr,met k (π 1 (M K )). Consider the following diagram α (z,χ) π 1 (M K ) Z H 1 (M K, Λ) Z H 1 (M K, Λ)/(t k 1) U(k) π 1 (N D ) Z H 1 (N D, Λ) Z H 1 (N D, Λ)/(t k 1) U(k) If χ vanishes on Ker{H 1 (M K, Λ)/(t k 1) H 1 (N D, Λ)/(t k 1)} then χ extends to χ N : H 1 (N D, Λ)/(t k 1) S 1 since S 1 is divisible. Furthermore, if χ is of prime power order, then χ N can be chosen to be of prime power order as well. Note that α (z,χn ) : π 1 (N D ) U(k) is an extension of α (z,χ) : π 1 (M K ) U(k). This proves the following. Lemma 4.3. (1) α (z,χ) extends to a metabelian representation of π 1 (N D ) if and only if χ vanishes on Ker{H 1 (M K, Λ)/(t k 1) H 1 (N D, Λ)/(t k 1)}. (2) If α (z,χ) P k (π 1 (M K )) extends to a metabelian representation of π 1 (N D ) then it extends to a representation in P k (π 1 (N D )). Theorem 4.4. Let K be a slice knot and D a slice disk. If α extends to a metabelian representation of π 1 (N D ) and if α P k (π 1 (M K )), then η α (M K ) = 0. Proof. By lemma 4.3 we can find an extension β P k (π 1 (N D )). We can decompose N D as N D = W 4 MO S 1 D 3 where M O = S 1 S 2 is the zero-framed surgery along the trivial knot in S 3 and W is a homology Z bordism between M K and M O. The statement now follows from theorem 4.2 and lemma 3.3 since (W, id) is a homology Z H 1 (W, Λ) bordism between (M K, i ) and (M O, i ) Main sliceness obstruction theorem. Theorem 4.5. Let K be a slice knot, k a prime power, then there exists a metabolizer P k T H 1 (M k ) for the linking pairing, such that for any representation α : π 1 (M K ) Z H 1 (M K, Λ)/(t k 1) U(k) vanishing on 0 P k and lying in P irr,met k (π 1 (M K )) we get η α (M K ) = 0. Proof. Let D be a slice disk. Let P k := Ker{H 1 (M K, Λ)/(t k 1) H 1 (N D, Λ)/(t k 1)}, this is a metabolizer for the linking pairing by proposition The theorem now follows from lemma 4.3 and theorem 4.4.

13 ETA INVARIANTS AS SLICENESS OBSTRUCTIONS 13 In the following we ll show that some eta-invariants of slice knots vanish for nonprime power dimensional irreducible representations. Let α 1 R k1 (G), α 2 R k2 (G), then we can form the tensor product α 1 α 2 R kl (G). Proposition 4.6. If k 1, k 2 are coprime, then α (k1,z 1,χ 1 ) α (k2,z 2,χ 2 ) = α (k1 k 2,z 1 z 2,χ 1 χ 2 ) If furthermore α (k1,z 1,χ 1 ) and α (k2,z 2,χ 2 ) are irreducible, then α (k1,z 1,χ 1 ) α (k2,z 2,χ 2 ) is irreducible as well. Proof. Denote by e 11,..., e k1 1 and e 12,..., e k2 2 the canonical bases of C l 1 and C l 2. Set f i := e i mod k1,1 e i mod k2,2 for i = 0,..., k 1 k 2 1. The f i s are distinct, therefore {f i } i=0,...,k1 k 2 1 form a basis for C k 1 C k 2. One can easily see that α 1 α 2 with respect to this basis is just α (k1 k 2,z 1 z 2,χ 1 χ 2 ). The last statement follows from the observation that if χ 1 χ 2 : H/(t k 1k 2 1) factors through H/(t k 1) for some k < k 1 k 2, then one of the χ i : H H/(t k i 1) factors through H/(t k 1) for some k < k i. For a prime number p and a Λ-module H denote by Pk,p irr (Z H) the set of representations α (z,χ) in Pk irr (Z H) where χ factors through a p-group. Define (π 1 (M K )) := Pk,p irr(z H 1(M K, Λ)). P irr,met k,p Theorem 4.7. Let K be a slice knot, k 1,..., k r pairwise coprime prime powers, then there exist metabolizers P ki T H 1 (M ki ), i = 1,..., r for the linking pairings, such that for any prime number p and any choice of irreducible representations α i : π 1 (M K ) Z H 1 (M K, Λ)/(t k i 1) U(k) vanishing on 0 P ki and lying in P irr,met k i,p (π 1 (M K )) we get η α1 α r (M K ) = 0. Proof. Let D be a slice disk and let P ki := Ker{H 1 (M K, Λ)/(t k i 1) H 1 (N D, Λ)/(t k i 1)}. All the representations α 1,..., α r extend to metabelian representations of N D, hence α 1 α r also extends to a metabelian representation of N D. Write α = α (zi,χ i ), then α 1 α r = α (z1...z r,χ 1 χ r) since the k i are pairwise coprime. This shows that α 1 α r P irr,met k (π 1 k r,p 1(M K )), therefore η α1 α r (M K ) = 0 by theorem 4.4. Proposition 4.6 shows that α 1 α r is irreducible since gcd(k 1,..., k r ) = 1, i.e. the theorem shows that certain non-prime-power dimensional irreducible eta invariants vanish for slice knots. Letsche [L00] first pointed out the fact that non prime power dimensional representations can give sliceness (ribbonness) obstructions. In fact his non prime power dimensional representations can be shown to be tensor products of prime power dimensional representations. We say that a knot K has zero slice eta obstruction (SE obstruction) if the conclusion of theorem 4.5 holds for all prime powers k, and K has zero slice tensor eta obstruction (STE obstruction) if the conclusion of theorem 4.7 holds for all pairwise coprime prime powers k 1,..., k r.

14 14 STEFAN FRIEDL Question 4.8. Are there examples of knots which have zero SE obstruction but non zero STE obstruction? Remark. It s easy to find examples of a knot K and one-dimensional representations α, β such that η α (M K ) = 0 and η β (M K ) = 0 but η α β (M K ) 0. This shows that in general η α β (M K ) is not determined by η α (M K ) and η β (M K ). Note that if T H 1 (M k ) = 0, then R irr,met k (π 1 (M K )) =, therefore theorem 4.7 only gives a non-trivial sliceness obstruction if T H 1 (M k ) 0 for some prime power k. This is not always the case, in fact Livingston [L01, thm. 0.5] proved the following theorem. Theorem 4.9. Let K be a knot. There exists a prime power k with T H 1 (M k ) 0 if and only if K (t) has a non-trivial irreducible factor that is not an n-cyclotomic polynomial with n divisible by three distinct primes. In chapter 8.4, we ll use this theorem to show that there exists a knot K with H 1 (L k ) = 0 for all prime powers k, but H 1 (L 6 ) 0. This shows that K has nontrivial irreducible U(6)-representations, but no unitary irreducible representations of prime power dimensions. In particular not all representations are tensor products of prime power dimensional representations. 5. Casson-Gordon obstruction 5.1. The Casson-Gordon obstruction to a knot being slice. We first recall the definition of the Casson-Gordon obstructions [CG86]. For m a number denote by C m S 1 the unique cyclic subgroup of order m. For a surjective character χ : H 1 (M k ) H 1 (L k ) C m, set F χ := Q(e 2πi/m ). Since Ω 3 (Z C m ) = H 3 (Z C m ) is torsion (cf. [CF64]) there exists (Vk 4, ɛ χ) and some r N such that (V k, ɛ χ) = r(m k, ɛ χ). The (surjective) map ɛ χ : π 1 (V k ) Z C m defines a (Z C m )-cover Ṽ of V. Then H 2 (C (Ṽ )) and F χ (t) have a canonical Z[Z C m ]-module structure and we can form H 2 (C (Ṽ ) Z[Z Cm] F χ (t)) =: H (V k, F χ (t)). Since F χ (t) is flat over Z[Z C m ] by Maschke s theorem (cf. [L93]) we get H (V k, F χ (t)) = H (C (Ṽ ) Z[Z Cm] F χ (t)) = H (Ṽ ) Z[Z Cm] F χ (t) which is a free F χ (t)-module. If χ is a character of prime power order, then the F χ (t)- valued intersection pairing on H 2 (V k, F χ (t)) is non-singular (cf. [CG86, p. 190]) and therefore defines an element t(v k ) L 0 (F χ (t)), the Witt group of non singular hermitian forms over F χ (t) (cf. [L93], [R98]). Denote the image of the ordinary intersection pairing on H 2 (V k ) under the map L 0 (C) L 0 (F χ (t)) by t 0 (V k ). Proposition 5.1. [CG86] If χ : H 1 (L k ) C m is a character with m a prime power, then τ(k, χ) := 1 r (t(v k) t 0 (V k )) L 0 (F χ (t)) Z Q

15 ETA INVARIANTS AS SLICENESS OBSTRUCTIONS 15 is a well-defined invariant of (M k, ɛ χ), i.e. independent of the choice of V k. Casson and Gordon prove the following theorem (cf. [CG86, p. 192]). Theorem 5.2. Let k be a prime power. If K S 3 is slice then there exists a metabolizer P k for the linking pairing, such that for any χ : T H 1 (M k ) C m, m a prime power, with χ(p k ) 0 we get τ(k, χ) = Interpretation of Casson-Gordon invariants as eta invariants of M k. For a numberfield F C we consider F with complex involution and F (t) with the involution given by complex involution and t = t 1. For z Σ 1 transcendental and τ L 0 (F (t)) we can consider τ(z) L 0 (z). Note that sign : L 0 (C) Z defines an isomorphism. Let K be a knot, k any number, m a prime power and χ : H 1 (L k ) C m a character and (Vk 4, ɛ χ : π 1(V k ) Z C m ) such that (V k, ɛ χ) = r(m k, ɛ χ). For a character χ : H 1 (L k ) C m define characters χ j by setting χ j (v) := χ(v) j, for z S 1 define β (z,χ j ) : π 1 (M k ) H 1 (M k ) = Z H 1 (L k ) S 1 = U(1) (n, v) z n χ j (v) Proposition 5.3. Let z S 1 transcendental, then sign(τ(k, χ)(z)) = η β(z,χ 1 ) (M k ) Proof. Let z S 1 transcendental, define θ : Z C m S 1 by θ(n, y) := z n y. Then (V k, θ (ɛ χ)) = r(m k, θ (ɛ χ)) = r(m k, β (z,χ 1 )) We view C as a Z[Z C m ]-module via θ (ɛ χ) and C as an F χ (t) module via evaluating t to z. Note that both modules are flat by Maschke s theorem, hence H β 2 (V k, C) = H 2 (C (Ṽ )) Z[Z Cm] C = (H 2 (C (Ṽ )) Z[Z Cm] F χ (t)) Fχ(t) C = H 2 (V k, F χ (t)) Fχ(t) C This also defines an isometry between the forms, i.e. sign β (V k ) = sign(v k (z)). This shows that rη β(z,χ 1 )(M k ) = sign β (V k ) sign(v k ) = sign(t(v k )(z)) sign(t 0 (V k )) = rsign(τ(k, χ)(z)) Remark. The eta invariant carries potentially more information than the function z sign(τ(k, χ)(z)), since for non-transcendental z S 1 the number τ z (K, χ) is not defined, whereas η β(z,χ1 ) (M k) is still defined. For example the U(1)-signatures for slice knots are zero outside the set of singularities, but the eta invariant at the singularities contains information about knots being doubly slice (cf. section 3.3).

16 16 STEFAN FRIEDL Proposition 5.4. Let K be a knot, k any number, m a prime power and χ : H 1 (L k ) C m a character, then τ(k, χ) = 0 L 0 (F χ (t)) Z Q sign(ρ(τ(k, χ))(z)) = 0 Q for all transcendental z S 1, ρ Gal(F χ, Q) η β(z,χ j ) (M k ) = 0 Z for all (j, m) = 1, all transcendental z S 1 Proof. The first equivalence is a purely algebraic statement, which will is proven in a separate paper (cf. [F03]) using results of Ranicki s [R98]. The second equivalence follows from proposition 5.3 and the observation that if ρ Gal(F χ, Q) sends e 2πi/m to e 2πij/m for some (j, m) = 1, then ρ(τ(k, χ)) = τ(k, χ j ) and hence sign(ρ(τ(k, χ))(z)) = sign(τ(k, χ j )(z)) = η β(z,χ j ) (M k ) Interpretation of Casson-Gordon invariants as eta invariants of M K. The goal is to prove a version of proposition 5.4 with eta invariants of M K instead of eta invariants of M k. Proposition 5.5. Let K be a knot, z S 1 and χ : H 1 (M k ) H 1 (L k ) C m a character. Let β := β (z,χ) : π 1 (M k ) U(1) and α = α (z,χ) : π 1 (M K ) U(k), then η α (M K ) η β (M k ) = k j=1 σ e 2πij/k(K) If K is algebraically slice and k such that H 1 (L k ) is finite, then η α (M K ) = η β (M k ). Proof. In [F03b] we show that if M G M is a G fold cover and α G : π 1 (M G ) U(1) is a representation then η αg (M G ) = η α (M) η α(g) (M) where α(g) : π 1 (M) G U(C[G]) = U(C G ) is given by left multiplication and α : π 1 (M) U(k) is the (induced) representation given by α : π 1 (M) Aut(Cπ 1 (M) Cπ1 (M G ) C) a (p v ap v) In our case G = Z/k and one can easily see that α = α (z,χ). Since Z/k is abelian it follows that α(g) = k i=1 α i where α i : π 1 (M K ) U(1) is given by α j (z) := e 2πij/k. The proposition now follows from propositions 2.7 and 3.4. We say that a knot K S 3 has zero Casson-Gordon obstruction if for any prime power k there exists a metabolizer P k T H 1 (M k ) for the linking pairing such that for any prime power m and χ : T H 1 (M k ) C m with χ(p k ) 0 we get τ(k, χ) = 0 L 0 (F χ (t)) Q. The following is an immediate consequence of propositions 5.4 and 5.5.

17 ETA INVARIANTS AS SLICENESS OBSTRUCTIONS 17 Theorem 5.6. Let K be an algebraically slice knot. Then K has zero SE obstruction if and only if K has zero Casson-Gordon obstruction. 6. Various obstruction theorems 6.1. Obstructions to a knot being ribbon. In the following we ll fix a collar S 3 [0, 1] D 4 of (D 4 ) = S 3 0. Definition. A knot K S 3 = S 3 {0} is called ribbon if there exists a smooth disk D in S 3 [0, 1] D 4 bounding K such that the projection map S 3 [0, 1] [0, 1] is a Morse map and has no local minima. Such a slice disk is called a ribbon disk. Proposition 6.1. ([G81, lemma 3.1] and [K75b, lemma 2.1]) If K is ribbon and D D 4 a ribbon disk, then the maps are surjective. i : π 1 (S 3 \ K) π 1 (D 4 \ D) H 1 (M K, Λ) H 1 (N D, Λ) It is a longstanding conjecture of Fox (cf. [F61, problem 25]) that slice knots are ribbon. We ll prove a ribbon-obstruction theorem (theorem 6.3) which ostensibly will be stronger than the corresponding theorem for slice knots (theorem 4.7), and therefore potentially gives a way to show that a given slice knot is not ribbon. Proposition 6.2. Assume that K is ribbon, D a ribbon disk, then H 1 (N D, Λ) is Z- torsion free, in particular P := Ker{H 1 (M K, Λ) H 1 (N D, Λ)} is a metabolizer for λ Bl. Proof. According to [L77, thm. 2.1 and prop. 2.4] there exists a short exact sequence 0 Ext 2 Λ(H 1 (N, M, Λ)) H 1 (N, Λ) Ext 1 Λ(H 2 (N, M, Λ)) 0 Here H 1 (N, Λ) denotes H 1 (N, Λ) with involuted Λ-module structure, i.e. t v := t 1 v. Furthermore Ext 1 Λ(H 2 (N, M, Λ)) is Z-torsion free (cf. [L77, prop. 3.2]). In order to show that H 1 (N, Λ) is Z-torsion free it is therefore enough to show that H 1 (N, M, Λ) = 0. Consider the exact sequence H 1 (M, Λ) H 1 (N, Λ) H 1 (N, M, Λ) H 0 (M, Λ) H 0 (N, Λ) 0 The last map is an isomorphism. By proposition 6.1 the first map is surjective. It follows that H 1 (N, M, Λ) = 0. The second part follows immediately from proposition 2.5. Theorem 6.3. Let K S 3 be a ribbon knot. Then there exists a Λ-subspace P H 1 (M K, Λ) such that P = P and such that for any α Pk irr(π 1(M K )) vanishing on 0 P we get η α (M K ) = 0.

18 18 STEFAN FRIEDL Proof. Let P := Ker{H 1 (M K, Λ) H 1 (N D, Λ)} where D is a ribbon disk for K. Then P = P by proposition 6.2. Let α P k (π 1 (M)) which vanishes on 0 P, then α extends to a metabelian representation of π 1 (N D ), hence η α(z,χ) (M K ) = 0 by lemma 4.3 and theorem 4.4. We say that K has zero eta invariant ribboness obstruction if the conclusion holds for K. Remark. One can show that if P is a metabolizer for λ Bl and k is such that H 1 (L k ) is finite, then P k := π k (P ) T H 1 (M k ) = H 1 (M, Λ)/(t k 1) is a metabolizer for λ L,k. In particular if K is and k any number such that H 1 (L k ) is finite then there exists a metabolizer P k for the linking pairing such that for all χ : T H 1 (M k ) S 1 of prime power order, vanishing on P k, and for all transcendental z S 1 we get η α(z,χ) (M K ) = 0. Comparing this result with theorem 4.7 we see that the ribbon obstruction is stronger in two respects. When K is ribbon (1) we can find metabolizers P k which all lift to the same metabolizer of the Blanchfield pairing, (2) the representations for non-prime power dimensions don t have to be tensor products. We ll make use of this in proposition Remark. (1) Note that the only fact we used was that for a ribbon disk H 1 (M K, Λ) H 1 (N D, Λ) is surjective. (2) Casson and Gordon [CG86, p. 154] prove a ribbon obstruction theorem which does not require the character to be of prime power order, but has a strong restriction on the fundamental group of π 1 (N D ) The Gilmer obstruction and the Letsche obstruction. Gilmer [G83], [G93] published a sliceness obstruction theorem which combines the Casson Gordon obstruction with the Seifert pairing. Letsche (cf. [L00, p. 313]) published a different obstruction theorem based on eta invariants which factor through Z Q(t)/Z[t, t 1 ]. Both papers contain gaps, in both cases the problem boils down to the fact for a slice disk one can not show that Ker{H 1 (M K, Λ) H 1 (N D, Λ)} is a metabolizer. The following can shown to be true. Theorem 6.4. Let K be a ribbon knot, then K has zero Gilmer obstruction for at least one F and K has zero Letsche obstruction. Furthermore one can show that if a knot K has zero eta invariant ribboness obstruction, then K has zero Gilmer obstruction and zero Letsche obstruction Obstructions to a knot being doubly slice. A knot K S 3 is called doubly slice (or doubly null-concordant) if there exists an unknotted smooth two-sphere S S 4 such that S S 3 = K. It is clear that a doubly slice knot is in particular slice.

19 ETA INVARIANTS AS SLICENESS OBSTRUCTIONS 19 Fox [F61] first posed the question which slice knots are doubly slice. Doubly slice knots have been studied by Levine [L89], Ruberman [R83] and Kim [K02]. We( say that ) knot K is algebraically doubly slice if K has a Seifert matrix of the 0 B form where B, C are square matrices of the same size. Sumners [S71] showed C 0 that if K S 3 is doubly slice, then K is algebraically doubly slice. We prove the following new doubly sliceness obstruction theorem. Theorem 6.5. Let K S 3 be a doubly slice knot. Then there exist metabolizers P 1, P 2 H 1 (M K, Λ) for the Blanchfield pairing such that (1) H 1 (M K, Λ) = P 1 P 2, (2) for any α Pk irr(z H 1(M K, Λ)) vanishing on 0 P i, i=1,2, we get η α (M K ) = 0. Proof. Let S S 4 be an unknotted two-sphere such that S S 3 = K. Intersecting S with {(x 1,..., x 5 ) R 5 x 5 0} and {(x 1,..., x 5 ) R 5 x 5 0} we can write write S = D 2 1 K D 2 2 and S 4 = D 4 1 S 3 D 4 2. Let N i = D 4 i \ D 2 i, then N 1 N 2 = S 3 \ K and N 1 N 2 = S 4 \ S. From the Mayer-Vietoris sequence we get H 1 (M K, Λ) = H 1 (N 1, Λ) H 1 (N 2, Λ) since H 1 (M K, Λ) = H 1 (S 3 \ K, Λ) and H 1 (S 4 \ S, Λ) = 0 since S is trivial. Now let P i := Ker{H 1 (M, Λ) H 1 (N i, Λ)}, the proof concludes as the proof of the main ribbon obstruction theorem 6.3 since H 1 (M K, Λ) H 1 (N i, Λ) is surjective for i = 1, 2. Remark. (1) The proof of theorem 6.5 shows in particular that if K is doubly slice we can find a slice disk D such that H 1 (N D, Λ) is Z-torsion free. (2) Comparing theorem 6.5 with theorems 4.7 and 6.3 we see that doubly slice knots immediately have zero (doubly) ribbon obstruction. Question 6.6. Using the notation of the proof we get from the van Kampen theorem that for a doubly slice knot Z = π 1 (D 4 1 \ D 2 1) π1 (S 3 \K) π 1 (D 4 2 \ D 2 2) Can we conclude that π 1 (S 3 \ K) π 1 (D 4 i \ D 2 i ) is surjective for at least one i? If yes, then this would show that a doubly slice knot is in fact homotopically ribbon. One can go further and ask whether doubly slice knots are in fact ribbon or doubly ribbon. The first part of the question is of course a weaker version of the famous slice equals ribbon conjecture. Remark. Taehee Kim [K03] introduced the notion of (n, m)-solvability (n, m 1 2 N) and introduced L 2 -eta invariants to find highly non trivial examples of non doubly slice knots.

20 20 STEFAN FRIEDL 7. The Cochran-Orr-Teichner-sliceness obstruction 7.1. The Cochran-Orr-Teichner-sliceness filtration. We give a short introduction to the sliceness filtration introduced by Cochran, Orr and Teichner [COT01]. For a manifold W denote by W (n) the cover corresponding to π 1 (W ) (n). Denote the equivariant intersection form H 2 (W (n) ) H 2 (W (n) ) Z[π 1 (W )/π 1 (W ) (n) ] by λ n, and the self-intersection form by µ n. An (n)-lagrangian is a submodule L H 2 (W (n) ) on which λ n and µ n vanish and which maps onto a Lagrangian of λ 0 : H 2 (W ) H 2 (W ) Z. Definition. [COT01, def. 8.5] A knot K is called (n)-solvable if M K bounds a spin 4-manifold W such that H 1 (M K ) H 1 (W ) is an isomorphism and such that W admits two dual (n)-lagrangians. This means that λ n pairs the two Lagrangians non-singularly and that the projections freely generate H 2 (W ). A knot K is called (n.5)-solvable if M K bounds a spin 4-manifold W such that H 1 (M K ) H 1 (W ) is an isomorphism and such that W admits an (n)-lagrangian and a dual (n + 1)-Lagrangian. We call W an (n)-solution respectively (n.5)-solution for K. Remark. (1) The size of an (n)-lagrangian depends only on the size of H 2 (N D ), in particular if K is slice, D a slice disk, then N D is an (n)-solution for K for all n, since H 2 (N D ) = 0. (2) By the naturality of covering spaces and homology with twisted coefficients it follows that if K is (h)-solvable, then it is (k)-solvable for all k < h. Theorem 7.1. K is (0) solvable Arf(K) = 0 K (t) ±1 mod 8 K is (0.5) solvable K is algebraically slice K is (1.5) solvable Casson-Gordon invariants vanish and K algebraically slice. The converse of the last statement is not true, i.e. there exist algebraically slice knots which have zero Casson-Gordon invariants but are not (1.5) solvable. Furthermore there exist (2.0) solvable knots The first statements are shown in [COT01]. Taehee Kim [K02] showed that there exist (1.0) solvable knots which have zero Casson-Gordon invariants, but are not (1.5) solvable (cf. also proposition 8.13) L 2 eta invariants as sliceness-obstructions. In this section we very quickly summarize some L 2 eta invariant theory. Let M 3 be a smooth manifold and ϕ : π 1 (M) G a homomorphism, then Cheeger and Gromov [CG85] defined an invariant η (2) (M, ϕ) = η (2) ϕ (M) R, the (reduced) L 2 (M, G) eta invariant. When it s clear which homomorphism we mean, we ll write η (2) G (M). for η (2) ϕ

21 ETA INVARIANTS AS SLICENESS OBSTRUCTIONS 21 Remark. If (W, ψ) = (M 3, ϕ), then (cf. [COT01, lemma 5.9 and remark 5.10]) η (2) ϕ (M) = sign (2) (W, ψ) sign(w ) where sign (2) (W, ψ) denotes Atiyah s [A76] L 2 signature. Let QΛ := Q[t, t 1 ]. Theorem 7.2. [COT01] (1) If K is (0.5) solvable, then η (2) Z (M K) = 0. (2) If K is (1.5) solvable, then there exists a metabolizer P Q H 1 (M K, QΛ) for the Blanchfield pairing λ Bl,Q : H 1 (M K, QΛ) H 1 (M K, QΛ) Q(t)/Q[t, t 1 ] such that for all x P Q we get η (2) (M K, β x ) = 0, where β x denotes the map π 1 (M K ) Z H 1 (M K, Λ) Z H 1 (M K, QΛ) id λ Bl,Q(x, ) Z Q(t)/Q[t, t 1 ] We say that a knot K has zero abelian L 2 eta invariant sliceness obstruction if η (2) Z (M K) = 0 and K has zero metabelian L 2 eta invariant sliceness obstruction if there exists a metabolizer P Q H 1 (M K, QΛ) for λ Bl,Q such that for all x P Q we get η (2) β x (M K ) = 0. Note that if K has zero metabelian L 2 eta invariant then by the above lemma and theorem 2.4 K is in particular algebraically slice. In [F03c] we show that if a knot K has zero metabelian eta invariant ribbon obstruction, then K has in particular zero metabelian L 2 eta invariant sliceness obstruction. In proposition 8.12 we ll see that the converse is not true. In this section we ll construct 8. Examples (1) a knot which has zero abelian L 2 eta invariant, but is not algebraically slice, (2) a (1.0) solvable knot which has zero metabelian L 2 eta invariant, but non-zero SE obstruction, (3) a knot which has zero metabelian L 2 eta invariant, zero STE obstruction, but is not ribbon, and (4) a knot which has zero ST E obstruction but non-zero metabelian L 2 eta invariant, (5) a ribbon knot K such that there exists no metabolizer P 5 of (H 1 (L 5 ), λ L ) with the property that τ(k, χ) = 0 for all (including non prime power) characters χ : T H 1 (M 5 ) S 1. The idea is in each case to start out with a slice knot K and make slight changes via a satellite construction, the change in the eta invariants can be computed explicitly.

22 22 STEFAN FRIEDL 8.1. Satellite knots. Let K, C S 3 be knots. Let A S 3 \K a curve, unknotted in S 3, note that S 3 \ N(A) is a torus. Let ϕ : (N(A)) (N(C)) be a diffeomorphism which sends a meridian of A to a longitude of C and a longitude of A to a meridian of C. The space S 3 \ N(A) ϕ S 3 \ N(C) is a 3 sphere and the image of K is denoted by S = S(K, C, A). We say S is the satellite knot with companion C, orbit K and axis A. Note that we replaced a tubular neighborhood of C by the torus knot K S 3 \ N(A). Proposition 8.1. If K S 1 D 2 S 3 and C are slice (ribbon), then any satellite knot with orbit K and companion C is slice (ribbon) as well. Proof. Let K S 1 D 2 S 3 and C be slice knots. Let φ : S 1 I S 3 I be a null-concordance for C, i.e. φ(s 1 0) = C and φ(s 1 1) is the unknot. We can extend this to a map φ : S 1 D 2 I S 3 I such that φ : S 1 D 2 0 is the zero-framing for C. Pick a diffeomorphism f : S 3 \ N(A) S 1 D 2 such that the meridian and longitude of A get sent to the longitude and meridian of S 1 0. Now consider ψ : S 1 I K I S 3 \ N(A) I f id S 1 D 2 I φ S 3 I Note that φ : S 1 D 2 1 is a zero framing for the unknot, since linking numbers are concordance invariants and φ : S 1 D 2 0 is the zero framing for C. This shows that ψ : S 1 1 S 3 1 gives the satellite knot of the unknot with orbit K, i.e. K itself. Therefore ψ gives a concordance between S = ψ(s 1 0) and K = ψ(s 1 1), but K is null-concordant, therefore S is null-concordant as well. Now assume that K, C are ribbon. Then we can find a concordance φ which has no minima under the projection S 1 [0, 1] S 3 [0, 1] [0, 1]. It is clear that ψ also has no minima, capping off with a ribbon disk for K we get a disk bounding S with no minima, i.e. S is ribbon. Proposition 8.2. [COT02, p. 8] Let K be an (n.0) solvable knot, C a (0) solvable knot, A S 3 \ K such that A is the unknot in S 3 and [A] π 1 (S 3 \ K) (n). Then S = S(K, C, A) is (n.0) solvable Eta invariants of satellite knots. Let S be a satellite knot with companion C, orbit K and axis A. Proposition 8.3. [L84, p. 337] If A π 1 (S 3 \ K) (1) then the inclusions S 3 \ N(K) S 3 \ N(K) \ N(C) (S 3 \ N(A) ϕ S 3 \ N(C)) \ N(K) = S 3 \ N(S) induce isometries of the Blanchfield pairing and the linking pairing of S and K for any cover such that H 1 (L K,k ) is finite.