aa r X i v : . [ m a t h . G T ] F e b FOUR MANIFOLDS WITH NO SMOOTH SPINES
IGOR BELEGRADEK AND BEIBEI LIU
Abstract.
Let W be a compact smooth 4-manifold that deformation retract toa pl embedded closed surface. One can arrange the embedding to have at mostone non-locally-flat point, and near the point the topology of the embedding isencoded in the singularity knot K . If K is slice, then W has a smooth spine, i.e.,deformation retracts onto a smoothly embedded surface. Using the obstructionsfrom the Heegaard Floer homology and the high-dimensional surgery theory, weshow that W has no smooth spines if K is a knot with nonzero Arf invariant,a nontrivial L-space knot, the connected sum of nontrivial L-space knots, or analternating knot of signature < −
4. We also discuss examples where the interiorof W is negatively curved. Introduction A spine is a topological (not necessarily locally flat), compact, boundaryless sub-manifold that is a strong deformation retract of the ambient manifold. A spine is smooth or pl if the submanifold has this property.Examples of 4-manifolds that are homotopy equivalent to closed sufaces but haveno pl spines can be found in [Mat75, MV79, LL19]. It is shown in [Ven98] thatan example in [MV79] does not even have a topological spine. Some 4-manifoldswith topological spines and no pl spines can be found in [KR]. The present paperconstructs 4-manifolds with pl spines and no smooth spines.Let W be a compact oriented smooth 4-manifold with a pl spine S homeomorphicto a closed oriented surface. By a standard argument S can be moved by a pl homeomorphism to a spine with at most one non-locally-flat point; henceforth weassume that S has this property. If S is locally flat, then the submanifold S issmoothable [RS68, Corollary 6.8]. Otherwise, S intersects the link of the non-locally-flat point in a singularity knot K . If K is smoothly slice, then replacing thecone on K in S with a smoothly embedded disk in W gives a smooth spine of W .Examples of non-slice singularity knot such that W has a smooth spine come fromexotic knot traces. Namely, if K r denotes the 4-manifold obtained by attaching a2-handle to the 4-ball along the knot K in its boundary with framing r , then K r has a pl spine homeomorphic to S with singularity knot K , and [Akb93, TheoremA] describes knots K , K such that K is slice, K is not slice, and K r , K r are Key words and phrases. spine, knot, homology cobordism, Heegaard-Floer, negative curvature.2020 MSC Primary 57K40, Secondary 20F67, 53C20, 57K18, 57Q35, 57Q40.Belegradek was partially supported by the Simons Foundation grant 524838.
IGOR BELEGRADEK AND BEIBEI LIU diffeomorphic for some r . We refer to [HMP] for a recent study of relations betweeninvariants of knot traces and knot concordance.Cappell and Shaneson [CS76] developed a surgery-theoretic criterion that can helpdecide when a manifold with a pl spine of dimension ≥ W × S we prove Theorem 1.1.
If the singularity knot K has nonzero Arf invariant, then the -manifold W has no smooth spines. If W be has two pl spines with regular neighborhoods R , R in the interior of W ,then there is a homology cobordism between the boundaries ∂R , ∂R obtained bygluing W \ Int( R ) and W \ Int( R ) along ∂W . The Heegaard-Floer d -invariants d top , d bot are preserved under homology cobordisms. Furthermore, one can expressthe d -invariants of ∂R , ∂R via singularity knots of their spines, and for someknots the d -invariants can be explicitly computed, which gives the following. Theorem 1.2.
If the singularity knot K is a nontrivial L-space knot, the nontrivialconnected sum of nontrivial L-space knots, or an alternating knot of signature < − ,then W contains no smooth spines. Recall that a knot K ⊂ S is an L-space knot if there is an integer n > n -framed surgery on the knot is an L-space [OS05, Definition 1.1]. For example,torus knots are L-space knots [OS05, p.1285].The d -invariants obstruction applies to some topologically slice knots in [HKL16],which gives Corollary 1.3.
For any g, e ∈ Z with g ≥ there exits a compact smooth -manifold W with no smooth spines and a topological locally flat spine that is an oriented closedgenus g surface with normal number e . We were led to the subject of this paper while thinking of examples in [GLT88,Kui88] of oriented hyperbolic 4-manifolds with pl spines. Each of these manifoldsis a quotient of the hyperbolic space H by a Kleinian group Γ which is a torsion-free finite index subgroup in a certain discrete group Γ of orientation preservingisometries of H . The group Γ is described via face-pairings of its fundamentaldomain F , which is obtained by removing from H a neighborhood of a nontrivialtorus knot T in the ideal boundary of H . In turn, the fundamental domain F forΓ is obtained by gluing k copies of F , where k is the index of Γ in Γ, and onecan describe F as the result of removing from H a neighborhood of the k -foldconnected sum of T . This k -fold connected sum is the singularity knot in a pl spine of H / Γ , and hence H / Γ has no smooth spines by Theorem 1.2.A related construction in [GLT88, Section 6] replaces the torus knot T by an ar-bitrary nontrivial knot K but the group Γ is now generated by reflections in the OUR MANIFOLDS WITH NO SMOOTH SPINES 3 codimension one faces of F . The resulting singularity knot of the pl spine of H / Γ is the k -fold connected sum of K r ¯ K , where r ¯ K is the reverse of the mirror im-age of K . (Here k is even because H / Γ is orientable and Γ does not preserveorientation). Since K r ¯ K is slice, the singularity knot is slice, and H / Γ has asmooth spine.An analog of these examples with variable pinched negative curvature is discussedin [Bel], which is based on Ontaneda’s Riemannian hyperbolization [Ont20]. Herethere is no need to pass to a finite index torsion-free subgroup, and for any knot K one gets pinched negatively curved 4-manifolds whose pl spine has K as asingularity knot. In particular, if K satisfies the assumptions of Theorems 1.1 or1.2, the negatively pinched 4-manifold has no smooth spines, while in the setting ofCorollary 1.3 there exists a topologically flat spine. Acknowledgments : We are grateful to Lisa Piccirillo for leading us to [Akb93]. Liuappreciates the support and hospitality of the Max Planck Institute for Mathematicsin Bonn, where she was a member when this work began.2.
Kervaire invariant of codimension two thickenings
Let W be a compact oriented pl manifold with a pl embedded spine S , a closedconnected oriented manifold of dim( S ) = dim( W ) −
2. Let ξ be an oriented planebundle over S whose Euler class is the normal Euler class of S in W , and let p ξ : D ξ → S be the associated 2-disk bundle. Then [CS76, Proposition 1.6] givesa homology isomorphism h : ( W, ∂W ) → ( D ξ , ∂D ξ ) such that h preserves the ori-entation class in the relative second cohomology, and p ξ ◦ h | S is homotopic to theidentity of S . The map h pulls α := p ∗ ξ ( ν W | S ) to the stable normal bundle ν W of W because h ∗ α and ν W are isomorphic over S to which W deformation retracts.This gives a normal map ( h, b W ) where b W : ν W → α is the above bundle map.Assuming, as we can, that h is transverse regular to the zero section S of D ξ , wesee that N := h − ( S ) is a closed surface which is locally flat in W with normalbundle h ∗ ξ . The stable normal bundle to N is ν N = ν W | S ⊕ h ∗ ξ = h ∗ ( α | S ⊕ ξ ). Thus h | N : N → S is covered by the bundle map b N : ν N → α | S ⊕ ξ . The orientation on ξ and W defines an orientation on N for which h | N : N → S has degree one, andhence ( h | N , b N ) is a normal map.The normal invariant of ( h | N , b N ) is the image of the normal invariant of ( h, b W )under the inclusion-induced map [ W, G/P L ] → [ S, G/P L ], which is a bijectionbecause
S ֒ → W is a homotopy equivalence. This standard fact is stated on [CS76,p.195] and in the appendix of [KR08], and the proof amounts to comparing variousdefinitions of the normal invariant.The Kervaire invariant of a normal map ( h | N , b N ) is by definition the Arf invariantof the intersection form restricted to the kernel of h N ∗ : H ( N ; Z ) → H ( S ; Z ).A normal map with nontrivial Kervaire invariant represents a nontrivial class in IGOR BELEGRADEK AND BEIBEI LIU [ S, G/P L ], see [RS71, Theorem 1.4(ii)], and in fact, the Kervaire invariant definesa group homomorphism [
S, G/P L ] → Z [RS71, Corollary 4.5].3. Kervaire and Arf invariants in dimension four
Let us adopt notations of Section 2 and suppose dim( W ) = 4. Then the group[ S, G/P L ] is isomorphic to H ( S ; Z ) ∼ = Z , see e.g. [KT01, Section 2], and hencethe Kervaire invariant defines an isomorphism [ S, G/P L ] → Z .Fix a triangulation of W for which S is a full subcomplex with only one non-locallyflat point. Its star is an embedded 4-ball B , and C := S ∩ B is the cone on theknot K = S ∩ ∂B , the singularity knot of S ⊂ W . Lemma 3.1.
The Kervaire invariant of W in [ S, G/P L ] is the Arf invariant ofthe knot K .Proof. Let V be the smallest subcomplex that contains a neighborhood of S in W .Since S is a full subcomplex, V is a regular neighborhood of S in W , to which W deformation retracts. Denote the relative interiors of B , C , V by ˚ B , ˚ C , ˚ V . Then V \ ˚ B is a trivial 2-disk bundle over S \ ˚ C . Give B the structure of a trivial 2-diskbundle over a 2-disk, whose zero section Z intersects ∂B in an unknot U . Glue V \ ˚ B and B by an orientation-preserving 2-disk bundle automorphism identifying ∂C with ∂Z = U so that the resulting 2-disk bundle D ξ → S has the same Eulerclass as S ⊂ W .Then the above-mentioned map h : W → D ξ can be chosen so that h | W \ ˚ V is adeformation retraction onto ∂V , the map h | V \ ˚ B is the identity, h takes ( B, ∂B, K )to (
B, ∂B, U ), and h maps a regular neighborhood R K of K homeomorphicallyonto a regular neighborhood R U of U . To define h | B apply [CS76, Proposition1.6] to the thickening B of C and use the fact that any homology equivalence( R K , ∂R K ) → ( R U , ∂R U ) is homotopic to a homeomorphism.Isotope Z to Z ⊂ ∂B and choose h to be transverse regular to Z . Then visibly h − ( Z ) is a Seifert surface S of K , and the kernel of h on the first homology canbe identified with H ( S ), and under this identification that Kervaire invariant of h corresponds to the Arf invariant of the knot K . (cid:3) Proof of Theorem 1.1.
The above thickening V of S is classified by the homotopyclass of a map f : S → BSRN . Let η : BSRN → G/P L be the normal invariantmap, see [CS76, p.182]. Then η ◦ f is the normal invariant of S ֒ → V . Since K has nonzero Arf invariant, by the above discussion the Kervaire invariant of η ◦ f isnonzero.It is easy to check that the thickening V ′ := V × S of S ′ := S × S is the pullbackof S ֒ → V under the coordinate projection p : S × S → S , see [CS76, pp.173–175]. Let i : S → S × S be a section of p , say, given by i ( m ) = ( m, OUR MANIFOLDS WITH NO SMOOTH SPINES 5 η ◦ f = η ◦ f ◦ p ◦ i is homotopically nontrivial, so is η ◦ f ◦ p . Hence S ′ ֒ → V ′ is athickening with nontrivial normal invariant.Arguing by contradiction suppose that W has a locally flat spine L . Then L ′ := L × S is a locally flat spine of W ′ . The restriction to L ′ of the deformationretraction W ′ → S ′ is homotopic to a diffeomorphism g : L ′ → S ′ , see e.g. [Lau74,p.5]. Hence the normal invariant of g is trivial.As we explain in [Bel, Appendix C], the pullback via g of the Poincar´e embeddinggiven by the inclusion S ′ ⊂ W ′ is isomorphic to the Poincar´e embedding of thelocally flat inclusion L ′ ⊂ W ′ . Since dim( S ′ ) is odd and ≥
3, Theorem 6.2 of [CS76]implies that the Poincar´e embedding for L ′ ⊂ W ′ can be realized by a locally flatembedding if and only if the normal invariants of g equals the normal invariant ofthe Poincar´e embedding S ′ ⊂ W ′ . This is a contradiction because these normalinvariants are different and L ′ ⊂ W ′ is locally flat. (cid:3) Remark 3.2.
The above argument can be reversed, namely, if K has zero Arfinvariant, then the the Poincar´e embedding induced by the inclusion S ֒ → W hastrivial normal invariant, and hence so does its product with a circle or more generally,with any closed manifold L , and if dim( L ) is odd, then W × L has a locally flatspine [CS76, Theorem 6.2].4. Heegaard Floer d -invariants and V -functions of knots Ozsv´ath and Szab´o introduced [OS03, OS04c, OS04b] Heegaard-Floer homology the-ories HF o ( M, t ) associated with a Spin c structure t on a closed oriented 3-manifold M . Here o is a decoration indicating the flavor of a Heegaard-Floer theory, and inthis paper o is ∞ or − . The homology groups HF − ( M, t ) and HF ∞ ( M, t ) aremodules over Z [ U ] and Z [ U, U − ], respectively, where U is a formal variable whoseaction lowers the relative homological degree by 2. Related invariants for knots andlinks in 3-manifolds were developed in [Ras03, OS04a, OS08a]. We refer to thesepapers for background.Henceforth, we assume that M has standard HF ∞ [OS03, p.240], and the Spin c structure t is torsion, i.e., their first Chern class has finite order in H ( M ).According to [OS04c, Section 4.2.5] the group H T ( M ) := H ( M ) / Tors acts onthe Heegaard-Floer chain complex CF o ( M, t ), and on the corresponding homologygroup HF o ( M, t ). Let HF o ( M, t ) bot and HF o ( M, t ) top denote the kernel and thecokernel of the H T ( M )-action on HF o ( M, t ). The d -invariants d top ( M, t ) and d bot ( M, t ) are the maximal homological degrees of a non-torsion class in HF − top ( M, t )and HF − bot ( M, t ), respectively, see [OS03, Section 9] and [LR14, Section 3]. If M is a rational homology sphere, the H T ( M )-action is trivial, so that HF − top ( M, t ) = HF − bot ( M, t ) = HF − ( M, t ), and d top ( M, t ) = d bot ( M, t ) is the usual d -invariant forrational homology spheres, as in [OS03]. The invariants d top ( M, t ), d bot ( M, t ) arepreserved under rational homology cobordisms [LR14, Proposition 4.5]. IGOR BELEGRADEK AND BEIBEI LIU
A null-homologous knot K in M gives rise to a Z ⊕ Z filtered chain complex CF K ∞ ( M, K, s ), which is a F [ U, U − ]-module, see [OS04a, Ras03]. The filtrationis indexed by the pair of integers ( i, j ), where i keeps track of the power of U , and j records the so called Alexander filtration . For s ∈ Z , let A − s ( K ) := A − s ( M, K, s )be the subcomplex of CF K ∞ ( M, K, s ) corresponding to max( i, j − s ) ≤ A − s ( K ) is the sumof one copy of F [ U ] and a U -torsion submodule.Following [NW15] we define the V -function V s ( K ) of an oriented knot K ⊂ S sothat − V s ( K ) is the maximal homological degree of the free part of H ∗ ( A − s ( K )).For one-component links the H -function for links of [BG18, Liu17] is the V -functionof knots. For example, the V -function of the unknot U is given by V s ( U ) = 0 for s ≥ V s ( U ) = − s for s <
0. The V -function takes values in nonnega-tive integers [BG18, Proposition 3.10], and furthermore, [BG18, Proposition 3.10]and [Liu17, Lemma 5.5] give Proposition 4.1.
The V -function of an oriented knot K ⊂ S satisfies V − s ( K ) = V s ( K ) + s and V s − ( K ) − V s ( K ) ∈ { , } . Example 4.2.
Let K be an alternating knot of signature σ ; recall that σ ∈ Z .By [HM17, Theorem 1.7] if σ >
0, then V s ( K ) = 0 for all s , and if σ ≤
0, thevalues V ( K ) are given in the table below: σ V ( K ) − k k − k − k + 1 − k − k + 1 − k − k + 25. Surgeries on knots and d -invariants For a positive integer g let C g := g S × S , the connected sum of 2 g copiesof S × S . As usual M n ( K ) denotes the n -framed surgery on a closed oriented3-manifold M along a knot K ⊂ M ; in what follows M is S or C g .If B ⊂ C g is the Borromean knot, then C gn ( B ) has the structure of an oriented circlebundle over the genus g oriented surface with Euler number n [OS08b, Section 5.2].For the unknot U ⊂ S it is well-known that S n ( U ) is an oriented circle bundleover S with Euler number n .It follows from [OS03, Propositions 9.3–9.4], cf. [Par14, Proposition 4.0.5], that C gn ( K B ) has standard HF ∞ for any oriented knot K ⊂ S . The same is truefor S n ( K ) [OS04b, Theorem 10.1]. Thus the d -invariants d top , d bot are definedfor C gn ( K B ) and S n ( K ), and moreover, for S n ( K ) they reduce to the usual d -invariants. They were computed by Ni-Wu [NW15, Proposition 1.6] for S n ( K ),and by Park [Par14, Theorem 4.2.3] for C gn ( B ), n = 0. Park’s argument extends to C gn ( K B ) as follows. OUR MANIFOLDS WITH NO SMOOTH SPINES 7
Theorem 5.1.
For n > , we have (5.2) d top ( C gn ( K B ) , k )) = g + (2 k − n ) − n n − a =0 , ··· ,g { a + V k − g +2 a ( K ) } . (5.3) d bot ( C gn ( K B, k )) = g + (2 k − n ) − n n − a =0 , ··· ,g { a + V k − g +2 a ( K ) } where k labels the torsion Spin c structures on C gn ( K B ) with − n/ < k ≤ n/ .The d -invariant of S n ( K ) is given by (5.4) d ( S n ( K ) , k ) = (2 k − n ) − n n − V k ( K ) . Proof.
As in the proof of [Par14, Theorem 4.1.1], a diagram chase in the surgerymapping cone formula [OS08b, Theorem 4.10] shows that the free part of H ∗ ( A − k )is isomorphic to the free part of HF − ( C gn ( K B ) , k ). The grading of the free partof H ∗ ( A − k ) can be found in [BHL17, Theorem 6.10]. (cid:3) Remark 5.5.
Theorem 5.1 extends to n < C gn ( K B ) = − C g − n ( ¯ K B ), where ¯ K is the mirror of K .Then [LR14, Proposition 3.7] gives d bot ( C gn ( K B ) , k ) = − d top ( C g − n ( ¯ K B ) , k ) d top ( C gn ( K B ) , k ) = − d bot ( C g − n ( ¯ K B ) , k ) . Remark 5.6.
A similar argument also computes d top and d bot for rational surgeries,i.e., when 0 = n ∈ Q .6. Spines, homology cobordisms, and d -invariants Let W be a compact, oriented, smooth 4-manifold with a pl spine S , an orientedgenus g surface with normal Euler number e . As before assume that S has at mostone non-locally-flat point with singularity knot K ⊂ S . If W also has a smoothspine S , then there is a homology cobordism C between the boundaries M , M of the regular neighborhoods of S , S . Namely, C is obtained by removing theinteriors of the regular neighborhoods from W and gluing the results along ∂W .Here M can be described as an e -surgery on C g = g S × S along the knot K B where B is the Borromean knot [BHL17, Theorem 3.1], while M is the circlebundle over S with Euler number e , which is the e -surgery on C g along B .Since H ( C ) ∼ = Z g ⊕ Z /e Z , every torsion Spin c structure on C can be thought ofan element of Z /e Z indexed by k ∈ ( − e/ , e/ M j , j ∈ { , } , gives a torsion Spin c structure on M j , which we denote t kj . Thus(6.1) d top ( M , t k ) = d top ( M , t k ) . IGOR BELEGRADEK AND BEIBEI LIU
Theorem 6.2. If e ≥ , and W contains a smooth spine, then the singularity knot K satisfies (6.3) min a =0 , ··· ,g { a + V − g +2 a ( K ) } = ⌈ g/ ⌉ , where ⌈ g/ ⌉ is the smallest integer that is ≥ g/ .Proof. Since V s ( U ) = | s | − s a =0 , ··· ,g { a + V − g +2 a ( U ) } = ⌈ g/ ⌉ . If e >
0, by Theorem 5.1 d top ( M , t k ) = g + s − a =0 , ··· ,g { a + V k − g +2 a ( K ) } and d top ( M , t k ) = g + s − a =0 , ··· ,g { a + V k − g +2 a ( U ) } where s = (2 k − e ) − e e . Hence(6.5) min a =0 , ··· ,g { a + V k − g +2 a ( K ) } = min a =0 , ··· ,g { a + V k − g +2 a ( U ) } . Combining (6.4) and (6.5) for k = 0 gives (6.3) in the case e > e = 0. Then M j is the 0-surgery on C g , where j = 1 ,
2. Let M ′ j denotethe 1-surgery on C g along the same knot as for M j . By the equality part of [OS03,Corollary 9.14], d top ( M j , t j ) −
12 = d top ( M ′ j , t ′ j ) , where t j , t ′ j are the trivial Spin c structures. Even though [OS03, Corollary 9.14] isstated for knots in S , it generalizes (with the same proof) to knots in 3-manifoldswith standard HF ∞ and trivial HF red , which is how we apply it.By (6.1) M , M have the same d top , and hence d top ( M ′ , t ′ ) = d top ( M ′ , t ′ ), andas before (6.4)–(6.5) imply (6.3), now for e = 0. (cid:3) Corollary 6.6. If W contains a smooth spine with normal Euler number e ≥ ,then the singularity knot K satisfies (6.7) V ( K ) = 0 if g is even and V ( K ) = 0 if g is odd. Proof. If g = 2 k , then by Theorem 6.2min k { V ( K ) + k, V ( K ) + k + 1 , · · · , V k ( K ) + 2 k } = k. Proposition 4.1 gives V s − ( K ) ≤ V s ( K ) + 1, and hence the minimum occurs for V ( K ) + k = k , which implies V ( K ) = 0. Similarly, if g = 2 k + 1, we havemin k { V ( k ) + k + 1 , · · · , V k +1 ( K ) + 2 k + 1 } = k + 1which means that V ( K ) + k + 1 = k + 1, and hence V ( K ) = 0. (cid:3) OUR MANIFOLDS WITH NO SMOOTH SPINES 9 Singularity knots and no smooth spines
As in Section 6 let W a compact, oriented, smooth 4-manifold with a pl spinewhich is an oriented genus g surface with normal Euler number e , and at most onenon-locally-flat point with singularity knot K . After changing orientation of W , ifneeded, we can and will assume that e ≥ Proof of Theorem 1.2.
By Corollary 6.6 V ( K ) = 0 or V ( K ) = 0 depending on theparity of g , and hence g ( K ) ≤ g ( K ) is the genus ofof K . If g ( K ) = 0, then K is the unknot. A genus-one L-space knot is the theright-handed trefoil [Ghi08, Corollary 1.5]. According to [NNU98] the Arf invariantfor the torus knot T ( p, q ) is ( p − q − /
24 (mod 2). Thus the Arf invariant of T (2 ,
3) is nonzero, which implies by Theorem 1.1 that W cannot contain a smoothspine. This completes the proof when K is an L-space knot.Suppose K is an alternating knot of signature < −
4. Hence V ( K ) ≥ V ( K ) ≥
1, which by Corollary 6.6 shows that W does not have a smooth spine.Finally, suppose that K is the connected sum of nontrivial L-space knots K , · · · , K n with n ≥
2. Thus g ( K ) = g ( K ) + · · · + g ( K n ). Since K i is nontrivial, g ( K i ) ≥ g ( K ) ≥ n . For j ∈ Z set R K i ( j ) := V g ( K i ) − j ( K i ) and R K ( j ) := min j ··· + jn = j R K ( j ) + · · · + R K n ( j n ) . By Proposition 4.1 the function R K i is nonnegative and nondecreasing, and com-bining the proposition with [Liu, Lemma 2.11] gives R K i (1) = V g ( K i ) − ( K i ) = 1.Hence R K i ( j ) ≥ j ≥ V j ( K )+ j = R K ( g ( K )+ j );the notations in [BL14] are different. Again, by Corollary 6.6 if V ( K ) and V ( K )are both nonzero, then W does not have a smooth spine.To see that V ( K ) = R K ( g ( K )) ≥ R K ( g ( K )) is attainedfor j + · · · + j n = g ( K ). Then j i ≥ i , and R K ( g ( K )) ≥ R K i ( j i ) ≥ ≤ V ( K ) = R K ( g ( K ) + 1) − R K ( g ( K ) + 1) is attained for j + · · · + j n = g ( K ) + 1. If j i ≥ g ( K i ) + 2, then R K ( g ( K ) + 1) ≥ R K i ( j i ) ≥ R K i ( g ( K i ) + 2) = V − ( K i ) = V ( K i ) + 2 ≥ j i ≥ g ( K i ) and j l = g ( K l ) + 1. Then R K i ( j i ) ≥ V ( K i ) ≥ R K l ( j l ) = V ( K l ) + 1 ≥
1, and hence R K ( g ( K ) + 1) ≥ (cid:3) Proof of Corollary 1.3.
For any m ≥
2, there is a topologically slice knot K m with V ( K m ) = m [HKL16, Proposition 6 and Theorem B.1]. The corresponding man-ifold W has a topologically flat spine. By Corollary 6.6 and Proposition 4.1 if W has a smooth spine, then V ( K ) ∈ { , } . (cid:3) References [Akb93] S. Akbulut,
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Email address : [email protected] URL : ∼ ib Beibei Liu, School of Mathematics, Georgia Tech, Atlanta, GA, USA 30332
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