aa r X i v : . [ m a t h . G T ] O c t HOMOTOPY PROPERTIES OF KNOTS IN PRIME MANIFOLDS
PRUDENCE HECK
Abstract.
We define homotopy-theoretic invariants, h -invariants, of knots in prime man-ifolds. Fix a knot J in a prime manifold M . Call a knot K ⊂ M concordant to J if itcobounds a properly embedded annulus with J in M × I , and call K J -characteristic ifthere is a degree-one map α : M → M throwing K onto J and mapping M − K to M − J .These h -invariants are invariants of concordance and of J -characteristicness when α in-duces the identity on π ( M ), and may be viewed as extensions of Milnor’s µ -invariants.We do not require the knots considered here to be rationally null-homologous or framed. Introduction
It is well known that closed, orientable, prime 3-manifolds come in three forms: thefirst type have finite fundamental group and universal cover S , the second type have in-finite, noncyclic fundamental group and contractible universal cover, and the third type is S × S . Manifolds of the first two types are irreducible, meaning that every embedded2-sphere bounds a ball. We define invariants for any knot K in a closed, orientable, prime3-manifold M ; no choice of framing for K is necessary. However, we orient K when itrepresents a nontrivial class in π ( M ).We consider concordance of knots in M , where knots K and J are concordant in M ifthey cobound a properly embedded cylinder in M × I . Concordant knots are freely ho-motopic, making the initial obstruction to concordance of knots the difference of their freehomotopy classes. Yet deeper homotopy theoretic obstructions underlie concordance, andwe investigate these here. We define new homotopy theoretic concordance invariants forknots in M . These invariants extend to knots in three manifolds the prior obstructions de-fined in [8, 6, 11, 10]. The invariants defined here are invariants of the Poincar´e embeddingtype of the knot. The term, while known to a few specialists, to my knowledge has notappeared in the literature in this context, so we elaborate below.Let γ ∈ π ( M ) be a fixed homotopy class. A knot K in the conjugacy class of γ deter-mines an inclusion of it’s sphere bundle in M , α K : T ֒ → E K . If K and J are concordantvia a concordance C and if E C ⊂ M × I is the exterior of C then the images of thesehomotopy classes, [ α K ] and [ α J ], agree in the set [ T , E C ]. Although the complement of aknot is not a concordance invariant, the knot determines the homology type of its exterior.In particular, as observed in Cappell and Shaneson [1], the exterior of a concordance isa Z [ π ( M )]-homology cobordism between E K and E J . Vogel defined a notion of homol-ogy localization of spaces, a functor ∧ on pointed CW-complexes which is initial amongfunctors that turn continuous maps between finite complexes with contractible cofiber intohomotopy equivalences, together with a natural transformation from the identity functor Date : October 19, 2018. to ∧ , [15, 5]. In particular, a homology equivalence f : Y ֒ → X inducing a normally surjec-tive homomorphism on fundamental groups induces a homotopy equivalence ˆ f : b Y → b X .Returning to concordance, it follows that the image of the homotopy class [ α K ] under theset map [ T , E K ] → [ T , d E K ] is a homotopy theoretic concordance invariant of K . Thisinvariant, a homological analogue for codimension two embeddings of the notion of Poincar´eembedding type that appears in Wall [16], first appeared in the context of concordance ofdisk links in work of Le Dimet [5]. Our invariants are invariants of the Poincar´e embeddingtype of the knot in this sense. However, localization of spaces is an obscure construction,for which very little is known and for which computations are currently out of reach. Bycontrast, we expect our invariants to be highly computable, reducing to computations inthree dimensional group homology.After fixing a Poincar´e embedding type one can try to construct a concordance betweentwo knots by choosing a cobordism rel boundary of the knot exteriors, and using surgerytechniques to modify this cobordism to a homology cobordism. This idea underpins Cap-pell and Shaneson’s approach to high-dimensional codimension-two embeddings [1], andwas refined in [2] to detect more subtle obstructions for knots in S . We refer the readerto Carolyn Otto’s thesis [12] for an illustration of why it makes sense to fix a Poincar´eembedding type before setting up a surgery problem. She considers the COT filtration ofthe string link concordance group, SL , which does not take Poincar´e embedding type intoaccount, and cleverly shows the non-triviality of SL n. / SL n +1 by constructing elementswith different Poincar´e embedding types (i.e. elements distinguished by their µ -invariants).In contrast, there is a unique Poincar´e embedding type for knots in S , and it is still anopen question if F n. / F n +1 is nontrivial for any n for the COT filtration F of the knot con-cordance group. Our invariants can be recognized as extensions of the homotopy theoreticinvariants of Orr for links in S [11, 10], which are equivalent to Milnor’s µ -invariants. Wenote that our invariants are not the first extension of homotop theoretic invariants to knotsin non-simply connected manifolds. In [7] Miller extends Milnor’s µ -invariants to knots inclosed, orientable, aspherical, irreducible, Seifert fibered 3-manifolds that are homotopic tothe Seifert fiber, although he considers knots up to Z (cid:20) π ( M ) Z (cid:21) -homology concordance.Fix a knot J in M . The inclusion of the exterior of J into M , E J ֒ → M , induces anepimorphism γ J : π J := π ( E J ) → G := π ( M ) with kernel Γ γ J . For n ≥ B n be themapping cylinder of the induced map ∂E J → K (cid:18) π J Γ n γ J , (cid:19) , where Γ n γ J is the n th -lowercentral subgroup of Γ γ J and K (cid:18) π J Γ n γ J , (cid:19) is an Eilenberg-MacLane space for π J Γ n γ J . Form X n , an approximation of the decomposition of M as M = E J ∪ ∂E J N ( J ), where N ( J ) isa closed tubular neighborhood of J , by gluing a solid torus to B n along ∂E J such that µ J bounds a disk. These { X n } n ≥ fit into a sequence up to homotopy / / X n +1 / / X n / / · · · / / X . OMOTOPY PROPERTIES OF KNOTS IN PRIME MANIFOLDS 3
Given a knot K in M homotopic to J , if there exists a map τ : E K → B n making thefollowing diagram commute µ K (cid:127) _ (cid:15) (cid:15) / / µ J (cid:127) _ (cid:15) (cid:15) ∂E K ∼ = / / τ " " EEEEEEEE ∂E J | | zzzzzzzz B n where µ J and µ K are the meridians of J and K , respectively, then we may extend τ to h n ( K, τ ) : M → X n . Our invariant, the homotopy class of h n ( K, τ ) in [
M, X n ], comparesthe decompositions M = E K ∪ ∂E K N ( K ) and M = E J ∪ ∂E J N ( J ) by obstructing the exis-tence of a map E K → B n that extends to M → X n .We call { Γ n γ A } n ≥ the G -lower central series of γ A . The following Stallings’-type result(see [14] for more on Stallings’ theorem), well known to experts, implies that concordantknots have isomorphic G -lower central series quotients: Proposition 2.4. If f : A → B is a homomorphism over G and f i : H i ( A ; Z [ G ]) → H i ( B ; Z [ G ]) is an isomorphism for i = 1 and an epimorphism for i = 2 then f inducesisomorphisms f : A Γ n γ A → B Γ n γ B for all n ∈ Z . Call a map E K → B n that extends over M as mentioned above extendable . We showthat if K admits an extendable map to B n then it induces an isomorphism of the first nG -lower central series quotients in a way that preserves certain peripheral data: Proposition 3.20. If τ : E K → B n is an extendable map for a knot K β then τ induces anisomorphism τ ∗ : π K Γ j +1 γ K ∼ = π J Γ j +1 γ J over G with τ ∗ ◦ † K ( H K ) = † j +1 ( H J ) and τ ∗ ( µ K ) = µ J for all j < n . An n -concordance between K and J is a properly embedded surface in M × I coboundedby J and K that “looks” like an annulus modulo its n th π ( M )-lower central series quotient. Theorem 3.29.
A knot K is n -concordant to J if and only if some j th h -invariant h j ( K, τ ) is defined and trivial for all j ≤ n . Since a concordance is an n -concordance for all n , we obtain the following corollary. Corollary 3.30. If K is concordant to J then for each n there is an extendable map τ suchthat h n ( K, τ ) is defined and trivial. Finally, recall that a knot K in M is J -characteristic if there is a continuous degree-onemap α : M → M such that α ( K ) = J and α ( M − K ) ⊆ M − J . Theorem 3.31.
Let K be a knot in a closed, orientable, aspherical 3-manifold. If K is J -characteristic via a map α : M → M that indices the identity on G then K has trivial h -invariants. PRUDENCE HECK
As a special case of this, let L ⊂ S be a knot and let η be a curve in E J that bounds adisk in M . Denote by J ( η, L ) the satellite of J formed by( M − N ( η )) ∪ − (cid:0) S − N ( L ) (cid:1) , where if λ η and λ L denote the longitudes of η and L , respectively, then µ L ∼ λ − η and λ L ∼ µ η . This manifold is diffeomorphic to M via a map that is the identity outside aregular neighborhood of the disk bounded by η . In particular, this diffeomorphism inducesthe identity homomorphism on G . Theorem 3.35.
Let J be a homotopically essential knot in a closed, orientable, aspherical3-manifold. Let η be a curve in E J that bounds an embedded disk in M and let L be a knotin S . Then J ( η, L ) has trivial h -invariants. While we expect these invariants to be highly nontrivial for knots in most closed, ori-ented, prime 3-manifolds, there are a few cases where we expect the invariants to be trivial.For example, if M = S × S and K generates π ( S × S ) ∼ = Z or if M is a lens spacewith prime-order fundamental group and K is homotopically essential then these invariantswill likely be trivial. Indeed, if H ( E K ; Z [ G ]) = Z these invariants will be trivial since Z has trivial lower central series. However, if M is a lens space with fundamental groupof composite order and K does not generate π ( M ), if K is a knot in a manifold of thesecond type, or if K is a knot in S × S representing a class with norm at least threein π ( S × S ) ∼ = Z then these invariants will likely be highly nontrivial. The reason isthat these invariants essentially measure the homotopy class of the boundary of a tubularneighborhood of the knot in the target space X n , and manifolds with nontrivial fundamen-tal groups allow knots to link themselves nontrivially, creating interesting toral embeddings.The paper is organized as follows: we review some basic definitions in section 2.1, andgive the analogous definitions for based spaces in section 2.2. We define the category ofgroups over G and prove Proposition 2.4 in section 2.3. In section 3.1 we define the h -invariants. We construct the spaces { X n } in section 3.2. We define extendable maps andprove Proposition 3.20 in section 3.3. We investingate the indeterminacy of the h -invariantsin section 3.4. We introduce n -concordance and prove Theorem 3.29 and Corollary 3.30 insection 3.5. Finally, in section 3.6 we consider J -characteristic knots in manifolds of thesecond type and prove Theorems 3.31 and 3.35. Acknowledgments.
This work of deducing a coherent model for studying free isotopyclasses of circles in general 3-manifolds started several years ago. I am greatly indebted to myadvisor Kent Orr for many helpful conversations and for his overwhelming encouragement.2.
Preliminaries
Henceforth we work in the smooth category and consider knots in closed, oriented, prime3-manifolds. In particular, we allow homologically nontrivial knots. In Section 2.1 wepresent some necessary definitions from knot theory. We give analogues of these samedefinitions for based knots in Section 2.2. In Section 2.3 we define the G -lower central series and prove a Stallings’ theorem for the G -lower central series. Recommended references are[3], [9], and [13]. OMOTOPY PROPERTIES OF KNOTS IN PRIME MANIFOLDS 5
Knots and concordance. A knot K in a 3-manifold M is an oriented one-dimensional closed submanifold. We willassume that M has base point p ∈ M and that K does not contain p . We also assume that K is based via an embedded path from p , although the choice of path will not matter for theresults of this paper. The exterior of K is the complement of an open normal neighborhoodof K , E K = M − N ( K ), based at p . Note that the inclusion i K : E K ֒ → M induces anepimorphism γ K : π ( E K , p ) → π ( M, p ). We denote the fundamental group of E K by π K and the fundamental group of M by G .A meridian µ K of K is an embedded curve in ∂E K that represents a primitive elementof H ( ∂E K ) and bounds a disk in N ( K ). It is uniquely determined up to isotopy in ∂E K .We will abuse notation by letting µ K denote the curve in M , its isotopy class in ∂E K , itshomology class in H ( E K ), and its homotopy class in π K , where in the last case we regard µ K as being based via the path basing K .A knot K in M is L -characteristic if there is a continuous degree-one map α : M → M such that α ( K ) = L and α ( M − K ) ⊆ M − L .Two knots K and L are said to be cobordant if there is an oriented submanifold V of M × I that meets the boundary of M × I transversely with V ∩ ( M × { } ) = L × { } and V ∩ ( M × { } ) = K × { } , and such that the orientations of L × { } and K × { } agreewith the orientations induced by V .If K and L are cobordant then we can always choose V to be disjoint from { p } × I .We call E V = M × I − N ( V ) the exterior of V in M × I , where N ( V ) is an open normalneighborhood of V that is disjoint from { p } × I . A meridian µ V of V is an embeddedcurve in ∂E V that is isotopic to µ K . The path ρ ( t ) := p × t induces an isomorphism ρ ∗ : π ( E V , p × → π ( E V , p ×
0) by ρ ∗ ( f ) = ρ ∗ f ∗ ρ − (where we read concatenation ofpaths from left to right). We therefore write π ( E V , p ×
1) = π ( E V , p ×
0) and denote bothof these groups by π V .If V is a concordance between K and L then the inclusions ι L : E L ֒ → E V and ι K : E K ֒ → E V induce homomorphisms ι L : π L → π V and ι K : π K → π V , respectively, and theinclusion i V : E V ֒ → M × I induces an epimorphism γ V : π V → G = π ( M ) such that thefollowing diagram commutes π L ι L / / γ L ! ! DDDDDDDD π Vγ V (cid:15) (cid:15) π Kι K o o γ K | | zzzzzzzz G. We call a cobordism V a G -homology concordance if the inclusions ι L : E L ֒ → E V and ι K : E K ֒ → E V induce isomorphisms on H ∗ ( − ; Z [ G ]). We call V a concordance if it ishomeomorphic to S × I . A Mayer-Vietoris sequences argument shows that a concordanceis a G -homology concordance (see also Proposition 2.1 of [4]). PRUDENCE HECK
Based knots and based concordance.
Recall from the previous section that p ∈ M is the base point of M .A based knot K α is a knot K together with a choice of embedded path α in M from p to K that intersects K only at the endpoint. We call α a basing of K . If K α and K β denote a knot K based along paths α and β , respectively, then conjugating by αβ − definesa homomorphism σ : π ( M, p ) → π ( M, p ) such that σ ([ K α ]) = [ K β ]. When the basing isclear from the context we will drop the subscript and simply write K .We may assume that a given basing α of K intersects the open normal neighborhood N ( K ) ⊂ M in an interval. Therefore, the exterior of K α is the exterior E K of the un-based knot K , with base point p . We abuse notation by writing ∂E K ∪ α for the space( ∂E K ∪ α ) − ( N ( K ) ∩ α ). Moreover, whenever we refer to ∂E K ∪ α we assume that this“truncated” part of alpha is reparameterized as a path with domain I and α (1) ∈ ∂E K .The inclusion ∂E K ∪ α ֒ → E K induces a homomorphism π ( ∂E K ∪ α, p ) → π K .A meridian of the based knot K α is a loop µ K α = α ∗ µ K ∗ α − in ∂E K ∪ α , where µ K is a meridian for the unbased knot K that passes through α (1). We will usually drop thesubscript α when referring to a meridian or longitude of a based knot.We say that L α and K β are based cobordant (resp. based concordant ) if there is a cobor-dism (resp. concordance) V between K and L such that µ K and µ L have the same imagein π V under π K → π V and π L → π V , respectively. If L α and K β are based cobordant (resp.based concordant) then they are cobordant (resp. concordant). However, the converse isnot true. It is often possible to find two different basing β and β of K such that K β isnot homotopic relative p to K β . We say that L α and K β are G -homology concordant ifthey are G -homology concordant as unbased knots. It can be shown that if L is concordantto K (as unbased knots) via a concordance C then given any basings α and β of L and K ,respectively, there is an inner automorphism σ of π C such that σ ( µ K ) = µ L .2.3. The G -lower central series. For more on the category G G see [4]. Definition 2.1.
Fix a group G . Let F G G denote the category whose objects are grouphomomorphisms γ A : A → G and whose morphisms f : γ A → γ B are group homomorphisms f : A → B making the following diagram commute: A f / / γ A (cid:31) (cid:31) @@@@@@@ B γ B ~ ~ ~~~~~~~ G Let G G denote the full subcategory of F G G whose objects are epimorphisms γ A : A → G .We call morphisms in F G G and G G homomorphisms over G . Definition 2.2.
For an object γ A ∈ F G G , defineΓ γ A = Ker { γ A } . OMOTOPY PROPERTIES OF KNOTS IN PRIME MANIFOLDS 7
Definition 2.3.
Define the G -lower central series of an object γ A ∈ F G G byΓ γ A = Γ γ A and Γ n +1 γ A = [Γ γ A , Γ n γ A ] . In particular, the G -lower central series of γ A is precisely the lower central series of Γ γ A .For convenience, we will drop the subscript of Γ and simply write Γ γ A for Γ γ A . It is clearthat for γ A : A → G this defines a normal series A ☎ Γ γ A ☎ Γ γ A ☎ · · · ☎ Γ n γ A ☎ · · · . Proposition 2.4. If f : γ A → γ B is a morphism in G G and f i : H i ( A ; Z [ G ]) → H i ( B ; Z [ G ]) is an isomorphism for i = 1 and an epimorphism for i = 2 then f induces isomorphisms f : A Γ n γ A → B Γ n γ B for all n ∈ Z .Proof. Because H i (Γ γ A ) = H i ( A ; Z [ G ]) and H i (Γ γ B ) = H i ( B ; Z [ G ]), Stallings’ Theoremimplies that f induces an isomorphism Γ γ A Γ n γ A → Γ γ B Γ n γ B for all n . The result now follows byapplying the Five Lemma to the following commutative diagram of short exact sequencesinduced by f : 0 / / Γ γ A Γ n γ A / / ∼ = (cid:15) (cid:15) A Γ n γ A γ A / / (cid:15) (cid:15) G / / = (cid:15) (cid:15) / / Γ γ B Γ n γ B / / B Γ n γ B γ B / / G / / (cid:3) Example . Consider the following short exact sequence0 / / F ι / / F × Z γ / / FF / / F is the free group on two generators x and y , Z is generated by t , and F is thecommutator subgroup of F . Define ι : F → F × Z by ι ([ x, y ] ω ) = [ x, y ] ω t − for any word ω ∈ F × Z . Then Γ γ ∼ = F is the free group on infinitely many generators and Γ n γ ∼ = ( F ) n .One can show that FF is the fundamental group of the Heisenberg manifold, a circle bundleover the torus, and that F × Z is the fundamental group of the complement of a fiber inthis manifold. 3. Homotopy invariants
Fix a based knot J α in M ; this is the knot to which we will compare all other basedknots. In Section 3.2 we will explicitly construct an infinite tower of spaces / / X n / / X n − / / · · · / / X that depends on J α . The X n can be thought of as successive approximations of E J . Ourgoal in Section 3.1 is to associate to any suitable unbased knot K a homotopy class h ∈ [ M, X n ]. To do this, we first associate to any suitable based knot K β a homotopy class h ∈ [ M, X n ] . We discuss in 3.4 the extent to which h depends on the basing β . Thedefinition of these invariants requires the notion of an extendable map , which, for brevity,we put off defining until Section 3.3. Roughly speaking, two knots are n -concordant if theycobound a surface V that “looks” like an annulus modulo Γ n γ V . We investigate the intimaterelationship between h -invariants and n -concordance in Section 3.5. Finally, we investigatethe relationship between h -invariants and J -characteristicness in Section 3.6.3.1. The invariant.Definition 3.1. A boundary condition is a pair ( † , µ ) consisting of a homomorphism † : H → π , where H is a free abelian group on two elements, and a primitive element µ ∈ H .Note that the boundary condition does not carry with it a particular choice of isomorphism H ∼ = Z , and † need not be injective. We abuse notation by simply writing µ for † ( µ ).Later, we will have a commutative diagram, H † / / @@@@@@@@ π (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) G making † a morphism in F G G . Definition 3.2 (Whitehead [17], p. 244) . Given a group π , an Eilenberg-MacLane space oftype ( π,
1) is a based space K ( π,
1) such that π n ( K ( π, n > π ( K ( π, → π . Remark . We now associate to the boundary condition ( † , µ ) a space that is well-definedup to based homotopy equivalence. Let T , a torus, be an Eilenberg-MacLane space for H ,and let ∗ denote the base point of T . With some abuse of notation, let µ be an embeddedcurve in T containing ∗ that represents the class µ ∈ H . Let K ( π,
1) be an Eilenberg-MacLane space for π , based at a point b . The homomorphism † determines a based map † : T → K ( π, B † be the mapping cylinder of † , B † = T × I ` K ( π, x, ∼ † ( x ) . We will abuse notation by denoting T × { } and µ × { } by T and µ , respectively. Define X † = B † ∪ ST, where the solid torus ST is attached to T via a homeomorphism ∂ ( ST ) ∼ = T so that µ bounds a disk in ST . Both B † and X † are based at the point b ∈ K ( π, T and µ to the base point b via the straight-line path along T × I from ( ∗ , ∼ b to ( ∗ , X † is well defined up to basedhomotopy type. Hence, we drop the subscript and simple write X for X † . Definition 3.4.
We call X the complex of the boundary condition ( † , µ ). Definition 3.5.
Let K β be a based knot. Let H K = π ( ∂E K ∪ β, p ) and let † K : H K → π K be the homomorphism induced by ∂E K ∪ β ֒ → E K . Call ( † K , µ K ) the boundary conditionassociated to K β . † K may be viewed as a morphism in F G G . OMOTOPY PROPERTIES OF KNOTS IN PRIME MANIFOLDS 9
Definition 3.6.
Fix a based knot J α ; this knot will remain fixed for the duration of thearticle. For n ≥
2, define † n : H J → π J → π J Γ n γ J to be † J followed by the canonical quotient homomorphism. Define the n th -knot complexof J α , denoted X n , to be the knot complex associated to the boundary condition ( † n , µ J ).Denote the mapping cylinder of T → K (cid:18) π J Γ n γ J , (cid:19) , as in Remark 3.3, by B n . Let T n and µ J denote T × { } and µ J × { } , respectively. Denote the base point of B n by b ,for all n . We abuse notation by letting γ J denote the epimorphism of fundamental groups π J Γ n γ J → π ( X n ) induced by the inclusion B n → X n .The spaces { X n } n ≥ were defined with respect to the fixed knot J and can be thoughtof as successive approximations of the decomposition M = E J ∪ ∂N ( J ) N ( J ). Given a knot K (freely) homotopic to J in M , the invariant below compares K to J by comparing thedecomposition M = E K ∪ ∂N ( K ) N ( K ) to X n . Definition 3.7.
Let K β be a based knot such that [ K β ] is conjugate to [ J α ] in G . Let τ : E K → B n be an extendable map for K β as in Definition 3.15, with epimorphism γ K : π K → G . Then it extends to a based map h n ( K, τ ) : M → X n by Proposition 3.17,and we obtain a based homotopy class [ h n ( K, τ )] ∈ [ M, X n ] . We refer to the based class[ h n ( K, τ )] ∈ [ M, X n ] as an n th h -invariant for the based knot K β . We define the freehomotopy class [ h n ( K, τ )] ∈ [ M, X n ]to be the n th h -invariant of the pair ( K, τ ). Let τ n : E J → B n be as in Example 3.18.We say that [ h n ( K, τ )] is J -trivial (or simply trivial when J is clear from the context) if[ h n ( K, τ )] = [ h n ( J, τ n )] in [ M, X n ]. That is, if h n ( K, τ ) is freely homotopic to h n ( J, τ n ).3.2. Boundary conditions and knot complexes.
We now construct the tower of spaces / / X n / / X n − / / · · · / / X . Definition 3.8.
Let ( † : H → π , µ ) and ( † : H → π , µ ) be boundary conditions. A homomorphism (resp. isomorphism) of boundary conditions p : ( † , µ ) → ( † , µ )is a homomorphism (resp. isomorphism) p : π → π with p ( µ ) = µ and which restrictsto an isomorphism p : † ( H ) ∼ = † ( H ). Remark . Since H and H are both isomorphic to Z , given any homomorphism p : π → π that restricts to an isomorphism † ( H ) ∼ = † ( H ) with p ( µ ) = µ , there exists anisomorphism H → H making the diagram H ∼ = / / † (cid:15) (cid:15) H † (cid:15) (cid:15) π p / / π commute. We will freely make use of this fact in the future without explicitly choosing suchan isomorphism. Proposition 3.10.
Let p : ( † , µ ) → ( † , µ ) be a homomorphism of boundary conditions,let X i be the complex of ( † i , µ i ) , and let T i and µ i denote T and µ in X i , respectively (seeRemark 3.3). Then p induces a based map P : X → X , well defined up to based homotopy,such that P ( K ( π , ⊂ K ( π , and P ( T ) = T with P ( µ ) = µ . Moreover, if p is anisomorphism then P is a homotopy equivalence.Proof. For i ∈ { , } , let † i : T → K ( π i ,
1) be a based map induced by † i , let µ i ⊂ T be an embedded curve representing µ i ∈ H i , and let X † i = B † i ∪ ST , as above. Thehomomorphism p : π → π induces a map Q : K ( π , → K ( π ,
1) up to based homotopy,and hence a map Q : X † → X Q ◦† , up to based homotopy, with Q ( T ) = T Q ◦† and Q ( µ ) = µ Q ◦† , and which is easily seen to be a homotopy equivalence if p is an isomorphism.Since Q ∗ = p : † ( H ) → † ( H ) isomorphically with p ( µ ) = µ , there is a based homotopyequivalence R : X Q ◦† → X that maps K ( π ,
1) to itself and T Q ◦† to T with µ Q ◦† µ .We define P := R ◦ Q. (cid:3) Lemma 3.11.
For each n , there is a canonical isomorphism π ( X n , b ) ∼ = G .Proof. Recall that X n = B n ∪ T n ST , where ST is attached so that µ J bounds a disk. TheSeifert-van Kampen Theorem implies that π ( X n , b ) is the pushout of the following diagram, H J † n / / (cid:15) (cid:15) π J Γ n γ J Z where the kernel of the vertical arrow is the subgroup generated by µ J . Hence, there is aunique homomorphism π ( X n , b ) → G making the following diagram commute, H J † n / / (cid:15) (cid:15) π J Γ n γ J (cid:15) (cid:15) (cid:21) (cid:21) Z / / , , π ( X n , b ) $ $ G where the bottom-left homomorphism Z → G sends the generator of Z to the homotopyclass of J α and π J Γ n γ J → G is the canonical epimorphism induced by γ J : π J → G . Since µ J normally generates Γ γ J Γ n γ J in π J Γ n γ J , π ( X n , b ) → G is an isomorphism. (cid:3) Remark . Henceforth we fix this isomorphism between π ( X n , b ) and G , and write π ( X n , b ) = G . Remark . The canonical epimorphisms q n +1 ,n : π J Γ n +1 γ J → π J Γ n γ J induce a tower ofboundary conditions / / ( † n +1 , µ J ) / / ( † n , µ J ) / / · · · / / ( † , µ J ) . OMOTOPY PROPERTIES OF KNOTS IN PRIME MANIFOLDS 11
By Proposition 3.10, this tower of boundary conditions induces a tower of knot complexes / / X n +1 Q n +1 ,n / / X n / / · · · / / X . Each map in the tower induces the identity homomorphism on G and is defined up to basedhomotopy equivalence. We therefore obtain maps of sets Q n +1 ,n : [ M, X n +1 ] → [ M, X n ] defined by Q n +1 ,n ([ f ]) := [ Q n +1 ,n ◦ f ].We place the following theorem here for completeness. Extendable maps are defined inDefinition 3.15. Theorem 3.14.
Let K β be a based knot and suppose there is an extendable map τ : E K → B n +1 for K β . Then there is an extendable map ˜ τ : E K → B n for K β such that Q n +1 ,n ([ h n +1 ( K, τ )]) = [ h n ( K, ˜ τ )] .Proof. Since T n +1 is a sub-complex of the CW-complex B n +1 we may assume that q n +1 ,n : (cid:0) † n +1 , µ J (cid:1) → ( † n , µ J ) induces a map ˜ Q : B n +1 → B n that restricts to a homeomorphismfrom T n +1 to T n with ˜ Q ( µ J ) = µ J and such that ˜ Q ( B n +1 − T n +1 ) ⊆ B n − T n . Then˜ τ := ˜ Q ◦ τ : E K → B n is the desired extendable map. (cid:3) extendable maps. We now define extendable maps and show that if E K → B n isextendable then it induces isomorphisms on the first n G -lower central series quotients,preserving peripheral data. Definition 3.15.
Let N be a manifold, possibly with boundary, based at q ∈ N , togetherwith an epimorphism γ N : π ( N, q ) ։ G . Let V ⊂ N be a codimension-two connected sub-manifold, not containing q , with trivial normal bundle and ∂V ⊂ ∂N , meeting transversely.Let V β denote V with a choice of basing β . Let N ( V ) be an open regular neighborhood of V that intersects β in a subinterval, let µ V be the meridian of V , let E V = N − N ( V ) basedat q , and let π V = π ( E V , q ). Let γ V : π V ։ G be the epimorphism of fundamental groupsinduced by the inclusion of E V into N followed by γ N . We call a based map f : E V → B n extendable if all of the following hold:(i) f ( ∂E V ) ⊆ T n and f ( µ V ) = µ J ,(ii) f ∗ ( π ( ∂E V ∪ β, q )) = † n ( H J ), and(iii) f ∗ : π V → π J Γ n γ J is a morphism of G G , as in the following diagram: π V f ∗ / / γ V (cid:30) (cid:30) ========= π J Γ n γ Jγ J ~ ~ }}}}}}}} G If V is not connected, we call a based map f : E V → B n extendable if it satisfies (i) - (iii)for each connected component of V . Remark . (1) The definition of extendable requires both a choice of basing for V and a choice ofepimorphism γ N : π ( N, q ) → G .(2) If V is not connected then a choice of basing is required for each connected compo-nent of V .(3) If f only satisfies (ii) and (iii) of Definition 3.15 with f ∗ ( µ V ) = µ J then f is basedhomotopic to an extendable map. Proposition 3.17. If f : E V → B is extendable then it extends to a based map f : N → X .Proof. We assume that V is connected because it is sufficient to prove the result for eachconnected component of V . Choose a homeomorphism N ( V ) ∼ = V × D , where N ( V ) isthe closure of N ( V ), such that {∗} × S is identified with µ V for some point ∗ ∈ V . Toprove the result we show that f : V × S → T n extends to a map F : V × D → ST . Let e = ∗ be the 0-cell of V and let { e kj } n k j =1 be the k -cells of V for 1 ≤ k ≤ n . Let f and f be the 0-cell and 1-cell of S , respectively, and let f be the 2-cell of D . Then V × S has a CW-decomposition with 0-cell e × f , 1-cells { e j × f } n j =1 and e × f , and 2-cells { e j × f } n j =1 and { e j × f } n j =1 . V × D has the same 0- and 1-cells, but one-additional2-cell, e × f , that is attached by a degree-one map to e × f . Since f ( e × f ) = µ , F extends to a map from V × S ∪ e × f D to ST . F extends f over the 2-skeleton of V × D .We may now extend F over the remainder of V × D because π k ( ST ) = 0 for all k ≥ (cid:3) Example . The canonical epimorphism q n : π J → π J Γ n γ J induces an extendable map τ n : E J → B n , well defined up to based homotopy. In this case the diagram in (iii) ofDefinition 3.15 is π J q n / / γ J (cid:30) (cid:30) <<<<<<<<< π J Γ n γ Jγ J ~ ~ }}}}}}}} G Proposition 3.19.
Let K be a knot in a closed, oriented, prime 3-manifold M . (1) If either M is irreducible or M ∼ = S × S and [ K ] = 0 ∈ π ( S × S ) ∼ = Z then theinclusion ∂E K ֒ → M = E K ∪ ∂E K N ( K ) induces an isomorphism H ( ∂E K ; Z [ G ]) ∼ = H ( E K ; Z [ G ]) ⊕ H (cid:16) N ( K ); Z [ G ] (cid:17) . (2) If [ K ] has infinite order in π ( M ) then H ( E K ; Z [ G ]) = H (Γ γ K ) = 0 . (3) If M is irreducible and [ K ] has infinite order in π ( M ) then H ( E K ; Z [ G ]) ∼ = H ( ∂E K ; Z [ G ]) ∼ = H (cid:16) µ K ; Z h G [ K ] Z i(cid:17) = L G [ K ] Z Z as Z [ G ] -modules, where [ K ] Z ⊂ G is the infinite cyclic subgroup generated by [ K ] and G [ K ] Z is the set of right cosets of [ K ] Z in G . OMOTOPY PROPERTIES OF KNOTS IN PRIME MANIFOLDS 13 (4)
For any knot K in M , if τ : E K → B n is extendible then the image of the homo-morphism H (Γ γ K ) → H (cid:18) Γ γ J Γ j γ J (cid:19) induced by τ equals the image of H (Γ γ J ) → H (cid:18) Γ γ J Γ j γ J (cid:19) induced by q j : π J → π J Γ j γ J for all j ≤ n .Proof. For the first statement consider the following long exact sequence induced from thedecomposition M = E K ∪ T N ( K ): H ( ∂E K ; Z [ G ]) / / H ( E K ; Z [ G ]) ⊕ H (cid:16) N ( K ); Z [ G ] (cid:17) / / H ( M ; Z [ G ]) ( ⋆ ) / / H ( ∂E K ; Z [ G ]) / / H ( E K ; Z [ G ]) ⊕ H (cid:16) N ( K ); Z [ G ] (cid:17) / / H ( M ; Z [ G ])If M is irreducible then H ( M ; Z [ G ]) = H ( M ; Z [ G ]) = 0 (being the homology of theuniversal cover of M ), establishing (1). If M ∼ = S × S then H (cid:0) S × S ; Z [ Z ] (cid:1) = 0, so weneed only consider H (cid:0) S × S ; Z [ Z ] (cid:1) → H ( ∂E K ; Z [ Z ]) . Since [ K ] = 0 the image of π ( ∂E K ) → π ( S × S ) ∼ = Z is trivial, so ∂E K lifts to an infinitenumber of tori (one for each element of Z ) and, if S is a sphere representing a generatorof H (cid:0) S × S ; Z [ Z ] (cid:1) ∼ = H (cid:0) R × S (cid:1) ∼ = Z then H (cid:0) S × S ; Z [ Z ] (cid:1) → H ( ∂E K ; Z [ Z ]) isgiven by the intersection of S with the lifts of ∂E K . Since K is null-homotopic in S × S these intersection curves must pair up so as to sum to zero in H ( ∂E K ; Z [ Z ]), making H (cid:0) S × S ; Z [ Z ] (cid:1) → H ( ∂E K ; Z [ Z ]) the zero map. This establishes (1).If [ K ] has infinite order in G then H ( ∂E K ; Z [ G ]) = 0. Moreover, H ( E K ; Z [ G ]) → H (Γ γ K ) is surjective by Hopf’s theorem. If M is irreducible then (2) follows. If M ∼ = S × S then H ( ∂E K ; Z [ Z ]) ∼ = Z (cid:2) Z [ K ] (cid:3) . By a similar argument to that of (1), if S is asphere representing a generator of H ( R × S ) and t generates Z [ K ] then H ( R × S ) ֒ → H ( ∂E K ; Z [ Z ])by [ S ] t + · · · + t [ K ] − , establishing (2).If M is irreducible and [ K ] has infinite order in G then H (cid:16) N ( K ); Z [ G ] (cid:17) = 0 and H ( ∂E K ; Z [ G ]) ∼ = H (cid:18) µ K ; Z (cid:20) G [ K ] Z (cid:21)(cid:19) = M G [ K ] Z Z , establishing (3).If [ K ] has infinite order in G then (4) follows from (2). Consider the commutativediagram, H ( ∂E K ; Z [ G ]) / / / / ∼ = (cid:15) (cid:15) H ( E K ; Z [ G ]) / / / / (cid:15) (cid:15) H (Γ γ K ) / / (cid:15) (cid:15) H (cid:18) Γ γ K Γ j γ K (cid:19) (cid:15) (cid:15) H ( T n ; Z [ G ]) / / H ( B n ; Z [ G ]) / / H (cid:18) Γ γ J Γ n γ J (cid:19) / / H (cid:18) Γ γ J Γ j γ J (cid:19) H ( ∂E K ; Z [ G ]) / / / / ∼ = O O H ( E K ; Z [ G ]) / / / / O O H (Γ γ K ) / / O O H (cid:18) Γ γ K Γ j γ K (cid:19) O O where H ( E K ; Z [ G ]) → H (Γ γ K ) is surjective by Hopf’s theorem. If M is irreducible then H ( ∂E K ; Z [ G ]) → H ( E K ; Z [ G ]) is surjective by the long exact sequence ( ⋆ ), establishing(4). Suppose that M ∼ = S × S and K is null-homotopic in M . Then from the long exactsequence ( ⋆ ) we obtain a short exact sequence as follows:0 / / H ( ∂E K ; Z [ Z ]) / / H ( E K ; Z [ Z ]) / / H (cid:0) S × S ; Z [ Z ] (cid:1) / / H (cid:0) S × S ; Z [ Z ] (cid:1) ∼ = Z is a free abelian group H ( E K ; Z [ Z ]) splits as a direct sum, H ( E K ; Z [ Z ]) ∼ = H ( ∂E K ; Z [ Z ]) ⊕ H (cid:0) S × S ; Z [ Z ] (cid:1) . Hence, we obtain the following commutative diagram: H ( ∂E K ; Z [ Z ]) ⊕ H (cid:0) R × S (cid:1) ∼ = / / (cid:15) (cid:15) (cid:15) (cid:15) H ( E K ; Z [ Z ]) (cid:15) (cid:15) (cid:15) (cid:15) H ( ∂E K ; Z [ Z ]) ⊕ H (cid:0) π ( R × S ) (cid:1) / / H ( π K ; Z [ Z ])As H (cid:0) π ( R × S ) (cid:1) = 0, H ( ∂E K ; Z [ Z ]) → H (Γ γ K ) is surjective and (4) follows. (cid:3) Proposition 3.20. If τ : E K → B n is an extendable map for a knot K β then τ induces anisomorphism τ ∗ : π K Γ j +1 γ K ∼ = π J Γ j +1 γ J over G with τ ∗ ◦ † K ( H K ) = † j +1 ( H J ) and τ ∗ ( µ K ) = µ J for all j < n .Proof. It follows from the definition of extendable map that τ ∗ ◦ † K ( H K ) = † j +1 ( H J ) for j < n and τ ∗ ( µ K ) = µ J .It remains to show that τ ∗ induces an isomorphism between π K Γ j +1 γ K and π J Γ j +1 γ J . It isclear that τ extends to a map τ : M → X n such that τ : ∂E K → T n homemorphically with OMOTOPY PROPERTIES OF KNOTS IN PRIME MANIFOLDS 15 τ ( µ K ) = µ J , and, since τ is extendable, H K id / / † K (cid:15) (cid:15) H K † K (cid:15) (cid:15) / / H J † j +1 (cid:15) (cid:15) π K / / γ K " " FFFFFFFFFFF π K Γ j +1 γ K / / γ K (cid:15) (cid:15) π J Γ j +1 γ Jγ J z z uuuuuuuuuu G commutes for j < n . Hence, we obtain the following commutative diagram of long exactsequences: / / H ( M ; Z [ G ]) / / (cid:15) (cid:15) H ( M, E K ; Z [ G ]) / / (cid:15) (cid:15) H ( E K ; Z [ G ]) / / (cid:15) (cid:15) / / H ( X n ; Z [ G ]) / / H ( X n , B n ; Z [ G ]) / / H ( B n ; Z [ G ]) / / / / H ( M ; Z [ G ]) / / O O H ( M, E J ; Z [ G ]) / / O O H ( E J ; Z [ G ]) / / O O H ( M ; Z [ G ]) = H ( X n ; Z [ G ]) = 0, being the homology of the universal covers of M and X n , respectively. By excision, for L = K, JH ( M, E L ; Z [ G ]) ∼ = H ( N ( L ) , ∂N ( L ); Z [ G ]) ∼ = H ( ST, T n ; Z [ G ]) ∼ = H ( X n , B n ; Z [ G ]) . Moreover, the image of H ( M ; Z [ G ]) → H ( M, E K ; Z [ G ]) ∼ = H ( X n , B n ; Z [ G ])equals that of H ( M ; Z [ G ]) → H ( M, E J ; Z [ G ]) ∼ = H ( X n , B n ; Z [ G ]) . Since H ( E J ; Z [ G ]) → H ( B n ; Z [ G ])is an isomorphism (recall, H ( E J ; Z [ G ]) = Γ γ J Γ γ J = H ( B n ; Z [ G ])),Γ γ K Γ γ K = H ( E K ; Z [ G ]) → H ( B n ; Z [ G ]) = Γ γ J Γ γ J is an isomorphism. Consider the following commutative diagram of five-term exact sequences for j < n , withvertical maps induced by τ : H (Γ γ K ) / / H (cid:18) Γ γ K Γ j γ K (cid:19) / / (cid:15) (cid:15) Γ j γ K Γ j +1 γ K / / (cid:15) (cid:15) H (Γ γ J ) / / H (cid:18) Γ γ J Γ j γ J (cid:19) / / Γ j γ J Γ j +1 γ J / / H (Γ γ K ) in H (cid:18) Γ γ J Γ j γ J (cid:19) is precisely the image of H (Γ γ J ) by Proposition3.19. We showed that Γ γ K Γ γ K ∼ = Γ γ J Γ γ J . Suppose that Γ γ K Γ j γ K ∼ = Γ γ J Γ j γ J for j < n . ThenΓ j γ K Γ j +1 γ K ∼ = Γ j γ J Γ j +1 γ J by the commutativity of the above diagram. It follows from the FiveLemma applied to 0 / / Γ j γ K Γ j +1 γ K / / (cid:15) (cid:15) Γ γ K Γ j +1 γ K / / (cid:15) (cid:15) Γ γ K Γ j γ K / / (cid:15) (cid:15) / / Γ j γ J Γ j +1 γ J / / Γ γ J Γ j +1 γ J / / Γ γ J Γ j γ J / / γ K Γ j +1 γ K ∼ = Γ γ J Γ j +1 γ J . Applying the Five Lemma to0 / / Γ γ K Γ j +1 γ K / / (cid:15) (cid:15) π K Γ j +1 γ K / / (cid:15) (cid:15) π K Γ j γ K / / (cid:15) (cid:15) / / Γ γ J Γ j +1 γ J / / π J Γ j +1 γ J / / π J Γ j γ J / / π K Γ j +1 γ K ∼ = π J Γ j +1 γ J for all j < n . (cid:3) indeterminacy of h -invariants. We now investigate the dependence of h n ( K, τ ) on β and τ . Proposition 3.21.
Suppose that τ : E K → B n is an extendable map for the knot K β .If β is another basing for K then there is a canonical choice, up to based homotopy, ofextendable map τ : E K → B n for K β such that [ h n ( K β , τ )] and [ h n ( K β , τ )] differ in [ M, X n ] by an inner automorphism of G . In particular, h n ( K β , τ ) is freely homotopic to h n ( K β , τ ) . OMOTOPY PROPERTIES OF KNOTS IN PRIME MANIFOLDS 17
Proof.
Let K i denote K β i for i = 1 ,
2, let H i = π ( E K ∪ β i , p ), and let µ i be the meridianof K i . Suppose that τ : E K → B n is extendable and let τ ∗ be the homomorphism offundamental groups induced by τ . If a = [ β ∗ β − ] then conjugation by τ ∗ ( a ) induces anisomorphism σ τ ∗ ( a ) : π ( B n ) = π J Γ n γ J → π J Γ n γ J such that σ τ ∗ ( a ) ( µ J ) = µ J and the followingdiagram commutes: H J † J (cid:15) (cid:15) ∼ = / / H J † J (cid:15) (cid:15) π J Γ n γ J σ τ ∗ ( a ) / / π J Γ n γ J Since B n is an Eilenberg-MacLane space, σ τ ∗ ( a ) induces a based homotopy equivalence f : B n → B n which is freely homotopic to the identity map. Moreover, we may assumethat f restricts to a homeomorphism from T n to T n with f ( µ J ) = µ J because T n and µ J are also Eilenberg-MacLane spaces and the inclusions µ J ֒ → T n ֒ → B n are cofibrations.By an obstruction theory argument we may extend f to a map F : X n → X n that isfreely homotopic to the identity map. The induced automorphism of G , F ∗ , is conjugationby γ J ( τ ∗ ( a )). Define τ := f ◦ τ , so τ : E K → B n is an extendable map for K and h ( K , τ ) = F ◦ h ( K , τ ). Since F is freely homotopic to the identity on X n , h n ( K , τ ) isfreely homotopic to h n ( K , τ ). (cid:3) Definition 3.22.
Define A n to be the group of automorphisms of the boundary condition( † n , µ J ) over G , A n := { p : ( † n , µ J ) ∼ = / / ( † n , µ J ) (cid:12)(cid:12)(cid:12) γ J ◦ p = γ J o . Proposition 3.23. If f, g : E K → B n are extendable maps for a based knot K β then thereis an element p ∈ A n inducing a based homotopy equivalence P : X n → X n such that P ∗ : π ( X n ) → π ( X n ) is the identity homomorphism and P ◦ h n ( K, f ) is based homo-topic to h n ( K, g ) . In particular, P induces a bijection P : [ M, X n ] → [ M, X n ] such that P ([ h n ( K, f )]) = [ h n ( K, g )] .Proof. By Proposition 3.20, f and g induce isomorphisms f ∗ , g ∗ : π K Γ n γ K ∼ = π J Γ n γ J over G with f ∗ ( µ K ) = g ∗ ( µ K ) = µ J that restrict to isomorphisms between † K ( H K ) and † n ( H J ).The automorphism p := g ∗ ◦ f − ∗ : π J Γ n γ J → π J Γ n γ J is therefore an element of A n . ByProposition 3.10, p induces a based homotopy equivalence P : X n → X n , and hence abijection P : [ M, X n ] → [ M, X n ] . Since P | B n ◦ f is based homotopic to g , P ([ h n ( K, f )]) =[ P ◦ h n ( K, f )] = [ h n ( K, g )]. By the construction of P in the proof of Proposition 3.10, P ∗ : π ( X n ) → π ( X n ) is the identity homomorphism. (cid:3) n -concordance. We define n -concordance and show that some n th h -invariant is de-fined and trivial if and only if K is n -concordant to J . As a consequence, if K is concordantto J then K admits trivial h -invariants for all n . Definition 3.24.
We say that a based knot K β is n -concordant to the based knot J α ifboth of the following hold: (1) There is a cobordism V ⊂ M × I from K to J . Note that the inclusion of i V : E V → M × I induces an epimorphism γ V : π V = π ( E V ) → G , and we consider E V as aspace equipped with this coefficient system.(2) When V is based via α the composition π ( ∂E J ) → π ( ∂E V ) → π V → π V Γ n γ V is an epimorphism onto the image of π ( ∂E V ) in π V Γ n γ V .We say that K β is based n -concordant to J α if there is an n -concordance V such that µ K β = µ J α in π V Γ n γ V . Remark . (1) Unless stated otherwise, we regard V as being based via α and we take µ V = µ J × { } .(2) Changing the basing of an n -concordance V induces an inner automorphism of π V .In particular, there is an inner automorphism σ : π V → π V such that σ ( µ J ) = µ K .(3) For a based n -concordance we do not require that µ K = µ J in π V ; we only requirethat they be equal in π V Γ n γ V . Proposition 3.26. If E V is an n -concordance between E K and E J then the inclusion ι L : E L → E V , for L = K, J , induces an isomorphism over Gπ L Γ j +1 γ L ∼ = π V Γ j +1 γ V for all j ≤ n .Proof. We will prove the result for K (the proof for J is analogous) and so we regard V as being based via the basing of K , since this makes sense when considering the inclusion ι = ι K . As in the proof of Proposition 3.20 we show that ι induces an isomorphismΓ γ K Γ γ K → Γ γ V Γ γ V and then prove the general result by induction.By construction, M = E K ∪ ∂E K N ( K ) and M × I = E V ∪ ∂E V N ( V ) such that M ֒ → M × I is a homology equivalence with any coefficients, ∂E K ֒ → ∂E V , given by inclusion, inducesa homeomorphism with one end of ∂E V , and N ( K ) ֒ → N ( V ) induces a homeomorphismwith one end of N ( V ). Consider the following diagram of long exact sequences, / / H ( M ; Z [ G ]) / / ∼ = (cid:15) (cid:15) H ( M, E K ; Z [ G ]) / / (cid:15) (cid:15) H ( E K ; Z [ G ]) / / (cid:15) (cid:15) ⋆⋆ ) / / H ( M × I ; Z [ G ]) / / H ( M × I, E V ; Z [ G ]) / / H ( E V ; Z [ G ]) / / OMOTOPY PROPERTIES OF KNOTS IN PRIME MANIFOLDS 19 where H ( M ; Z [ G ]) = H ( M × I ; Z [ G ]) = 0, being the homology of the universal covers of M and M × I , respectively. By excision, H ( M, E K ; Z [ G ]) ∼ = H (cid:16) N ( K ) , ∂N ( K ); Z [ G ] (cid:17) and H ( M × I, E V ; Z [ G ]) ∼ = H (cid:16) N ( V ) , ∂N ( V ); Z [ G ] (cid:17) . From the inclusion N ( K ) ֒ → N ( V ) we obtain0 / / H (cid:16) N ( K ) , ∂N ( K ); Z [ G ] (cid:17) / / (cid:15) (cid:15) H ( ∂N ( K ); Z [ G ]) / / (cid:15) (cid:15) H (cid:16) N ( K ); Z [ G ] (cid:17) / / (cid:15) (cid:15) / / H (cid:16) N ( V ) , ∂N ( V ); Z [ G ] (cid:17) / / H ( ∂N ( V ); Z [ G ]) / / H (cid:16) N ( V ); Z [ G ] (cid:17) / / H (cid:16) N ( K ); Z [ G ] (cid:17) = H (cid:16) N ( V ); Z [ G ] (cid:17) = 0 since N ( K ) and N ( V ) each have thehomotopy type of a 1-complexe. If V × D ∼ = N ( V ) is a framing such that {∗} × S ∼ = µ K for some ∗ ∈ K ⊂ V then, under this framing, the above commutative diagram becomesthe following:0 / / H (cid:16) N ( K ) , ∂N ( K ); Z [ G ] (cid:17) / / (cid:15) (cid:15) H ( K ; Z [ G ]) ⊕ H ( µ K ; Z [ P ]) / / (cid:15) (cid:15) H (cid:0) K × D ; Z [ G ] (cid:1) / / (cid:15) (cid:15) / / H (cid:16) N ( V ) , ∂N ( V ); Z [ G ] (cid:17) / / H ( V ; Z [ G ]) ⊕ H ( µ K ; Z [ P ]) / / H ( V × D ; Z [ G ]) / / P = GC [ K ] , the set of right cosets of the cyclic subgroup C [ K ] ≤ G generated by [ K ],and the coefficients in H ( µ K ; Z [ P ]) are untwisted. Since the second rightmost horizontalhomomorphisms are induced by inclusion of K × S and V × S into the boundary of K × D and V × D , respectively, H ( K ; Z [ G ]) → H ( K × D ; Z [ G ])and H ( V ; Z [ G ]) → H ( V × D ; Z [ G ])are isomorphisms. Moreover, µ K bounds in K × D ⊂ V × D , so these short exact sequencesare split. It follows that H (cid:16) N ( K ) , ∂N ( K ); Z [ G ] (cid:17) ∼ = H ( µ K ; Z [ P ]) ∼ = H (cid:16) N ( V ) , ∂N ( V ); Z [ G ] (cid:17) . Returning to the commutative diagram of long exact sequences ( ⋆⋆ ), we see from the Fivelemma that Γ γ K Γ γ K = H ( E K ; Z [ G ]) ∼ = H ( E V ; Z [ G ]) = Γ γ V Γ γ V . We want to proceed exactly as in the end of the proof of Proposition 3.20. For that, inthe commutative diagram H (Γ γ K ) / / ι ∗ (cid:15) (cid:15) H (cid:18) Γ γ K Γ j γ K (cid:19) / / ι ∗ (cid:15) (cid:15) Γ j γ K Γ j +1 γ K / / ι ∗ (cid:15) (cid:15) (cid:15) (cid:15) H (Γ γ V ) / / H (cid:18) Γ γ V Γ j γ V (cid:19) / / Γ j γ V Γ j +1 γ V / / H (Γ γ K ) and H (Γ γ V ) have the same image in H (cid:18) Γ γ V Γ j γ V (cid:19) . From theprevious discussion we have a commutative diagram of exact sequences:0 / / H ( ∂E K ; Z [ G ]) / / (cid:15) (cid:15) H ( E K ; Z [ G ]) a K / / (cid:15) (cid:15) H ( M ; Z [ G ]) / / ∼ = (cid:15) (cid:15) H ( µ K ; Z [ P ]) ∼ = (cid:15) (cid:15) / / H ( ∂E V ; Z [ G ]) / / H ( E V ; Z [ G ]) a V / / H ( M × I ; Z [ G ]) / / H ( µ K ; Z [ P ])If M is irreducible or if M ∼ = S × S and [ K ] has infinite order in π ( S × S ) then a K and a V are both the zero map (in the second case this is proved as in the proof of Proposition3.19 (2)). Hence, H ( ∂E K ; Z [ G ]) / / / / (cid:15) (cid:15) H ( E K ; Z [ G ]) / / / / (cid:15) (cid:15) H (Γ γ K ) / / (cid:15) (cid:15) H (cid:18) Γ γ K Γ j γ K (cid:19) (cid:15) (cid:15) H ( ∂E V ; Z [ G ]) / / / / H ( E V ; Z [ G ]) / / / / H (Γ γ V ) / / H (cid:18) Γ γ V Γ j γ V (cid:19) is commutative, with two epimorphisms in each row, as indicated. It follows from thedefinition of n -concordance that H ( ∂E K ; Z [ G ]) and H ( ∂E V ; Z [ G ]) have the same imagein H (cid:18) Γ γ V Γ j γ V (cid:19) , as desired. If M ∼ = S × S and K is null-homotopic then a K and a V are onto (see the proof of Proposition 3.19 (1)). As in the proof of Proposition 3.19 (4), H ( ∂E K ; Z [ G ]) → H (Γ γ K ) and H ( ∂E V ; Z [ G ]) → H (Γ γ V ) are epimorphisms.We now proceed exactly as in the end of the proof of Proposition 3.20. It was shownearlier that ι : E K ֒ → E V induces an isomorphism Γ γ K Γ γ K ∼ = Γ γ V Γ γ V . Suppose by inductionthat for some j ≤ n , ι induces an isomorphism Γ γ K Γ j γ K ∼ = Γ γ V Γ j γ V . From the above diagram it OMOTOPY PROPERTIES OF KNOTS IN PRIME MANIFOLDS 21 follows that Γ j γ K Γ j +1 γ K ∼ = Γ j γ V Γ j +1 γ V . Applying the Five Lemma to0 / / Γ j γ K Γ j +1 γ K / / ∼ = ι ∗ (cid:15) (cid:15) Γ γ K Γ j +1 γ K / / ι ∗ (cid:15) (cid:15) Γ γ K Γ j γ K / / ∼ = ι ∗ (cid:15) (cid:15) / / Γ j γ V Γ j +1 γ V / / Γ γ V Γ j +1 γ V / / Γ γ V Γ j γ V / / ι induces an isomorphism Γ γ K Γ j +1 γ K → Γ γ V Γ j +1 γ V . Applying the Five Lemmato 0 / / Γ γ K Γ j +1 γ K / / ∼ = ι ∗ (cid:15) (cid:15) π K Γ j +1 γ K / / ι ∗ (cid:15) (cid:15) G / / ∼ = ι ∗ (cid:15) (cid:15) / / Γ γ V Γ j +1 γ V / / π V Γ j +1 γ V / / G / / ι induces an isomorphism π K Γ j +1 γ K → π V Γ j +1 γ V for all j ≤ n . (cid:3) Corollary 3.27. If K β is n -concordant to J α then there is some basing β ′ such that K β ′ isbased n -concordant to J α .Proof. Let V be an n -concordance between K β and J α . Then π K Γ n γ K ∼ = π V Γ n γ V by Proposition3.26. Since changing the basing of V corresponds to an inner automorphism of π V , there isan element g ∈ π K such that g − µ K g µ J under π K → π V → π V Γ n γ V . Let c be a simpleclosed curve representing g − and define β ′ = c ∗ β . Then V is a based n -concordancebetween K β ′ and J α . (cid:3) For the next proposition recall from Example 3.18 that the canonical epimorphism q n : π J → π J Γ n γ J induces an extendable map τ n : E J → B n up to based homotopy. Proposition 3.28.
Let V be an n -concordance. For all j ≤ n there exists an extendablemap f : E V → B j . Moreover, the restriction of f to E J × { } is based homotopic to τ j .Proof. By Proposition 3.26 the inclusion ι = ι J : E J ֒ → E V induces an isomorphism ι ∗ : π J Γ j γ J ∼ = π V Γ j γ V for j ≤ n + 1. We therefore get a homomorphism f ∗ : π V → π J Γ j γ J as inthe following commutative diagram: π J q j / / ι ∗ (cid:15) (cid:15) π J Γ j γ Jι ∗ ∼ = (cid:15) (cid:15) π V / / f ∗ ? ? π V Γ j γ V . As B j is an Eilenberg-MacLane space, f ∗ induces a map f : E V → B j up to based homotopy.Since V is an n -concordance, µ V = µ J × { } and π ( ∂E J ) / / π ( ∂E V ) / / π V / / π V Γ n γ V / / π V Γ j γ V is an epimorphism onto the image of π ( ∂E V ) in π V Γ j γ V for j ≤ n . In particular, f ∗ ( π ( ∂E V ∪ α, p × † j ( H J ) . The inclusion ∂E V ֒ → E V is a cofibration, so we may take f : E V → B j such that f ( ∂E V ) ⊆ T j and f ( µ V ) = µ J . Finally, f ∗ ◦ ι ∗ = q j , so f restricted to E J × { } is based homotopic to τ j . It follows that f is the desired map. (cid:3) Theorem 3.29.
A knot K is n -concordant to J if and only if some j th h -invariant h j ( K, τ ) is defined and trivial for all j ≤ n .Proof. Suppose that K is n -concordant to J via V . We may assume by Corollary 3.27 andProposition 3.21 that V is a based n -concordance. By Proposition 3.28 there exists for each j ≤ n an extendable map f j : E V → B j that when restricted to E J × { } is based homotopicto τ j . Restricting f j to E K × { } defines an extendable map E K → B j , so a j th h -invariant h j ( K, f j ) of K exists. Extending f j to F j : M × I → X n defines a homotopy from h j ( K, f j )to h j ( J, τ j ), so [ h j ( K, f j )] is trivial.Suppose now that an n th h -invariant h n ( K, τ ) of K is defined and trivial. Then there isa homotopy H : M × I / / X n with H ( − , s ) = (cid:26) h n ( J, τ n )( − ) if s = 0 h n ( K, τ )( − ) if s = 1 . Choose a framing t : S × D ∼ = ST with {∗} × ∂D ∼ = µ J for some ∗ ∈ S and let S = t ( S × { } ). We may assume that H is transverse to ST and contains S ⊂ ST in itsimage. Pulling back S yields a surface V = V ` · · · ` V m in M × I , transverse to the boundary, such that V ∩ ( M × { } ) = J ×{ } and V ∩ ( M × { } ) = K × { } , and each V i is connected. If V is connected (and based via the basing of J ) then H | E V : E V → B n is an extendable map and V is an n -concordance.Suppose that V is not connected. We now show how to connect the disjoint pieces of V and redefine H such that the new map is extendable, thus obtaining an n -concordancefrom K to J . Let q ∈ S and for i ∈ { , } let q i ∈ V i be points such that H ( q i ) = q . Let B ( q ) ∼ = I × D be a closed regular neighborhood of a subinterval of S containing q with I × { } ⊂ I × D a zero-section. Let B ( q i ) ∼ = D × D be a regular neighborhood of a closeddisk in V i containing q i such that H ( B ( q i )) = B ( q ) and such that D × { } ⊂ D × D isthe zero-section. Attach a one-handle to ( M × I ) × I along ( M × I ) × { } to obtain a newmanifold N = ( M × I ) × I ∪ D × D × I. We can arrange that D × D × { i } is attached to B ( q i ) × { } such that OMOTOPY PROPERTIES OF KNOTS IN PRIME MANIFOLDS 23 (1) D × { } × { i } is identified with the zero-section of B ( q i ) × { } ,(2) The resulting 3-dimensional submanifold W = V × I ∪ D × { } × I is orientable,and(3) N is orientable.We give N the orientation of ( M × I ) × I .Since E J (cid:16) p ! ! CCCCCCCC / / B n E V = = {{{{{{{{ commutes, H ∗ : π ( E V × I, p × × → π J Γ n γ J is surjective. Define meridians µ and µ for V and V , respectively, as follows: Choose a path α i , for i ∈ { , } , from p × × v i × e i ∈ D × S ⊂ B ( q i ) and let µ i = { v i } × S ∪ α i . Since H ∗ is surjective, we canchoose α i such that H ∗ ( µ i ) = µ J . Recall that W = V × I ∪ D × { } × I, and let N ( W ) be an open regular neighborhood of W . Then π ( N − N ( W )) = π ( M × I − N ( V × I )) ∗ Z h µ ∼ a − µ a i , where a generates Z and if y : I → N runs along the one-handle from v × e to v × e ,without intersecting ( D × { } ) × I , then a = [ α ∗ y − ∗ α − ]. Define an epimorphism h : π ( N − N ( W )) → π J Γ n γ J over G by defining h ( a ) to be trivial (that is, h ( a ) is the identity element) and defining h tobe ( H | M × I − N ( V × I ) ) ∗ on π ( M × I − N ( V × I )). This is well-defined since we chose µ and µ such that H ∗ ( µ ) = H ∗ ( µ ) = µ J . Since H ( ∂N ( V × I )) ⊆ T n and h ( µ ) = h ( µ ) = µ J ,it follows that h ( π ( ∂N ( W ))) = † n ( H J ), where we base W via the basing of ( J ×{ } ) ×{ } .Since B n is an Eilenberg-MacLane space, h extends H to a based map H : N − N ( W ) → B n such that H ( ∂N ( W )) ⊆ T n . In particular, H restricted to N − N ( W ) is an extendablemap to B n , after choosing suitable basings for the components of W (which can be donesince, as we observed earlier, H ∗ is surjective).Our goal now is to add a two-handle to cancel the previously attached one-handle. Let c , : I → E V × I ⊂ ( M × I ) × I be an embedded path from ∂B ( q ) ×{ } to ∂B ( q ) ×{ } such that H ( c , (0)) and H ( c , (1))lie in the image of the basing path α in B n . Recall that H ∗ : π ( E V × I, p × × → π J Γ n γ J is surjective. We may therefore assume that H ( c , ) ∪ α , the path in B n given by following α until it reaches H ( c , (0)), following H ( c , ) until it reaches H ( c , (1)), and then goingbackward down α , is homotopically trivial in B n . Let c : I → E W ⊂ N be a curve that travels from c , (0) ∈ ∂B ( q ) × { } to c , (1) ∈ ∂B ( q ) × { } throughthe one-handle and geometrically intersects the belt of the one-handle once. Since h : π ( N − N ( W )) → π J Γ n γ J is surjective and h ( a ) is trivial, we may assume that H ( c ) ∪ α is ho-motopically trivial in B n . Then c , ∗ c − is a closed embedded curve in N and H ( c , ∗ c − ) ∪ α is homotopically trivial in B n . We may isotope c , ∗ c − to lie embedded on the bound-ary of N , still geometrically intersecting the belt of the previously attached one-handleexactly once, and then attach a two-handle to cancel this one-handle. Since c , ∗ c − isnull-homotopic in B n , H extends over this two-handle. The result is a manifold diffeomor-phic to ( M × I ) × I together with a submanifold W with one less connected componentthan V × I . Moreover, H restricts to an extendable map on the exterior of W , after anappropriate choice of basings for its components.After connecting the components of V we will have a cobordism between two knots, butwe still need to see that those two knots are J and K . However, we can choose the dif-feomorphism between ( M × I ) × I and the manifold we obtained by adding the one- andtwo-handles to be the identity on ( M × I ) × { } . Hence, the knots W ∩ ( M × { } ) × { } and W ∩ ( M × { } ) × { } are isotopic to J and K , respectively.Let W m ⊂ ( M × I ) × I be the manifold obtained by repeating this procedure m times toconnect the components of V = V ` · · · ` V m . The connected surface W m ∩ ( M × I ) × { } ,based via the basing of ( J × { } ) × { } in ( M × { } ) × { } , is our n -concordance. (cid:3) Corollary 3.30. If K is concordant to J then for each n there is an extendable map τ suchthat h n ( K, τ ) is defined and trivial.Proof. A concordance is an n -concordance for all n . (cid:3) Characteristic knots.
We now restrict our attention to knots in manifolds of thesecond type (i.e. closed, orientable, aspherical 3-manifolds) and investigate the relationshipbetween h -invariants and J -characteristicness.Recall that a knot K in M is J -characteristic if there is a continuous degree-one map α : M → M such that α ( K ) = J and α ( M − K ) ⊆ M − J . Theorem 3.31.
Let K be a knot in an orientable, aspherical 3-manifold. If K is J -characteristic via a map α : M → M that indices the identity on G then K has trivial h -invariants.Proof. Up to based homotopy, α induces a map τ : E K → E J that restricts to a homeo-morphism ∂E K ∼ = ∂E J with τ ( µ K ) = µ J . Hence, the composition τ n ◦ τ : E K → B n isextendable for each n such that the following diagram commutes: M h ( K,τ n ◦ τ ) ! ! CCCCCCCC α / / M h ( J,τ n ) } } {{{{{{{{ X n Since α indices the identity on G and M is an Eilenberg-MacLane space, α is based homo-topic to the identity map on M . In particular, h ( K, τ n ◦ τ ) is homotopic to h ( J, τ n ). (cid:3) Remark . Let K be a knot in M , let η ⊂ E K be an embedded curve that bounds anembedded disk in M , and let N ( η ) be a regular neighborhood of η in the interior of E K . OMOTOPY PROPERTIES OF KNOTS IN PRIME MANIFOLDS 25
Let L be a knot in S . There is a canonical choice of longitude for both η and L , which wedenote λ η and λ L , respectively; let µ η and µ L be their respective meridians. Then( M − N ( η )) ∪ − (cid:0) S − N ( L ) (cid:1) , where µ L ∼ λ − η and λ L ∼ µ η , is diffeomorphic to M via a diffeomorphism that is the identityoutside a regular neighborhood of the disk bounded by η . In particular, this diffeomorphisminduces the identity homomorphism on G . Definition 3.33.
Define K ( η, L ) to be the image of K in ( M − N ( η )) ∪ − (cid:0) S − N ( L ) (cid:1) ∼ = M . That is, the image of K under the inclusion K ֒ → M − N ( η ) ֒ → ( M − N ( η )) ∪ − (cid:0) S − N ( L ) (cid:1) . For a special case of the following lemma see Proposition 2.2 of [4]. The proof is thesame.
Lemma 3.34. If K is a homotopically essential knot in a closed, orientable, aspherical3-manifold then E K is an Eilenberg-MacLane space. Theorem 3.35.
Let J be a homotopically essential knot in a closed, orientable, aspherical3-manifold. Let η be a curve in E J that bounds an embedded disk in M and let L be a knotin S . Then J ( η, L ) has trivial h -invariants.Proof. Since E J ( η,L ) = ( E J − N ( η )) ∪ − (cid:0) S − N ( L ) (cid:1) , where µ L ∼ λ − η and λ L ∼ µ η , it follows from the Seifert-van Kampen theorem that π J ( η,L ) := π ( E J ( η,L ) ) = π ( E J − N ( η )) ∗ Z π ( S − N ( L )) , as in the following pushout diagram, Z / / (cid:15) (cid:15) π ( S − N ( L )) (cid:15) (cid:15) π ( E J − N ( η )) / / π J ( η,L ) where Z = π ( ∂N ( η )). Also, the image of µ J in π J ( η,L ) is a meridian, µ J ( η,L ) . Let f : π ( E J − N ( η )) → π J be the epimorphism induced by inclusion E J − N ( η ) ֒ → E J and let g : π ( S − N ( L )) → Z → π J be abelianization followed by the homomorphism that sends µ L to λ − η . Then the pushout property of π J ( η,L ) induces a homomorphism h : π J ( η,L ) → π J over G as in the following diagram: Z / / (cid:15) (cid:15) π ( S − N ( L )) (cid:15) (cid:15) g (cid:25) (cid:25) π ( E J − N ( η )) / / f - - π J ( η,L ) h & & π J Since f ( µ J ) = µ J , the commutativity of the diagram implies that h ( µ J ( η,L ) ) = µ J . More-over, the inclusion ∂E J ֒ → E J − N ( η ) induces a commutative diagram: π J ( η,L ) h (cid:15) (cid:15) H J / / π ( E J − N ( η )) llllllllllllll ) ) SSSSSSSSSSSSSSSS π J Because E J is an Eilenberg-MacLane space by Lemma 3.34 and µ J ֒ → ∂E J ֒ → E J arecofibrations, h induces a map τ : E J ( η,L ) → E J that extends to M → M and inducesthe identity on G . As in the proof of Theorem 3.31, we see that J ( η, L ) has trivial h -invariants. (cid:3) References [1] S. E. Cappell and J. L. Shaneson. The codimension two placement problem and homology equivalentmanifolds.
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