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Alexander Duality, Gropes and Link Homotopy 53 ISSN 1364-0380 51 Geometry Topology G T T G T T G T G T olume V G T G T G T Published Octob er G T G T G T G T G G T G G T Duality Grop es and Link Homotopy Alexander acheslav S Krushkal Vy Peter Teichner Department of Mathematics Michigan State University East Lansing MI USA Current address MaxPlanckInstitut fur Mathematik D Bonn Germany GottfriedClarenStrasse and Department of Mathematics University of California in San Diego La Jolla CA USA Email krushkalmathmsuedu and teichnereucliducsdedu Abstract We prove a geometric renement of Alexander duality for certain complexes the socalled gropes emb edded into space This renement can b e roughly formulated as saying that dimensional Alexander duality preserves the dis joint Dwyer ltration In addition we give new pro ofs and extended versions of two lemmas of Freed which are of central imp ortance in the ABslice problem the man and Lin main op en problem in the classication theory of top ological manifolds Our metho ds are group theoretical rather than using Massey pro ducts and Milnor invariants as in the original pro ofs Classication numb ers Primary M M AMS Secondary M N N ords Alexander duality manifolds grop es link homotopy Milnor Keyw group Dwy er ltration osed Robion Kirby Received June Prop Michael Freedman Ronald Stern Revised Octob er Seconded Copyright Geometry and Topology 52 Vyacheslav S Krushkal and Peter Teichner Intro duction n Consider a nite complex X PLemb edded into the n dimensional sphere S n integer homology H S r X with Alexander duality identies the reduced i 1i n X This implies that the homology or even the sta the cohomology H ble homotopy typ e of the complement cannot distinguish b etween p ossibly n t emb eddings of X into S Note that there cannot b e a duality for ho dieren motopy groups as one can see by considering the fundamental group of classical 1 knot complements ie the case X S and n Ho wever one can still ask whether additional information ab out X do es lead to n additional information ab out S r X For example if X is a smo oth closed n i dimensional manifold then the cohomological fundamental class is dual n to a spheric al class in H S r X Namely it is represented by any meridional i i sphere which by denition is the b oundary of a normal disk at a p oint in X This geometric picture explains the dimension shift in the Alexander duality theorem n By reversing the roles of X and S r X in this example we see that it is not n true that H X b eing spherical implies that H S r X is spherical i n1i However the following result shows that there is some kind of improved duality es in if one do es not consider linking dimensions One should think of the Grop our theorem as means of measuring how spherical a homology class is 4 Grop e Duality If X S is the disjoint union of closed em Theorem 4 b edded Grop es of class k then H S r X is freely generated by r disjointly 2 Here r is the rank of H X Moreover emb edded closed Grop es of class k 1 4 H S r X cannot b e generated by r disjoint maps of closed grop es of class 2 k As a corollary to this result we show in that certain Milnor invariants of 3 a link in S are unchanged under a Grope concordance The Gropes ab ove are framed thickenings of very simple complexes called gropes which are inductively built out of surface stages see Figure and Sec a surface with a single b oundary tion For example a grop e of class is just comp onent and grop es of bigger class contain information ab out the lower cen tral series of the fundamental group Moreover every closed grop e has a fun damental class in H X and one obtains a geometric denition of the Dwyer 2 ltration X X X X H X 2 3 2 2 k Geometry and Topology, Volume 1 (1997) Alexander Duality, Gropes and Link Homotopy 53 by dening X to b e the set of all homology classes represented by maps k of closed grop es of class k into X Theorem can thus b e roughly formulated as saying that dimensional Alexander duality preserves the disjoint Dwyer ltration Figure A grop e of class is a surface two closed grop es of class Figure shows that each grop e has a certain typ e which measures how the surface stages are attached In Section this will b e made precise using certain ro oted trees compare Figure In Section we give a simple algorithm for obtaining the trees corresp onding to the dual Grop es constructed in Theorem The simplest application of Theorem with class k is as follows Consider 2 4 the standard emb edding of the torus T into S which factors through the 2 3 usual unknotted picture of T in S Then the b oundary of the normal bundle 2 of T restricted to the two essential circles gives two disjointly emb edded tori 2 4 2 Z Since b oth of these tori may b e representing generators of H S r T 2 4 2 surgered to emb edded spheres H S r T is in fact spherical However 2 it cannot b e generated by two maps of spheres with disjoint images since a map of a sphere may b e replaced by a map of a grop e of arbitrarily big class This issue of disjointness leads us to study the relation of grop es to classical link homotopy We use Milnor group techniques to give new pro ofs and improved namely the Grope Lemma and the Link versions of the two central results of Comp osition Lemma Our generalization of the grop e lemma reads as follows 3 Two n comp onent links in S are link homotopic if and only if Theorem 3 I they cob ound disjointly immersed annuluslike grop es of class n in S This result is stronger than the version given in where the authors only make a comparison with the trivial link Moreover our new pro of is considerably shorter than the original one Geometry and Topology, Volume 1 (1997) 54 Vyacheslav S Krushkal and Peter Teichner Our generalization of the link comp osition lemma is formulated as Theorem in Section The reader should b e cautious ab out the pro of given in It turns out that our Milnor group approach contributes a b eautiful feature to algebraization of link homotopy He proved in that by forgetting Milnors one comp onent of the unlink one gets an ab elian normal subgroup of the Milnor group which is the additive group of a certain ring R We observe that the Magnus expansion of the free Milnor groups arises naturally from considering the conjugation action of the quotient group on this ring R Moreover we show in Lemma that comp osing one link into another corresp onds to multiplication in that particular ring R This fact is the key in our pro of of the link comp osition lemma Our pro ofs completely avoid the use of Massey pro ducts and Milnor invariants and we feel that they are more geometric and elementary than the original pro ofs This might b e of some use in studying the still unsolved ABslice problem which is the main motivation b ehind trying to relate grop es their du ality and link homotopy It is one form of the question whether top ological surgery and scob ordism theorems hold in dimension without fundamental group restrictions See for new developments in that area It is a pleasure to thank Mike Freedman for many imp or Acknow ledgements tant discussions and for providing an inspiring atmosphere in his seminars In particular we would like to p oint out that the main construction of Theorem is reminiscent of the metho ds used in the linear grope height raising pro cedure of The second author would like to thank the Miller foundation at UC Berkeley for their supp ort Preliminary facts ab out grop es and the lower central series The following denitions are taken from A grope is a sp ecial pair complex circle A grop e has a Denition class k For k a grop e is dened to b e the pair circle with a circle For k a grop e is precisely a compact oriented surface single b oundary comp onent For k nite a k grope is dened inductively as i genusg follow Let f b e a standard symplectic basis of circles i i k and p q k for for For any p ositive integers p q with p q i i i i i i 0 0 at least one index i a k grop e is formed by gluing p grop es to each and 0 i i q grop es to each i i Geometry and Topology, Volume 1 (1997) Alexander Duality, Gropes and Link Homotopy 55 The imp ortant information ab out the branching of a grop e can b e very well captured in a ro oted tree as follows For k this tree consists of a single which is called the root For k one adds vertex v genus edges 0 to v and may lab el the new vertices by Inductively one gets the 0 i i tree for a k grop e which is obtained by attaching p grop es to and q i i i grop es to by identifying the ro ots of the p resp ectively q grop es with i i i the vertices lab eled by resp ectively Figure b elow should explain the i i corresp ondence b etween grop es and trees leaves ro ot Figure A grop e of class and the asso ciated tree Note that the vertices of the tree which are ab ove the ro ot v come in pairs 0 corresp onding to the symplectic pairs of circles in a surface stage and that such ro oted paired trees corresp ond bijectively to grop es Under this bijection the valent vertices of the tree corresp ond to circles on the grop e which leaves freely generate its fundamental group We will sometimes refer to these circles as the tips of the grop e The b oundary of the rst stage surface will b e referred to as the bottom of the grop e k Given a group we will denote by the k th term in the lower central series 1 k k 1 of dened inductively by and the characteristic subgroup of k fold commutators in Lemma
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