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where knot theory studies embeddings of circles into @xmath0 , spatial graph theory studies embeddings of arbitrary graphs . let @xmath1 be a finite abstract graph . let @xmath2 be an embedding of @xmath1 into the @xmath3-sphere . we will call @xmath4 a _ spatial embedding of @xmath1 _ or simply a _ spatial graph_. since it is the map @xmath2 that matters in this construction , we will often refer to @xmath2 as a spatial embedding . we will often use the notation @xmath5 to emphasize that we are thinking of the spatial graph as a subcomplex of @xmath0 . several analogues of link homotopy have been proposed for spatial graphs , such as the edge homotopy and vertex homotopy of taniyama @xcite . two spatial embeddings of a graph @xmath1 are called _ edge - homotopic _ if they are transformed into each other by _ self crossing changes _ and ambient isotopies , where a self crossing change is a crossing change between two arcs of the same edge , and _ vertex - homotopic _ if they are transformed into each other by crossing changes between arcs of two adjacent edges and ambient isotopies . each of these notions reduces to link homotopy in the case when the graph is a disjoint union of circles , and invariants of these relations have been produced @xcite . classification of the embeddings of some graphs under these relations is possible . for example , spatial embeddings of @xmath6 up to edge - homotopy have been completely analyzed by nikkuni @xcite . milnor invariants are sufficient to classify two and three component links up to link homotopy @xcite and contributed to the eventual classification of all links @xcite . so far , no analogue of these invariants exists for the spatial graph relations listed above . since milnor invariants are a useful tool for studying link homotopy , our goal is to introduce a generalization of link homotopy for spatial graphs that allows a reasonable generalization of milnor s link homotopy invariants . the equivalence relation we will use is called _ component homotopy_. let @xmath7 denote the set of all spatial embeddings of an abstract graph @xmath1 . two embeddings @xmath8 are called component homotopic if @xmath9 can be transformed to @xmath10 by ambient isotopy and crossing changes between edges that belong to the same component of @xmath1 . note that if @xmath1 is connected , then any two embeddings are component homotopic . this is the analogue of the fact that any knot is link homotopic to the unknot , though for most graphs there is no canonical choice of embedding to call the `` unknot . '' however , up to component homotopy , there is a clear choice for the analogue of the unlink , which we will call a _ completely split _ embedding . a spatial embedding @xmath9 of @xmath1 is completely split if each component of @xmath9 can be separated from all others by an embedded @xmath11 . note that any two completely split spatial embeddings @xmath12 are component homotopic , and thus up to component homotopy , there is a canonical `` unlink . '' if milnor s link homotopy invariants vanish for @xmath13 , then @xmath13 is link homotopic to the unlink @xcite . the vanishing of milnor s link homotopy invariants is equivalent to the milnor group of the link , @xmath14 , being isomorphic to the free milnor group @xmath15 . choosing the correct analogue of these groups , which we will call @xmath16 and @xmath17 , we are able to prove the following theorems , which are direct generalizations of theorem 8 and corollary 2 of milnor s original paper @xcite .    * theorem [ mainthm ] * _ let @xmath9 be an embedding of a graph @xmath1 into @xmath0 . then @xmath18 where the isomorphism preserves generators ( up to conjugacy ) if and only if @xmath9 is component homotopic to a completely split embedding . _    * theorem [ itriv ] * _ let @xmath9 be an embedding of a graph @xmath1 into @xmath0 with components @xmath19 . if the map @xmath20 from @xmath17 to @xmath16 sends generators to conjugates of generators , and @xmath21 is the induced map on the groups for @xmath22 , then @xmath23 if and only if @xmath9 is component homotopic to an embedding where @xmath24 can be separated from the rest of @xmath9 by an embedded @xmath11 . _    a spatial subgraph of @xmath9 that is homeomorphic to @xmath25 is called a _ constituent link_. the following theorem shows that the link homotopy classes of certain constituent links of @xmath9 determine whether @xmath9 is component homotopic to a completely split embedding . note that this is in contrast to edge homotopy and vertex homotopy , where there are infinite families of examples of non - split embeddings , all of whose constituent links are link homotopic to trivial links @xcite . * theorem [ splitthm ] * _ let @xmath13 be a constituent link of @xmath9 , where at most one component of @xmath13 is contained in each component of @xmath9 . every such constituent link of @xmath9 is link homotopic to the trivial link if and only if @xmath9 is component homotopic to a completely split embedding . _ it is also possible to extract numerical invariants from the @xmath16 by looking at successive nilpotent quotients . milnor s numbers can be interpreted as arising from the elements in the kernel of the map @xmath26 , and so by examining the kernel of the map @xmath27 we will be able to produce similar ( though less subtle ) numerical invariants . naturally , any invariant of component homotopy is an invariant of edge homotopy and vertex homotopy , so these numerical invariants are invariants of those relations as well . we would like to thank peter teichner for helpful conversations , and akira yasuhara for an engaging discussion that led to example [ e : akiraeg ] . in this chapter , we will study embeddings of graphs up to component homotopy . that is , up to colored edge homotopy , where every edge in a given component of a graph is labeled with the same color , and each component of the graph is given a distinct color . notice that for the case @xmath28 , this is simply link homotopy .    in an analogue to the colored milnor group @xmath29 of @xcite , for an embedding @xmath30 we can define a `` free '' group @xmath17 associated to @xmath1 , and a group @xmath16 associated to the embedding . the meridians of the edges of @xmath9 form a normally generating set for @xmath31\gg$ ] , where the generator @xmath32 is the meridian of the @xmath33th edge in the @xmath34th component , the @xmath35 are elements of @xmath36 , and @xmath37 denotes the subgroup normally generated by the set @xmath38 . in fact , we need only one generator per loop in @xmath9 , but by contracting a spanning tree , we may think of these generators as meridians for a subset of the edges . let @xmath17 denote the free colored milnor group corresponding to @xmath1 , defined in the following way . fix a spanning tree @xmath39 of @xmath1 . label the edges of component @xmath34 in @xmath40 by @xmath41 . define @xmath42 , where @xmath43 is the free group with generators @xmath41 , and @xmath44\gg $ ] , with @xmath45 arbitrary words in the @xmath41 . we will often refer to the relations induced by @xmath46 as the _ milnor relations_. when @xmath17 is generated by @xmath41 with @xmath47 and @xmath48 then @xmath49 is generated by meridians @xmath32 with @xmath47 and @xmath48 , for the same values of @xmath50 and @xmath51 .    for an edge @xmath41 in @xmath1 , there is a unique minimal path @xmath52 in @xmath39 connecting the end points of @xmath41 . let @xmath53 . note that the free colored milnor group @xmath17 is a colored milnor group in the sense of freedman and teichner , and hence is nilpotent by lemma 3.1 of @xcite . as @xmath16 is a quotient of @xmath17 , it is nilpotent as well . this is important , as it allows us to make use of the following theorem of stallings @xcite . let @xmath54 and @xmath55 be nilpotent groups , and @xmath56 a homomorphism . if the induced map @xmath57 is an isomorphism , and the induced map @xmath58 is surjective , then @xmath59 is an isomorphism . [ stallingsthm ]    with stallings theorem at our disposal , it is possible to show that the group @xmath16 is invariant under component homotopy on the spatial graph @xmath9 in the same way as for the colored milnor group in @xcite . roughly , the idea is this : given @xmath60 a spatial embedding of @xmath1 , and @xmath61 be a component homotopy from @xmath60 to @xmath62 , let @xmath63 be the track of this homotopy in @xmath64 , and @xmath65 . consider the maps @xmath66 induced by inclusion . the inclusions carry meridians of @xmath24 to meridians of @xmath63 , so the @xmath67 clearly induce isomorphisms @xmath68 . by using finger moves to introduce self intersections on @xmath63 , we can arrange @xmath69 . then by alexander duality , @xmath70 is generated by the handelbodies dual to @xmath24 , and the linking tori at the self intersection points . these linking tori simply realize the relations in @xmath46 , so @xmath71 is surjective . thus , by stallings theorem , @xmath72 and @xmath73 are isomorphisms , and hence @xmath74 . it will be necessary later to understand the structure of the groups @xmath16 in more detail . given a spatial embedding @xmath9 , a regular neighborhood of @xmath24 is a handlebody , and the 2-cell of this handlebody induces the relation @xmath75 = 1 $ ] in @xmath36 and hence in @xmath16 . we will call this relation the _ surface relation _ induced by @xmath24 , and the element @xmath76 $ ] the _ surface element_.    just as is shown for the classical milnor group in @xcite , it is possible to show that @xmath16 has the presentation below . the fact that the only relations in @xmath16 are the surface relations and the milnor relations will be important in later arguments . the group @xmath16 has the presentation @xmath77 , where the @xmath32 are meridians to the edges @xmath41 , @xmath51 is the surface relation given by @xmath24 , and @xmath78 denotes the relations arising from @xmath46 . [ presentation ]    we begin with a wirtinger presentation for @xmath79 , which is calculated from a projection of the spatial graph . before projecting , we may first contract a spanning tree of @xmath9 , as this does not affect @xmath36 . given a projection , choose an arbitrary orientation for each edge , and label the generators corresponding to each arc of that edge in the diagram by @xmath80 , @xmath81 in the order the arcs are met as the edge is traversed . at each vertex we have the relation that the product of the meridians to all the arcs meeting that vertex is trivial . in a fixed component , each edge begins and ends at the vertex , so we may choose a diagram such that the relation induced by that vertex is @xmath82 . the relations in the wirtinger presentation for this diagram are those coming from the vertices and those of the form @xmath83 induced by the crossings , where the @xmath84 are arbitrary words in the @xmath85 . using the milnor relations , the relations from the crossings , and a double induction as in @xcite , we can eliminate the relations from the crossings and reduce the set of generators to only the @xmath86 . the only remaining relations are those induced by the vertices , which are of the form @xmath87 = 1 $ ] . these are precisely the surface relations discussed above .    recall that a spatial embedding @xmath9 of @xmath1 is completely split if each component of @xmath9 can be separated from all others by an embedded @xmath11 . we may now formulate an analogue of theorem 8 of @xcite . let @xmath9 be an embedding of a graph @xmath1 into @xmath0 . then @xmath18 where the isomorphism preserves meridinal generators ( up to conjugacy ) if and only if @xmath9 is component homotopic to a completely split embedding . [ mainthm ]    suppose that @xmath9 is component homotopic to a completely split embedding . since @xmath16 is invariant under component homotopy , we may assume that @xmath9 is completely split . the group @xmath16 is a free group modulo the set @xmath46 and the surface relations generated by the handlebodies forming the boundaries of neighborhoods of each component of @xmath9 . the surface relations in @xmath16 are of the form @xmath88 = 1 $ ] for fixed @xmath34 . since each component is separated from all others , for a fixed @xmath34 , each @xmath89 is a word in the @xmath32 . since @xmath32 commutes with all other generators with index @xmath34 , @xmath90 = 1 $ ] for each pair @xmath91 . thus , in @xmath16 the surface relations are trivial , so we have only the relations introduced by @xmath46 , and the map @xmath92 is an isomorphism @xmath27 . suppose that an isomorphism @xmath93 exists . we will induct on the number of edges in the final component . clearly if @xmath9 has one component , it is component homotopic to a split embedding , so the base case is trivial . suppose @xmath94 is a spatial graph of @xmath50 components , and the first betti number @xmath95 . then there are @xmath96 generators of @xmath16 arising from @xmath97 , specifically @xmath98 , @xmath99 . if deleting an edge of @xmath97 does not change @xmath100 , that edge is contractible and so the argument is trivial . so suppose that deleting an edge @xmath101 does change @xmath100 . we may choose the spanning tree @xmath39 to avoid @xmath101 , so let @xmath102 be the generator of @xmath16 represented by a meridian of @xmath101 . let @xmath103 and @xmath10 be the restriction of @xmath2 to @xmath104 . there is an obvious surjection @xmath105 given by @xmath106 , and the kernel of this map is clearly @xmath107 . let @xmath108 be the restriction of @xmath20 to this kernel and @xmath109 be the map induced by @xmath20 . we want to show that @xmath110 is an isomorphism . by the 5 lemma , this is equivalent to showing that @xmath108 is an isomorphism . @xmath111_{\theta_{e } } \ar@{^{(}->}[r ] & cmf(g ) \ar[d]_{\theta}^{\cong } \ar@{->>}[r]^{x_{nr}=1 } & cmf(g ' ) \ar[d]_{\theta ' } \ar[r ] & 1 \ar[d]\\ \ll m_{nr } \gg \ar@{^{(}->}[r ] & cmf(\phi ) \ar@{->>}[r]^{m_{nr}=1 } & cm(\phi ' ) \ar[r ] & 1}\ ] ]    since @xmath20 sends generators to conjugates of generators , @xmath112 and so @xmath113 . clearly , as @xmath108 is the restriction of an isomorphism it is injective , so it suffices to show that it is surjective . given @xmath114 , we have @xmath115 . thus @xmath108 is an isomorphism , and so is @xmath110 . as @xmath10 has fewer edges , and @xmath110 is an isomorphism carrying generators to conjugates of generators , we may apply the induction hypothesis and use component homotopy to convert @xmath10 to a completely split embedding . if we perform this component homotopy on @xmath9 simply carrying the extra edge @xmath101 along , we arrive at the embedding shown in figure [ setup ] . ( 360,144 )     after the inductive hypothesis . the edge labeled @xmath101 winds through some or all of the components . ]    since the embedding @xmath10 is completely split , in @xmath116 the loop @xmath117 is a word in the @xmath98 . thus , in @xmath16 , @xmath117 is a word in @xmath98 , @xmath118 and @xmath119 for any @xmath120 . now , since @xmath121 = 1 $ ] in @xmath16 , we know that @xmath122=1 \in cm(\phi)$ ] for all @xmath123 . we have the relation @xmath124 = 1 $ ] in @xmath16 from the 2-cell of the neighborhood of @xmath97 , but as we just saw that @xmath122=1 $ ] , @xmath123 , we may conclude that @xmath125 = 1 $ ] in @xmath16 .    using the isomorphism @xmath20 , we have that @xmath126 = 1 $ ] in @xmath17 . in @xmath17 , the only elements that commute with a generator @xmath41 are conjugates of generators of the same color @xcite . thus , we know @xmath127 for some @xmath128 , and hence has the same form in @xmath16 . this means that @xmath129 $ ] in @xmath130 . we may eliminate the commutators through component homotopy as shown in figure [ linkcancel ] . to eliminate the terms of @xmath131 , we use the component homotopy move as in figure [ graphcancel ] . we may reduce to @xmath132 in @xmath130 . thus the loop @xmath133 is contractible and may be isotoped into the sphere containing @xmath97 , so @xmath9 is component homotopic to a completely split embedding . ( 360,144 )    $ ] . ] ( 360,144 )    . ] let @xmath134 denote the colored milnor group of @xmath135 , the spatial graph obtained from @xmath9 by removing the @xmath34th component @xmath24 . that is @xmath136 for a fixed @xmath34 . given a map @xmath93 , we have the sequence    @xmath137 & cmf(g ) \ar[r]^{\theta } & cm(\phi)}\ ] ]    let @xmath138 , and let @xmath21 denote the restriction of the map @xmath20 to @xmath139 . this induces the sequence    @xmath140 & cmf(g^{i } ) \ar[r]^{\theta^{i } } & cm(\phi^{i})}\ ] ]    we are now able to produce a theorem analogous to milnor s corollary 2 of @xcite . let @xmath9 be an embedding of a graph @xmath1 into @xmath0 . suppose the map @xmath141 sends generators to conjugates of generators . then @xmath142 if and only if @xmath9 is component homotopic to an embedding where @xmath24 can be separated from the rest of @xmath9 by an embedded @xmath11 . [ itriv ]    this proof is similar to that of theorem [ mainthm ] . recall that the only relations in @xmath16 are those induced by @xmath46 and the surface relations . under @xmath20 , the set @xmath46 in @xmath16 is hit only by elements of @xmath46 in @xmath17 , so @xmath143 is generated by the preimages of the surface relations . suppose that @xmath9 is component homotopic to an embedding where @xmath62 can be separated from the other components by an embedded @xmath11 . splitting along this @xmath11 , van kampen s theorem implies that @xmath144 , where @xmath145 . as the surface element @xmath146 $ ] lies in @xmath147 and the surface elements @xmath148 $ ] lie in @xmath130 for @xmath149 , the surface relation given by @xmath62 is trivial in @xmath16 , and the surface elements given by the @xmath24 , @xmath150 can be written without the generators @xmath151 . thus , @xmath152 under the map taking @xmath153 to 1 and @xmath41 to @xmath41 for @xmath150 . now suppose that the component of interest is @xmath97 and @xmath154 . we now induct on the number of edges in @xmath97 as before . since @xmath155 , we have that the surface relation from @xmath97 is trivial , that is @xmath156 = 1 $ ] in @xmath17 . however , by induction we know that @xmath156 = [ x_{nr},l_{nr } ] = 1 $ ] in @xmath17 . we may now use the same techniques as in the proof of theorem [ mainthm ] to alter @xmath9 by component homotopy so that @xmath97 can be separated from the other components . recall that a _ constituent link _ is a subgraph of @xmath9 that is homeomorphic to a disjoint union of circles . let @xmath13 be a constituent link of @xmath9 , where at most one component of @xmath13 is contained in each component of @xmath9 . every such constituent link of @xmath9 is link homotopic to the trivial link if and only if @xmath9 is component homotopic to a completely split embedding . [ splitthm ]    let @xmath9 be a spatial graph component homotopic to a completely split embedding @xmath10 , and let @xmath13 be a constituent sublink of @xmath9 , with at most one component of @xmath13 contained in each component of @xmath9 . the component homotopy from @xmath9 to @xmath10 induces a link homotopy carrying @xmath13 to @xmath157 . each component of @xmath10 can be separated from the others by an embedded @xmath11 . each separating 2-sphere contains a single component of @xmath157 , which by additional component homotopy moves ( contained in that sphere ) can be unknotted . thus , @xmath13 is link homotopic to the unlink . assume that all the relavent constituent links of @xmath9 are link homotopically trivial . we will work by induction on the number of components in @xmath9 . for a one component graph , all spatial embeddings trivially satisfy the condition to be completely split . suppose @xmath158 is an @xmath50 component graph and @xmath5 is not split up to component homotopy . then by theorem [ mainthm ] , no map from @xmath17 to @xmath16 preserving generators ( up to conjugacy ) is an isomorphism . so given the map @xmath159 , there must exist a nontrivial element in the kernel . relabeling the components of @xmath9 if necessary , it must be of the form @xmath160 $ ] . so some @xmath161 \neq 1 $ ] in @xmath17 . delete all edges in @xmath54 except for the edges forming @xmath162 to produce @xmath163 , and form the embedding @xmath164 by restricting @xmath2 to @xmath165 . we have reduced @xmath62 to a circle , so we may now think of @xmath162 as the zero framed pushoff . suppose @xmath161 = 1 $ ] in @xmath166 , then @xmath167 in @xmath166 by @xcite . the map @xmath168 is given by @xmath169 for @xmath170 , so the kernel is normally generated by those @xmath153 . so , in @xmath17 , @xmath171 , and thus @xmath161 = [ x_{1j_{1 } } , \pi x_{1j}^{g_{j } } ] = 1 $ ] in @xmath17 . this is a contradiction , so @xmath161 \neq 1 $ ] in @xmath166 , and thus @xmath172 given by the restriction of @xmath20 is not an isomorphism . so by theorem [ mainthm ] , @xmath164 is not completely split . suppose @xmath173 = 1 $ ] in @xmath166 for all @xmath33 . then as discussed in the proof of theorem [ mainthm ] @xmath174 , and so @xmath175 $ ] in @xmath176 . we may now do component homotopy as in figure [ linkcancel ] and figure [ graphcancel ] to separate @xmath177 from the rest of @xmath164 . let @xmath178 , and @xmath179 . then , @xmath164 is component homotopic to @xmath180 . the component homotopy induces a link homotopy on the relevant constituent links , so by assumption , these constituent links of @xmath181 are link homotopic to trivial links . the spatial graph @xmath181 has @xmath182 components , so by induction , it is completely split up to component homotopy , and hence so is @xmath164 . this is a contradiction . thus , there exists some @xmath183 with @xmath184 \neq 1 $ ] in @xmath166 . now , as before , delete edges of @xmath55 to reduce it to the circle defined by @xmath185 , forming @xmath186 . again , restricting @xmath2 , we obtain an embedding @xmath187 , and @xmath188 \neq 1 $ ] in @xmath189 . repeat the argument as above until each component of @xmath1 has been reduced to a circle , forming @xmath190 . form @xmath10 by restricting @xmath2 to @xmath104 . notice that @xmath191 \neq 1 $ ] in @xmath192 . now , @xmath10 is an @xmath50 component link @xmath13 where each component of @xmath13 came from a distinct component of @xmath1 . thus @xmath116 is @xmath14 , and the map @xmath193 induces a map @xmath194 , where @xmath15 is the free milnor group on @xmath50 generators . the commutator @xmath191 \neq 1 $ ] in @xmath15 , but is an element of the kernel of @xmath110 . since the map @xmath195 takes generators to conjugates of generators but is not an isomorphism , by theorem [ mainthm ] @xmath13 is not link homotopically trivial .    to determine when a spatial graph @xmath9 is completely split up to component homotopy , theorem [ splitthm ] implies that it is sufficient to check that the constituent links ( of a certain type ) are all split instead of checking the conditions of theorem [ mainthm ] . note that a similar statement is not true for theorem [ itriv ] . as the following example shows , it is possible for a every constituent link of @xmath9 to be @xmath34-trivial , but @xmath9 to not be component homotopic to @xmath196 . [ e : akiraeg ] let @xmath9 be the spatial graph shown in figure [ akiraeg ] . then clearly every constituent link is @xmath3-trivial . however , we have the relation @xmath197 = 1 \in cm(\phi)$ ] , but @xmath198 \neq 1 \in cm(g)$ ] . thus , for the map @xmath199 , we have @xmath200 , and so by theorem [ itriv ] the embedding @xmath9 is not component homotopic to @xmath201 . teichner and freedman introduced the notion of colored link homotopy in @xcite . in this notion , each component of link is assigned a color , and crossing changes are allowed between components of the same color . thereom [ splitthm ] can be easily extended to the statement that @xmath9 is component homotopic to a completely split embedding if and only if every constituent link of @xmath9 is colored link homotopic to the unlink , where two components of the link have the same color if and only if they are contained in the same component of @xmath9 . this shows that in some sense , the group @xmath16 contains information about colored link homotopy classes of all the constituent links . it is well known that the milnor group of a link is nilpotent , and the group @xmath16 is also nilpotent . the kernel of the map @xmath27 is generated by surface elements , which are products of commutators of elements . this suggests that examining successive quotients by all commutators of a fixed length may be productive . let @xmath63 be a subgraph of @xmath1 , and @xmath5 a spatial embedding . let @xmath202 be the spatial embedding of @xmath203 that is induced by @xmath2 . let @xmath204 denote the colored milnor group of @xmath202 , given a map @xmath205 , we have the sequence below :              since the groups @xmath16 and @xmath204 are invariant under component homotopy , so is @xmath209 . here we consider @xmath211 as a subgroup of @xmath17 , and @xmath212 $ ] denotes all commutators of length @xmath50 whose entries are elements of @xmath17 . notice that if @xmath28 , then @xmath213 is the length of the index of the shortest nonvanishing milnor homotopy invariant whose index contains @xmath34 . the first nonvanishing milnor invariant of the borromean rings is @xmath214 . if @xmath13 is the borromean rings , then the kernel of the map @xmath215 is generated by the element @xmath216 $ ] . when the first component is removed , however , the resulting link is the unlink , so @xmath217 is trivial . thus , @xmath218 . notice that while the surface relation given by @xmath223 contains multiple commutators , only the shortest matters for the computation of @xmath224 . also , the presence of the length two commutator involving @xmath223 and @xmath225 does not affect the computation of @xmath226 . in fact , by the proof of theorem [ splitthm ] , when @xmath227 the value of @xmath209 will be the length of the shortest non - vanishing ( colored ) milnor invariant of any link that contains a cycle from @xmath228 .
link homotopy has been an active area of research for knot theorists since its introduction by milnor in the 1950s . 
 we introduce a new equivalence relation on spatial graphs called component homotopy , which reduces to link homotopy in the classical case . 
 unlike previous attempts at generalizing link homotopy to spatial graphs , our new relation allows analogues of some standard link homotopy results and invariants .    in particular we can define a type of milnor group for a spatial graph under component homotopy , and this group determines whether or not the spatial graph is splittable . 
 more surprisingly , we will also show that whether the spatial graph is splittable up to component homotopy depends only on the link homotopy class of the links contained within it . 
 numerical invariants of the relation will also be produced .