Transverse fundamental group and projected embeddings
TTRANSVERSE FUNDAMENTAL GROUP ANDPROJECTED EMBEDDINGS
SERGEY A. MELIKHOV
Abstract.
For a generic degree d smooth map f : N n → M n we introduce its “trans-verse fundamental group” π ( f ) , which reduces to π ( M ) in the case where f is a cover-ing, and in general admits a monodromy homomorphism π ( f ) → S | d | ; nevertheless, weshow that π ( f ) can be non-trivial already for rather simple degree maps S n → S n .We apply π ( f ) to the problem of lifting f to an embedding N (cid:44) → M × R : forsuch a lift to exist, the monodromy π ( f ) → S | d | must factor through the group ofconcordance classes of | d | -component string links. At least if | d | < , this requires π ( f ) to be torsion-free. Introduction
A generic C ∞ map f : N → M is called a k -projected embedding , or a k -prem if thereexists a g : N → R k such that f × g : N → M × R k is a smooth embedding (cf. [3], [2]).A necessary condition for a generic smooth map f : N n → M m to be a k -prem is theexistence of a Z / -equivariant map from the double point set ∆ f = { ( x, y ) ∈ N × N | f ( x ) = f ( y ) , x (cid:54) = y } (endowed with the restriction of the factor exchanging involutionon N × N ) to the sphere S k − (endowed with the antipodal involution x (cid:55)→ − x ). Indeed,given a g : N → R k as above, we define ϕ : ∆ f → S k − by ϕ ( x, y ) = g ( x ) − g ( y ) || g ( x ) − g ( y ) || ; clearly, ϕ ( y, x ) = − ϕ ( x, y ) . One can show that when n − m ≤ k and m + k ≥ n +1)2 , thisnecessary condition is also sufficient [16].Using this result, it is not hard to see that every generic smooth map f betweenorientable smooth n -manifolds is an n -prem for all even n > ; indeed, an equivariantmap ∆ f → S n − exists for all even n (including n = 2 ) as observed in the proof ofTheorem 1.4 below. Problem 1.1.
Is every generic smooth map between orientable surfaces a -prem? It is known that the answer is affirmative in the following cases: for maps of any -manifold into R [20]; for maps of S into any orientable -manifold (Yamamoto–Akhmetiev [15]); and for maps S × S → S × S [14].Petersen proved that the answer is also affirmative for all regular coverings of degree < [19]. This is established as follows: a group of order < is solvable; a solvablecovering is a composition of abelian coverings; an abelian covering over a compact poly-hedron with free abelian H is induced from a covering over a torus S × . . . × S ; acovering over a torus is equivalent by a change of coordinates to a product of coverings This work is supported by the Russian Science Foundation under grant 14-50-00005. a r X i v : . [ m a t h . G T ] F e b RANSVERSE FUNDAMENTAL GROUP AND PROJECTED EMBEDDINGS 2 over S ; a product of coverings over S is a composition of coverings over the torusinduced from coverings over S ; and a covering over S is obviously a -prem.Petersen also proved that if a composition of two coverings is a -prem, then each ofthem is a -prem [19]. Let us note that the regular covering corresponding to the kernelof the monodromy homomorphism of a covering p factors through p . Consequently,every covering with solvable monodromy group (hence in particular every covering ofdegree < ) between orientable surfaces is a -prem.Let us also note that a covering over a connected sum of tori is a 2-prem if it is inducedfrom a covering of the wedge of these tori; indeed, any covering induced from a coveringover a wedge of tori is a 2-prem (see Theorem 2.12 below and the subsequent remarks).Finally, we should note that one motivation of Problem 1.1 is that its affirmativesolution would yield an affirmative answer to the following Problem 1.2. [5] , [15] Does every inverse limit of orientable -manifolds embed in R ? P. M. Akhmetiev proved that an inverse limit of stably parallelizable n -manifoldsembeds in R n for n (cid:54) = 1 , , , [1] (see [15] for an explicit proof). It is well-known thatthe p -adic solenoid, which is an inverse limit of copies of S , does not embed in theplane. Akhmetiev also constructed inverse limits of - and -dimensional parallelizablemanifolds that do not embed in R , resp. R (see [18]). . . Content of the paper It is clear from Theorem 1.4 below that Problem 1.1 is a typical problem of four-dimensional topology in that there is no lack of potential counterexamples (such as5-fold coverings and regular 60-fold coverings, not to mention generic approximations ofvarious branched coverings) but an obviuous lack of invariants/obstructions capable ofdetecting actual counterexamples.The present note develops one approach to constructing such an obstruction in thecase of generic maps other than coverings. The “transverse fundametal group” π ( f ) of ageneric smooth map f : M → N between manifolds of the same dimension is introducedin §2. In the case where f is a covering, π ( f ) specializes to π ( M ) and so gives nothingnew. On the other hand, we compute, for instance, that π ( f ) contains an infinite cyclicsubgroup for a certain fold map f : S → S , which is a generic C -approximation of thesuspension of the double covering S → S . This and other examples are studied in §3.The following is a special case of Corollary 2.15. Theorem 1.3.
Let f be a generic smooth map of degree < between orientable -manifolds. If π ( f ) contains torsion, then f is not a -prem. The author does not know if π ( f ) can contain torsion when π ( M ) is torsion-free;generally speaking, nothing seems to preclude from this. Keeping in mind, say, the 4-dimensional PL Poincar´e conjecture, the Andrews–Curtis conjecture, theproblem of PL embeddability of acyclic and contractible 2-polyhedra in R , etc. RANSVERSE FUNDAMENTAL GROUP AND PROJECTED EMBEDDINGS 3
From the viewpoint of algebraic topology, the elements of π ( f ) are analogous, orrather dual, to spherical classes in the -homology of a -manifold (see Remark 2.5).Even though the technique enabling us to show that π ( f ) is well-defined was originallydeveloped in the course of a study of projected embeddings [15], the present paper waswritten with hope that π ( f ) may also find other applications. . . A motivation: The double point obstruction One may look at the following straightforward obstruction to 1.1. Let us consider, moregenerally, a generic smooth map f : N n → M n . Take a generic lift ¯ f : N → M × R n of f and pick some basepoint b ∈ N . Each (necessarily isolated) double point z =¯ f ( x ) = ¯ f ( y ) of ¯ f has a sign ε z = ± determined by comparing the orientations of thetwo sheets of N with the orientation of M × R n . Let us pick a path p x joining b and x and a path p y joining y and b . Then f ( p x p y ) is an f ( b ) -based loop in M . The class g z ∈ G := π ( M, f ( b )) of this path is well defined up to multiplication on both sides byelements of H := f ∗ ( π ( N )) . Let θ ( ¯ f ) be the algebraic sum (cid:88) z ε z Hg z H ∈ Z [ H \ G/H ] of the resulting double cosets. If ¯ f (cid:48) is another generic lift of f , a generic homotopybetween ¯ f and ¯ f (cid:48) over f yields an oriented bordism between the set of double points of ¯ f and that of ¯ f (cid:48) . The critical levels of this bordism consist of cancellations/introductions ofpairs ( z, z (cid:48) ) such that ε z = − ε z (cid:48) and g z = g z (cid:48) ; and (unless f is a covering) of births/deathsof individual double points z such that g z ∈ H . Hence θ ( f ) := θ ( ¯ f ) = θ ( ¯ f (cid:48) ) is welldefined. Obviously, if f is an n -prem, θ ( f ) = 0 . Theorem 1.4. If n is even, θ ( f ) = 0 for every generic smooth f : N n → M n .Proof. Since the dimensions of N and M have the same parity, ∆ f /T is orientable,where T is the factor exchanging involution on N × N (see e.g. [15; Akhmetiev’s Lemma(preceding Example 5) or the beginning of §3]). If λ is the line bundle associated withthe double covering ∆ f → ∆ f /T , its Euler class e ( λ ) is an element of order two in thecohomology group H (∆ f /T ; Z T ) with local coefficients (see e.g. [17; §2]). Since ∆ f /T is orientable, whereas the coefficients Z ⊗ nT are constant when n is even, H n (∆ f /T ; Z ⊗ nT ) is free abelian, and therefore e ( λ ) n = 0 . This yields an equivariant oriented null-bordism W of the oriented -manifold ∆ ¯ f in ∆ f (see e.g. [15; Lemma 7] or [17; §3, subsections “Geometric definition of ϑ ( f ) ” and“Cohomological sectional category”]). Without loss of generality W has no componentswithout boundary. By the definition of ∆ f , we have f pT ( a ) = f ( y ) = f ( x ) = f p ( a ) foreach a = ( x, y ) ∈ ∆ f , where p projects N × N onto the first factor. Since W ⊂ ∆ f , itfollows that f pT | J = f p | J for each component J of W . This immediately implies the existence of an equivariant map ∆ f → S n − (see e.g. [17; Alternativeproof of Theorem 3.2]). RANSVERSE FUNDAMENTAL GROUP AND PROJECTED EMBEDDINGS 4
Let c = ( x, y ) and d = ( x (cid:48) , y (cid:48) ) be the endpoints of a compact component J correspond-ing to double points z = ¯ f ( x ) = ¯ f ( y ) and z (cid:48) = ¯ f ( x (cid:48) ) = ¯ f ( y (cid:48) ) of ¯ f , so that ε z = − ε z (cid:48) .Let p x be a path joining b to x = p ( c ) and p y a path joining y = pT ( c ) to b . Then p x followed by p | J is a path p x (cid:48) joining b to x (cid:48) = p ( d ) and the inverse of pT | J followed by p y is a path p y (cid:48) joining y (cid:48) = pT ( d ) to b . Now f pT | J = f p | J implies that f ( p x (cid:48) p y (cid:48) ) ishomotopic to f ( p x p y ) , whence g z (cid:48) = g z .Similarly if J is noncompact and so has only one endpoint a = ( x, y ) correspondingto a double point z = ¯ f ( x ) = ¯ f ( y ) , then g z ∈ H . (cid:3) The author is grateful to P. Akhmetiev and M. Yamamoto for very valuable remarks.The paper also benefited from stimulating conversations with N. Brodskiy, V. Chernov,J. Keesling, E. Kudryavtseva, S. Maksymenko, R. Mikhailov and R. Sadykov.2.
In search of non- -prems . . Transverse fundamental group Definition 2.1 (Pullback) . If L , M and N are smooth manifolds and L g −→ M f ←− N aresmooth maps, we say that g is transverse to f and write g (cid:116) f if f × g : N × L → M × M is transverse to ∆ M . In this case P := ( f × g ) − (∆ M ) is a smooth submanifold of N × L ,and consequently the composition of the inclusion P (cid:44) → N × L and the projection N × L → N is a smooth map; if additionally f is generic, then so is this composition.This composition is called the pullback (or the “base change map”) of f along g and isdenoted g ∗ f , and its domain P (also known as the pullback of the diagram L g −→ M f ←− N )may be denoted ( g ∗ f ) − ( L ) . Note that if g is an embedding, then so is f ∗ g , whichtherefore performs a homeomorphism between ( g ∗ f ) − ( L ) = ( f ∗ g ) − ( N ) and f − ( g ( L )) . Definition 2.2 (Coherent homotopy) . Let f : N → M be a generic smooth map betweenclosed oriented connected n -manifolds, n ≥ , and let b ∈ M be its regular value(in particular, b is a value of f , i.e. b ∈ f ( N ) .) Consider f -transverse based loops l , l : ( S , pt ) → ( M, b ) . A based homotopy h : ( S × I, pt × I ) → ( M, b ) between l and l will be called ( b, f ) -coherent if it is f -transverse and every connected component of thepullback ( h ∗ f ) − ( S × I ) that intersects ( h ∗ f ) − ( pt × I ) is an annulus with one boundarycomponent in ( l ∗ f ) − ( S ) and another in ( l ∗ f ) − ( S ) . Note that some individual levels h t : S → M of a ( b, f ) -coherent homotopy may be non- f -transverse, and the number ofcomponents in ( h ∗ t f ) − ( S ) may vary depending on t . Definition 2.3 ( π ( f ) : The case of unfolded basepoint) . Suppose first the cardinality | f − ( b ) | equals the absolute value | deg f | (so in particular deg f (cid:54) = 0 , since we areassuming that b ∈ f ( N ) ). The set π ( f, b ) of b -based f -transverse loops in M up to ( b, f ) -coherent homotopy is clearly a group with respect to the usual product (i.e. theconcatenation) of loops and the usual inverse of a loop. Example 2.4.
If the generic map f : N → M is a covering, π ( f, b ) (cid:39) π ( M ) sincecoverings enjoy the covering homotopy property. RANSVERSE FUNDAMENTAL GROUP AND PROJECTED EMBEDDINGS 5
On the other hand, there exists, for instance, a fold map f : S → S with four foldcurves and with | f − ( b ) | = 1 such that π ( f ) (cid:54) = 1 (see Examples 3.5, 3.9). Remark 2.5.
Note that the Pontryagin construction identifies f -transverse framed loopsin M with stable maps of the mapping cylinder of f into S n − that are transverse to pt ∈ S n − . Of course, homotopies between such maps, transverse to pt , are identifiedwith arbitrary f -transverse framed bordisms, which are not necessarily homotopies (notto mention coherent homotopies). Thus from the viewpoint of algebraic topology, thequestion of existence of a coherent homotopy is a question of representability of a gener-alized cohomology class by a genus zero cocycle extending a given representation on theboundary. (By a cocycle we mean a pseudo-comanifold, i.e. an embedded mock bundlewith codimension two singularities — see [4].) Definition 2.6 ( π ( f ) : The general case) . Without loss of generality we may assumethat d := deg( f ) ≥ . Let j : I → M be an f -transverse path, and let J = ( j ∗ f ) − ( I ) .Since M is oriented, ∆ M is co-oriented in M × M , hence J is co-oriented in I × N . Since I and N are oriented, J is oriented.A component C of J is called a positive (negative) arc if j ∗ f | C : ( C, ∂C ) → ( I, ∂I ) hasdegree +1 (resp. − ). Else C could be a circle or an arc with both endpoints mappingonto the same endpoint of I , with ( j ∗ f )( C ) (cid:54) = I . Note that the signs of the arcs reverse(along with the sign of deg( f ) ) when the orientation of M or N is reversed; but remainunchanged when the orientation of I is reversed. Lemma 2.7. [15; §2, proof of Observation 2]
Let f : N → M be a generic smooth mapbetween closed oriented connected n -manifolds, n ≥ , with deg( f ) ≥ . Then thereexists an f -transverse path (cid:96) : I → M such that ( (cid:96) ∗ f ) − ( I ) contains no negative arcs. Without loss of generality a := (cid:96) (0) and b := (cid:96) (1) are f -regular values. (In fact,since any f -transverse path (cid:96) + containing (cid:96) is again such that ( (cid:96) ∗ + f ) − ( I ) contains nonegative arcs, a and b could have been any f -regular values given in advance.) Let L = ( (cid:96) ∗ f ) − ( I ) , and let D be a bijection between [ d ] := { , , . . . , d − } and the set ofendpoints in ( (cid:96) ∗ f ) − (0) of the d positive arcs in L .Let j : ( I, ∂I ) → ( M, b ) be any f -transverse loop. Then the product ˆ j of the paths (cid:96) , j and the inverse path ¯ (cid:96) (defined by ¯ (cid:96) ( t ) = (cid:96) (1 − t ) ) is again such that ˆ J := (ˆ j ∗ f ) − ( I ) contains no negative arcs; moreover, each positive arc of L is contained in a uniquepositive arc of ˆ J .We define π ( f, (cid:96) ) to be the set of all f -transverse b -based loops up to b -based homotopy j t such that the a -based homotopy ˆ j t is ( a, f ) -coherent. Clearly this is a group withrespect to the usual product of loops and the usual inverse of a loop.Furthermore, assigning to an endpoint of a positive arc in ˆ J the other endpoint of thisarc, we get a bijection h j,D : [ d ] → [ d ] . If [ j (cid:48) ] = [ j ] ∈ π ( f, (cid:96) ) , clearly h j (cid:48) ,D = h j,D . Hence [ j ] (cid:55)→ h j,D defines a homomorphism ϕ f,(cid:96),D : π ( f, (cid:96) ) → S d . RANSVERSE FUNDAMENTAL GROUP AND PROJECTED EMBEDDINGS 6
The following theorem says in particular that the transverse fundamental group π ( f ) := π ( f, (cid:96) ) is well-defined up to an inner automorphism, and its monodromy map ϕ f := ϕ f,(cid:96),D : π ( f ) → S | deg f | is well-defined up to an inner automorphism of the target group. Theorem 2.8.
Let f : N → M be a generic smooth map between closed oriented con-nected n -manifolds. Then π ( f, b ) := π ( f, (cid:96) ) does not depend on the choice of (cid:96) . More-over, every path p joining b (cid:48) and b induces an isomorphism H p : π ( f, b ) → π ( f, b (cid:48) ) and apermutation h p,D,D (cid:48) ∈ S d such that h p,D,D (cid:48) ϕ f,(cid:96),D = ϕ f,(cid:96) (cid:48) ,D (cid:48) H p . The original motivation of this result was to distil some algebraic topology from thegeometric part of the proof of the main theorem of [15].
Proof.
The bijection h p,D,D (cid:48) is defined similarly to h j,D . The isomorphism H p is definedby assigning to any b -based loop j the b (cid:48) -based loop j (cid:48) defined as the product of thepaths (cid:96) p , j and ¯ (cid:96) p , where (cid:96) p is in turn the product of (cid:96) (cid:48) , p , ¯ (cid:96) and (cid:96) . (cid:3) Proposition 2.9.
The image of the monodromy map is transitive. In particular, theorder of π ( f ) is divisible by deg f . This follows easily from
Lemma 2.10. [15; §2, proof of Lemma 2]
Let f : N → M be a generic smooth mapbetween closed oriented connected n -manifolds, n ≥ . If x, y ∈ N are such that f ( x ) and f ( y ) are f -regular values, any f -transverse path joining f ( x ) and f ( y ) extends(with respect to a fixed inclusion I (cid:44) → S ) to an f -transverse loop l : S → M suchthat ( f ∗ l ) − ( x ) and ( f ∗ l ) − ( y ) are singletons and lie in the same connected componentof ( l ∗ f ) − ( S ) . . . 2-prems and string links Proposition 2.11. If f is a -prem, then the monodromy map ϕ f factors through theprojection T | deg( f ) | → S | deg( f ) | , where T k is the group of concordance classes of stringlinks of multiplicity k . By a string link of multiplicity k we mean an embedding g : ( Z /k ) × I (cid:44) → C × I sending ( Z /k ) × { j } ⊂ C × { j } to itself for j = 0 , . If moreover g sends each (( e πi/k ) n , j ) toitself for j = 0 , , then g is a pure string link . Thus the group C k of concordance classesof pure string links of multiplicity k is the kernel of the projection T k → S k . Proof.
Let p : M × R → M be the projection, and let us consider an f -transverse loop j : ( I, ∂I ) → ( M, b ) . Since f factors through an embedding of N into M × R , its pullback j ∗ f factors through an embedding of ( j ∗ f ) − ( I ) into the pullback ( j ∗ p ) − ( M × R ) = I × R . This embedding is a string link. Similarly a coherent homotopy gives rise to aconcordance. (cid:3) Proposition 2.11 is analogous to the “only if” implication in
RANSVERSE FUNDAMENTAL GROUP AND PROJECTED EMBEDDINGS 7
Theorem 2.12 (Hansen [10], [11]) . A d -fold covering f : N → M is a -prem if andonly if the monodromy π ( M ) → S d factors through the projection B d → S d , where B d denotes the braid group on d strands. The “if” implication can be proved as follows (compare [6; statement of Theorem 2]).Let D be an open disk in M ; then π ( M \ D ) is the free group (cid:104) x , y , . . . , x g , y g (cid:105) , where g is the genus of M , and the inclusion M \ D (cid:44) → M induces a homomorphism π ( M \ D ) → π ( M ) whose kernel is the normal closure of [ x , y ] . . . [ x g , y g ] . Let ϕ : π ( M ) → B d bethe given homomorphism, and let r : M \ D → W be a deformation retraction ontoa wedge of g copies of S . Then the braids ϕ ( x ) , ϕ ( y ) , . . . , ϕ ( x g ) , ϕ ( y g ) combine toyield the desired lift ¯ f : f − ( W ) → W × R of the restriction of f over W , and itspullback r ∗ ( ¯ f ) is the desired lift of the restriction of f over M \ D . Now over ∂D thelatter partial lift restricts to the braid ϕ ([ x , y ] . . . [ x g , y g ]) , which is trivial, since ϕ is ahomomorphism; hence the lift extends over D .The “only if” part of 2.12 along with Petersen’s results discussed in §1 have the follow-ing group-theoretic consequence: every homomorphism G → S d , where G is a finitelygenerated free abelian group, factors through B d . In particular, since B d → S d factorsthrough T d , we get Corollary 2.13.
Let f be a generic smooth map between compact connected oriented n -manifolds, n ≥ . If the monodromy ϕ f factors through a free product of finitelygenerated free abelian groups then it also factors through T | deg( f ) | . Another immediate thing to note about 2.11 and 2.12 is that every f : N → M factorsinto the composition of the embedding Γ f : N (cid:44) → N × M and the projection N × M → M .Hence Proposition 2.14. (a) The monodromy π ( M ) → S d of every d -fold covering N f −→ M between surfaces factors through the group B d ( N ) of braids in N × I .(b) The monodromy π ( f ) → S | deg( f ) | of every generic map N f −→ M between orientablesurfaces factors through the group T | deg( f ) | ( N ) of concordance classes of string links in N × I . Le Dimet showed that the natural map B d → T d is injective, i.e. concordant braidsare isotopic (cf. [12; p. 312]). Indeed, the Artin representation B d → Aut( F d ) is injectiveand agrees with the representation T d → Aut( F d /γ n ) , which is well-defined for each n bythe Stallings Theorem on the lower central series γ n (see [9; §1]). But (cid:84) ker[Aut( F d ) → Aut( F d /γ n )] = 1 since (cid:84) γ n = 1 in F d .On the other hand, B d and T d have a common quotient, the homotopy braid group HB d (see [7], where the difference between B d and HB d is explained). Indeed, everystring link is link homotopic to a braid (see [8]) and concordance implies link homotopyby a well-known result of Giffen and Goldsmith. The latter also follows from the injec-tivity of HB d → Aut( F d /µ ) [8], where µ is the product of the commutator subgroupsof the normal closures of the generators of F d , which contains γ d +1 . RANSVERSE FUNDAMENTAL GROUP AND PROJECTED EMBEDDINGS 8
Similarly to Artin’s combing P d (cid:39) ( . . . ( F (cid:110) F ) (cid:110) . . . ) (cid:110) F d − of the pure braid group,the kernel HP d of the projection HB d → S d admits the combing HP d (cid:39) ( . . . ( F /µ (cid:110) F /µ ) (cid:110) . . . ) (cid:110) F d − /µ [7], [8]. Hence HP d is torsion-free. Using this, Humphries provedthat HB d is torsion-free for d < ; in fact he showed that α ∈ HB d has infinite order ifits image in S d has order divisible by , , or [13]. Corollary 2.15.
Let f be a generic smooth map between compact connected oriented n -manifolds, n ≥ . If π ( f ) contains an element α of finite order whose monodromy ϕ f ( α ) ∈ S | deg( f ) | is of order divisible by , or , then f is not a -prem. Note that for the hypothesis to hold, the group HT | deg( f ) | ( N ) of link homotopy classesof string links in N × I must contain torsion, by Proposition 2.14(b).Taking into account the Yamamoto–Akhmetiev Theorem [15], we have Corollary 2.16. If f is a generic smooth map from S to a closed orientable surfacethen π ( f ) contains no torsion with monodromy of order divisible by , or . An interesting question is whether already some lower central series quotient of HB d istorsion free. For instance, the abelianization of HB is not: the braid [ σ , σ ] projectsnontrivially to S , but it is easy to check that the three strands of the pure braid [ σ , σ ] are pairwise unlinked. This means, in particular, that it would not be a good idea tosimplify the definition of ( b, f ) -coherent homotopy into “link map bordism”, i.e. to allowthe positive components of the preimage to change by pairwise disjoint bordisms. Remark 2.17.
It was shown by Habegger and Lin [9; §1] that(i) the image of the group C d of concordance classes of pure string links in Aut( F d /γ n ) is the subgroup Aut ( F d /γ n ) that depends on the chosen set { x , . . . , x n } of freegenerators of F d and consists of those automorphisms that send the coset ¯ x i ofeach x i to a conjugate of ¯ x i and fix the product ¯ x · · · ¯ x d ;(ii) Aut ( F d /γ ) = 1 , and each Aut ( F d /γ n +1 ) is a central extension of Aut ( F d /γ n ) by a free abelian group, which they denote K n − ; in particular, for n > , Aut ( F d /γ n ) is torsion-free and nilpotent of class n − .It follows easily from these that(i (cid:48) ) the image of T d in Aut( F d /γ n ) is the subgroup Aut ( F d /γ n ) consisting of thoseautomorphisms that send the coset ¯ x i of each x i to a conjugate of some ¯ x j andfix the product ¯ x · · · ¯ x d ;(ii (cid:48) ) Aut ( F d /γ ) (cid:39) S d , and for n > , each Aut ( F d /γ n +1 ) is a central extension of Aut ( F d /γ n ) by the same free abelian group K n − .Thus the homomorphism T d → S d factors through the limit of the inverse sequence · · · → Aut ( F d /γ ) → Aut ( F d /γ ) (cid:39) S d . If some term of this inverse sequence or theinverse limit is torsion-free (for each d ) — or if HB d is torsion-free for all d — then therestriction on the order of the monodromy is superfluous in Corollary 2.15. RANSVERSE FUNDAMENTAL GROUP AND PROJECTED EMBEDDINGS 9 Some computations of π ( f ) . . A certain fold map S n → S n of degree d Let f be the degree d map f : S n → S n defined by picking d + 1 disjoint n -disks in S n and sending each of them homeomorphically to its own exterior in S n . Let b be a pointwith | f − ( b ) | = d (i.e. a point in the interior of one of the disks).It is easy to see that f lifts to an embedding f × g : S n (cid:44) → S n × R . Namely, fixan embedding of pt ∗ [ d + 1] (the cone over [ d + 1] = { , . . . , d } ) into R , and let g : S n → pt ∗ [ d + 1] ⊂ R send the interior of the i th n -disks into pt ∗ { i } \ pt ∗ ∅ , andthe exterior of the disks into pt ∗ ∅ . Example 3.1 (the case d = 2 ) . Let α ∈ π ( f, b ) be the class of a loop intersectingeach disk along its diameter (compare Example 6 in [15]). Then α is nontrivial sinceit is easily seen to have a nontrivial monodromy ϕ f ( α ) ∈ S . On the other hand, byProposition 2.11 the monodromy map ϕ f lifts to ˆ ϕ f : π ( f, b ) → T . Since T → S factors through HB , we conclude that the image of ˆ ϕ f ( α ) in HB (cid:39) Z is nontrivial.Hence α has infinite order, and the composition Z (cid:39) (cid:104) α (cid:105) ⊂ π ( f, b ) → HB (cid:39) Z is anisomorphism. Thus π ( f ) contains a direct summand isomorphic to Z . Example 3.2 (the case d = 2 , n > ) . We will now show that if n > (and still d = 2 ),then π ( f ) is isomorphic to Z .Let A , B and C denote the disks, with b ∈ B . A loop representing an elementof π ( f, b ) gives rise to a word in the alphabet { A, B, C } (starting and ending with theletter B ), which encodes the sequence of disks intersected by the loop. If two loopsgive rise to the same word, then (using that n ≥ ) they represent the same element of π ( f, b ) . Furthermore, it is easy to see that XX = X and XY X = X in π ( f, b ) for any X, Y ∈ {
A, B, C } .Let F be the free monoid (=semi-group with ) on the alphabet A, B, C (where theproduct of words is given by concatenation), and let
BF B be the submonoid of F consisting of all words of the form BwB , where w ∈ F . Let G be the quotient of BF B by the relations
XY X = X = XX , where X, Y ∈ {
A, B, C } . Then G is agroup with unit B and with the inverse given by BX . . . X n B (cid:55)→ BX n . . . X B . Indeed, BX n . . . X BBX . . . X n B = BX n . . . X BX . . . X n B = BX n . . . X . . . X n B = · · · = B .In fact, G is nothing but the group of simplicial loops in the triangle ∂ ∆ (withvertices A , B , C ) under the relation of simplicial homotopy. Thus G (cid:39) Z , with n ∈ Z corresponding to the class of B ( ABC ) n B (where ( ABC ) − = CBA ). By construction,we have an epimorphism G → π ( f, b ) sending a generator onto α . Hence π ( f ) (cid:39) Z . Example 3.3 ( n = 2 , d = 2 ) . In the case n = 2 , d = 2 , π ( f ) is larger than Z , forsimilarly to Example 3.6 below it can be shown that there are loops giving rise to thewords B , BAB and
BCB yet representing elements of infinite order in π ( f ) / Z . The author is grateful to P. M. Akhmetiev for pointing out these relations.
RANSVERSE FUNDAMENTAL GROUP AND PROJECTED EMBEDDINGS 10
Example 3.4 ( n > , d > ) . In the case d > , n > , the same considerations asin 3.2 show that π ( f ) is a quotient of π ((∆ d +1 ) (1) ) (that is, of the free group on d ( d − letters), and admits an epimorphism onto the homotopy braid group HB d . . . Fold maps of geometric degree with embedded spherical folds Let f : S n → S n be a generic fold map that embeds its fold surface Σ f := { x ∈ S n | ker df x (cid:54) = 0 } and is such that | f − ( b ) | = 1 ; in particular, f has degree ± . ( Fold map means that every point of Σ f is a fold point, rather than a point of a higher singularitytype.) In the case n > , let us additionally assume that each component S i of Σ f is asphere.The following notation will be used. If S i is a component of Σ f , let B i be the n -ballbounded by S i in S n \ f − ( b ) . On the other hand, let D i be the n -ball bounded by f ( S i ) in S n \ b . Each f ( B i ) is connected, hence coincides with some D ρ ( i ) , таким что D i ⊂ D ρ ( i ) . Note that ρρ = ρ ; in other words if j is of the form ρ ( i ) , then f ( B j ) = D j .In particular, in this case S j is outer , i.e. f sends a neighborhood of S j in B j into D j .Accordingly, we call an S i inner if f sends such a neighborhood into the closure of S \ D i . Example 3.5 (a two-dimensional example) . There exists a generic fold map f : S → S such that f embeds Σ f and | f − ( b ) | = 1 , yet π ( f, b ) is non-trivial.Namely, f is the unique (up to reparametrization) map such that Σ f is the union offour curves S , S , S = S ρ (1) and S = S ρ (2) such that D ∩ D = ∅ and D ∪ D ⊂ D ⊂ D .Let l : ( S , pt ) → ( S , b ) be a loop intersecting D by a diameter that separates D from D within D , and intersecting D by some diameter (which contains the formerdiameter). Then the pullback ( l ∗ f ) − ( S ) consists of three components P , P and P ,with P containing ( l ∗ f ) − ( pt ) and with ( f ∗ l )( P ) and ( f ∗ l )( P ) contained respectivelyin B and B . By a homotopy h from l to an ˜ l with values in S \ ( D ∪ D ) one cannoteliminate either P or P ; that is, neither P nor P bounds a disk in ( h ∗ f ) − ( S × I ) .On the other hand, any such homotopy h with values not only in S \ ( D ∪ D ) joinseither P or P to P ; that is, either P or P (or both) belongs to the component Q of ( h ∗ f ) − ( S × I ) containing P . Hence Q has at least two boundary components in ( l ∗ f ) − ( S ) , and so h is not coherent. Thus one cannot eliminate either P or P by acoherent homotopy. So there exists no coherent null-homotopy of l .Similar considerations show that no power of l is trivial in π ( f, b ) . Example 3.6 (a mild generalization) . Let us generalize Example 3.5 to show that if(under the hypothesis of § . ) there exists a pair ( i, j ) such that D i ⊂ f ( B j ) and D j ⊂ f ( B i ) (for brevity, we shall call such a pair ( i, j ) linked ) and additionally B ∩ B = ∅ ,then π ( f, b ) is nontrivial as long as n = 2 .Indeed, up to renumbering we may assume that ( i, j ) = (1 , . Since (1 , is linked, ρ (1) (cid:54) = 1 . Then, in particular, S ρ (1) is outer. Hence if S is also outer, then there existsan i such that B ⊃ B i ⊃ B ρ (1) and S i is inner. Then ρ ( i ) = ρ (1) and ( i, is a linkedpair. Thus without loss of generality we may assume that S is inner; similarly for S . RANSVERSE FUNDAMENTAL GROUP AND PROJECTED EMBEDDINGS 11
The construction of Example 3.5 will apply here once we show that there is a loop l : ( S , pt ) → ( S \ ( D ∪ D ) , b ) such that the component L x of L := ( l ∗ f ) − ( S ) thatcontains ( l ∗ f ) − ( pt ) represents, via f ∗ l , a nontrivial element of H ( S \ ( B ∪ B )) . Let B be the union of all disks bounded by the collection of circles f − ( f ( S ∪ S )) in thecomplement to x := f − ( b ) . Let S +1 be a pushoff of S into the complement of B , and let y be a point of S +1 . Since S is inner, y / ∈ B ; and since S \ B is connected, x and y canbe joined by a path p in S \ B . By Lemma 2.10 there exists a loop l : ( S , pt ) → ( S , b ) such that x := ( l ∗ f ) − ( pt ) = ( f ∗ l ) − ( x ) and y := ( f ∗ l ) − ( y ) are singletons and lie inthe same component L x of L := ( l ∗ f ) − ( S ) . Moreover, by the proof of Lemma 2.10(found in [15]) we may assume that l has values in a neighborhood of f ( p ( I )) , hencein S \ ( D ∪ D ) . Then f ∗ l sends L x into S \ ( B ∪ B ) . Now amend l by cuttingit open at f ( y ) and inserting a loop circling around f ( S +1 ) . Then the resulting loop l : ( S , pt ) → ( S \ ( D ∪ D ) , b ) is such that the component L x of L := ( l ∗ f ) − ( S ) that contains x := ( l ∗ f ) − ( pt ) differs from L x by a loop circling around S . Thus f ∗ l | L x and f ∗ l | L x represent distinct elements of H ( S \ ( B ∪ B )) ; so at least one ofthem is non-trivial. Example 3.7 (a higher-dimensional proposition) . We shall show that the phenomenonexhibited in Example 3.5 does not occur in higher dimensions; more specifically, that(under the hypothesis of § . ) for n > , a b -based f -transverse loop crossing each S i atmost twice represents the trivial element of π ( f, b ) .Indeed, let l : ( S , pt ) → ( S n , b ) be such a loop. Since n > , it may be assumedto be embedded. (This assumption will not be essentially used, but allows to simplifynotation.) Since l is f -transverse, it is transverse to the codimension one submanifold f (Σ f ) . Up to a renumbering of S i ’s, we may assume that D meets l ( S ) and is innermostamong all the balls D i in S n \ b that meet l ( S ) . Write A = l ( S ) ∩ D ; by our hypothesis,it is an arc. Then ∂A bounds an arc A (cid:48) in f ( S ) so that A (cid:48) meets l ( S ) only in ∂A (cid:48) = ∂A .The circle A ∪ A (cid:48) bounds a -disk D in D , meeting ∂D only in A (cid:48) . Without loss ofgenerality D meets l ( S ) only in A . Since n > , we may assume that D is disjoint fromall D j that lie in the interior of D . Let us homotop l across D , from A to A (cid:48) , to a loop l such that l ( S ) ∩ D has fewer components than l ( S ) ∩ D . Proceeding inductively,we obtain a pointed homotopy h t , t ∈ [0 , . . . , N ] , from l := l , through loops l , l , . . . ,to a loop l N disjoint from every D i . Since n > , the latter is pointed null-homotopicwith values in the complement to all D i ’s and so represents the trivial element of π ( f ) .Let us show that the constructed null-homotopy of l is coherent. Let L i = ( l ∗ i f ) − ( S ) .If S i +1 is outer, then L i +1 is obtained from L i by removing one component, disjoint from ( l ∗ i f ) − ( pt ) . Else (i.e. if S i +1 is inner) L i +1 is obtained from L i by splitting one of thecomponents into two. (Here we are using our hypothesis, implying by induction that l i crosses S i +1 just twice.) The component of L i being split may contain ( l ∗ i f ) − ( pt ) , inwhich case of the two resulting components P , Q of L i +1 one (say, P ) would contain ( l ∗ i +1 f ) − ( pt ) . Then for the constructed null-homotopy to be coherent, the other compo-nent Q has to be glued up by a disk in ( h ∗ [ i +1 ,N ] f ) − ( S × I ) , where h [ i,j ] : S × I → N RANSVERSE FUNDAMENTAL GROUP AND PROJECTED EMBEDDINGS 12 denotes the interval from l i to l j in the constructed null-homotopy h t . Indeed, by con-struction, Q will be glued up by a disk already in ( h ∗ [ i +1 ,ρ ( i )] f ) − ( S × I ) . Example 3.8 (a two-dimensional proposition) . Let us show that the proposition inExample 3.7 remains valid in dimension two under the additional hypothesis that thereare no linked pairs ( i, j ) .Indeed, let us examine the argument of 3.7. It contains only two essential applicationsof the condition n > : to conclude that D is disjoint from all D j ’s that lie in the interiorof D and to coherently null-homotop l N . Now if n = 2 and D meets some D j , thenit follows from our assumption of (1 , j ) being unlinked that D contains f ( B j ) . (Forif it doesn’t contain, then taking into account our assumption that D is innermostamong all D i ’s meeting l ( S ) , the only possibility is that f ( B j ) contains D . However D j ⊂ D ⊂ f ( B ) , so (1 , j ) is linked.) Thus D contains each D j together with its f ( B j ) ,which implies that f − ( D ) is homeomorphic to a disjoint union of disks, each boundedby a component of f − ( ∂D ) . Thus the homotopy of l along D will be coherent. Similarlythe null-homotopy of l N will be coherent. Fig. 1.
A map f : S n → S n of geometric degree with nontrivial π ( f ) . Example 3.9 (a higher-dimensional example) . In fact, π ( f, b ) need not be trivial for n > (under the hypothesis of § . ) as shown by the map depicted in Figure 1. Thesix thick colored (or grayscale, depending on the reader’s medium) ellipses depict thefolds; the thick black curve is the loop l in question; and the thin curves illustrate thepullback l ∗ f . The arrows mark those of the two components of ( l ∗ f ) − ( S ) that passesthrough ( l ∗ f ) − ( pt ) . RANSVERSE FUNDAMENTAL GROUP AND PROJECTED EMBEDDINGS 13
The curve l ( S ) meets each of the three n -balls D i bounded by the images of the threeinner spheres of folds S i , i = 1 , , , in two arcs J i , J (cid:48) i . A key feature of this picture isthat the marked component has a fold over one endpoint of each of the six arcs, andthe other (unmarked) component has folds over their opposite endpoints. Thus if any ofthe six arcs is eliminated as in Example 3.7, this would result in the marked componentbeing joined to the unmarked one, whence the eliminating homotopy would fail to becoherent (cf. Example 3.5).Moreover, no preliminary tampering with the six arcs by a coherent homotopy of l is going to help. Indeed, if H : D → S n is a coherent null-homotopy of l , then thepullback H ∗ f of f has a fold curve over each H − ( S i ) . Each H − ( D i ) is a codimensionzero submanifold in D , whose boundary contains the two arcs l − ( J i ) and l − ( J (cid:48) i ) . If j (cid:54) = i , then these arcs alternate with l − ( J j ) and l − ( J (cid:48) j ) with respect to the cyclic orderon S , whereas H − ( D i ) and H − ( D j ) are disjoint. Hence either l − ( J i ) and l − ( J (cid:48) i ) are contained in different components of H − ( D i ) , or l − ( J j ) and l − ( J (cid:48) j ) are containedin different components of H − ( D j ) (or both assertions hold). By symmetry, we mayassume the former. Since the component of H − ( D i ) containing l − ( J i ) does not contain l − ( J (cid:48) i ) , its boundary component containing the arc l − ( J i ) otherwise contains only pointsof H − ( S i ) , which therefore must constitute an arc. Thus the two endpoints of the arc l − ( J i ) belong to the same component of H − ( S i ) . Hence the fold curve of H ∗ f overthis component constitutes a path in ( H ∗ f ) − ( D ) starting on the marked componentof ( l ∗ f ) − ( S ) and ending on the other (unmarked) component. Therefore the latter twocomponents are joined into one in the null-homotopy, which is therefore non-coherent.Thus l is not coherently null-homotopic. References [1] P. M. Akhmet’ev,
On an isotopic and a discrete realization of mappings of an n -dimensionalsphere in Euclidean space , Mat. Sb. (1996), no. 7, 3–34; Mathnet; English transl., Sb. Math. (1996), 951–980. ↑ Projected embeddings and near-projected embeddings .preprint. ↑ Obstructions to approximating maps of n -manifolds into R n by embeddings , Topology Appl. (2002), 3–14. Journal. ↑ A geometric approach to homology theory ,London Math. Soc. Lecture Note Series, vol. 18, Cambridge Univ. Press, 1976. ↑ Embedding solenoids , Fund. Math. (2004), 111–124. Clark’s home-page. ↑ Embedding finite covering spaces into bundles , Topology Proc. (1979), 361–370. Journal. ↑ . [7] D. L. Goldsmith, Homotopy of braids — in answer to a question of E. Artin , Topology Conference(Virginia Polytech. Inst. and State Univ., Blacksburg, Va. 1973), Lecture Notes in Math. vol. 375,Springer, 1974, pp. 91–96. ↑ . , . [8] N. Habegger and Xiao-Song Lin, The classification of links up to link-homotopy , J. Amer. Math.Soc. (1990), 389–419. Lin’s homepage. ↑ . , . , . RANSVERSE FUNDAMENTAL GROUP AND PROJECTED EMBEDDINGS 14 [9] ,
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