Knotted n-spheres K = f(S n) c Sn+2 with n ~ 2 make what seems to be their first appearance in a famous paper by E. Artin published in 1925, where he describes a construction which produces examples of non-trivial n-knots for arbitrary n ~ 2. (Detailed reference data are provided at the end of the survey). In today's terminology, introduced by E.C. Zeeman (1959), the construction is called spinning and it goes as follows.

§ 2. Some historical landmarks. D = { ~~in+=4I xi 2 = I, Xn+ 3 ~ O, Xn+ 4 : 0 ) in Sn+3 C R n+4 is an (n+2)dimensional disk which we identify v,ith D n+2. Thus, B C D . Now, the sphere Sn+3 can be obtained by rotating this disk D in Rn+4 around the (n+2)-plane P = {Xn+3 = 0, Xn~ 4 = O) Note that P contains the unknotted boundary sphere bD = Sn+~ c Sn+3 which thus remains point_ wise fixed during the rotation. In the process, the set B c D v.ill sweep out a smooth (n+~)-dlmensional sDhere embedded in Sn+3. This is the spun knot Zz c Sn+3 of the knot K c Sn+2 E.Artin observed in his paper that

~(s n+3 - z K) ~ ~(s n+2 - K) .

Thus, EK C Sn+] is certainly knotted if w~(S n+2 - K) ~ Z .

The objective of multidimensional knot theory is, as for classical knots, to perform classification, ultimately (and ideally) with respect to isotopy, and meanwhile with respect to weaker eauivalence relations. There is however with higher dimensional knots the additional difficulty that the construction of a knot cannot merely be described by the simple-minded drawing up of a knot projection. Thus, efforts at classification (i.e. finding invariants) now have to be complemented by construction methods (i.e. showing that the invariants are realizable). This is why Artin's paper is so significant. It gives the first construction showing that the groups of classical knots are all realizable as fundamental groups of the complement of n-knots for arbitrary n.

After Artin's paper, multidimendional knot theory went into a long sleep. Strangely enough, the theory awoke subseauently to PapaPs proof of the sphere theorem. One of the consequences of this famous result is that classical knots have aspherical complements, i.e. : wi(S3 - f(S3)) = 0 for i > ~. Hence a natural euestion : What about multidimensional knots ? The answer came Quickly : In ~959, J.J. Andrews and M. L.Curtis showed that the complement of the spun trefoil has a non-vanlshing second homotopy group. In fact their result is more general and also better : there is an embedded 2-sphere which represents a non-zero element.

One thus began to suspect that multidimensional knots would behave quite differently from the classical ones. The major breakthrough came from the development of surgery techniques which made it possible to get a general method of constructing knots with prescribed properties of their complements. In a perhaps subtler way, surgery techniques were also decisive in classification problems. See our chapters III and IV.

Here is an illustration of the power of surgery techniques. A common feature to the examples (all based on spinning) kno~ in ~960 was that w 2 was ~ 0 because w! g Z . It was thus natural to ask : Can one produce an n-knot with 72 J 0 but wl = ~ ? Clearly such an example cannot be obtained by spinning a classical knot. Hovewer, J. Stallings (see M. Kervaire's paper (1963), p. ~J5) and C.T.C. Wall (see his book : Surgery on compact manifolds, p.18 ) proved in 4963 that for all n ~ 3, there exist many knots K c Sn+2 with w~(sn+2-K~Z but w2(S n+2 - K) J 0 . The construction is an easy exercise in surgery. called twist-spinning. We shall talk about it in chap. V § 4.

Hovewer, one cannot expect Kinoshita's nor Fox's level curve methods to be applicable in higher dimensions because they still rely on drawings and intuitive descriptions in the next lower-dimensional 3- spac e.

As a conclusion, let us make a few remarks : i) The use of surgery techniques showed that multidimensional knot theory could do well without direct appeal to 3-dlmensional goemetric intuition nor immediate computability. There resulted a useful kickback for classical knot theory which benefited much, since 1965, from the use of geometrical tools borrowed from higher dimensional topology and from a partial rellnquisment of computational methods. 2) Around 1964, it became generally accepted that the theory of imbeddings in codimensions ~ 3 was well understood. Piecewise linear imbeddings Sn Sn+q with q ~ 3 are all unknotted by a theorem of E.C. Zeeman of 1962, published in Unknotting combinatorial balls, Ann. of Math. 78 (1963)P.501-526. The differentiable theory was in good shape with the works, both in 1964, of J. Levine, "A classification of Differentiable Knots", Ann. of Math. 82 (1965), 15-50 on the one hand, and A. Haefliger, "Differentiable Embeddings of Sn in Sn+q for q > 2", Ann. of Math. 83(1966)p.402-436 on the other.

These impressive pieces of work provided a decisive encouragement to take up the certainly less tractable codimension 2 case. A lot of effort went into it and since then the growth of the subject has been so important that we cannot follow a chronological presentation. We have chosen instead to talk about articles published after 1964 in the chapters corresponding to their subject as listed in the table of contents below. Of course, at some points, whenever convenient, we did go back again to papers which appeared before this date.

For the same reason we had to delete from this survey the mention of many beautiful papers. In particular, we have mostly disregarded the papers centering around a discussion of the equivalence (or nonequivalence) of various possible definitions. We have rather tried to emphasize the moving aspect of the subject. § 2 . Some definitions and notations.

Usually an n-knot is a codimension 2 submanifold K in Sn+2. Most of the time Sn+2 will be the standard (n+2)-dimenslonal smooth sphere. However, in some cases, one is forced to relax this condition. (For instance, when n+2 = 4, in order to get the realization theorems for w I. (See Chap. II, § 3).

What K should be is a little harder to make definite. For us, it will be a locally flat, oriented, PL-submanifold of Sn+2, PL-homeomorphic to the standard n-sphere or a differential submanifold homeomorphic (or diffeomorphic) to the standard n-sphere.

The reason for such hesitations can easily be explained. The proof of the algebraic properties of the various knot invariants usually does not require a very restrictive definition of a knot. In some cases, Sn+2 could as well be replaced by a homotopy sphere and K by a homology sphere, or even less (see chap. V, § 5), sometimes not even locally flat.

On the other hand, to be able to perform geometrical constructions we usually need more restrictions. For instance the proof of the existence of a Seifert surface requires local flatness in order to get a normal bundle (which will be trivial).

Moreover, when one wants to prove realization theorems for the algebraic invariants, the str~nger the restrictions on the knot definition, the better the theorems.

So we decided to let a little haze about the definition of a knot, leaving to the reader the task to get to the original papers whenever needed and see what is really required (or used).

Therefore, in this spirit, ~) We have often written up in some detail elementary arguments which are well known to people working in the field, but perhaps not so easy to find in the literature.

2) We did not attempt to talk about everything in the subject, but rather ~ried to emphasize what seems to be its most exciting aspects. 3) The latest news is often not here. Other parts of this book should fill this gap and provide references.

On the other hand, we have assumed that the reader knows some algebraic and geometric topology, and even sometimes that he is moderately familiar with classical knot theory.

Chap. I :

OF C O N T E N T S CHAPTER

§ I : Completeness theorems

However, the question to decide Just how close the complement comes to be a complete invariant of the knot does not seem to have occured (for higher knots) before the paper of H. Gluck in 1962.

Actually it is technically advantageous to replace the complement X ° = Sn+2 - K by the so-called exterior , that is the complement X = Sn+2 - N of an open tubular neighborhood N of K. Observe that K has trivial normal bundle v so that N is diffeomorphic to Sn × D 2 and a trivialization of v will give an identification N ~ Sn × D 2. Observe also that X o is diffeomorphic to the interior of the compact manifold X and that bX = bN ~ Sn × S i. Thus X determines X o . The converse is true at least for n > 3 ? Since this point seems left in the dark in the printed literature, the following explanations may perhaps he helpfu Suppose X and X' are knot exteriors and let F o : X o ~ X o ' be a diffeomorphism. Take a neighborhood U of bX of the form U ~ bX x[0,1], i.e. a collar. Look at the submanifold M = X' - Fo(X-U ) .

U

X - U X

F o -~

If U has been taken narrow enough, M is contained in a collar around bX' and it is easy to construct continuous retractions of M onto each of its two boundary components M ° = bX' and M I = bFo(X-U ). Thus M is an h-cobordlsm between M ° and M I. Now, M ° and M I are both dlffeomorphic to Sn x S I, and wIM o = E . A basic theorem of differential topology, the s-cobordism theorem, now states that under these conditions tlons and if dim M ~ 6, then the diffeomorphism F ° : b(X-U) ~ M 1 can be extended to a diffeomorphism F : X > X' (For a proof of the s-cobordism theorem see M. Kervaire, Comm. Math. Helv. 40 (1965),3142. Here one also needs the fact that the Whitehead group of the infinite cyclic group is trivial. For this fact, see H. Bass, A. Heller and R.G. Swan, Publications math@matiques, I.H.E.S. No 22. In the case n = 3, the s-cobordims theorem does not apply and one needs Theorem 16.1 in C.T.C. Wall's book, Surgery on compact manifolds, p. 232).

Sketch of proof. The group~(S n × S I) projects onto the group of concordance classes of homeomorphisms~(S n × S i) and it turns out that the extendability question for the above f : Sn × S I ~ Sn × S I depends only on its image in ~S n × Si).

The method of S. Cappe~ and J. Shaneson is general and should yield examples of non-equivalent knots with diffeomorphic complements for all n ~ 3. It stumbles for n ~ 6 on the following purely algebraic open problem : does there exist for all n an automorphism A of Z n+] without any real-negative eigenvalue and with determinant + I such that for all exterior powers kiA, i = l,...,n , the endomorphism kiA - I : kiz n+l > kiz n+l is again an automorphism ?

Such an A can be concocted fairly easily for n = 3,4 and if one finds other values of n for which A exists with the required properties, it can be fed into the machinery of S. Cappell and J. Shameson to produce new examples of inequivalent n-knots with diffeomorphic complements.

§ 2 . Unknotting theorems

There is one case where one would certainly lime the complement X O = Sn+2 - K to determine the knot. That is the case where X O has the homotopy type of S I, i.e. the homotopy type of the complement of the trivial, unknotted imbedding K O = Sn C Sn+2 . Is it then true that K is isotopic to K ° 9.

For n = 2, this problem is still unsolved today, as far as we know.

For n > 3, it was solved by J. Stallings in 1962 for topological knots. If K c Sn+2 is a locally flat, topologically imbedded n-sphere with n > 3 and if Sn+2 - K has the homotopy type of S I, then there exists a homeomorphism h : Sn+2 e Sn+2 such that hK = K o.

From the point of view of differential topology however, the major problem is whether a smooth knot K c Sn+2 with Sn+2 - K ~ S 1 is smoothly unknotted, i.e. whether there exists a diffeomorphism h : Sn+2 Sn+2 such that hK = K O

J. Levine's paper proving this and a little more in 1964 certainly played a decisive role in getting multidimensional knot theory off the ground.

His precise result is ~follows.

L~VI~E!:S~~I~TX~)~EM: - Let K C Sn+2 be a smooth n-knotwith n > 4 and let X o = Sn+2 K. Suppose that wi(Xo) ~ wi(S I) for i < ½(n+1) Then there is a dlffeomorphlsm h of Sn+2 onto itself such that hK is the standard n-sphere Sn in Sn+2

Remark. The reader has perhaps noticed that we have slided from the homeomorphism type of the complement to its homotopy type, in the beginning of this paragraph. The invariants we are going to talk about in the next paragraph are invarlants of the homotopy type of the complement. So, the question arises whether the homotopy type determines the topology of the complement. There are several results in this direction. See S. Cappell(1969) for a discussion. Here are some striking results :

Let us treat the exterior as a pair (X, bX). Then, the homotopy type of (X, bX) determines the homeomorphism type :

i) For classical knots. This is a beautiful result due to F. Waldhausen : "On irrreductible 3-manifolds which are sufficiently large" Annals of Math. 87 (1968) p. 56-88.

Wl(X ) was easy to understand once surgery techniques were
available to perform the necessary knot constructions. (See M. Kervalre
(1963))~ The fundamental group w of the complement of an n-knot, n > 3,
is characterized by the following properties :
(1) w is finitely presented.
(

Surgery techniques (for instance ) enable one to construct an (n+2)
dimensional oriented manifold M with wl(M ) ~ w , and HI(M ) = 0 for
i ~ O, i, n+l, n+2 . (For this the properties (1) and (

Then one takes an imbedding ? : S I x D n+l , M representing am element ~ ~ w whose conjugates generate w. One constructs a new manifold E by removing from M the interior of the image o(S 1 x Dn~l), say X = M - int ~(S 1 x Dn+1), and replacing it by D 2 x Sn. Since D 2 x Sn and S I × D n+1 have the same boundary S I × Sn, it follows that D 2 × Sn can be glued to X along S ~ × Sn by the map m . The resulting manifold Z = X U m(D 2 x Sn) has the homotopy type of Sn+2, and for n ~ 3 is therefore homeomorphic to Sn+2 by the theorems of S. Smale (Annals of Math. 74(1961)p.391-406). Actually, with some patching up one can even assume that Z is diffeomorphlc to Sn+2. By construction E contains a beautifully imbedded n-sphere, namely the core K = (0} × Sn in the subspace N = D 2 × Sn C Z . The subspace X = E - N is just the exterior of the obtained n-knot K c Z and Wl(X) ~Wl(M) ~w .

The construction of Z from M is one of the simplest examples of surgery.

For a discussion of the case n = 2, see M. Kervaire (1963) as well as J. Levine's article : "Some results on higher- dimensional knot groups" in this volume.

These references also contain some analysis of the above algebraic
conditions (i), (

As to the higher wi, i > I, we have already mentionned in the introduction the papers of J.J. Andrews and M.L. Curtis ($959) and D.B.A. Epstein (1959).

More recently the subject has been taken up again. See E. Dyer and A. Vasquez (1972) and B. Eckmann (1975). Their result is that for n > i, the space X o = Sn+2 - K is never aspherical unless the knot is trivial.

Nevertheless, a complete understanding of the higher homotopy groups of knot complements seems out of reach today.

The most gratifying invariants at present are the homology modules of coverings of X and in particular those of the maximal abelian cover X corresponding to the kernel of the surjection Wl(X) .2 HI(X ).

These are simple enough to be tractable and yet non-trivial enough to provide a beautiful theory.

Observe that ~I(X) is Just the commutator subgroup G = [~,~] of the knot group ~ = w1(X ). Therefore HI(X ) is G/G' viewed as a group with operators from HI(X ) vla the extension I ) G/G'--~ ~/G' ~ HI(X)~-~ I. Thus AI(K ) is determined by the knot group ~ .

In the classical case, AI(K ) is the only (non-zero) Alexander module. It possesses a square presentation matrix (over A ) whose determinant is the familiar Alexander polynomial.

The fundamental group w = wl(X ) influences A2(K ) also .

Since G = w~(X), there is an exact sequence , X 2 ( ~ ) , H2(G ) , 0 by a celebrated theorem of H. Hopf (Fundamentalgruppe und zweite Betti' sche Gruppe, Comm. Math. Helv. 14 (1941), 257-309) and thus A2(K ) must surject onto H2(G ).

It may then perhaps be more appropriate to ask : what set {~, A 1, A 2 . . . . . An} with A I = HI(G ) and surJection A2---, H2(G), G = [z,~], is realizable with ~ the knot group and A q the knot modules for q = i, ..., n ?

A start on this question with ~ infinite cyclic was made by M. Kervaire (1964). The formulation (in terms of the homotopy modules of the knot complement) was however very ackward. The decisive breakthrough was accomplished by J. Levine (1974) which we now follow.

Let X again be the exterior of a knot K c Sn+2. Assume X is triangulated as a finite complex and let X be the infinite cyclic covering of X with the natural triangulation (such that X , X is a simplicial map). We denote by C the multiplicative infinite cyolic group with generator t . C operates on X without fixed point and the chain groups Cq(X) are finitely generated free ZC-modules.

Since A has no divisors of zero, the multiplication by l-t induces an injection l-t : C.(X) , C.(X). The quotient module is (canonically) isomorphic to the chain group of X (regarded as A-module with trivial action) and we get an exact sequence of complexes : o , c . ( ~ ) 1 - t c . ( ~ ) , c . ( x )

, o .

Passing to the associated long homology sequence gives first where p runs over all rational primes. Now, Cg(~p) has been evalua ted above since F P is a perfect field, and it can be calculated explicitly. Wequote Corollary 2.9. of N. Stoltzfus' paper (1976) : c~(~p) = ez/2z e w~(rp),

CHAP. V : FIBERED KNOTS § I : General properties

In this chapter we study the very important special case of fibered knots. At least two reasons make this special case worth of study :

I) Knots which appear as local singularities of complex hypersurfaces are fibered knots.

2) The geometry of the complement of fibered knots can be made quite explicit and thus many knot invariants get a very nice geometrical interpretation.

Let us start with the definition. Racall that HI(X;Z)~ Z and let t be a chosen generator. One says that K is a fibered knot if one is given a representative p : X >S1 for t, which is a locally trivial (differentiable) fibration.

Remark : It is often nice to add the further restriction that plbX~ S I (which is, by hypothesis, a fibration) should be the projection onto the fiber associated with a trivialization of the sphere normal bundle to K in Sn+2. A useful remark due to S. Cappell shows that whenever n / 2,3, one can always change p such that this further requirement is satisfied. See Cappell (1969). In the sequel, we shall usually make this assumption.

The fiber of p is a codimension one submanifold W of Sn+2. It is connected because p represents a generator of HI(X;Z). (To see that, consider the end of the homotopy exact sequence of the libration : ~I(X) > ~ ( s I) , %(F) , o .)

If we add to W a collar inside the normal bundle to K in Sn+2, we get a Seifert surface V for K.

Looking at things a bit differently, we see that we can think of X as being obtained from W x [0, i] by W x{0} and W x{1} identified together via a homeomorphism h : W ~ W. More precisely, X is the quotient of W x [0,1] by the equivalence relation (x,0) N (h(x),~).

h is called "the" monodromy of the fibration, p being given, h is well defined up to isotopy. If we insist that p satisfies the restriction condition on bX, we shall get a monodromy map which is the identity on bW. § 2. The infinite cyclic covering of a fibered knot.

Let us consider the product W x R and the equivalence relation (x,a) ~ (hi(x), a+j) for any j ~ Z . It is immediate to verify that the quotient space is homeomorphic to X. Moreover, the quotient map W × ~ ~X is a regular covering map, whose Galois group is C. So thls is the infinite cyclic covering of X. We deduce from that : I) X has the homotopy type of W, which is a compact C.W. complex. 2) The generator t of the Galois group C acts by the map (x,a)I ~ (h(x), a+l). So t acts on H.(X) as h acts on H.(W).

As before, let us denote by Fk(X ) the torsion-free quotient of Hk(X~Z ). By I), Fk(X ) is a finitely generated free abelian group and it is also a ZC-module. Under these circumstances,a theorem of algebra says that a generator k of the first elementary ideal of the ZC-module Fk(X ) is just the characteristic polynomial of t @ Moreover, it is not hard to see that k is Just the Alexander polynomial A I of Hk(X;Z ). (The lazy reader can look at Weber's paper in this book). Recalling that t acts like ~ we get the folklore theorem :

When a knot fibers, the Alexander polynomial of Hk(X;Z)is just the characteristic polynomial of the monodromy hk acting on Fk(W),

As it is a characteristic polynomial, its leading coefficient is +I ; as hk is an isomorphism on the finitely generated free abelian group Fk(~ ) its last coefficient is ~I Remark : A simplified version of the above argument gives the following : Let F be a field. Then the order of the PC-module Hk(X~;F) is just the characteristic polynomial of the automorphism h k : Hk(W;F )

Using some more algebra, it is not hard to see that the minimal polynomial of the action of h k on Fk(W ) is &I/&2 , A i being the g.c.d, of the ith elementary ideal of Hk(~; Z).

We saw in this chapter § I that a knot fibers if and only if one can find a Seifert surface V such that Y is homeomorphic to W × [O,i]. One can choose a homeomorphism which is the "identity" from W+ to W × (0) . The homeomorphism we get from W to W × (I} is just h .

Moreover i+ and i are homotopy equivalences. So, (i+) k and (i) k : Hk(W ~ Hk(Y ) are isomorphisms and : hk = (i+)k-I ? (i_) k for all k .

If the fibered knot is (2m-i)-dimensional, the Seifert pairing A : Fm(W ) × Fm(W ) ~ Z associated with the fiber W is unimodular, because (i+) m is an isomorphism.

Suppose now that, for a given (2m-1)-knot, we can find an (m-i)connected Seifert surface W such that its Seifert pairing is unimodular. Then, if m > 3, by the h-cobordism theorem Y is homeomorphic to the product W x [0, I] and so the knot fibers. Moreover, using notations introduced in chap. III§ i and § 3, the matrix for hm is given by (_I) m÷l A -I , A T

For classical knots (m = I), the unimodularlty of a Seifert matrix is necessary for a knot to fiber, but it is not sufficient. See R. Crowel! and D. Trotter (1962). The correct condition, due to L. Neuwirth and J. Stallings is that one should find a Seifert surface such that i+ and i induce isomorphisms on the fundamental group.

It is harder to get useful fibration theorems for non-simple knots. However, we saw in § 2 that a necessary condition for a knot to fiber is that the extremal coefficients of the Alexander polynomial for Hk(X;Z ) should be ± I for all k ~ I. A theorem due to D.W. Sumnets says that the converse is true if Wl(X) = Z and n ~ 4. See Sumners (1971).

If one spins a fibered knot, one gets again a fibered knot. This fact has been used by J.J. Andrews and D.W. Sumners (1969). § 4. Twist-spinning.

An important and striking way to construct a fibered knot is E.C. Zeaman's twist-spinning.

We give a sketched description of the twist-spinning construction and for more details, we refer the reader to Zeeman's original paper (1963), where the geometry of the construction is beautifully described.

Look at the unit closed ball En+2 as being the product En x E 2. In E 2 use polar coordinates, (p,@) being mapped onto pe 2iw@, 0 <p41,O(~l. So, a point in En+2 will be described by a triple (x,p,@). Also, S 1 is the unit circ~in E 2, with angular coordinate e , 0 ~ e < i .

Suppose now that we have a subspace A c En+2. Let r ~ Z be given. The full r twist of A is the subspace A r ~ En+2 x S 1 consisting of the quadruples : (x,p,@ + re,e) for all (x,p,@) ~ A, e ~ [0, i]. It is obvious that A r is abstractly homeomorphic to A × S I.

Now, let an n-knot K c Sn+2 be given. Choose a small open (n+2)disc neighborhood of a point belonging to K such that : 1) The intersection of the disc with K is an open n-disc. 2) The small disc pair thus obtained is standard.

Let us take the complementary pair (Dn+2, B). Identify Dn+2 with En+2. r ~ Z being given, look at the pair (Dn+2 x S I, Br ). On the boundary it is the standard (Sn+l x S 1, Sn-I x S I ), because via the identification bB goes to bE n x (0} . Glue along the boundary the standard (Sn+1 x D 2, Sn-1 x D 2) and you get an (n+1)-knot; because, abstractly for any k ~ 0 (Dk x SI)-L(S k-1 × D 2) glued along Sk-1 x S 1 yields Sk+1. This is the r-twist spinning of the original knot. Zeeman's theorem : Suppose r ~ 0 . Then : i) The exterior of the r-twist spun knot fibers on S I, in the sense of § I. 2) The fiber W is the r-th cyclic branched covering of the original knot, minus an open (n+2)-disc. 3) Let f be a correctly chosen generator of the Galois action on the unbranched r-th cyclic covering of the exterior of the original knot. f extends to an automorphism T of the branched cyclic covering (the knot being fixed) and T restricts to an automorphism h of the punctered branched cyclic covering W. h is of order r and can be taken as the monodromy of the fibration. Beware : h is not quite the identity on bW. 4) There is an action of S 1 on Sn+3 leaving the twist-spun knot invariant, and acting freely outside the knot. But the action on the knot is not the identity.

Comments : a) Because of point 4), one is very close to counter-examples to the Smith conjecture, for multidimensional knots. Soon after Zeeman's paper, C.H. Giffen (1964) was able to produce such counter-examples; by using as a start the twist-spinnlng operation. Several other counter-examples are now known (all for non-classical knots ~ ) . d) As the l-branched cyclic covering of an n-knot is the (n+1)sphere, l-twist spun knots bound a disc and are thus trivial. § 5. Isolated singularities of complex hypersurfaces.

Let f : Cm+1 ~ C be a C-polynomial map, such that f(O) = 0 and that 0 ~ Cm+l is an isolated singularity of f. (This means that the C-gradient of f does not vanish in a neighborhood of 0 except at 0). J. Milnor (1968b) shows : i) The intersection K of the hypersurface f-l(O) = H with sufficiently s m a l l s p h e r e s S c 2m+1 i n Cm+l, c e n t e r e d i n O, i s t r a n s v e r s a l . Thus K i s a ( r e a l ) c o d i m e n s i o n two s u b m a n i f o l d o f S C 2m+1, b u t n o t n e c e s s a r i l y a sphere. 2) The exterior of K in S 2m+1 fibers in the strong sense, i.e. the restriction of the fibration to bX is the projection onto S I associated to a trivialization of the normal bundle of K in S 2m+I. 3) The fiber W has the homotopy type of a wedge of m-dimensional spheres.

If we look at the homology exact sequence of the pair V mod.K, we see that K is not too far from being a homology sphere. Its only (possibly) non-vanishing homology greups are in dimensions (m-i) and m .Their vanishing depends on the intersection pairing on Hm(V ) = Hm(W ). Moreover one can prove that if m ~ 3, K is slmply-connected .

It is clear that there is a Seifert pairing for W, and that, if we agree to call "knots" submanifolds such as K, we have got an odd dimensional, fibered, simple knot. One can check that Levine's S-equivalence theory works in that case also. So, from a topological point of view, the situation is rather well understood. See A.F. Durfee (1973) for detail. Remarks : a) It is known (see Milnor's book) that locally around 0, the pair (cm+I,H) is homeomorphic to the cone on the pair (s2m+I,K). Thus, topologically, the singularity is determined by the knot. c) Define two holomorphlc germs fi : U i- ~ C , i = 1,2, U i open neighborhood of 0 in Cm+l, fi(O) = 0, to be topologically equivalent if there exist a germ ~ of homeomorphism at 0 ~ Cm+], ~(0)= 0 and a germ ~ of homeomorphism at 0 E C, ~(0) = 0 such that : f2 = ~ " fl " ~

in a suitable neighborhood of 0 ~ Cm+l.

A recent theorem of H.C. King (1977) Says that, if m ~ 2, two holomorphic germs with isolated singularities at 0 are topologically equivalent if and only if the knots they determine are isotopic. So, roughly speaking, the knot determines the topological type of the germ f, a result much stronger than the classical one stated in a) above.

The main question in the topological study of isolated singularities of complex hypersurfaces is to relate the topological invariants coming from knot theory and the invarlants coming from algebraic geometry. (Here, when we say "topological" we mean as well "differential" as opposed to "algebraic" or "analytic"). For instance, the differential structure on K is an interesting invariant. More precisely : I) One would like to compute the knot invariants from the algebraic data. Historically the whole story began (after O. Zariski's work in the thirties) when F. Pham (]965) and subsequently E. Brieskorn (1966) studied the sigularities :

f(Zo ' zl .... , Zk ) = (Zo) io + (zl) i i + ... + (Zk) ik In that case, computations can be done. For other results, see P. Orlik and J. Milnor (1969). 2) One would also like to know when a given knot is obtained from a singularity. This can be first attacked by trying to determine which restrictions are imposed on the knot invariants when it is "algebraic", besides the fact that it is a fibered knot. Striking examples of such restrictions are : a) The monodromy theorem, which says that the roots of the Alexander polynomial of Hm(X;Z ) (which is the characteristic polynomial of hm) are all roots of unity. See E. Brieskorn(1969). b) There exists a basis for Hm(W;Z)such that the Seifert matrix is triangular. See A.F. Durfee (1973). c) The trace of hm is equal to (-i) m+i. See N.A'Campo (1972b) and, more generally, N. A'Campo(1974).

All these are very deep results about singularities.

Since the beginning of the theory, a lot of work has been spent to get nice geometrical descriptions of some singularities. For recent results, look at L.H.Kauffman (1973) and also at L.H. Kauff~an and W.D. Neumann(i976).

For a more detailed exposition and more references about the whole subject, the reader should see J. Milnor's book (1968b), A.H. Durfee (1975), M. Demazure (1974).

Historical remark : The theory of isolated singularities began in the late twenties by the study of singularities of complex plane curves, approximately at the same time as knot theory really started (exception being made for M. Dehn's papers). In fact, progresses were made in knot theory to understand O. Zariski's results about curves and conversely, algebraic geometers found beautiful applications of J.W. Alexander and K. Reidemeister's work. It is amusing to note that a remark (due to W. Wirtinger) about the singularity z12 + z23 = 0 being locally the cone on the trefoil knot appears already in E. Artin's paper (1925). This permits us to close this paper at the point where we started it.

B I B L I O G R A P H Y

This bibliography contains the articles on knot theory quoted in this paper. The articles indirectly pertaining to the subject, or of a more technical nature have been mentioned in the text and will not be listed here again.

The reader should be aware that this bibliography is incomplete.

The articles are dated either according to the year in which the work was done, or in case of insufficient information, according to the year of submission to the journal. The publication year appears in the reference.

: "Intersection theory of manifolds with operators, with applications to knot theory". Annals of Math.

65 (

1959 : J.J. ANDREWS and M.L. CURTIS : "Knotted 2-spheres in 4-spaces".

Annals of Math. 70 (1959) p. 565-571. 1960 :

"Homology of group systems with applications to knot theory". Annals of Math. 76 (1962) p. 464-498. a) R.H. FOX : "A quick trip through knot theory". Topology of 3manifolds (M.K. Fort Jr Editor). Prentice Hall (1962) p. 120-167. b) R.H. FOX : "Some problems in knot theory". Topology of 3-manifolds (M.K. Fort Jr Editor). Prentice Hall (1962) p. 168-176. H. GLUCK : " The embeddings of the two-sphere in the four-sphere".

Transactions AMS 104 (1962) p. 308-333.

R.H. CROWELL and. H. TROTTER : "A class of pretzel knots". Duke Math.

Jour. 30 (1963) p. 373-377. : "On topologically unknotted spheres". Annals of Math.

77 (1963) p. 490-503.

M. KERVAIRE : "On higher dimensional knots". Differential and combinatorial topology. (S. Cairn edit). Princeton Univ.

Press (1965) p. 105-120.

M.W. HIRSCH and L. NEwJWIRTH : "On piecewise regular n-knots". Annals of Math. 80 (1964) p. 594-612. J. LEVINE : "Unknotting spheres in codimension two". Topology 4 (1965) p. 9-16.

C.T.C. WALL : "Unknotting tori in codimension one and spheres in codimension two". Proc. Camb. Phil. Soc. 61 (1965) p. 659-664.

PHAM : "Formules de Picard-Lefschetz gdn4ralis4es et ramification des int@grales". Bull. Soc. Math. de France 93 (1965) p. 333-367. : "Beispiele zur Differentialtopologie yon Singularitgten". Invent. Math. 2 (1966) p. 1-14. : "Diffeomorphisms of one-connected manifolds". Transac

tions AMS 128 (1967) p. 155-163.

J. LEVINE : "Polynomials invariants of knots of codimension two".

Annals of Math. 84 (1966) p. 537-554. 1965 : 1968: R.K. LASHOF and J.L. SHANESON : "Classification of knots in codimension two". Bulletin AMS 75 (1969) p. 171-175.

J. LEVINE : "Knot cobordism in codimension two". Comment. Math. Helv.

44 (1969) p. 229-244.

S.J. LOMONAC0 Jr : "The second homotopy group of a spun knot". Topo~ logy 8 (1969) p. 95~98 . a) J. MILNOR : "Infinite cyclic coverings". Conference on the Topology of Manifolds. Prindle, Weber and Schmidt. (1968). b) J. MILNOR : "Singular points of complex hypersurfaces". Annals of Math. Studies vol. 61 (1968). 1970 : M. KERVAIRE : "Knot cobordism in codimension two". Springer Lecture

Notes vol. 197 (1970) p. 83-105. 1972 : G. BREDON : Regular O(n)-manifolds, suspensions of knots and knot periodicity, 79 (1973), p. 87-91.

S. CAPPELL and J. SHANESON: "Submanifolds, group actions and knots I".

Bulletin AMS 78 (1972), p. 1045-1048.

E. DYER and A.T. VASQUEZ :"The sphericity of higher dimensional knots". Can. J. Math. 25 (1973) p. 1132-1136.

H. TROTTER : "On S-equivalence of Seifert matrices ". Invent. Math.

20 (1973) P. 173-207. 1973 : 1974 : A.H. DURFEE : "Fibered knots and algebraic singularities". Topology 13 (1974) p. 47-59.

L. H. KAUFFMAN : "Branched cyclic coverings, open books and knot periodicity". Topology 13 (1974) p. 143-160.

N. A'CAMPO : "La fonction zeta d'une monodromie". Comment. Math. Helv. 50 (1975) p. 233-248. M. DEMAZURE : "Classification des germes & points critiques isol@s et h nombre de modules 0 ou 1" Sgmlnaire Bourbaki,

F@vrier 1974, expos@ 443.

M. FARBER : "Linking coefficients and two-dimensional knots"

Soviet Math. Dokl. 16 (1975) p. 647-650.

C. KEARTON : "Blanchfield duality and simple knots". Transactions

AMS 202 (1975) p. 141-160.

J. LEVINE : "Knot modules I". Transactions AMS 229 (1977) p. 1-50. 1975: S. CAPPELL and J.L. SHANESON : "There exist inequivalent knots with the same complement". Annals of Math. 103 (1976)

P. 349-353. L.H. KAUFFMAN and D.W. NEUMANN : "Products of knots, branched flbratlons and sums of singularities". Preprlnt (1976) 92 p. 1977 : : "Unraveling the integral knot concordance group"

Memoirs AMS vol. 192 (1977), 91 p.

W.C. KING : "Topological type of isolated singularities". Preprint (1977) 22 p.

A. RANICKI : "The algebraic theory of surgery". Preprint (1977). 322 pages.