Products of knots, branched fibrations and sums of singularities

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INTRODUCTION A KNOT X = (S“, K) is an oriented k-sphere with an oriented 2codimensional submanifold K c Sk. We say that a knot 2 = (S’, L) is fibered if a fibration S’ L L S’ is given such that the closure Ft = b-‘(t) of any fiber of b is a closed submanifold of S’ with boundary aF, = L. We shall discuss a product operation that associates to two knots X and 2, one of which is fibered, a new knot X @ 5’ = (Sk+‘+‘, K @ L). This construction has many useful properties. It is commutative (up to orientation), associative, and distributes over connected sums. The product of fibered knots is again fibered. Algebraic information about a product may be deduced easily from the factors. For example, the monodromy of a product of fibered knots is the tensor product of the monodromy of the factors. The Seifert pairing for a product is the tensor product of the Seifert pairings for the factors. A particularly interesting class of fibered knots is the class of links of isolated complex polynomial singularities. That is, let f: (C”, O)+(C, 0) be a polynomial mapping such that 0 E C” is an isolated critical point of f. Then the link off is the knot 5?v) = (S?+‘, L(f)), where E > 0 is sufficiently small and L(f) = ST+’ fl f-‘(O). Milnor [ 171 showed that Yu) has a natural fibered structure. If g: Cm+’ +C is another such mapping, then so is f + g: C”+’ x Cm+’ +C given by cf + g)(x, y) = f(x) + g(y). We show that Zcf + g) and 5!‘m @ 5?(g) are isomorphic as fibered knots. Thus the product operation gives a geometric construction for sum of singularities. We show the corresponding result also for isolated singularities of real polynomial maps f: Rk+’ + R*, g: RI+’ +R*, except that in certain low dimensions it is still an open question whether the isomorphism of 6pcf+ g) and Z’u) @ Z’(g) preserves fibered structure. This isomorphism is also proved in a yet more general situation“tame” singularities (Definition 1.3)except that in low dimensions we now only get an h-cobordism of fibered knots. The properties of knot product generalize and put in a clearer perspective many known results about complex polynomial singularities. In particular, Thorn and Sebastiani[20] showed that the monodromy for Zu+ g) is the tensor product of the monodromies of f and g. The same result was conjectured for Seifert pairings by A. Durfee and first proved by Sakamoto [ 193. Our results generalize the Thorn-Sebastiani theorem and the Sakamoto theorem to (and in fact beyond) the real polynomial case. Note that the low dimensional problems mentioned above are irrelevant here, since these homological invariants only depend on the h-cobordism class of a fibered knot. It is well known that the class of links of real polynomial singularities is a much more extensive class than that of links of complex polynomial singularities (see [15] and [17]). Every fibered knot is the link of a tame singularity. Thus these generalizations are very non-empty. G. Bredon in [2] gave a suspension construction for knots, using 0 (n)-manifolds, which he used to give a geometric version of knot cobordism periodicity. His results also generalized results of Hirzebruch[9] and Erle[7] about K. Jtinich’s knot manifolds[lO]. We observe that Bredon’s construction corresponds in our context to forming X @ Z(z,* + . . + + 2,‘) and our results generalize Bredon’s results. Our construction generalizes also results [ 121 and [ 181. Some of the results were announced in [13]. In a final section we indicate how the product construction, its properties, and its relation to isolated singularities, generalizes to arbitrary codimension. The paper is organized as follows. 91 discusses fibered knots, open books, and branched