Laurence R. Taylor

Signature of links

Let L be an oriented tame link in the three sphere S3. We study the Murasugi signature, a(L), and the nullity, r(L). It is shown that a(L) is a locally flat topological concordance invariant and that r(L) is a topological concordance invariant (no local flatness assumption here). Known results about the signature are re-proved (in some cases generalized) using branched coverings. 0. Introduction. Let L be an (oriented) tame link of multiplicity ,u in the three-sphere S3. That is, L consists of,u oriented circles K1, ... , Kg disjointly imbedded in S3. Various authors have investigated a numerical invariant, the signature of L (notation: u(L)). The signature was first defined for knots (, = 1) by H. Trotter [21]. J. Milnor found another definition for this knot signature (see [1 2]) in terms of the cohomology ring structure of the infinite cyclic cover of the knot complement. In [2], D. Erle showed that the definitions of Milnor and Trotter are equivalent. In [15], K. Murasugi formulated a definition of signature for arbitrary links. In this paper we investigate the Murasugi signature in the context of branched covering spaces. To be specific, let D4 denote the four dimensional ball with 3D4 = S3, and let L C S3 be a link and F C D4 a properly imbedded, orientable, locally flat surface with 3F = L C S3. Let M denote the double branched cover of D4 along F. Then we show that u(L) is the signature of the four manifold M (see Lemma 1.1 and Theorem 3.1). Our proof of Theorem 3.1 contains the technicalities necessary to show this in the topological category. Using this viewpoint we are able to prove that u(L) is a topological concordance invariant (Theorem 3.8). We also rederive many of Murasugis results, generalizing some of them (see Theorems 3.9-3.16). The paper is organized as follows: ? 1 contains the classical definitions of the signature and nullity of a link. It also deals with necessary background concerning branched coverings. Received by the editors November 4, 1974 and, in revised form, December 4, 1974 and April 11, 1975. AMS (MOS) subject classifications (1970). Primary SSA25.

Pin structures on low-dimensional manifolds

Pin structures on vector bundles are the natural generalization of Spin structures to the case of nonoriented bundles. Spin(n) is the central Z/2Z extension (or double cover) of SO(n) and Pin−(n) and Pin(n) are two different central extensions of O(n), although they are topologically the same. The obstruction to putting a Spin structure on a bundle ξ (= R → E → B) is w2(ξ) H(B;Z/2Z); for Pin it is still w2(ξ), and for Pin − it is w2(ξ) + w 1(ξ). In all three cases, the set of structures on ξ is acted on by H(B;Z/2Z) and if we choose a structure, this choice and the action sets up a one–to–one correspondence between the set of structures and the cohomology group. Perhaps the most useful characterization (Lemma 1.7) of Pin± structures is that Pin− structures on ξ correspond to Spin structures on ξ ⊕ det ξ and Pin to Spin structures on ξ ⊕ 3 det ξ where det ξ is the determinant line bundle. This is useful for a variety of “descent” theorems of the type: a Pin± structure on ξ ⊕ η descends to a Pin (or Pin− or Spin) structure on ξ when dim η = 1 or 2 and various conditions on η are satisfied. For example, if η is a trivialized line bundle, then Pin± structures descend to ξ (Corollary 1.12), which enables us to define Pin± bordism groups. In the Spin case, Spin structures on two of ξ, η and ξ ⊕ η determine a Spin structure on the third. This fails, for example, for Pin− structures on η and ξ ⊕ η and ξ orientable, but versions of it hold in some cases (Corollary 1.15), adding to the intricacies of the subject. Another kind of descent theorem puts a Pin± structure on a submanifold which is dual to a characteristic class. Thus, if V m−1 is dual to w1(TM ) and M is Pin±, then V ∩| V gets a Pin± structure and we have a homomorphism of bordism groups (Theorem 2.5),

ON THE HOMOLOGY OF CONFIGURATION SPACES

Configuration spaces appear in various contexts such as algebraic geometry, knot theory, differential topology or homotopy theory. Although intensively studied their homology is unknown except for special cases, see for example [ 1, 2, 7, 8, 9, 12, 13, 14, 18, 261 where different terminology and notation is used. In this article we study the Betti numbers of

DETECTION THEOREMS FOR K-THEORY AND L-THEORY

Suppose G is a p-hyperelementary group and R is a commutative ring such that the order of G is a unit in R. Suppose J is either one of Quillen’s K-theory functors or one of Wall’s oriented L-theory functors. We show that J(RG) can be detected by applying J(R?) to the subquotients of G such that all normal abelian subgroups are cyclic. In 3.A.6 we show that such subquotients have a quite simple structure. We also show how to detect more general L-theory functors, in particular unoriented ones and those that arise in the study of codimension one submanifolds.

James maps, Segal maps, and the Kahn-Priddy theorem

The standard combinatorial approximation C(R n, X) to Qn~nx is a filtered space with easily understood filtration quotients Di Rn, X). Stably, C( Rn, X) splits as the wedge of the Di Rn, X). We here analyze the multiplicative properties of the James maps which give rise to the splitting and of various related combinatorially derived maps between iterated loop spaces. The target of the total James map } := (}q): Qn~nx -,) X Q2nq~2nqDq( Rn, X)

Relative Rochlin invariants

Abstract The Rochlin invariant of a compact 3-manifold with a fixed spin structure can be regarded as the signature (mod 16) of any solution to a certain surgery problem. This paper explores this remark in some detail. The relative Rochlin invariants arise from consideration of other surgery problems. We work out the general theory and apply it with RP 3 replacing S 3 to study free involutions on 3-manifolds. The Morgan-Sullivan linking cycle theory gives new insight into the relation between spin structures on the 3-manifold and how circles in the manifold link. From the algebra which expresses this relation one can calculate the relative Rochlin invariants mod 8, and can often recover the spin structure on the manifold.

An invariant of smooth 4–manifolds

We dene a dieomorphism invariant of smooth 4{manifolds which we can estimate for many smoothings of R 4 and other smooth 4{manifolds. Using this invariant we can show that uncountably many smoothings of R 4 support no Stein structure. Gompf [11] constructed uncountably many smoothings of R 4 which do support Stein structures. Other applications of this invariant are given.

Birationality of étale maps via surgery

Abstract We use a counting argument and surgery theory to show that if D is a sufficiently general algebraic hypersurface in , then any local diffeomorphism F : X → of simply connected manifolds which is a d-sheeted cover away from D has degree d = 1 or d = ∞ (however all degrees d > 1 are possible if F fails to be a local diffeomorphism at even a single point). In particular, any étale morphism F : X → of algebraic varieties which covers away from such a hypersurface D must be birational.

Rational permutation modules for finite groups

By the Artin Induction theorem, C(G) is a finite abelian group with exponent dividing the order ofG. Some work on this sequence has already been done. In [14] and [16], Ritter and Segal proved that C(G) = 0 for G a finitep–group. Serre [17, p. 104] remarked that C(G) / = 0 for G = Z/3 × Q8 (the direct product of a cyclic group of order 3 and a quaternion group of order 8). Berz [2] gave a nice description of P (G) for G metabelian or supersolvable. To describe the result, recall that R(G) additively is a free abelian group with basis given by the irreducible rational representations of G. The subgroupP (G) is generated by the induced representations IndG(1H ) = Q[G/H], whereH runs over the subgroups of G. If aφ denotes thegcd over all H of the numbers〈φ, IndG(1H )〉, thenaφ divides〈φ, χ〉 wheneverχ is a virtual permutation representation. Let αφ = aφ 〈φ,φ〉 .

Generalized splitting theorems

In (5), we and Fred Cohen gave some quite general splitting theorems. These described how to decompose the suspension spectra of certain filtered spaces C X as wedges of the suspension spectra of their successive filtration quotients D q X. The spaces C X were of the form C r × X r /(˜) for suitable sequences of spaces { C r } and { X r }, and the construction C X was intended to be a reworking in ‘proper generality’ of the constructions introduced in (9).