Robion Kirby

Foundational essays on topological manifolds, smoothings, and triangulations

Since Poincares time, topologists have been most concerned with three species of manifold. The most primitive of these--the TOP manifolds--remained rather mysterious until 1968, when Kirby discovered his now famous torus unfurling device. A period of rapid progress with TOP manifolds ensued, including, in 1969, Siebenmanns refutation of the Hauptvermutung and the Triangulation Conjecture. Here is the first connected account of Kirbys and Siebenmanns basic research in this area. The five sections of this book are introduced by three articles by the authors that initially appeared between 1968 and 1970. Appendices provide a full discussion of the classification of homotopy tori, including Cassons unpublished work and a consideration of periodicity in topological surgery.

Branched covers of surfaces in 4-manifolds

We give an algorithm for describing, as a framed link, the p-fold branched cover of (i) B 4 branched along the Seifert surface F of a link with int F pushed into int/34 (see Sect. 2); (ii) /34 union handles branched along F C B 4 (see Sect. 3); (iii) S 4 branched along a surface which, except for a trivial 2-ball, lies in S 3 (see Sect. 4); (iv) C P 2 branched along a nice surface such as a non-singular complex curve (see Sect. 5). Along the way we show how to describe the p-fold branched cover of B 4 along the ribbon disk of a ribbon link (Sect. 3), prove that the p-fold branched cover of B 4 along a Seifert surface for the unknot is trivial (Theorem 4.1, Sect. 4), show that the Mitnor fiber and various other complex surfaces can be built without 1 and 3-handles (Theorem 5.1 and corollaries), and draw the framed links for the complex surfaces, the cubic, quintic and Kummer (see Sect. 5). In Sect. I we fix conventions and notations.

Dedekind sums, μ-invariants and the signature cocycle

0 Introduction The Dedekind eta function, defined by o~ ~(z) = e ~z/12 l-I(1-e 2€ n=l for z in the upper half plane H, plays a central role in number theory. It is a modular form of fractional weight whose 24 th power is proportional to the fundamental discriminant cusp form of weight 12. In particular ~?(z)24dz 6, is invariant under the modular group PSL(2, Z), and so there is a function r PSL(2, Z)-* Z given by (0.1) r) for A= 7r~ c where #(A) = ~ log if c =~ 0, and #(A) = 0 if c = 0. Dedekind [D] gave a formula \ i sign c~ (0.2) r = 2 if c = 0 a+d 12sign(c)s(a,c) if c~0 C in terms of certain arithmetic sums s(a, c) defined for coprime integers a and c by Lcl-I k=l where ((x)) = x-[x]-1/2. These sums, now called Dedekind sums, arise in many contexts and have been intensively studied during the past hundred years. Many of their fundamental properties were discovered by Rademacher, and the function r is

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),

A potential smooth counterexample in dimension 4 to the Poincare conjecture, the Schoenflies conjecture, and the Andrews-Curtis conjecture

WE DESCRIBE a homotopy 4-sphere X4, built with the usual zero and 4-handle and two lhandles and two 2-handles (Figure 28). Of course X4 is homeomorphic to S4 [Freedman] but considerable effort has not led to a proof that X4 is diffeomorphic to S4. X4 has the following virtues: (A) Although it is easy to construct smooth homotopy 4-spheres (e.g. the Gluck construction on knotted 2-spheres or via non-trivial presentations of the trivial group), this is the only (except S4) example we know without 3-handles and with so few handles altogether. (B) The presentation of the trivial group arising from X4 (see

Constructing symplectic forms on 4–manifolds which vanish on circles

2) is {x, y lxyx = yxy, x5 = y4}; it is easy to show that this group is trivial, but it seems difficult to do so using Andrews-Curtis moves ([l] or [lo] 5.1)

Constructing Lefschetz-type fibrations on four-manifolds

Given a smooth, closed, oriented 4-manifold X and α ∈ H2(X, Z) such that α � α > 0, a closed 2-form ω is constructed, Poincare dual to α, which is symplectic on the complement of a finite set of unknotted circles Z. The number of circles, counted with sign, is given by d = (c1(s) 2 −3σ(X)−2χ(X))/4, where s is a certain spin C structure naturally associated to ω.

Trisecting 4–manifolds

We show how to construct broken, achiral Lefschetz fibrations on arbitrary smooth, closed, oriented 4‐manifolds. These are generalizations of Lefschetz fibrations over the 2‐sphere, where we allow Lefschetz singularities with the non-standard orientation as well as circles of singularities corresponding to round 1‐handles. We can also arrange that a given surface of square 0 is a fiber. The construction is easier and more explicit in the case of doubles of 4‐manifolds without 3‐ and 4‐handles, such as the homotopy 4‐spheres arising from nontrivial balanced presentations of the trivial group. 57M50; 57R17

AN EXOTIC INVOLUTION OF S4

We show that any smooth, closed, oriented, connected 4‐manifold can be trisected into three copies of \ k .S 1 B 3 /, intersecting pairwise in 3‐dimensional handlebodies, with triple intersection a closed 2‐dimensional surface. Such a trisection is unique up to a natural stabilization operation. This is analogous to the existence, and uniqueness up to stabilization, of Heegaard splittings of 3‐manifolds. A trisection of a 4‐manifold X arises from a Morse 2‐function GW X! B 2 and the obvious trisection of B 2 , in much the same way that a Heegaard splitting of a 3‐manifold Y arises from a Morse function gW Y ! B 1 and the obvious bisection of B 1 .

EIGHT FACES OF THE POINCARE HOMOLOGY 3-SPHERE

CAPPELL and Shaneson [1] construct a family of smooth 4-manifolds which are simple homotopy equivalent to real projective 4-space RP4, but not even smoothly h- cobordant to RP4. (It is possible they are homeomorphic to RP4.) It is natural to ask whether their double covers are S4 or not.