A geometrically bounding hyperbolic link complement

A finite-volume hyperbolic 3-manifold geometrically bounds if it is the geodesic boundary of a finite-volume hyperbolic 4-manifold. We construct here an example of non-compact, finite-volume hyperbolic 3-manifold that geometrically bounds. The 3-manifold is the complement of a link with eight components, and its volume is roughly equal to 29.311.Comment: 23 pages, 19 figure

Higher order intersection numbers of 2-spheres in 4-manifolds

This is the beginning of an obstruction theory for deciding whether a map f:S^2 --> X^4 is homotopic to a topologically flat embedding, in the presence of fundamental group and in the absence of dual spheres. The first obstruction is Wall's self-intersection number mu(f) which tells the whole story in higher dimensions. Our second order obstruction tau(f) is defined if mu(f) vanishes and has formally very similar properties, except that it lies in a quotient of the group ring of two copies of pi_1(X) modulo S_3-symmetry (rather then just one copy modulo S_3-symmetry). It generalizes to the non-simply connected setting the Kervaire-Milnor invariant which corresponds to the Arf-invariant of knots in 3-space. We also give necessary and sufficient conditions for moving three maps f_1,f_2,f_3:S^2 --> X^4 to a position in which they have disjoint images. Again the obstruction lambda(f_1,f_2,f_3) generalizes Wall's intersection number lambda(f_1,f_2) which answers the same question for two spheres but is not sufficient (in dimension 4) for three spheres. In the same way as intersection numbers correspond to linking numbers in dimension 3, our new invariant corresponds to the Milnor invariant mu(1,2,3), generalizing the Matsumoto triple to the non simply-connected setting.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-1.abs.htm

New hyperbolic 4-manifolds of low volume

We prove that there are at least 2 commensurability classes of minimal-volume hyperbolic 4-manifolds. Moreover, by applying a well-known technique due to Gromov and Piatetski-Shapiro, we build the smallest known non-arithmetic hyperbolic 4-manifold.Comment: 21 pages, 6 figures. Added the Coxeter diagrams of the commensurability classes of the manifolds. New and better proof of Lemma 2.2. Modified statements and proofs of the main theorems: now there are two commensurabilty classes of minimal volume manifolds. Typos correcte

Homotopy versus isotopy: spheres with duals in 4-manifolds

David Gabai recently proved a smooth 4-dimensional "Light Bulb Theorem" in the absence of 2-torsion in the fundamental group. We extend his result to 4-manifolds with arbitrary fundamental group by showing that an invariant of Mike Freedman and Frank Quinn gives the complete obstruction to "homotopy implies isotopy" for embedded 2-spheres which have a common geometric dual. The invariant takes values in an Z/2Z-vector space generated by elements of order 2 in the fundamental group and has applications to unknotting numbers and pseudo-isotopy classes of self-diffeomorphisms. Our methods also give an alternative approach to Gabai's theorem using various maneuvers with Whitney disks and a fundamental isotopy between surgeries along dual circles in an orientable surface.Comment: Included into section 2 of this version is a proof that the operation of `sliding a Whitney disk over itself' preserves the isotopy class of the resulting Whitney move in the current setting. Some expository clarifications have also been added. Main results and proofs are unchanged from the previous version. 39 pages, 25 figure

On equivalence of Floer's and quantum cohomology

(In the revised version the relevant aspect of noncompactness of the moduli of instantons is discussed. It is shown nonperturbatively that any BRST trivial deformation of A-model which does not change the ranks of BRST cohomology does not change the topological correlation functions either) We show that the Floer cohomology and quantum cohomology rings of the almost Kahler manifold M, both defined over the Novikov ring of the loop space LM of M, are isomorphic. We do it using a BRST trivial deformation of the topological A-model. As an example we compute the Floer = quantum cohomology of the 3-dimensional flag space Fl_3.Comment: 28 pages, HUTP-93/A02