Calculus III: Taylor Series

We study functors from spaces to spaces or spectra that preserve weak homotopy equivalences. For each such functor we construct a universal n-excisive approximation, which may be thought of as its n-excisive part. Homogeneous functors, meaning n-excisive functors with trivial (n-1)-excisive part, can be classified: they correspond to symmetric functors of n variables that are reduced and 1-excisive in each variable. We discuss some important examples, including the identity functor and Waldhausen's algebraic K-theory.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper19.abs.htm

The smooth Whitehead spectrum of a point at odd regular primes

Let p be an odd regular prime, and assume that the Lichtenbaum-Quillen conjecture holds for K(Z[1/p]) at p. Then the p-primary homotopy type of the smooth Whitehead spectrum Wh(*) is described. A suspended copy of the cokernel-of-J spectrum splits off, and the torsion homotopy of the remainder equals the torsion homotopy of the fiber of the restricted S^1-transfer map t: SigmaCP^infty--> S. The homotopy groups of Wh(*) are determined in a range of degrees, and the cohomology of Wh(*) is expressed as an A-module in all degrees, up to an extension. These results have geometric topological interpretations, in terms of spaces of concordances or diffeomorphisms of highly connected, high dimensional compact smooth manifolds.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper4.abs.htm

Canonical decompositions of 3-manifolds

We describe a new approach to the canonical decompositions of 3-manifolds along tori and annuli due to Jaco-Shalen and Johannson (with ideas from Waldhausen) - the so-called JSJ-decomposition theorem. This approach gives an accessible proof of the decomposition theorem; in particular it does not use the annulus-torus theorems, and the theory of Seifert fibrations does not need to be developed in advance.Comment: 20 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol1/paper3.abs.htm

A chain rule in the calculus of homotopy functors

We formulate and prove a chain rule for the derivative, in the sense of Goodwillie, of compositions of weak homotopy functors from simplicial sets to simplicial sets. The derivative spectrum dF(X) of such a functor F at a simplicial set X can be equipped with a right action by the loop group of its domain X, and a free left action by the loop group of its codomain Y = F(X). The derivative spectrum d(E o F)(X)$ of a composite of such functors is then stably equivalent to the balanced smash product of the derivatives dE(Y) and dF(X), with respect to the two actions of the loop group of Y. As an application we provide a non-manifold computation of the derivative of the functor F(X) = Q(Map(K, X)_+).Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol6/paper25.abs.htm

The fibered isomorphism conjecture for complex manifolds

In this paper we show that the fibered isomorphism conjecture of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for the fundamental groups of a large class of complex manifolds. A consequence of this result is that the Whitehead group, reduced projective class groups and the negative K-groups of the fundamental group of these manifolds vanish whenever the fundamental group is torsion free. We also prove the same results for a class of real manifolds.Comment: accepted for publication in Acta Mathematica Sinica, English Serie

Unexpected local minima in the width complexes for knots

In "Width complexes for knots and 3-manifolds," Jennifer Schultens defines the width complex for a knot in order to understand the different positions a knot can occupy in the 3-sphere and the isotopies between these positions. She poses several questions about these width complexes; in particular, she asks whether the width complex for a knot can have local minima that are not global minima. In this paper, we find an embedding of the unknot that is a local minimum but not a global minimum in its width complex. We use this embedding to exhibit for any knot K infinitely many distinct local minima that are not global minima of the width complex for K.Comment: 9 pages, 4 figure

Finiteness of classifying spaces of relative diffeomorphism groups of 3-manifolds

The main theorem shows that if M is an irreducible compact connected orientable 3-manifold with non-empty boundary, then the classifying space BDiff(M rel dM) of the space of diffeomorphisms of M which restrict to the identity map on boundary(M) has the homotopy type of a finite aspherical CW-complex. This answers, for this class of manifolds, a question posed by M Kontsevich. The main theorem follows from a more precise result, which asserts that for these manifolds the mapping class group H(M rel dM) is built up as a sequence of extensions of free abelian groups and subgroups of finite index in relative mapping class groups of compact connected surfaces.Comment: 19 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol1/paper7.abs.htm

Heegaard splittings and the pants complex

We define integral measures of complexity for Heegaard splittings based on the graph dual to the curve complex and on the pants complex defined by Hatcher and Thurston. As the Heegaard splitting is stabilized, the sequence of complexities turns out to converge to a non-trivial limit depending only on the manifold. We then use a similar method to compare different manifolds, defining a distance which converges under stabilization to an integer related to Dehn surgeries between the two manifolds.Comment: This is the version published by Algebraic & Geometric Topology on 11 July 200

CCD photometry in the region of NGC 6994: the remains of an old open cluster

We present the results of BV(RI)_KC CCD photometry down to V=21 mag in the region of NGC 6994. To our knowledge, no photometry has previously been reported for this object and we find evidences that it is a poor and sparse old open cluster, with a minimum angular diameter of 9 arcmin, i.e. larger than the 3 arcmin originally assigned to it. We obtain a color excess E(B-V) = 0.07 +/- 0.02 mag by means of the BVI_(C) technique. Based on the theoretical isochrones from VandenBergh (1985) that are in better agreement with our data, we estimate for this cluster a distance from the Sun of 620 pc (Vo-Mv = 9 +/- 0.25 mag) and an age lying within the range of 2 - 3 Gyr, adopting solar metallicity. Thus, the corresponding cluster's Galactocentric distance is 8.1 kpc and is placed at about 350 pc below the Galactic plane. According to this results, NGC 6994 belongs to the old open cluster population located in the outer disk and at large distances from the Galactic plane, and must have suffered significant individual dynamical evolution, resulting in mass segregation and evaporation of low mass stars.Comment: 10 pages including 11 figures and 1 table. Accepted for publication in Astronomy & Astrophysic

Blanchfield and Seifert algebra in high-dimensional boundary link theory I: Algebraic K-theory

The classification of high-dimensional mu-component boundary links motivates decomposition theorems for the algebraic K-groups of the group ring A[F_mu] and the noncommutative Cohn localization Sigma^{-1}A[F_mu], for any mu>0 and an arbitrary ring A, with F_mu the free group on mu generators and Sigma the set of matrices over A[F_mu] which become invertible over A under the augmentation A[F_mu] to A. Blanchfield A[F_mu]-modules and Seifert A-modules are abstract algebraic analogues of the exteriors and Seifert surfaces of boundary links. Algebraic transversality for A[F_mu]-module chain complexes is used to establish a long exact sequence relating the algebraic K-groups of the Blanchfield and Seifert modules, and to obtain the decompositions of K_*(A[F_mu]) and K_*(Sigma^{-1}A[F_mu]) subject to a stable flatness condition on Sigma^{-1}A[F_mu] for the higher K-groups.Comment: This is the version published by Geometry & Topology on 2 November 200