Obstruction theory in model categories

Many examples of obstruction theory can be formulated as the study of when a lift exists in a commutative square. Typically, one of the maps is a cofibration of some sort and the opposite map is a fibration, and there is a functorial obstruction class that determines whether a lift exists. Working in an arbitrary pointed proper model category, we classify the cofibrations that have such an obstruction theory with respect to all fibrations. Up to weak equivalence, retract, and cobase change, they are the cofibrations with weakly contractible target. Equivalently, they are the retracts of principal cofibrations. Without properness, the same classification holds for cofibrations with cofibrant source. Our results dualize to give a classification of fibrations that have an obstruction theory.     

Refinements of Milnor's fibration theorem for complex singularities

Let X be an analytic subset of an open neighbourhood U of the origin in Cn. Let be holomorphic and set V=f−1(0). Let Bε be a ball in U of sufficiently small radius ε>0, centred at . We show that f has an associated canonical pencil of real analytic hypersurfaces Xθ, with axis V, which leads to a fibration Φ of the whole space (X∩Bε)?V over S¹. Its restriction to (X∩Sε)?V is the usual Milnor fibration , while its restriction to the Milnor tube f−1(∂Dη)∩Bε is the Milnor-Lê fibration of f. Each element of the pencil Xθ meets transversally the boundary sphere Sε=∂Bε, and the intersection is the union of the link of f and two homeomorphic fibres of ? over antipodal points in the circle. Furthermore, the space obtained by the real blow up of the ideal (Re(f),Im(f)) is a fibre bundle over RP¹ with the Xθ as fibres. These constructions work also, to some extent, for real analytic map-germs, and give us a clear picture of the differences, concerning Milnor fibrations, between real and complex analytic singularities.     

A theory of PWD-structures

A general study is undertaken of product-wedge-diagonal (=PWD) structures on a space. In part this concept may be viewed as arising from G.W. Whitehead's fat-wedge characterization of Lusternik-Schnirelmann category. From another viewpoint PWD-structures occupy a distinguished position among those structures that provide data allowing Hopf invariants to be defined. Indeed the Hopf invariant associated with a PWD-structure is a crucial component of the structure. Our overall theme addresses the basic question of existence of compatible structures on X and Y with regard to a map X→Y. A principal result of the paper uses Hopf invariants to formulate a Berstein-Hilton type result when the space involved is a double mapping cylinder (or homotopy pushout). A decomposition formula for the Hopf invariant (extending previous work of Marcum) is provided in case the space is a topological join U*V that has PWD-structure defined canonically via the join structure in terms of diagonal maps on U and V.     

A Kummer-type construction of self-dual 4-manifolds

Supported in part by NSF grant DMS-9204093     

Bouquet and join theorems for disentanglements

The disentanglement of certain augmentations is shown to be the topological join of a disentanglement and a Milnor fibre. The kth disentanglement of a finite map is defined and for corank 1 maps from ℂ ⁿ to ℂ ^{n +1} it is shown that they are homotopically equivalent to a wedge of spheres. Applications to the Mond conjecture are given. Oblatum 24-VII-2000 & 5-VII-2001?Published online: 12 October 2001     

Lʼindice des champs de vecteurs sur les courbes de Cohen–Macaulay

In this paper a new method for computing the topological index of a vector field at Cohen–Macaulay curves is described. It is based on properties of regular meromorphic differential forms which are used for computing the homological index of vectors fields introduced by X. Gómez-Mont. In particular, we show how to compute the index at quasihomogeneous Gorenstein curves and complete intersections, at monomial curves, at Cohen–Macaulay space curves, and others. In contrast to previous articles on this subject we do not use the technique of spectral sequences, or computer algebra systems for symbolic calculations.     

Boundary manifolds of projective hypersurfaces

We study the topology of the boundary manifold of a regular neighborhood of a complex projective hypersurface. We show that, under certain Hodge-theoretic conditions, the cohomology ring of the complement of the hypersurface functorially determines that of the boundary. When the hypersurface defines a hyperplane arrangement, the cohomology of the boundary is completely determined by the combinatorics of the underlying arrangement and the ambient dimension. We also study the LS category and topological complexity of the boundary manifold, as well as the resonance varieties of its cohomology ring.     

Hirzebruch–Milnor classes of complete intersections

We prove a new formula for the Hirzebruch–Milnor classes of global complete intersections with arbitrary singularities describing the difference between the Hirzebruch classes and the virtual ones. This generalizes a formula for the Chern–Milnor classes in the hypersurface case that was conjectured by S. Yokura and was proved by A. Parusiński and P. Pragacz. It also generalizes a formula of J. Seade and T. Suwa for the Chern–Milnor classes of complete intersections with isolated singularities.     

Characteristic classes of complex hypersurfaces

The Milnor-Hirzebruch class of a locally complete intersection X in an algebraic manifold M measures the difference between the (Poincaré dual of the) Hirzebruch class of the virtual tangent bundle of X and, respectively, the Brasselet-Schürmann-Yokura (homology) Hirzebruch class of X. In this note, we calculate the Milnor-Hirzebruch class of a globally defined algebraic hypersurface X in terms of the corresponding Hirzebruch invariants of vanishing cycles and singular strata in a Whitney stratification of X. Our approach is based on Schürmann's specialization property for the motivic Hirzebruch class transformation of Brasselet-Schürmann-Yokura. The present results also yield calculations of Todd, Chern and L-type characteristic classes of hypersurfaces.     

Local structure of Koszul-Vinberg and of Lie algebroids

In this short note we continue our study of Koszul-Vinberg algebroids which form a subcategory of the category of Lie algebroids, and which appear naturally in the study of affine structures, affine and transversally affine foliations [N. Nguiffo Boyom, R. Wolak, J. Geom. Phys. 42 (2002) 307-317]. We prove a local decomposition theorem for KV-algebroids. Using the notion of KV-algebroids we introduce a new class of singular foliations: affine singular foliations. In the last section we study the holonomy of these foliations and prove a stability theorem.     

Configuration spaces are not homotopy invariant

We present a counterexample to the conjecture on the homotopy invariance of configuration spaces. More precisely, we consider the lens spaces L_7,1 and L_7,2, and prove that their configuration spaces are not homotopy equivalent by showing that their universal coverings have different Massey products.     

Purity and homotopy theory of coalgebras

We construct a combinatorical monoidal model category on simplicial flat cocommutative coalgebras over a Prüfer domain. The cofibrations are the morphisms which are pure as module maps.     

Waldhausen's theory of k-fold end structures: a survey

Let R⁺ be the space of nonnegative real numbers. F. Waldhausen defines a k-fold end structure on a space X as an ordered k-tuple of continuous maps x_f:X → R⁺, 1 ? j ? k, yielding a proper map x:X → (R⁺)^k. The pairs (X,x) are made into the category E^k of spaces with k-fold end structure. Attachments and expansions in E^k are defined by induction on k, where elementary attachments and expansions in E⁰ have their usual meaning. The category E^k/Z consists of objects (X, i) where i: Z → X is an inclusion in E^k with an attachment of i(Z) to X, and the category E^k6Z consists of pairs (X,i) of E^k/Z that admit retractions X → Z. An infinite complex over Z is a sequence X = {X₁ ? X₂ ? … ? X_n …} of inclusions in E^k6Z. The abelian grou p S₀(Z) is then defined as the set of equivalence classes of infinite complexes dominated by finite ones, where the equivalence relation is generated by homotopy equivalence and finite attachment; and the abelian group S₁(Z) is defined as the set of equivalence classes of X₁, where X ∈ E^k/Z deformation retracts to Z. The group operations are gluing over Z. This paper presents the Waldhausen theory with some additions and in particular the proof of Waldhausen's proposition that there exists a natural exact sequence 0 → S₁(Z × R)→^πS₀(Z) by utilizing methods of L.C. Siebenmann. Waldhausen developed this theory while seeking to prove the topological invariance of Whitehead torsion; however, the end structures also have application in studying the splitting of a noncompact manifold as a product with R[1].     

Generic linear sections of complex hypersurfaces and monomial ideals

Let f:(Cn,0)→(C,0) be an analytic function germ. Under the hypothesis that f is Newton non-degenerate, we compute the μ^?-sequence of f in terms of the Newton polyhedron of f. This sequence was defined by Teissier in order to characterize the Whitney equisingularity of deformations of complex hypersurfaces.     

Holomorphic functions and quasiconformal mappings with smooth moduli

We discuss the implication , where f is a holomorphic function (resp., a quasiconformal mapping) on a domain (resp., ) and Λ_ω(G) is the Lipschitz space associated with a majorant ω.     

Multiplier ideals, V-filtrations and transversal sections

We show that the restriction to a smooth transversal section commutes to the computation of multiplier ideals and V-filtrations. As an application we prove the constancy of the jumping numbers and the spectrum along any stratum of a Whitney regular stratification.     

Tangent circle bundles admit positive open book decompositions along arbitrary links

In the present paper we generalize the divide lying in the unit disk, introduced by A'Campo, to compact, oriented, smooth surfaces, and prove a fibration theorem for generalized divides. As a consequence, we will show that, for any link L in the tangent circle bundle Y to the compact surface, there exists an additional knot K such that the link L∪K is the binding of a “positive” open book decomposition of Y.