Fibrewise completion and unstable Adams spectral sequences

We describe a tower of spaces whose inverse limit is a “fiberwise completion” of a fibrationE →B, and study the resulting spectral sequence converging to the homotopy groups of the space of lifts of a mapX →B. This is used to give a proof of the “generalized Sullivan conjecture”. All three authors were supported in part by the National Science Foundation.     

On topological invariants of stratified maps with non-Witt target

The Cappell-Shaneson decomposition theorem for self-dual sheaves asserts that on a space with only even-codimensional strata any self-dual sheaf is cobordant to an orthogonal sum of twisted intersection chain sheaves associated to the various strata. In sharp contrast to this result, we prove that on a space with only odd-codimensional strata (not necessarily Witt), any self-dual sheaf is cobordant to an intersection chain sheaf associated to the top stratum: the strata of odd codimension do not contribute terms. As a consequence, we obtain formulae for the pushforward of characteristic classes under a stratified map whose target need not satisfy the Witt space condition. To prove these results, we introduce a new category of superperverse sheaves, which we show to be abelian. Finally, we apply the results to the study of desingularization of non-Witt spaces and exhibit a singular space which admits a PL resolution in the sense of M. Kato, but no resolution by a stratified map.

     

On convergence of sections of sequences in Banach spaces

An elementary proof of the (known) fact that each element of the Banach spaceℓ _w ^p (X) of weakly absolutelyp-summable sequences (if 1≤p<∞) in the Banach spaceX is the norm limit of its sections if and only if each element ofℓ _w ^p (X) is a norm null sequence inX, is given. Little modification to this proof leads to a similar result for a family of Orlicz sequence spaces. Some applications to spaces of compact operators on Banach sequence spaces are considered.     

Taylor formula for homogenous groups and applications

In this paper, we provide a Taylor formula with integral remainder in the setting of homogeneous groups in the sense of Folland and Stein (Hardy spaces on homogeneous groups. Mathematical notes, vol 28. Princeton University Press, Princeton, 1982). This formula allows us to give a simplified proof of the so-called ‘Taylor inequality’. As a by-product, we furnish an explicit expression for the relevant Taylor polynomials. Applications are provided. Among others, it is given a sufficient condition for the real-analiticity of a function whose higher order derivatives (in the sense of the Lie algebra) satisfy a suitable growth condition. Moreover, we prove the ‘L-harmonicity’ of the Taylor polynomials related to a ‘L-harmonic’ function, when L is a general homogenous left-invariant differential operator on a homogeneous group. (This result is one of the ingredients for obtaining Schauder estimates related to L).     

Compactification of Limit ?-Net Spaces

 Limit ?-net spaces are defined as convergence spaces whose convergence is expressed by using generalized nets, the so-called ?-nets (where ? is a construct). For limit ?-net spaces we study compactifications, especially those ones that are analogous to the Alexandrov and Čech-Stone compactifications known for topological spaces.     

Compactification of Limit ?-Net Spaces

 Limit ?-net spaces are defined as convergence spaces whose convergence is expressed by using generalized nets, the so-called ?-nets (where ? is a construct). For limit ?-net spaces we study compactifications, especially those ones that are analogous to the Alexandrov and Čech-Stone compactifications known for topological spaces. (Received 24 February 2000)     

Simple Hironaka resolution in characteristic zero

Building upon work of Villamayor and Bierstone-Milman we give a proof of the canonical Hironaka principalization and desingularization. The idea of ``homogenized ideals" introduced in the paper gives a priori the canonicity of the algorithm and radically simplifies the proof.

     

Quasi-contractions on a nonnormal cone metric space

Ilić and Rakočević [6] proved a fixed point theorem for quasi-contractive mappings on cone metric spaces when the underlying cone is normal. Recently, Z. Kadelburg, S. Radenović, and V. Rakočević obtained a similar result without using the normality condition but only for a contractive constant λ ∈ (0, 1/2) [8]. In this note, using a new method of proof, we prove this theorem for any contractive constant λ ∈ (0, 1).     

On the homology of the Harmonic Archipelago

We calculate the singular homology and Čech cohomology groups of the Harmonic Archipelago. As a corollary, we prove that this space is not homotopy equivalent to the Griffiths space. This is interesting in view of Eda’s proof that the first singular homology groups of these spaces are isomorphic.     

Whitehead’s problem and reverse mathematics

Our main result establishes functorial desingularization of noetherian quasi-excellent schemes over Q with ordered boundaries. A functorial embedded desingularization of quasi-excellent schemes of characteristic zero is deduced. Furthermore, a standard simple argument extends these results to other categories including, in particular, (equivariant) embedded desingularization of the following objects of characteristic zero: qe algebraic stacks, qe formal schemes, complex and non-archimedean analytic spaces. We also obtain a semistable reduction theorem for formal schemes.     

Genus-one stable maps, local equations, and Vakil–Zinger’s desingularization

We describe an algebro-geometric approach to Vakil–Zinger’s desingularization of the main component of the moduli of genus one stable maps to \mathbbPⁿ{\mathbb{P}^{n}} (Vakil and Zinger in Res Announc Am Math Soc 13:53–59, 2007; Geom Topol 12(1):1–95, 2008). Our approach is based on understanding the local structure of this moduli space; it also gives a partial desingularization of the entire moduli space. The results proved should extend to higher genera.     

Counting metric spaces

If κ is a cardinal number, then any class of mutually non-homeomorphic metric spaces of size κ must be a set whose cardinality cannot exceed 2 ^κ . Our main result is a vivid construction of 2 ^κ mutually non-homeomorphic complete and both path connected and locally path connected metric spaces of size κ for each cardinal number κ from continuum up. Additionally we also deal with counting problems concerning countable metric spaces and Euclidean spaces.     

Associated families of pluriharmonic maps¶and isotropy

Like minimal surface immersions in 3-space, pluriharmonic maps into symmetric spaces allow a one-parameter family of isometric deformations rotating the differential (“associated family”); in fact, pluriharmonic maps are characterized by this property. We give a geometric proof of this fact and investigate the “isotropic” case where this family is constant. It turns out that isotropic pluriharmonic maps arise from certain holomorphic maps into flag manifolds. Further, we also consider higher dimensional generalizations of constant mean curvature surfaces which are K?hler submanifolds with parallel (1,1) part of their soecond fundamental form; under certain restrictions there are also characterized by having some kind of (“weak”) associated family. Examples where this family is constant arise from extrinsic K?hler symmetric spaces. Received: 8 July 1997     

Weights on polytopes

Theg-theorem describes the possible face-vectors of a simple polytopeP. Much of the author's proof of the necessity of its conditions, while working within the polytope algebra π, in fact only used the spaces of weights onP. Even though this proof was conceptually easier than the original, which employed techniques from algebraic geometry, nevertheless the properties of π which are needed still require some effort to establish, despite a recent simpler approach to π itself. In the earlier paper, doubt was expressed about whether two basic results could be proved directly for weights; later, it appeared that there might also be a possible problem concerning an alternative definition of the product of certain weights. In this paper these questions are settled, in the context of developing an independent theory of an algebra Ω of weights on polytopes. Since the construction of Ω is more approachable than that of π, a yet easier proof of theg-theorem is now available.     

Two remarks on the relationship between BMO-regularity and analytic stability of interpolation for lattices of measurable functions

We study in this paper Hardy-type spaces on a measure space ( \mathbbT \mathbb{T} , m) × (Ω, μ), where ( \mathbbT \mathbb{T} , m) is the unit circle with Lebesgue measure. There is a characterization of analytic stability for real interpolation of weighted Hardy spaces on \mathbbT \mathbb{T} × Ω, a complete proof of which was present in the literature only for the case where μ is a point mass. Here this gap is filled, and a proof of the general case is presented. In a previous work by Kislyakov, certain results concerning BMO-regular lattices on ( \mathbbT \mathbb{T} × Ω, m × μ) were proved under the assumption that the measure μ is discrete. Here this extraneous assumption is lifted. Bibliography: 9 titles.     

Uniformly convex spaces,bead spaces,and equivalence conditions

The notion of a metric bead space was introduced in the preceding paper (L.Pasicki: Bead spaces and fixed point theorems, Topology Appl., vol. 156 (2009), 1811–1816) and it was proved there that every bounded set in such a space (provided the space is complete) has a unique central point. The bead spaces themselves can be considered in particular as natural extensions of convex sets in uniformly convex spaces. It appears that normed bead spaces are identical with uniformly convex spaces. On the other hand the “metric” approach leads to new elementary conditions equivalent to the uniform convexity. The initial part of the paper contains the proof that discus spaces (they seem to have a richer structure) are identical with bead spaces.     

A Frobenius Theorem for Locally Convex Global Analysis

 We prove a Frobenius theorem for Banach space complemented subbundles of the tangent bundle of a manifold modelled on locally convex spaces. The proof is based on an implicit function theorem for maps from locally convex spaces to Banach spaces proved in a recent paper of the author.     

Various Slicing Indices on Banach Spaces

We give a short and direct proof for the computation of the Szlenk index of the C(K) spaces, when K is a countable compact space and determine their Lavrientiev indices. We also compute the Szlenk index of certain C(α) spaces, where α is an uncountable ordinal. Finally, we show that if the Szlenk index of a Banach space is ω (first infinite ordinal), then its weak*-dentability index is at most ω² and that this estimate is optimal. The first author was supported by the grants: Institutional Research Plan AV0Z10190503, A100190502, GA ČR 201/04/0090.     

Pseudoradial Spaces: Results and Problems

In this paper it is given a survey of principal results (old and new) concerning the class of pseudoradial spaces. In this class cardinal invariants and their inequalities are considered. The behaviour of pseudoradial spaces under the operations of taking topological products and subspaces are examined and a typical proof is given. A particular attention is dedicated to the so called “small cardinals” in connection with pseudoradiality. Pseudoradiality of 2^{ω 2} is also examined. It is proved that pseudoradiality can be ω¹ productive for spaces of weight at most ω². Finally, several open problems are presented. This work was supported by the National Group “Real Analysis, Measure Theory with Applications to Economy” of the Italian Ministery of Education, University and Research.