5,209 research outputs found
A note on the homotopy type of the Alexander dual
We investigate the homotopy type of the Alexander dual of a simplicial complex. It is known that in general the homotopy type of K does not determine the homotopy type of its dual K∗ . We construct for each finitely presented group G, a simply connected simplicial complex K such that Ï€1(K∗ ) = G and study sufficient conditions on K for K∗ to have the homotopy type of a sphere. We extend the simplicial Alexander duality to the more general context of reduced lattices and relate this construction with Bier spheres using deleted joins of lattices. Finally we introduce an alternative dual, in the context of reduced lattices, with the same homotopy type as the Alexander dual but smaller and simpler to compute.Fil: Minian, Elias Gabriel. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santalo"; ArgentinaFil: RodrÃguez, Jorge Tomás. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santalo"; Argentin
Discrete Morse theory for moment-angle complexes of pairs (D^n,S^{n-1})
For a finite simplicial complex K and a CW-pair (X,A), there is an associated
CW-complex Z_K(X,A), known as a polyhedral product. We apply discrete Morse
theory to a particular CW-structure on the n-sphere moment-angle complexes
Z_K(D^{n}, S^{n-1}). For the class of simplicial complexes with
vertex-decomposable duals, we show that the associated n-sphere moment-angle
complexes have the homotopy type of wedges of spheres. As a corollary we show
that a sufficiently high suspension of any restriction of a simplicial complex
with vertex-decomposable dual is homotopy equivalent to a wedge of spheres.Comment: Corollary 1.2 and 1 reference added. Some formulations and arguments
made more precis
Alexander duality, gropes and link homotopy
We prove a geometric refinement of Alexander duality for certain 2-complexes,
the so-called gropes, embedded into 4-space. This refinement can be roughly
formulated as saying that 4-dimensional Alexander duality preserves the
disjoint Dwyer filtration. In addition, we give new proofs and extended
versions of two lemmas of Freedman and Lin which are of central importance in
the A-B-slice problem, the main open problem in the classification theory of
topological 4-manifolds. Our methods are group theoretical, rather than using
Massey products and Milnor \mu-invariants as in the original proofs.Comment: 19 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTVol1/paper5.abs.htm
Spectral pairs, Alexander modules, and boundary manifolds
Let f: \CN \rightarrow \C be a reduced polynomial map, with ,
\U=\CN \setminus D and boundary manifold M=\partial \U. Assume that is
transversal at infinity and has only isolated singularities. Then the only
interesting non-trivial Alexander modules of \U and resp. appear in the
middle degree . We revisit the mixed Hodge structures on these Alexander
modules and study their associated spectral pairs (or equivariant mixed Hodge
numbers). We obtain upper bounds for the spectral pairs of the -th Alexander
module of \U, which can be viewed as a Hodge-theoretic refinement of
Libgober's divisibility result for the corresponding Alexander polynomials. For
the boundary manifold , we show that the spectral pairs associated to the
non-unipotent part of the -th Alexander module of can be computed in
terms of local contributions (coming from the singularities of ) and
contributions from "infinity".Comment: comments are very welcom
Moment-angle complexes, monomial ideals, and Massey products
Associated to every finite simplicial complex K there is a "moment-angle"
finite CW-complex, Z_K; if K is a triangulation of a sphere, Z_K is a smooth,
compact manifold. Building on work of Buchstaber, Panov, and Baskakov, we study
the cohomology ring, the homotopy groups, and the triple Massey products of a
moment-angle complex, relating these topological invariants to the algebraic
combinatorics of the underlying simplicial complex. Applications to the study
of non-formal manifolds and subspace arrangements are given.Comment: 30 pages. Published versio
- …