415 research outputs found
Resonance varieties and Dwyer-Fried invariants
The Dwyer-Fried invariants of a finite cell complex X are the subsets
\Omega^i_r(X) of the Grassmannian of r-planes in H^1(X,\Q) which parametrize
the regular \Z^r-covers of X having finite Betti numbers up to degree i. In
previous work, we showed that each \Omega-invariant is contained in the
complement of a union of Schubert varieties associated to a certain subspace
arrangement in H^1(X,\Q). Here, we identify a class of spaces for which this
inclusion holds as equality. For such "straight" spaces X, all the data
required to compute the \Omega-invariants can be extracted from the resonance
varieties associated to the cohomology ring H^*(X,\Q). In general, though,
translated components in the characteristic varieties affect the answer.Comment: 39 pages; to appear in "Arrangements of Hyperplanes - Sapporo 2009,"
Advanced Studies in Pure Mathematic
Fundamental groups, Alexander invariants, and cohomology jumping loci
We survey the cohomology jumping loci and the Alexander-type invariants
associated to a space, or to its fundamental group. Though most of the material
is expository, we provide new examples and applications, which in turn raise
several questions and conjectures.
The jump loci of a space X come in two basic flavors: the characteristic
varieties, or, the support loci for homology with coefficients in rank 1 local
systems, and the resonance varieties, or, the support loci for the homology of
the cochain complexes arising from multiplication by degree 1 classes in the
cohomology ring of X. The geometry of these varieties is intimately related to
the formality, (quasi-) projectivity, and homological finiteness properties of
\pi_1(X).
We illustrate this approach with various applications to the study of
hyperplane arrangements, Milnor fibrations, 3-manifolds, and right-angled Artin
groups.Comment: 45 pages; accepted for publication in Contemporary Mathematic
The spectral sequence of an equivariant chain complex and homology with local coefficients
We study the spectral sequence associated to the filtration by powers of the
augmentation ideal on the (twisted) equivariant chain complex of the universal
cover of a connected CW-complex X. In the process, we identify the d^1
differential in terms of the coalgebra structure of H_*(X,\k), and the
\k\pi_1(X)-module structure on the twisting coefficients. In particular, this
recovers in dual form a result of Reznikov, on the mod p cohomology of cyclic
p-covers of aspherical complexes. This approach provides information on the
homology of all Galois covers of X. It also yields computable upper bounds on
the ranks of the cohomology groups of X, with coefficients in a prime-power
order, rank one local system. When X admits a minimal cell decomposition, we
relate the linearization of the equivariant cochain complex of the universal
abelian cover to the Aomoto complex, arising from the cup-product structure of
H^*(X,\k), thereby generalizing a result of Cohen and Orlik.Comment: 38 pages, 1 figure (section 10 of version 1 has been significantly
expanded into a separate paper, available at arXiv:0901.0105); accepted for
publication in the Transactions of the American Mathematical Societ
When does the associated graded Lie algebra of an arrangement group decompose?
Let \A be a complex hyperplane arrangement, with fundamental group G and
holonomy Lie algebra \H. Suppose \H_3 is a free abelian group of minimum
possible rank, given the values the M\"obius function \mu: \L_2\to \Z takes on
the rank 2 flats of \A. Then the associated graded Lie algebra of G decomposes
(in degrees 2 and higher) as a direct product of free Lie algebras. In
particular, the ranks of the lower central series quotients of the group are
given by \phi_r(G)=\sum_{X\in \L_2} \phi_r(F_{\mu(X)}), for r\ge 2. We
illustrate this new Lower Central Series formula with several families of
examples.Comment: 14 pages, accepted for publication by Commentarii Mathematici
Helvetic
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
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