5,528 research outputs found
A spectral sequence for polyhedral products
The purpose of this paper is to exhibit fine structure for polyhedral
products Z(K;(X,A) and polyhedral smash products .
(Moment-angle complexes are special cases for which (X,A) = (D^2,S^1)). There
are three main parts. The first defines a natural filtration of the polyhedral
product and derives properties of the resulting spectral sequence. This is
followed with applications. The second part uses the first to give a
homological decomposition of the polyhedral smash product. Finally there are
applications to the ring structure of H*(Z(K;(X,A))) for CW-pairs (X,A)
satisfying suitable freeness conditions
Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension
We elaborate on the interpretation of some mixed finite element spaces in
terms of differential forms. First we develop a framework in which we show how
tools from algebraic topology can be applied to the study of their
cohomological properties. The analysis applies in particular to certain
finite element spaces, extending results in trivial topology often referred to
as the exact sequence property. Then we define regularization operators.
Combined with the standard interpolators they enable us to prove discrete
Poincar\'e-Friedrichs inequalities and discrete Rellich compactness for finite
element spaces of differential forms of arbitrary degree on compact manifolds
of arbitrary dimension.Comment: 26 page
Linking combinatorial and classical dynamics: Conley index and Morse decompositions
We prove that every combinatorial dynamical system in the sense of Forman,
defined on a family of simplices of a simplicial complex, gives rise to a
multivalued dynamical system F on the geometric realization of the simplicial
complex. Moreover, F may be chosen in such a way that the isolated invariant
sets, Conley indices, Morse decompositions, and Conley-Morse graphs of the two
dynamical systems are in one-to-one correspondence
Covariantly functorial wrapped Floer theory on Liouville sectors
We introduce a class of Liouville manifolds with boundary which we call
Liouville sectors. We define the wrapped Fukaya category, symplectic
cohomology, and the open-closed map for Liouville sectors, and we show that
these invariants are covariantly functorial with respect to inclusions of
Liouville sectors. From this foundational setup, a local-to-global principle
for Abouzaid's generation criterion follows.Comment: Final version to appear in Publ Math IHES, 115 pages, 15 figure
Toric Differential Inclusions and a Proof of the Global Attractor Conjecture
The global attractor conjecture says that toric dynamical systems (i.e., a
class of polynomial dynamical systems on the positive orthant) have a globally
attracting point within each positive linear invariant subspace -- or,
equivalently, complex balanced mass-action systems have a globally attracting
point within each positive stoichiometric compatibility class. A proof of this
conjecture implies that a large class of nonlinear dynamical systems on the
positive orthant have very simple and stable dynamics. The conjecture
originates from the 1972 breakthrough work by Fritz Horn and Roy Jackson, and
was formulated in its current form by Horn in 1974. We introduce toric
differential inclusions, and we show that each positive solution of a toric
differential inclusion is contained in an invariant region that prevents it
from approaching the origin. We use this result to prove the global attractor
conjecture. In particular, it follows that all detailed balanced mass action
systems and all deficiency zero weakly reversible networks have the global
attractor property.Comment: Version 2 corrects some typos and makes other minor improvement
E-infinity obstruction theory
The space of E-infinity structures on an simplicial operad C is the limit of
a tower of fibrations, so its homotopy is the abutment of a Bousfield-Kan
fringed spectral sequence. The spectral sequence begins (under mild
restrictions) with the stable cohomotopy of the graded right Gamma-module
formed by the homotopy groups of C ; the fringe contains an obstruction theory
for the existence of E-infinity structures on C. This formulation is very
flexible: applications extend beyond structures on classical ring spectra to
examples (in references) in motivic homotopy theory.Comment: 33 page
Locally standard torus actions and sheaves over Buchsbaum posets
We consider a sheaf of exterior algebras on a simplicial poset and
introduce a notion of homological characteristic function. Two natural objects
are associated with these data: a graded sheaf and a graded
cosheaf . When is a homology manifold, we prove the
isomorphism which can be
considered as an extension of the Poincare duality. In general, there is a
spectral sequence , where is the
local homology stack on . This spectral sequence, in turn, extends
Zeeman--McCrory spectral sequence. This sheaf-theoretical result is applied to
toric topology. We consider a manifold with a locally standard action of a
compact torus and acyclic proper faces of the orbit space. A principal torus
bundle is associated with , so that . The orbit type
filtration on is covered by the topological filtration on . We prove
that homological spectral sequences associated with these two filtrations are
isomorphic in many nontrivial positions.Comment: 23 pages. Several typos were corrected and the numbering of theorems
change
Supersymmetric Euclidean Field Theories and K-theory
We construct spaces of 1-dimensional supersymmetric Euclidean field theories
and show that they represent real or complex K-theory. A noteworthy feature of
our bordism category is that the identity bordism of a point is connected to
intervals of positive length.Comment: 24 pages, 4 figure
Persistent homology of the sum metric
Given finite metric spaces and , we investigate the
persistent homology of the Cartesian product
equipped with the sum metric . Interpreting persistent homology as a
module over a polynomial ring, one might expect the usual K\"unneth short exact
sequence to hold. We prove that it holds for and , and we
illustrate with the Hamming cube that it fails for . For , the prediction for from the expected
K\"unneth short exact sequence has a natural surjection onto . We compute the nontrivial kernel of this surjection for the splitting of
Hamming cubes . For all ,
the interleaving distance between the prediction for and the
true persistent homology is bounded above by the minimum of the diameters of
and . As preliminary results of independent interest, we establish an
algebraic K\"unneth formula for simplicial modules over the ring
of polynomials with coefficients in a field and
exponents in , as well as a K\"unneth formula for
the persistent homology of -filtered simplicial sets -- both of
these K\"unneth formulas hold in all homological dimensions .Comment: To appear in Journal of Pure and Applied Algebr
Homological stability of revisited
We give another proof of a theorem of Hatcher and Vogtmann stating that the
sequence satisfies integral homological stability. The paper is for
the most part expository, and we also explain Quillen's method for proving
homological stability.Comment: To appear in the proceedings of the 7th Seasonal Institute of the
Mathematical Society of Japan (MSJ-SI) on Hyperbolic Geometry and Geometric
Group Theory held July 30th - August 5th 2015 at the University of Toky
- β¦