25,266 research outputs found
Subdivisional spaces and graph braid groups
We study the problem of computing the homology of the configuration spaces of
a finite cell complex . We proceed by viewing , together with its
subdivisions, as a subdivisional space--a kind of diagram object in a category
of cell complexes. After developing a version of Morse theory for subdivisional
spaces, we decompose and show that the homology of the configuration spaces
of is computed by the derived tensor product of the Morse complexes of the
pieces of the decomposition, an analogue of the monoidal excision property of
factorization homology.
Applying this theory to the configuration spaces of a graph, we recover a
cellular chain model due to \'{S}wi\k{a}tkowski. Our method of deriving this
model enhances it with various convenient functorialities, exact sequences, and
module structures, which we exploit in numerous computations, old and new.Comment: 71 pages, 15 figures. Typo fixed. May differ slightly from version
published in Documenta Mathematic
Decomposition into pairs-of-pants for complex algebraic hypersurfaces
It is well-known that a Riemann surface can be decomposed into the so-called
pairs-of-pants. Each pair-of-pants is diffeomorphic to a Riemann sphere minus 3
points. We show that a smooth complex projective hypersurface of arbitrary
dimension admits a similar decomposition. The n-dimensional pair-of-pants is
diffeomorphic to the complex projective n-space minus n+2 hyperplanes.
Alternatively, these decompositions can be treated as certain fibrations on
the hypersurfaces. We show that there exists a singular fibration on the
hypersurface with an n-dimensional polyhedral complex as its base and a real
n-torus as its fiber. The base accomodates the geometric genus of a
hypersurface V. Its homotopy type is a wedge of h^{n,0}(V) spheres S^n.Comment: 35 pages, 9 figures, final version to appear in Topolog
Analytic cell decomposition and analytic motivic integration
The main results of this paper are a Cell Decomposition Theorem for Henselian
valued fields with analytic structure in an analytic Denef-Pas language, and
its application to analytic motivic integrals and analytic integrals over
\FF_q((t)) of big enough characteristic. To accomplish this, we introduce a
general framework for Henselian valued fields with analytic structure, and
we investigate the structure of analytic functions in one variable, defined on
annuli over . We also prove that, after parameterization, definable analytic
functions are given by terms. The results in this paper pave the way for a
theory of \emph{analytic} motivic integration and \emph{analytic} motivic
constructible functions in the line of R. Cluckers and F. Loeser
[\emph{Fonctions constructible et int\'egration motivic I}, Comptes rendus de
l'Acad\'emie des Sciences, {\bf 339} (2004) 411 - 416]
Crossed simplicial groups and structured surfaces
We propose a generalization of the concept of a Ribbon graph suitable to
provide combinatorial models for marked surfaces equipped with a G-structure.
Our main insight is that the necessary combinatorics is neatly captured in the
concept of a crossed simplicial group as introduced, independently, by
Krasauskas and Fiedorowicz-Loday. In this context, Connes' cyclic category
leads to Ribbon graphs while other crossed simplicial groups naturally yield
different notions of structured graphs which model unoriented, N-spin, framed,
etc, surfaces. Our main result is that structured graphs provide orbicell
decompositions of the respective G-structured moduli spaces. As an application,
we show how, building on our theory of 2-Segal spaces, the resulting theory can
be used to construct categorified state sum invariants of G-structured
surfaces.Comment: 86 pages, v2: revised versio
A jigsaw puzzle framework for homogenization of high porosity foams
An approach to homogenization of high porosity metallic foams is explored.
The emphasis is on the \Alporas{} foam and its representation by means of
two-dimensional wire-frame models. The guaranteed upper and lower bounds on the
effective properties are derived by the first-order homogenization with the
uniform and minimal kinematic boundary conditions at heart. This is combined
with the method of Wang tilings to generate sufficiently large material samples
along with their finite element discretization. The obtained results are
compared to experimental and numerical data available in literature and the
suitability of the two-dimensional setting itself is discussed.Comment: 11 pages, 7 figures, 3 table
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