36 research outputs found
An update on domineering on rectangular boards
Domineering is a combinatorial game played on a subset of a rectangular grid
between two players. Each board position can be put into one of four outcome
classes based on who the winner will be if both players play optimally. In this
note, we review previous work, establish the outcome classes for several
dimensions of rectangular board, and restrict the outcome class in several
more.Comment: 9 pages. References fixe
Homotopically trivializing the circle in the framed little disks
This paper confirms the following suggestion of Kontsevich. In the
appropriate derived sense, an action of the framed little disks operad and a
trivialization of the circle action is the same information as an action of the
Deligne-Mumford-Knudsen operad. This improves an earlier result of the author
and Bruno Vallette.Comment: 36 pages. This version accepted for publication by the Journal of
Topolog
A criterion for existence of right-induced model structures
Suppose that is a functor whose target is a
Quillen model category. We give a succinct sufficient condition for the
existence of the right-induced model category structure on in the
case when admits both adjoints. We give several examples, including
change-of-rings, operad-like structures, and anti-involutive structures on
infinity categories. For the last of these, we explore anti-involutive
structures for several different models of -categories, and show
that known Quillen equivalences between base model categories lift to
equivalences
Cones in homotopy probability theory
This note defines cones in homotopy probability theory and demonstrates that
a cone over a space is a reasonable replacement for the space. The homotopy
Gaussian distribution in one variable is revisited as a cone on the ordinary
Gaussian.Comment: 8 pages. Missing reference adde
The minimal model for the Batalin-Vilkovisky operad
The purpose of this paper is to explain and to generalize, in a homotopical
way, the result of Barannikov-Kontsevich and Manin which states that the
underlying homology groups of some Batalin-Vilkovisky algebras carry a
Frobenius manifold structure. To this extent, we first make the minimal model
for the operad encoding BV-algebras explicit. Then we prove a homotopy transfer
theorem for the associated notion of homotopy BV-algebra. The final result
provides an extension of the action of the homology of the
Deligne-Mumford-Knudsen moduli space of genus 0 curves on the homology of some
BV-algebras to an action via higher homotopical operations organized by the
cohomology of the open moduli space of genus zero curves. Applications in
Poisson geometry and Lie algebra cohomology and to the Mirror Symmetry
conjecture are given.Comment: New section added containing applications to Poisson geometry, Lie
algebra cohomology and to the Mirror Symmetry conjecture. [36 pages, 4
figures
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