8 research outputs found
The open dihypergraph dichotomy for generalized Baire spaces and its applications
The open graph dichotomy for a subset of the Baire space
states that any open graph on either admits a coloring in
countably many colors or contains a perfect complete subgraph. This strong
version of the open graph axiom for was introduced by Feng and
Todor\v{c}evi\'c to investigate definable sets of reals. We first show that its
recent generalization to infinite dimensional directed hypergraphs by Carroy,
Miller and Soukup holds for all subsets of the Baire space in Solovay's model,
extending a theorem of Feng in dimension .
The main theorem lifts this result to generalized Baire spaces
in two ways.
(1) For any regular infinite cardinal , the following holds after a
L\'evy collapse of an inaccessible cardinal to .
Suppose that is a -dimensional box-open directed hypergraph on a
subset of such that is definable from a -sequence
of ordinals. Then either admits a coloring in many colors or there
exists a continuous homomorphism from a canonical large directed hypergraph to
.
(2) If is a Mahlo cardinal, then the previous result extends to all
box-open directed hypergraphs on any subset of that is
definable from a -sequence of ordinals.
We derive several applications to definable subsets of generalized Baire
spaces, among them variants of the Hurewicz dichotomy that characterizes
subsets of sets, an asymmetric version of the Baire property, an
analogue of the Kechris-Louveau-Woodin dichotomy that characterizes when two
disjoint sets can be separated by an set, the determinacy of
V\"a\"an\"anen's perfect set game for all subsets of , and an
analogue of the Jayne-Rogers theorem that characterizes functions which are
-continuous with closed pieces.Comment: 115 pages, 11 figures. Added new results in Section 6.2.2 which
strengthen and replace the results in Section 6.3 of the previous version.
Improved results in Section 5.3. Various other minor corrections. Comments
are welcom
Four-manifolds, geometries and knots
The goal of this book is to characterize algebraically the closed 4-manifolds
that fibre nontrivially or admit geometries in the sense of Thurston, or which
are obtained by surgery on 2-knots, and to provide a reference for the topology
of such manifolds and knots. The first chapter is purely algebraic. The rest of
the book may be divided into three parts: general results on homotopy and
surgery (Chapters 2-6), geometries and geometric decompositions (Chapters
7-13), and 2-knots (Chapters 14-18). In many cases the Euler characteristic,
fundamental group and Stiefel-Whitney classes together form a complete system
of invariants for the homotopy type of such manifolds, and the possible values
of the invariants can be described explicitly. The strongest results are
characterizations of manifolds which fibre homotopically over S^1 or an
aspherical surface (up to homotopy equivalence) and infrasolvmanifolds (up to
homeomorphism). As a consequence 2-knots whose groups are poly-Z are determined
up to Gluck reconstruction and change of orientations by their groups alone.
This book arose out of two earlier books "2-Knots and their Groups" and "The
Algebraic Characterization of Geometric 4-Manifolds", published by Cambridge
University Press for the Australian Mathematical Society and for the London
Mathematical Society, respectively. About a quarter of the present text has
been taken from these books, and I thank Cambridge University Press for their
permission to use this material. The book has been revised in March 2007. For
details see the end of the preface.Comment: This is the revised version published by Geometry & Topology
Monographs in March 200
Superperfect pairs of trees in graphs
SIGLEAvailable from British Library Document Supply Centre- DSC:7769.09285(WU-DCS-RR--239) / BLDSC - British Library Document Supply CentreGBUnited Kingdo