394 research outputs found
Cohomotopy sets of 4-manifolds
Elementary geometric arguments are used to compute the group of homotopy
classes of maps from a 4-manifold X to the 3-sphere, and to enumerate the
homotopy classes of maps from X to the 2-sphere. The former completes a project
initiated by Steenrod in the 1940's, and the latter provides geometric
arguments for and extensions of recent homotopy theoretic results of Larry
Taylor. These two results complete the computation of all the cohomotopy sets
of closed oriented 4-manifolds and provide a framework for the study of Morse
2-functions on 4-manifolds, a subject that has garnered considerable recent
attention.Comment: 20 pages, 6 figures; this version to appear in the FreedmanFest (G&T
Monographs, Volume 18
On metrics of curvature 1 with four conic singularities on tori and on the sphere
We discuss conformal metrics of curvature 1 on tori and on the sphere, with
four conic singularities whose angles are multiples of pi/2. Besides some
general results we study in detail the family of such symmetric metrics on the
sphere, with angles (pi/2,3pi/2,pi/2,3pi/2).Comment: 25 pages, 5 figure
The geometry of dynamical triangulations
We discuss the geometry of dynamical triangulations associated with
3-dimensional and 4-dimensional simplicial quantum gravity. We provide
analytical expressions for the canonical partition function in both cases, and
study its large volume behavior. In the space of the coupling constants of the
theory, we characterize the infinite volume line and the associated critical
points. The results of this analysis are found to be in excellent agreement
with the MonteCarlo simulations of simplicial quantum gravity. In particular,
we provide an analytical proof that simply-connected dynamically triangulated
4-manifolds undergo a higher order phase transition at a value of the inverse
gravitational coupling given by 1.387, and that the nature of this transition
can be concealed by a bystable behavior. A similar analysis in the
3-dimensional case characterizes a value of the critical coupling (3.845) at
which hysteresis effects are present.Comment: 166 pages, Revtex (latex) fil
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