2,055 research outputs found
On Extension Of Functors
A.Chigogidze defined for each normal functor on the category Comp an
extension which is a normal functor on the category Tych. We consider this
extension for any functor on the category Comp and investigate which properties
it preserves from the definition it preserves from the definition of normal
functor. We investigate as well some topological properties of such extension
Extending Topological Surgery to Natural Processes and Dynamical Systems
Topological surgery is a mathematical technique used for creating new
manifolds out of known ones. We observe that it occurs in natural phenomena
where a sphere of dimension 0 or 1 is selected, forces are applied and the
manifold in which they occur changes type. For example, 1-dimensional surgery
happens during chromosomal crossover, DNA recombination and when cosmic
magnetic lines reconnect, while 2-dimensional surgery happens in the formation
of tornadoes, in the phenomenon of Falaco solitons, in drop coalescence and in
the cell mitosis. Inspired by such phenomena, we introduce new theoretical
concepts which enhance topological surgery with the observed forces and
dynamics. To do this, we first extend the formal definition to a continuous
process caused by local forces. Next, for modeling phenomena which do not
happen on arcs or surfaces but are 2-dimensional or 3-dimensional, we fill in
the interior space by defining the notion of solid topological surgery. We
further introduce the notion of embedded surgery in for modeling
phenomena which involve more intrinsically the ambient space, such as the
appearance of knotting in DNA and phenomena where the causes and effect of the
process lies beyond the initial manifold, such as the formation of black holes.
Finally, we connect these new theoretical concepts with a dynamical system and
we present it as a model for both 2-dimensional 0-surgery and natural phenomena
exhibiting a `hole drilling' behavior. We hope that through this study,
topology and dynamics of many natural phenomena, as well as topological surgery
itself, will be better understood.Comment: 54 pages, 34 figure
On Operadic Actions on Spaces of Knots and 2-Links
In the present work, we realize the space of string 2-links as
a free algebra over a colored operad denoted (for "Swiss-Cheese
for links"). This result extends works of Burke and Koytcheff about the
quotient of by its center and is compatible with Budney's
freeness theorem for long knots. From an algebraic point of view, our main
result refines Blaire, Burke and Koytcheff's theorem on the monoid of isotopy
classes of string links. Topologically, it expresses the homotopy type of the
isotopy class of a string 2-link in terms of the homotopy types of the classes
of its prime factors.Comment: Comments are welcom
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