67,936 research outputs found
A uniqueness theorem for stable homotopy theory
In this paper we study the global structure of the stable homotopy theory of
spectra. We establish criteria for when the homotopy theory associated to a
given stable model category agrees with the classical stable homotopy theory of
spectra. One sufficient condition is that the associated homotopy category is
equivalent to the stable homotopy category as a triangulated category with an
action of the ring of stable homotopy groups of spheres. In other words, the
classical stable homotopy theory, with all of its higher order information, is
determined by the homotopy category as a triangulated category with an action
of the stable homotopy groups of spheres. Another sufficient condition is the
existence of a small generating object (corresponding to the sphere spectrum)
for which a specific `unit map' from the infinite loop space QS^0 to the
endomorphism space is a weak equivalence
Homotopy limits of model categories and more general homotopy theories
Generalizing a definition of homotopy fiber products of model categories, we
give a definition of the homotopy limit of a diagram of left Quillen functors
between model categories. As has been previously shown for homotopy fiber
products, we prove that such a homotopy limit does in fact correspond to the
usual homotopy limit, when we work in a more general model for homotopy
theories in which they can be regarded as objects of a model category.Comment: 10 pages; a few minor changes made. arXiv admin note: text overlap
with arXiv:0811.317
Milnor Invariants for Spatial Graphs
Link homotopy has been an active area of research for knot theorists since
its introduction by Milnor in the 1950s. We introduce a new equivalence
relation on spatial graphs called component homotopy, which reduces to link
homotopy in the classical case. Unlike previous attempts at generalizing link
homotopy to spatial graphs, our new relation allows analogues of some standard
link homotopy results and invariants.
In particular we can define a type of Milnor group for a spatial graph under
component homotopy, and this group determines whether or not the spatial graph
is splittable. More surprisingly, we will also show that whether the spatial
graph is splittable up to component homotopy depends only on the link homotopy
class of the links contained within it. Numerical invariants of the relation
will also be produced.Comment: 11 pages, 5 figure
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