8,173 research outputs found
Knots, operads and double loop spaces
We show that the space of long knots in an euclidean space of dimension
larger than three is a double loop space, proving a conjecture by Sinha. We
construct also a double loop space structure on framed long knots, and show
that the map forgetting the framing is not a double loop map in odd dimension.
However there is always such a map in the reverse direction expressing the
double loop space of framed long knots as a semidirect product. A similar
compatible decomposition holds for the homotopy fiber of the inclusion of long
knots into immersions. We show also via string topology that the space of
closed knots in a sphere, suitably desuspended, admits an action of the little
2-discs operad in the category of spectra. A fundamental tool is the
McClure-Smith cosimplicial machinery, that produces double loop spaces out of
topological operads with multiplication.Comment: 16 page
Configuration spaces on the sphere and higher loop spaces
We show that the homology over a field of the space of free maps from the
n-sphere to the n-fold suspension of X depends only on the cohomology algebra
of X and compute it explicitly. We compute also the homology of the closely
related labelled configuration space on the n-sphere with labels in X and of
its completion, that depends only on the homology of X. In many but not all
cases the homology of the configuration space coincides with the homology of
the mapping space. In particular we obtain the homology of the unordered
configuration spaces on a sphere.Comment: 11 page
The topological cyclic Deligne conjecture
Let O be a cyclic topological operad with multiplication. In the framework of
the cosimplicial machinery by McClure and Smith, we prove that the totalization
of the cosimplicial space associated to O has an action of an operad equivalent
to the framed little 2-discs operad.Comment: An appendix comparing with the construction by McClure and Smith has
been adde
Configuration spaces are not homotopy invariant
We present a counterexample to the conjecture on the homotopy invariance of
configuration spaces. More precisely, we consider the lens spaces and
, and prove that their configuration spaces are not homotopy
equivalent by showing that their universal coverings have different Massey
products.Comment: 6 page
On the notions of upper and lower density
Let be the power set of . We say that a
function is an upper density if, for
all and , the following hold: (F1)
; (F2) if ;
(F3) ; (F4) , where ; (F5)
.
We show that the upper asymptotic, upper logarithmic, upper Banach, upper
Buck, upper Polya, and upper analytic densities, together with all upper
-densities (with a real parameter ), are upper
densities in the sense of our definition. Moreover, we establish the mutual
independence of axioms (F1)-(F5), and we investigate various properties of
upper densities (and related functions) under the assumption that (F2) is
replaced by the weaker condition that for every
.
Overall, this allows us to extend and generalize results so far independently
derived for some of the classical upper densities mentioned above, thus
introducing a certain amount of unification into the theory.Comment: 26 pp, no figs. Added a 'Note added in proof' at the end of Sect. 7
to answer Question 6. Final version to appear in Proc. Edinb. Math. Soc. (the
paper is a prequel of arXiv:1510.07473
Upper and lower densities have the strong Darboux property
Let be the power set of . An upper density (on
) is a non\-decreasing and subadditive function such that and for all and , where .
The upper asymptotic, upper Banach, upper logarithmic, upper Buck, upper
P\'olya, and upper analytic densities are examples of upper densities.
We show that every upper density has the strong Darboux property,
and so does the associated lower density, where a function is said to have the strong Darboux property if, whenever and , there is a set such
that and . In fact, we prove the above under
the assumption that the monotonicity of is relaxed to the weaker
condition that for every .Comment: 10 pages, no figures. Fixed minor details and streamlined the
exposition. To appear in Journal of Number Theor
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