19,845 research outputs found
On several varieties of cacti and their relations
Motivated by string topology and the arc operad, we introduce the notion of
quasi-operads and consider four (quasi)-operads which are different varieties
of the operad of cacti. These are cacti without local zeros (or spines) and
cacti proper as well as both varieties with fixed constant size one of the
constituting loops. Using the recognition principle of Fiedorowicz, we prove
that spineless cacti are equivalent as operads to the little discs operad. It
turns out that in terms of spineless cacti Cohen's Gerstenhaber structure and
Fiedorowicz' braided operad structure are given by the same explicit chains. We
also prove that spineless cacti and cacti are homotopy equivalent to their
normalized versions as quasi-operads by showing that both types of cacti are
semi-direct products of the quasi-operad of their normalized versions with a
re-scaling operad based on R>0. Furthermore, we introduce the notion of
bi-crossed products of quasi-operads and show that the cacti proper are a
bi-crossed product of the operad of cacti without spines and the operad based
on the monoid given by the circle group S^1. We also prove that this particular
bi-crossed operad product is homotopy equivalent to the semi-direct product of
the spineless cacti with the group S^1. This implies that cacti are equivalent
to the framed little discs operad. These results lead to new CW models for the
little discs and the framed little discs operad.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-13.abs.htm
Enumeration of m-ary cacti
The purpose of this paper is to enumerate various classes of cyclically
colored m-gonal plane cacti, called m-ary cacti. This combinatorial problem is
motivated by the topological classification of complex polynomials having at
most m critical values, studied by Zvonkin and others. We obtain explicit
formulae for both labelled and unlabelled m-ary cacti, according to i) the
number of polygons, ii) the vertex-color distribution, iii) the vertex-degree
distribution of each color. We also enumerate m-ary cacti according to the
order of their automorphism group. Using a generalization of Otter's formula,
we express the species of m-ary cacti in terms of rooted and of pointed cacti.
A variant of the m-dimensional Lagrange inversion is then used to enumerate
these structures. The method of Liskovets for the enumeration of unrooted
planar maps can also be adapted to m-ary cacti.Comment: LaTeX2e, 28 pages, 9 figures (eps), 3 table
Groups, cacti and framed little discs
Let G be a topological group. Then the based loopspace of G is an algebra
over the cacti operad, while the double loopspace of the classifying space of G
is an algebra over the framed little discs operad. This paper shows that these
two algebras are equivalent, in the sense that they are weakly equivalent
E-algebras, where E is an operad weakly equivalent to both framed little discs
and cacti. We recover the equivalence between cacti and framed little discs,
and Menichi's isomorphism between the BV-algebras obtained by taking the
homology of the loopspace of G and of the double loopspace of BG.Comment: 40 page
Cells and cacti
Let be a Coxeter system, let be a weight function on
and let denote the associated {\it cactus group}.
Following an idea of I. Losev, we construct an action of on which has nice properties with respect to the
partition of into left, right or two-sided cells (under some hypothesis,
which hold for instance if is constant or if is finite of rank
\textless{} 5). It must be noticed that the action depends heavily on
.Comment: 23 pages. This new version extends the scope of validity of the main
result by removing an hypothesis. For this purpose, we have slightly extended
some results of arXiv:1502.0166
McClure-Smith cosimplicial machinery and the cacti operad
McClure and Smith constructed a functor that sends a topological
multiplicative operad O to an E_2 algebra TotO. They define in fact an operad
D_2 (acting on the totalization TotO) weakly equivalent to the little 2-disks
operad. On the other hand, Salvatore showed that D_2 is isomorphic to the cacti
operad MS, which has a nice geometric description. He also built a geometric
action of MS on TotO. In this paper we detail the McClure-Smith action and the
cacti action. Our main result says that they are compatible in the sense that
some squares must commute.Comment: 16 pages, 3 figure
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