#### Algebraic and Geometric Topology 5 (2005),
paper no. 13, pages 237-300.

## On several varieties of cacti and their relations

### Ralph M. Kaufmann

**Abstract**.
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.
**Keywords**.
Cacti, (quasi-)operad, string topology, loop space, bi-crossed product, (framed) little discs, quasi-fibration

**AMS subject classification**.
Primary: 55P48, 18D40.
Secondary: 55P35, 16S35.

**DOI:** 10.2140/agt.2005.5.237

**E-print:** `arXiv:math.QA/0209131`

Submitted: 9 January 2004.
(Revised: 16 March 2005.)
Accepted: 30 March 2005.
Published: 15 April 2005.

Notes on file formats
Ralph M. Kaufmann

Department of Mathematics, University of Connecticut

196 Auditorium Road, Storrs, CT 06269-3009, USA

Email: kaufmann@math.uconn.edu

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