On several varieties of cacti and their relations

By R. Kaufmann. 2002

53 citations recorded by the source corpus.

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 Fiedorow- icz, 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. AMS Classification 55P48, 18D40; 55P35, 16S35

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/af5f30becba0791ac23fb27eb015765806ad82f8

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In Algebra and Number Theory

1. R. Kaufmann
2. M. Livernet
3. R. Penner
4. K. Costello

In Geometry and Topology

1. R. Kaufmann
2. M. Livernet
3. R. Penner
4. K. Costello

In Mathematical Physics

1. R. Kaufmann
2. M. Livernet
3. R. Penner
4. K. Costello

Citations in the corpus

• Hochschild homology of structured algebras (2011, unknown): “The chain complex of [1]–Sullivan diagrams of topological type a surface of genus 0 with p + 1 boundary components is a cellular complex for the pth space of the normalized cactus operad [60, 26].”
• A proof of a cyclic version of Deligne’s conjecture via Cacti (2004, methodology): “S n action permuting the labels turn the collection {Cacti1(n)} into a homotopy associative quasi-operad Cacti1. The collection {Cact1(n)} forms a homotopy associative quasi-suboperad Cact1 of Cacti1 [K1]. 1.7. Relations among cacti. Theorem 1.12. [K1] Normalized cacti are homotopy equivalent throug”
• On spineless cacti, Deligne's conjecture and Connes-Kreimer's Hopf algebra (2003, methodology): “Background on spineless cacti and cacti In [17] we introduced the operad of spineless cacti.”
• Moduli space actions on the Hochschild co-chains of a Frobenius algebra I: cell operads (2006, methodology): “ns of subspaces and operads to known operads. It will be convenient for the reader to list the known equivalences of sub-operads Suboperad isomorphic operad equivalent operad DT reecp Cacti [KLP] fD2 [K1] DLTreecp Cact [KLP] D2 [K1] DCorolcp SCC [K1] A∞ [K1] The operads in the third column are the f”
• Arc Operads and Arc Algebras (2002, background): “Section 3 is devoted to the inclusion of the cacti operad into the arc operad both in its original version [1, 16] as well as in its spineless version [7].”