Higher genus surface operad detects infinite loop spaces

Abstract

The operad studied in conformal field theory and introduced ten years ago by G. Segal [S] is built out of moduli spaces of Riemann surfaces. We show here that this operad which at first sight is a double loop space operad is indeed an infinite loop space operad. This leads to a new proof of the fact that the classifying space of the stable mapping class group Z × BΓ + ∞ , is an infinite loop space after plus construction [T2]. This new approach has various advantages. In particular, the infinite loop space structure is more explicid. (1991): 55P47, 32G15, 81T40, 55R35 Mathematics Subject Classification

On the quotients of mapping class groups of surfaces by the Johnson subgroups

“…Here + is the Quillen plus-construction and Γ ∞ := colim g Γ g,1 is the stable mapping class group. Tillmann [14] proved that this is in fact an infinite loop-space. We can likewise put a monoid structure on the space X k :=: g 0 BΓ g,1 (k), and the obvious maps X k → X k ′ for 1 k ′ k ∞ are monoid homomorphisms.…”

“…These functors almost define the structure of an operad on the sequence of spaces { g BE g,n } n 0 -they are associative and equivariant with respect to the appropriate actions of symmetric groups, but there is no unit in g BE g,1 . This was remedied by Tillmann [14] (with a minor mistake corrected by Wahl [15]): Theorem 6.1. There are full subgroupoids S g,n ֒→ E g,n , retractions R : E g,n → S g,n and functors γ which make the diagrams…”

A graphical category for higher modular operads

“…Define a collection of groupoids {S g,n } where S g,n has objects S where S is a surface of genus g with n boundary components constructed by gluing atomic surfaces as in [Til00, 2.2]. Morphisms in S g,n are given by isotopy classes of homeomorphisms which fix the boundary components pointwise and preserves the orderings (modulo the identifications imposed in [Til00]). Notice that the automorphism group of an object S in S g,n is the mapping class group Γ g,n .…”

From Mapping Class Groups to Automorphism Groups of Free Groups

“…Only Section 5 does not have a straightforward extension. Let M denote the mapping class group operad of [24]. One can define M in terms of the cobordism category by taking M(k) = S(k, 1) and the operad composition induced by composition in S. It is shown in [24] that M-algebras are infinite loop spaces after group completion.…”

“…Let M denote the mapping class group operad of [24]. One can define M in terms of the cobordism category by taking M(k) = S(k, 1) and the operad composition induced by composition in S. It is shown in [24] that M-algebras are infinite loop spaces after group completion. The pair of pants multiplication does not define a symmetric monoidal structure on D p (0, 1), but it extends to an action of M. One can adapt the proof of Theorem 5.1 (or rather the proof of the main theorem of [25]) to show the equivalence between the infinite loop space structure of Ω BD, induced by disjoint union on D, and that of Ω BD p (0, 1), induced by the M-algebra structure of D p (0, 1).…”