Combinatorial models for real configuration spaces and En operads

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Combinatorial models for real configuration spaces and En-operads Clemens Berger Abstract. We define several partially ordered sets with the equivariant ho- motopy type of real configuration spaces F (Rn, p). The main tool is a general method for constructing En-suboperads of a given E∞-operad by appropriate cellular subdivision. Introduction The configuration space F (R∞, p) of p-tuples of pairwise distinct points of R∞ can serve as universalSp-bundle, the symmetric group acting freely by permutation of the p points. The main result of this paper is a combinatorial construction of the natural filtration of F (R∞, p) induced by the finite-dimensional configuration spaces. More generally, an E∞-operad with some extra cell structure has a combina- torially defined filtration by En-suboperads. As a byproduct, we obtain several partially ordered sets with the equivariant homotopy type of F (Rn, p). In particu- lar, we rediscover the Smith-filtration [19] of Barratt-Eccles' ?-functor [4] and also Milgram's permutohedral models of F (Rn, p) [17], [3]. We have tried to concentrate here on the combinatorial aspects of En-operads and to trace connections to other similar developments (cf. [1], [11]) when we were aware of them.

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Berger

CombinatorialmodelsforrealconﬁgurationspacesandEn-operadsClemensBergerAbstract.Wedeﬁneseveralpartiallyorderedsetswiththeequivariantho-motopytypeofrealconﬁgurationspacesF(Rn,p).ThemaintoolisageneralmethodforconstructingEn-suboperadsofagivenE∞-operadbyappropriatecellularsubdivision.IntroductionTheconﬁgurationspaceF(R∞,p)ofp-tuplesofpairwisedistinctpointsofR∞canserveasuniversalSp-bundle,thesymmetricgroupactingfreelybypermutationoftheppoints.ThemainresultofthispaperisacombinatorialconstructionofthenaturalﬁltrationofF(R∞,p)inducedbytheﬁnite-dimensionalconﬁgurationspaces.Moregenerally,anE∞-operadwithsomeextracellstructurehasacombina-toriallydeﬁnedﬁltrationbyEn-suboperads.Asabyproduct,weobtainseveralpartiallyorderedsetswiththeequivarianthomotopytypeofF(Rn,p).Inparticu-lar,werediscovertheSmith-ﬁltration[19]ofBarratt-Eccles’Γ-functor[4]andalsoMilgram’spermutohedralmodelsofF(Rn,p)[17],[3].WehavetriedtoconcentratehereonthecombinatorialaspectsofEn-operadsandtotraceconnectionstoothersimilardevelopments(cf.[1],[11])whenwewereawareofthem.WecompletelyleftouttheapplicationofEn-operadston-folditeratedloopspacesandrefertheinterestedreaderto[15],[8],[5].ThecombinatorialaspectsofthetheoryofEn-operadshaveperhapsbeenun-derestimatedforsometime.Thisisquitesurprising,ifoneconsidersF.Cohen’salreadyclassicalcomputation[8]ofthehomologyandcohomologyofF(Rn+1,p)whichamongothersidentiﬁes(inmodernlanguage)thecohomologyringwiththeOrlik-Solomonn-algebraofthecompletegraphonpverticesandthehomologywiththemultilinearpartofthefreePoissonn-algebraonpgenerators(cf.[11]).Itwouldbenicetohaveapurelycombinatorialproofofthisresult(possiblyalongtheselines)relatingittosomesurprisingcombinatorialwork(cf.[2]).Wehavedividedourexpositionintotwoparts:1991MathematicsSubjectClassiﬁcation.18B3506A0720B30.Keywordsandphrases.Conﬁgurationspaces,cellularEn-operads,symmetricgroups,poset-models,permutohedra.1

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