Combinatorial models for real configuration spaces and En operads

Clemens Berger

Description

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.

• cell


• graph preoperad


• dimensional configuration spaces


• into simplicial


• colimit lim??a c?


• cellular e∞


• group sp


• interior o˘


• fq ?

Sujets

• Berger

Informations

• Publié par: pefav
• Nombre de lectures: 7
• Langue: English

Extrait

Combinatorialmodelsfor realconfigurationspacesandE n -operads ClemensBerger Abstract. Wedefineseveralpartiallyorderedsetswiththeequivariantho- motopytypeofrealconfigurationspacesF( R n ,p).Themaintoolisageneral methodforconstructingE n -suboperadsofagivenE ∞ -operadbyappropriate cellularsubdivision. Introduction TheconfigurationspaceF( R ∞ ,p)ofp-tuplesofpairwisedistinctpointsof R ∞ canserveasuniversalS p -bundle,thesymmetricgroupactingfreelybypermutation oftheppoints.Themainresultofthispaperisacombinatorialconstructionof thenaturalfiltrationofF( R ∞ ,p)inducedbythefinite-dimensionalconfiguration spaces. Moregenerally,anE ∞ -operadwithsomeextracellstructurehasacombina- toriallydefinedfiltrationbyE n -suboperads.Asabyproduct,weobtainseveral partiallyorderedsetswiththeequivarianthomotopytypeofF( R n ,p).Inparticu- lar,werediscovertheSmith-filtration[19]ofBarratt-Eccles’Γ-functor[4]andalso Milgram’spermutohedralmodelsofF( R n ,p)[17],[3]. WehavetriedtoconcentratehereonthecombinatorialaspectsofE n -operads andtotraceconnectionstoothersimilardevelopments(cf.[1],[11])whenwewere awareofthem.WecompletelyleftouttheapplicationofE n -operadston-fold iteratedloopspacesandrefertheinterestedreaderto[15],[8],[5]. ThecombinatorialaspectsofthetheoryofE n -operadshaveperhapsbeenun- derestimatedforsometime.Thisisquitesurprising,ifoneconsidersF.Cohen’s alreadyclassicalcomputation[8]ofthehomologyandcohomologyofF( R n+1 ,p) whichamongothersidentifies(inmodernlanguage)thecohomologyringwiththe Orlik-Solomonn-algebraofthecompletegraphonpverticesandthehomology withthemultilinearpartofthefreePoissonn-algebraonpgenerators(cf.[11]). Itwouldbenicetohaveapurelycombinatorialproofofthisresult(possiblyalong theselines)relatingittosomesurprisingcombinatorialwork(cf.[2]). Wehavedividedourexpositionintotwoparts: 1991MathematicsSubjectClassification.18B3506A0720B30. Keywordsandphrases.Configurationspaces,cellularE n -operads,symmetricgroups,poset- models,permutohedra. 1