RESEARCH SUMMARY BENJAMIN WALTER

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1 RESEARCH SUMMARY BENJAMIN WALTER 1. Introdution My generl re of reserh is lgeri topology. Brodly speking, this mens tht I m interested in onnetions etween strt lger nd topologil spes. Modern lgeri topology egins with the ttempt to lssify ll possile topologil spes y lgorithmilly onstruting ertin strt groups from eh topologil spe. Mnipultions of nd reltions etween spes trnslte into mnipultions of nd reltions etween strt groups. To pture different struturl spets, multiple onstrutions moving etween topologil spes nd strt lgeri ojets hve een identified. Questions out struture or existene of topologil spes re trnslted vi these methods to questions out struture or existene of lgeri ojets. We my lso run reverse onstrutions uilding topologil spes from strt lgeri ojets. This llows us to onvert lgeri questions into topologil questions, sometimes hnging omplited lgeri prolems into topologil ones whih re more trtle. Very si exmples of this re the proofs tht sugroups of free groups re free, s well s the fundmentl theorem of lger. My urrent reserh is divided into three projets. My primry projet is ontinuing joint work with Dev Sinh t the University of Oregon on Lie olgers nd relted strutures in lgeri topology. Relted to this work, my seond projet is ontinuing work giving new formultion of ooperds nd olgers in generl symmetri monoidl tegories with prtiulr pplitions in the tegories of ungrded ring modules nd topologil spes. My third projet is joint with Brin Munson t Wellesley College onstruting lulus of grph funtors whih we hope to use to stremline omputtions of hromti numers of grphs à l Bson nd Kozlov. More long-term, I m interested in investigting further new, deep onnetion etween topologil spes nd lgeri ojets vi spetr hinted t y reent work in Goodwillie s lulus of funtors nd topologil operds. The three projets ove eh feed into this long term gol in different mnner. Following is very rief overview of these projets, with in-depth disussions in lter setions Lie olgers nd relted strutures in lgeri topology. My joint work with Dev Sinh on Lie olgers nd pplitions in topology egn in lte Our foundtionl work fed mny ostles whih hve only reently een overome. Now tht the foundtions re omplete, rod vist of pplitions hs opened. In our first pper Lie olgers nd rtionl homotopy theory, I: grph olgers we define new tegory of olgers whih we ll grph olgers, we introdue new pproh to Lie olgers s quotients of grph olgers, nd we pply this to lssil rtionl homotopy theory. We show tht the Hrrison homology funtor from ommuttive lgers to Lie olgers nturlly ftors through our tegory of grph olgers, nd we use this ftoriztion to expliitly disply the dulity etween Hrrison homology nd homotopy vi piring of grphs nd trees. Our work leds to new view of rtionl homotopy theory s well s more omputtionlly-friendly frmework for Hrrison homology nd Lie olgers. In our seond pper Lie olgers nd rtionl homotopy theory, II: Hopf invrints we revisit the geometry ehind the Hrrison/homotopy piring of the first pper in order to give new, geometri 1

2 2 BENJAMIN WALTER nswer to the lssil question how n ohin dt determine homotopy groups? Expliitly we show tht grphs deorted y ohin elements give omplete set of homotopy funtionls omptile with Whitehed produts, nd we give simple, geometri method for mking lultions. Our work unifies nd generlizes onstrutions given y Bordmn-Steer, Sullivn, Hefliger, Hin nd Novikov, using our grph olgeri viewpoint to inorporte oth formlism nd geometry. Our work on grph olgers gives rdilly different nd highly omputle view of n extremely fundmentl ojet, Lie olgers. Initil work indites numer of fruitful pplitions nd extensions of our ides rnging from rtionl homotopy theory with infinite fundmentl group, to the word prolem in group theory, to generlized Hopf invrint one question Cooperds nd Colgers. Considering grph olgers nd the grph ooperd hs led me to work on the foundtions of ooperd theory. Operds re ojets whih enode lger strutures. For exmple there operds Lie, Com, As enoding Lie, ommuttive, nd ssoitive lger struture, s well s A nd E operds enoding homotopy ssoitive nd homotopy ommuttive lger struture. Reent work generlizes these notions to the topologil setting nd eyond. Dully, ooperds enode olger strutures. Tht is, operds tell how things re multiplied, ooperds tell how they re ftored. Operd theory is ided y the ft tht in ll tegories of interest, operdi omposition is ssoitive. However, the dul notion of ooperdi oomposition is ssoitive in lmost none of the stndrd tegories of interest. One striking effet of this is the diffiulty of desriing generl ofree olgers, question whih hs grnered interest in the theoretil omputer siene ommunity (see e.g. work of Berstel nd Reutenuer nd lso Blok nd Griffin). In my pper Cofree olgers over ooperds I ly foundtions for new pproh to ooperd theory using n ssoitive universl omposition produt whose left nd right Kn extensions give the operdi nd ooperdi omposition produts. This is used to give new onstrution of ofree olgers over ooperds improving the existing onstrutions of Smith, Fox, Hzewinkel, nd Blok-Griffin. Furthermore I give exmples showing tht my new foundtions give reltively esy wy to give expliit desriptions of ooperd strutures nd mps etween ooperds nd mke diret omputtions. In further work, I pln to extend these onstrutions to give new desription of the ofree ooperd funtor, vlid in the tegories of ungrded ring modules nd lso topologil spes. This would yield new onstrution of the ooperdi or onstrution whih preliminry lultions indite would extend tht of Mihel Ching in the topologil setting. There re numer of further projets, in my joint work with Sinh desried ove, nd lso investigting Goodwillie s homotopy lulus of funtors, where suh onstrution ould e useful Clulus of grph funtors. My disserttion work ws on Goodwillie s homotopy lulus of funtors. In my disserttion I onstruted new homotopy lulus of funtors in the rtionl homotopy tegories of differentil grded ommuttive olgers nd differentil grded Lie lgers. In similr, joint projet with Brin Munson t Wellesley College we re onstruting new lulus of grph funtors (funtors from the tegory of grphs to the tegory of topologil spes). Our gol is to use this lulus of grph funtors to stremline nd etter understnd the method of Lovász generlized y Bson nd Kozlov in 2007 for omputing ounds for hromti numers of grphs using lgeri topology. We hve urrently solved out hlf of the prolems required to omplete our lulus of grph funtors onstrution, nd expet to e le to omplete our work nd hve it redy for pulition within the next yer. We expet our work to e of gret interest oth to homotopy theorists wnting to etter understnd Weiss nd Goodwillie s luli of funtors nd to pplied mthemtiins nd omputer sientists interested in hromti numers of grphs. 2. Lie olgers nd relted strutures in lgeri topology 2.1. A rief introdution to Lie olgers nd generlized Hopf invrints.

3 RESEARCH SUMMARY 3 Definition 2.1. Let V e vetor spe. Define G(V ) to e the spn of the set of oriented yli grphs with verties leled y elements of V modulo multilinerity in the verties. G(V ) hs n nti-ommuttive oprodut given y ]G[ = e (Gê1 Gê2 Gê2 Gê1 ), where e rnges over the edges of G, nd Gê1 nd Gê2 re the onneted omponents of the grph otined y removing e, whih points to Gê2. For exmple, ] [ ( ) ( ) ( ) ( ) = +. We ll G(V ) the ofree grph olger on V. The numer of verties in grph is lled its weight. Grph olgers pir with non-ssoitive inry lgers vi the onfigurtion piring etween oriented yli grphs nd inry trees desried elow. Definition 2.2. Given G nd T, n oriented yli grph with verties V = {1,..., n} nd inry tree emedded in the upper hlf-plne with leves L = {1,..., n}, define β G,T : { edges of G } { internl verties of T } y sending n edge from vertex i to j in G to the vertex t the ndir of the shortest pth in T etween the leves i nd j. Use β G,T to define the onfigurtion piring of G nd T y sgn ( β G,T (e) ) if β is surjetive, G, T = e n edge of G 0 otherwise where sgn ( β G,T (e) ) = ±1 depending on whether the diretion of e grees with the ordering of the leves of T indued y its plnr emedding. Exmple 1. Following is the mp β G,T for two different inry trees T. 1 2 e 1 e 2 β(e 1) β(e 2) 1 2 e e 1 2 β(e1) β(e 2) In the first exmple, sgn ( β(e 1 ) ) = 1 nd sgn ( β(e 2 ) ) = 1. In the seond exmple, sgn ( β(e 1 ) ) = 1 nd sgn ( β(e 2 ) ) = 1. The onfigurtion piring is so nmed euse it nturlly rises in the ohomology-homology piring for onfigurtion spes. The grph oprodut nd inry lger produt re dul in the onfigurtion piring. Further, the kernel of the onfigurtion piring is preisely the Joi nd nti-symmetry suspe on the inry lger side, nd the Arnold nd rrow-reversing suspe on the grph olger side. Definition 2.3. Define E(V ) to e the quotient of G(V ) y Arn(V ), where Arn(V ) is the suspe generted y rrow-reversing nd Arnold omintions of grphs: (rrow-reversing) + (Arnold) + + Aove,, nd re verties of grph whih is fixed outside of the indited re. Theorem 2.4 (Sinh nd W ). If V is grded in positive degrees, then E(V ) is isomorphi to the ofree Lie olger on V, with the grph oprodut on G(V ) desending to the Lie orket on E(V ). Indeed, if V nd W re linerly dul, then the onfigurtion piring gives perfet piring etween E(V ) nd the free Lie lger on W with grph oprodut dul to Lie rket.

4 4 BENJAMIN WALTER Definition 2.5. Let (A, d A, µ A ) e one-onneted differentil grded lger. The grph olgeri r onstrution G(A) is the totl omplex of ( G(s -1 Ā), da, d µ ). Here s -1 Ā is the desuspension of the idel of positive-degree elements of A, d A is the extension of d A y Leiniz, nd d µ (g) is sum ontrting eh edge of g in turn, multiplying endpoints. If A is ommuttive, the Lie olgeri r onstrution E(A) is the quotient G(A) Arn(s -1 Ā). Theorem 2.6 (Sinh nd W ). E(A) is well defined nd gives model for the Hrrison omplex, or equivlently the André-Quillen homology of ommuttive lger. Thus, these onstrutions ftor through the tegory of grph olgers. The origin of these grph omplexes ws the study of generlized Hopf invrints. Let X e simpliil set, A(X) e the PL-forms on X, E(X) = E(A(X)), nd H n E (X) = H n(e(a(x))). Define G(X) nd H n G (X) similrly. Lemm 2.7. H n 1 E (S n ) is rnk one, generted y weight one oyle. Definition 2.8. Given oyle γ E n 1 S n we let τ(γ) γ e ny ohomologous oyle of weight one. Cll τ(γ) Hopf oyle ssoited to γ. Write E(S n ) for the mp from oyles in En 1 (S n ) to Q given y E(S n ) γ = S n τ(γ). Lemm 2.9. The mp E(S n ) is well defined nd indues the isomorphism Hn 1 E (S n ) = Q. Definition Given oyle γ E n 1 S n nd f : S n X, the Hopf invrint η γ (f) of f with respet to γ is E(S n ) f (γ). Just s E ftors through grph olgers, so too does E(S n ) ftor through (whih is defined G(S n ) in the sme wy s 2.8). In prtie we ompute Hopf invrints y piking grph representtives nd lulting ηγ G(f) = G(S n ) f γ. Exmple 2. Let ω e generting 2-oyle on S 2 nd f : S 3 S 2. The grph γ = ω ω is oyle in G(S 2 ) with f γ = hoie of oounding ohin. Then. Beuse is losed nd of degree two on S 3, it is ext. Let d 1 e d ( d 1 ) = + ( d 1 ). Thus f γ is homologous to d 1. The orresponding Hopf invrint is S 3 d 1, whih is the lssil formul for Hopf invrint given y Whitehed in The formule get more omplited in higher weights, ut they re still omputle nd hve the sme si ingredients pulling k ohins, tking d -1 nd produts. In prtiulr, if γ is defined using Thom forms of disjoint sumnifolds W i of mnifold X, then η γ (f) is generlized linking numer of the f 1 (W i ). Theorem 2.11 (Sinh nd W ). Let X e simply onneted. Then the mp H n E (X) Hom( π n+1 (X), Q ), whih sends γ to the Hopf invrint ssoited to γ, is n isomorphism of Lie olgers. Furthermore the mp ove is desended from olger mp H n G (X) Hom( π n+1 (X), Q ).

5 RESEARCH SUMMARY Future work. First, we pln to extend our pproh to rtionl homotopy theory eyond the simply onneted setting in hrteristi zero. In preliminry lultions, it seems tht our tehniques will pply eyond the nilpotent setting, possily even to infinite fundmentl groups (extending work of Gómez-Tto, Hlperin, nd Thoms). Further, pplying our tehniques to K(π, 1) s ould yield new insight into the lower entrl series nd new lgorithm to ttk the word prolem for residully nilpotent groups. Hopf invrints in this se n e understood s sort of generlized linking numer of elements. A seond re of inquiry is the deeper setting of hrteristi p. The min strting point for our Lie olgeri pproh to homotopy theory ws studying expliit Hopf invrints, nd relizing tht their vlues on Whitehed produts were governed y the sme piring s governs the homology nd ohomology of ordered onfigurtion spes, phenomenon explined y iterted loopspe theory. At p, loopspe theory leds to studying the piring etween the ohomology nd homology of symmetri groups. Another losely relted prolem is our generliztion of the Hopf invrint one question. Hopf invrints over the integers re defined for ny spe nd give full rnk sugroup of Hom ( π n (X), Z ). Clulting the okernel of the inlusion of this sugroup generlizes the lssil Hopf invrint one question nd ould help us onnet our pproh in hrteristi zero to topology in hrteristi p. 3. Cooperds nd olgers 3.1. Universl ompositions nd ofree olgers. Fix symmetri monoidl tegory (C, ), nd write Σ for the tegory whose ojets re surjetions S from finite set to singleton set nd whose morphisms re omptile set isomorphisms (note: this is equivlent to just the tegory of finite sets nd set isomorphisms, whih is equivlent to the symmetri groups viewed s tegory). Write Σ Σ for the tegory of whose ojets re set surjetions S 1 S 2 nd whose morphisms re omptile set isomorphisms. More generlly write the iterted wreth produt Σ Σ }{{} n for the tegory of surjetions S 1 S n nd omptile set isomorphisms. There re funtors Σ n Σ m given y omposing surjetions if n > m, nd inserting isomorphisms if m > n. These fit together to give n ugmented simpliil tegory with extr degeneries. Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ The dotted, leftwrds rrows ove re setions of their neighoring solid rightwrds rrows. Write γ n for the omposition γ n : Σ Σ Σ. A symmetri sequene in C is funtor A : Σ C. Given two symmetri sequenes A, B : Σ C, we define their universl omposition produt A B : Σ Σ C y f g g ( (A B)(S 1 S2 ) = A(S2 ) B ( f 1 (s) f {s} ) ). s S 2 We define A B C : Σ Σ Σ C nd et. similrly. Note tht A B is not gin symmetri sequene. However we my onstrut symmetri sequene from A B y either left or right Kn extension over γ 2 : Σ Σ Σ. The left Kn extension Lkn γ2 A B is the usul operdi omposition produt from the literture. In this lnguge, n operd is symmetri sequene equipped with nturl trnsformtion µ : (A A) Aγ 2 suh tht the two indued nturl trnsformtions µ(µ Id), µ(id µ) : (A A A) Aγ 3 re equl. Dully, ooperd is symmetri sequene with nturl trnsformtion : Aγ 2 (A A) so tht the two indued nturl trnsformtions Aγ 3 (A A A) re equl. The universl omposition produt is ssoitive, ut its Kn extensions my not e. In most tegories of interest (e.g. ungrded ring modules nd topologil spes), ssoitivity is lost on right Kn extension; though not on left Kn extension. Write A B = Rkn γ2 (A B), A B C = Rkn γ3 (A B C), et.

6 6 BENJAMIN WALTER In generl A (B C) (A B) C. However there re prenthesiztion mps from eh of these to A B C. Muh of the diffiulty of olgers n e tred diretly to this point. Using this frmework I hve proven new method for uilding ofree olgers extending nd simplifying tht of Justin Smith. Theorem 3.1 (W ). Let C e ooperd nd V O(C). The ofree C-olger on V is given y the tegoril limit of the digrm C (C V ) C ( C (C V ) ) C V C C V C C C V 3.2. Future work. My desription of ooperds nd olgers vi universl omposition produts hs lredy proven useful in omputtions with the grph ooperd nd its reltionship with the Lie nd ssoitive ooperds. Coneptully, it hs proven onvenient to e le to work on the other side of Kn extension from the literture, nd nottionlly it is muh more len to work with universl omposition produts rther thn the ooperdi omposition produt. Further this work leds to ler understnding of Mihel Ching s onstrution of the topologil ooperdi or onstrution. In this vein, I pln to ontinue reorgniztion of ooperd theory round universl omposition produts. In prtiulr this leds to new wy of desriing the ofree ooperd funtor whih n e used for topologil operdi r onstrution. Applitions would inlude onstrutions useful for my joint work with Sinh on Lie olgers, understnding the topologil grph ooperd nd its liner nd Moore duls, s well s revisiting the work of Ching understnding Goodwillie s homotopy lulus of funtors. 4. Clulus of grph funtors 4.1. Chromti numers vi lgeri topology. By grph G we will men n undireted grph with no multiple edges or loops. A grph mp G H is mp of vertex sets whih preserves djeny. A grph hs n n-oloring if its verties n e olored with n olors suh tht no djent verties re olored the sme. Note tht n n-oloring of grph G is equivlent to grph mp G K n to the omplete grph on n verties. Write χ(g) for the miniml numer of olors required to olor G. In 1978 Lovász onstruted homology ostrutions to the existene of mps G K n, nd more reently his work ws generlized y Bson, Kozlov, nd others through the introdution of ifuntor Hom(, ) whih ssigns topologil spe to pir of grphs. Definition 4.1. Let H e grph. Define P(H), the power grph of H, to e the grph whose vertex set is P 0 (V (H)) with n edge etween S nd T if n only if there is n edge etween every element of S nd every element of T in H. Given grphs G nd H define the poset hom(g, H) to e the set of grph mps G P(H) with poset struture inherited from P(H). Write Hom(G, H) for the reliztion of the poset hom(g, H). Hom(, ) is ifuntor, ontrvrint in the first vrile nd ovrint in the seond. Bson nd Kozlov uild ostrutions to grph mps using funtorility of Hom(, ) s follows. A grph mp f : G K n indues mp of spes Hom(K 2, f) : Hom(K 2, G) Hom(K 2, K n ). Furthermore the ntipodl mp on K 2 gives free Z 2 -tion on the ove spes nd mkes Hom(K 2, f) Z 2 -equivrint mp. Considering Hom(K 2, f) s mp of Z 2 -undles, the generlized Borsuk-Ulm theorem tells us tht the imge of the first nonvnishing Stiefel-Whitney lss of Hom(K 2, G)/Z 2 must e nontrivil in Hom(K 2, K n )/Z 2. However y diret omputtion, Hom(K 2, K n ) is (n 3)-onneted. Theorem 4.2 (Lovász). For ny grph G, χ(g) onnetivity ( Hom(K 2, G) ) + 3.

7 RESEARCH SUMMARY 7 There is nothing prtiulrly importnt out the test grph K 2 used ove side from it hving free group tion. Bson nd Kozlov repled it y the yli grph C 2k+1 with the following result. Theorem 4.3 (Bson nd Kozlov). For ny grph G, χ(g) onnetivity ( Hom(C 2k+1, G) ) + 4. In prtie, the ove onstrutions seem to give firly shrp lower ound for hromti numers. The min ostle to quik hromti numer lultions ppers to e tht it is very omputtionlly intensive to extrt onnetivity informtion for generl grphs. The work required to ompute Hom(C 2k+1, K n ) ove is triumph of ferless omintoril mthemtis A lulus of grph funtors. Brin Munson nd I hve egun the onstrution of lulus of grph funtors pproximting ovrint funtors F : Grphs Top whih re ontinuous in the sense tht the evlution mp Hom(G, H) F(G) F(H) is ontinuous mp. The funtor Hom(K, ) is one suh. The onstrution of our lulus is inspired y Weiss orthogonl lulus of funtors from vetor spes to topologil spes whih re ontinuous in the sense tht mor(v, W) F(V) F(W) is ontinuous. Our work relies on the ft tht hom(, ) is essentilly n enrihment of the tegory of grphs over the tegory of posets. The end result of our work is, for every grph funtor F, n pproximting tower of firtions of grph funtors P 3 F P 2 F P 1 F where the fiers ll hve prtiulr simple form, nd going up the tower etter pproximtes good funtors F in the sense tht nturl trnsformtion F P n F eomes highly onneted. In this se, we should e le to red off onnetivity informtion for Hom(K, G) from the onnetivity of the fiers in the ottom of the pproximting tower of P n Hom(K, ) evluted t G. Working nlogous to Weiss, we hve ompleted roughly hlf of the onstrutions required to reh this gol. At present, we hve worked out the universl property whih should define the n-th polynomil pproximtion P n F (whih involves poset-enrihed homotopy olimit), we n show tht polynomil of degree n implies polynomil of degree n 1, nd we n show tht pproximting towers for funtors exist. We hve lso ompleted the initil onstrutions towrds understnding fiers of pproximting towers. Middle Est Tehnil University North Cyprus Cmpus E-mil ddress: enjmin@metu.edu.tr URL: