Cohomology operations and the Deligne conjecture

Martin Markl

The aim of the talk, which raises more questions than it answers, will be tostudy natural operations acting on the cohomology of various types of algebras.It will contain a lot of very surprising partial results and examples.

1

1..1111111111-" ~l . '. }

. M .0 JVlA R K L~M~ .\f4,~f.,~ ~~1:.ort\,) ~ftt,~8Q.4\1l.(.L. t.~ 'r'~, '~A.."':l:llrJ.~;(A. ~",1,

~~~ 01 c.o!f.M,~~, t(..o)~(mffp~~ ":~ S. Ih'l"'r1...~'\f~~~. r~~: .,."..t,,(,~:

'---

.) "C~~?ia.t(' tk:~pfJ.8 ( kAJ~\~&t"\rJ,, ~: .

.) ~QdJA,e.: ~p. \)! R. ~1J,~\frAJ..LS '

.) QAlt.,Wfp\,. . \j j'\J.L" ~u~.~ =.:.~ .\~(,-j)'1.i. (1 II).,.,~ I/~ (\" t:"",f /):(.\ ii,

,) ru..p'\I. ~~t"J \. ~ ~a ol~ L- ~ 1.Ct..\ : 'rf ~I .. ,

1.: IJJ ,,) 'II I. till.. '.~t 1,(. (~~,(,j""', ,',,'I ;' ' ( 1 ,j.(, \

- - . ;,~.. I". ~.J,'})1J, a.. .1':;\~',11.A,I'1 "'c""J.]

..) ~~~.A)w..t. "'3. .~('t'l fftt. t{~~ie4 ('QR.om\.oft,~\t-f' ctl.~f'... ;),(1(. \)11 l)t., I

l~tCt+1~ C'.,()()\4,pflX J\\t\J..1-.S ~ L ~ !r\l., ,i f)\.rr)-,. (_o.f. f~! t, ~ ~ .- .

:1!',.~.,,~--~ CA. ~t.rr\AA.oJA,~~l\.o(\\' (WirJ.},(. fA 8"'~hj(.C':.' ~., ,:t" ~ ,.,)).~...

8) H~eo\t~ 'PROP ~tJOJi~ b\\ ~t1MC~wu.,~ -t.\.c. (~I:.t.t . I:' (,..,,;f))1'):.1 l/, .

(~1~r~ (;()~r~t~ o~ tftt ~()\('y)1. t rtE.~~I"fi - (. -~..~"J). 1~l.f I:~. ryj:j,,(,'!/( ~ 1':!

.b~l ~~-~~-- '

~ 1;-\~~c,..(1~ t.~ ,i,""\r-<lk~(AJ\1"b ()~~~,.f)~ 1A'f." (~i~(f ~ ~wC)tU.':;lJ1 (:)!).('.~~

+,.() (~~D1)~~ ~ r ( E.) ~ ) . f

I ~ Jo J l "...o_.~A-cl) TRiJf tuIJ,:(!/JJJ.'{ ('r'J,~'. :t!:'/I/~..t 'lU- Ulr.J>\.L~ .'N~, .'!! "

. d- \~ l~~_!~ ~~~--~~--~~~~~-~~~~~ .

~~(\)~ p\,. htio(\\ t \"\.4~. (v~~. ~tt ~). CoO'\o'~o#;.61J'-t ()(),_. () "1~l'()'lN:'.\'!A')!!,J.,

I _..II~I~~I~~j(l at crJlCJAe~k1..C\ V~rt}wr-,XJ.(., £J' If'{'~'!i, It j\I"\,,li\'t.'.(J)l.J... Cfr'..L-vh.-~~ U (, . -- -",t " '

LI

,.

~~~~~--~~~ ~ w. ~ c..o ("m, p ~ ?. ~ -= (OJ"') f())

~ '=. ~~"r'1;(~/L, wrtrtlbi/rIAtilO~s o~ 1'L/A.r~S ~t

~11 ~~ Q,- 0 '. ~ 8 ~(;4..ck~JJt III : . ) ~ tW W, t t:o (\1, Q j \t1), ~ u. ". ~ ~ I\A xt =' ~ - ()UIt ~tl\M;6tli,u

(] v~..", ~-/ : I

op.s ~ *' ii~ ~ PROfs

.) oo_eo~~~. . .) C}vUlct~OM oj 1d~{s} LW . ,

. + ~ a.SIl4,(Vk,p~OM8: .) 0 01, " CQ~/&()":1J"i.L~. .) (Yt,O cw¥IA.d S J .too.

..~ Mt8 ~~'- 'J'

d . I ~ L:::t {I. 3 ~1l,

~t

~o ~ cL.t~~~~ C~~ ~ts om, 1J(A,tLCt~

~ 'RL~3~ht(,) ptOpfA, (\(,aR,,;ll.d eol oj ~1..(A.p~ CiJ('rYj,~~ {.!

~ (-Q. ("P)) d) } 'r. fA. IrAMc\ b~ 0pfl'l,Cl~ Q1. 'P'RO'P J

_Cl-~-) Q. 3~tC'\,bf<, bQJ\, CoC),.,lh.

~Vtt"L~ t. j ~ ~ 1II.0l~~ tLc & 1<01 ~e (\r1.L s a r ~ l S .

t 1. """;1;,'4 """

~~ D~ iliJ t~pl. 1&l (\4..Q,t~ t,k'Q..fm,pk tk ()y~ @, 01 ~~ b~Q,e~(,b1,~ ~'RO-P ) 1~t.A.~~ lo(M.L Jc,rn.o~ bt!O1L

b~ mol 1.A..unU}(tJ. a.s ~(..(.cj.,. EUtrt1/ jo" {~1t. ()'nJ,t:.~cl t

~~i~ (fmI., ~Vow('nt>V') H~(P'r) ..) t

~M ~~lS Wt s~w f~~l ~1..obfAIIWII G~ }'~iJ'I,1,~ ("h;Q,tw~

. ~~Q.ti,0("r\! oCr\ c,o~o~~~ ~as.f" ~. c. o~ ('\;O('t\ - c.o~QI\.,

baFt. ~'Q ~fWIII &~orw R(f-.,) ~RA, ~~ ~O\ b! ~.c. 1k8~jQ.u..~ .l~ t.A./t1.,tIl~l.,C~ 8~~Q.tvorns_, /

--- ~ - - ~ ~ -- )( ---~=--"'--

~~.2~~~~,: +-\ l-{ ~~ o,1S. Ctf.«"Atb\Q. A ~ CN(,~. iIM, .t,t~t.e~..

'. ~ t '

+\+to~ ~ A) A) =- ~'* \ C +loci. (A) A) )J-I-1o~)~ ~OC dIIIOr /

'. 'C('f1, l R J A) ~ ~cn ( A~~fn~1) I A)

~o~ '<fn..,WS ~. .

~t Q/l;(. ~~~~~e '1~~~t~~~ ocyv C "', ( f) , C1 ) .

~t 1 e~ ~ ~ (Y)\AAttl~~M. (Y)a,~~ f)(t\ Q. t p(A.~

~!~ OM ~SIO~~\'11{, ("mM,e,~pe,;catW()\f ~

~) 0", ", ~flAQ.tl8(k1, "°1, 1 - . 11 ahId. tLfAP-,lt.

.) (YY\.At1ttfA"f~c.Q.t~o~

'2f" l ~ V'. 8, ,,~ aRt Crt - (YnAA. t t~~~ ~t,t.(lta..t '

() pVt,Ot -lL 0 (".-\ t I . rr1, ~ O.

:~~~g~ ~ECAtAJA)J~~C~(A)PJ\ W

=-"') lIA \ ~)~) clA.~rt\..(,rA ~d ~ t1) ~) (a.)'o, C)~ t.) ~ ~ J (() b) # ~ tc.) d,)Q.)

~~ rn.~w"'a1. f)p~t\,Ofr'\, I CU e. ~(~) #

. ,---

I ~!i.caR1: ,aMq ~ flA(~)~) = j ~,

Fo\, C«. {'tII.~&O~l1--: .~t1)~)-:'(olA°-t~)o1J :~(aAAo1~)'O3~.

. II,l ~= h ' I

. . crf..ls is ~ ~ tn\h\Q ..l cuJL) ~ 0 f.., 1M ~ /:aM tt., ,

3 I .

dLrml~1Il £ ~ (4) - -

'Ll(~)d)l)e) tQ.)b)t)c\)~J !~1)::

t ( Q.) bc~) . ct (d ~ e ,) ~ ) ) ~ { 1)) . .

~~J~ : f6 t M) -= ~ ~ t>! reQ/\t\IOJ\, 4 'Ut,1:

(w\\~.) r.1, N-cAt\tlt eCi'b~c..\ -'\) ~- ) ('t\., ~ :unt""t ~~kM.t

.) SO("P1A. f1\M.CW8btA o! . .. - :un~~t i.tfAQ.ttd. crU'

.

11818;,

. b o~ttO()\' @.) ~:'= t ~~~)~ ~~& ~~ o~c\,} ~ t.Prtr'fJ

0\ (VL~tWl.at ofs. .qt.o<'tru.t!IA:<:alld 'Mtvc 1fv\Stit,{iOM rn.o~ ~1..Q.! \;.~1.

.) I d-}\o~ 1JhdM.u.~ S bd '

,~ ~ t ~ ) ~ ) --) ~ at 'U ( ! ) ~I ~ -! ij (c~ 5 ) d ) -,) t 'U l ~ ) ~ ~) . - . )so ~~~ fi = t ~lM) IS \ .\.~ Q. d.o - ()pfl\.a.a ,l .

. 'F' , J ~";:: - ~~ v!.::;.:..J :

~~/ll.Q.d 21(h~~al OpfA,Q,tl:OlkS ~ +t. tI1~ w~ p~ .

. .-H*\ fb~ ~~MQ..~a1J~ b~ {-\~O<J..~f\;A)

~ ~ ~t-'\I1.1t opttJ,Q.ct t>! (n.~~Ql, OptJl,Q.~O~l f)~ +!. tok('rr'tO~'1.

1_. ?~b~: W~~ i~ t~ ~('M-okp~ ~~t... bd ~~ {~(v\,-O\.bt~ ~~-) ~bt~. bJ VF(/\..~cl~?

~~ PQhh.C,w..lQltt ~ "k)~t 1,~ }l '-t g) ~ -

"

W~ ~po~~b ~ U(h~ttQM&~.i . ~11~;~~~:i~~~~~~~i~ ~ ~ '" quat - .

~ QAt, II p1..ooj~" {~., ~,~-1Q(1nQ/L~Mo\ b~t ~ ~ mot~ h@ II!f.:IAbYA,. -' ': - . ¥""" ~~

Ob3fAAeL ~~~.:.l.;::j~~~~~!- ~r.~Cht., GO~~ ~~ ~tA ~,t,

b£t ~lw~f4, .' t\

~~~~~~~~j ='> I' ~ 9lKAA rt\Oa.~oJ. °\JIA.Ct.UolM." ~~~ U & C - , - j .

II1II'~,

"";I!,I!~~tj._.;ft~;

-~ '

qt~~~~1~f4~. @6' ... ~ IAQ.dI14~ k .1 J I , t4.a.c:l,

~ OS~ o~ (CO~) ~) a,s J 'PoiuJ -.. ) I A -. fjJ -at t.bt.Q.

e, (A, A) .. - tt. "&LftA_-'- _I 1 -j.-. ., 3~!!~~ ~s~~~ dL ,,- . r.ii4:~q+I; ( A; A) , - -. , r a

I Ex: ~"' ~ ~f -. lilt , I ~. -A~ ~.~) ""CofW\ ': "ttQAl1.) H ~ ~ CE) -~. /0 ('rIL ~ dL ~ tit(, (~::1~~~~~;~ iO -= ( 8

(~ ) t ) :IP- n. u II /0' '>0 ~ . -..'1_,-"tftl. ": ass ~l, t (\t - - d (1~c,~~

~~~ t>l ~Q.tfAf1.al t>pUAf,LOt'\t.1 ()(M, c~ (A;f-')- -

~~~=~I~~ ." ~ ~ ~(.. =. '> '

~f -= C~E:) ..w 1,1 .... ) .a\f '" )Cf-:(Lj L ~ ~ (/\ L, L

& ~~L ~.rIM1Q..\ to Bass b~ g~lt1~~~ "iJk~tta.cl o! -

!;~~~, t.'UL!..., {\I'IIlv'J~t b)/VvL (YYW1...(..., (J i,fJ':,CJ.AJ.h ~t"Q4\1. I), ~ ,\ (.I&,~. ~

rtW\8_~/1'_~jJ.AJ (J \9, It I fl(). ~.:.-r~--~~ ~ { ~ O(k~ ~iL. b1.Q.C~

3 ~ w.d.. .l +I 0 { ~ lA:.c-) a ~L .11~i~~~~~ 1 b fA. t eo ~ b j c.. P ~ e..- cc., f.io (\\ t .~ bt, af.~~>Ql6~~~~fi;J :-tJ. t , ~~~!~~Q4.. VlWV\, l TJ- \ V~I ': V'_-=--' .:::'~-- ('W')fl ~,

-==-

~ ~

~~, ~Co~.]GC~t lot. Qt~to.e 6>.

~. q~1/{J\1, fJ, fi)llcM,~ & ~,t. r>! BOJ. ~ t,Wnt,~ bJ f.

~ ftJ(A,ti~: d..Ll~bl. R.t. DI ~~ Q.l1.d1 al3(...b~ Bf(i\)h."..-l,..O "aI~\..."I\... nnt"4,'nt, 1.1"4( ~.t'(O)'

..I"

~

~~I~~~~~~. ()1- ~~DtJ,. Q: k ko ~ ~ \Ai = '> ~ 6> tV B 6' ! .t~~aJ~ co.~t.A~(\4,l. ~ l~ & 1 II. -- t.O""~~~

.#

8, (0) (\.c.p\L&~~S frI.().~o.I." tJLrrt~t" ~ c, (A)~)~ ~!:!* (8,(01) .- ('nQ,bMQt c,trAs8u t(Jk -H~ (A)"A).

.'§~~~~}~.: ((!>a.llto)6as~) "'~(A,~ '

~ ,& ~ ~Ik ~ Ik [~",J -'>- f«.):,~ 1 --.') 1< (~\t 1. -')0 --

rwRJAL £5' ~s o.Jl r ~~ t " u.~~\4J~ It ~~ ~ ~t\al ~ '

~ ~w~ +\ . ( 6 atS ( 0)) &o.t~) 'C o.- -

-. S~~~:"e~: t ~~(. ( 0) } ~ ~t) Q.t 1A.a.Q.~.It) p 0 -1l ~~ -4 ~ -+ ~ '--~ l -"\.. ,-

. "*-tAw! O'\L -H ~ e f\c., l 0) ) &~ ) '": O.

~ -

. ~ c.aX1 +t ~ t B 6 { 0 ) ~ tf) -1&.,~~, ~ {Vu, ~~.JX~.~~)~ 1 .

{~~~~i~;~~(~'> S<X.Le ~ ~ - ~O1., "J1.(.a.W~~L\

oJ,~(,~! ~1 ~1A,1rial- I CoS- ~fei~ J°tfnlf,) \(,o~~f("Y\.(,t /.

1-J~~~~~~.. I~A( ~Q~S tQg,s l 0)) ~~u *Q.~c. ) ~ lk J '. .

.,,', -

~~ &~~ tW f)p f'dOt~ (Mt ('nLU-&~td ~~~ ~,tl

..:1~:J==I~~OP~~~ ~o~fJ ~1II.~k ~~,~ ~n~~.;. h/~1lL 0 pl.{,l>-d ~ ~ QJu ~ ~u ~ {Ctc.,b

~ 6> 1\.1 f4~ut 'kottwf

I ~~~~~ K6 O~'M~e 2.0~.(, Atlus tla.tV ~ ,Ko.1\uf tf7 ) 8(/ '1,1 d°l.~ O/t\cl,

. . ~ f tV Qf r

~ Q ~ ~L(D.k ~M Ico &It.l.l ,

H9t(,'tsQ.tiD",,1: fJa.&l .'. ..,,' qVtll- f3 ~ .," ~ ~-" ~'.t.

& ... ~. . , .COt.." , .~- .'

~ ~~O'"' g~~ U ~ Roft 1WA.8 IW'L O~ .

~~~c- W~t Qd.~w~al ~top".,t1 fYM.Q.Jc.c.~ f\,OpL

!Wt~ UQ,~,(,3 Jp,:.,~~~a£ (),~okt at~)~(, a.t.cI &~ ~

IIiI

~~~0fM3 b~ ~\&~f.! 6~:) '::' ~L , (\

" . .) ~j oj (1) tok.ts about ~S ~Q.~l.1 .~ . . ) (1) ~~ ~l"'~~\lial. 11141 pfA:cc.. .uOnc.t .

.I I@ () '~~ , ..,;J ~~ (f¥") - -+ e ~f,.(~)

i

I' t cL I ,J ~

. ~1..L-~t (X",) -.J Xc.-.) ~ fJ1L-Z:t (x~) ) ~~) tJ)

d, ~L.\'(~ ~ ~NM.rfL rk~~l !o~A.eC(,.

'. G)<=-> ~ ~ ~ ~ tx~I-.~)X~)

! (1) ~ '> Q, CAOI1,Q-ct ~ \ ~ . () ~ t1 ~ ~ to oiL~JW;l1 :w,

m - ~ oR Lb~.. ~ ~ b! c9LQJ~cb~\G.~ ,,! ~ e.l.-Lf'fflI,,~t t i1.i. 1't v )

PI 1/rIo 'tVtSYttt, ~ ~ ~ ~! ~ ~ 1. ~ 0 '~ t;I)IO ~ ~ d ~ UA1. ~ fr\..o 4\;0 k .

(1)\ 'tK~t cl..:M - OpUQ.tt>,.1 ('\c>D'lk ~ p'LoQtLt.,sg) ~~.~~I ~ SH. i . . 0

F; 1~ OL4,t p...b~ \! r.Ik~~r..Q,ttAi\a..t ~ij~ ~

~~ f)p~t 6f,t~(.tJ~;,:, 1U(.c~ bWlJ1L1 (W.dA..tt.L~) L1. ~IM'J

.) ~~~~~QMl ~O1, ~Rt6~. ) -1YnI.t. r..od. ~ 'I) ~ J f. t ~ krc. tiP 114. ~ CI '.

to co.£CAA. ~k~~ ~ ~ ~ ~) J "t, "'frill- ~- C<>/'/VL I~ !CJt ~~~ ~l {.o vc.. "1 -=~~~t~:

@'i=YflMllb~: X "E- ~ ( H) 0~~~ ) ,. } "f'Y\

~~~~b~~\ ~ ~.DA~ t ~ £, at f ~ t tMat 0 r rAQ;t OI,.! tJa. Q. t

t.o1IL X*,) ..#)X~ ~ t\ttCto\' r'ttd~ ~ r.-t>;l.t q,tlW

I ~- -~~;~~-"'~-;:~~"d:[~~XJ ~~ .' ~~t ,:~~,~ 0

~\ ~\ ¥iI.oW\4., ~J;. ~ -= ~ I ~ [ X~ [. ')(.\ ~111 4 ptltfAl\., 0 '

~~ :---; ~ J--

~t -t14,~ Okt i~t ~\, "",,{Q.tu!. V {~S ~~b~~ ~S

(~d~) \!'...~~.=~=~. (!) 1hn 0(3 oR!~' . f/1,Q.t~~-- . ~ aM, ,ilfA.Q.itd. C-} -] .

~ ~ 'La.{'r'k. : fV\O.vM at '

~ t\4J ~ b oa& ~ ~-I.I."" di 1 i. () p "- ~ tr;t, 1

foi ~3~\.tU) ~kI., JOt}(, Q.pp~ ~i~ oc. ~~ p5 0 ~

O~ ~~ ~~a.th"e.o.t ,tA'ttrm.l.I\1,~ .fI.o.~ ~ PC'kt1 ~~~tMaJ.

;u\1.\/QMCIJ'II.t~-~~~~~t~ 'QII-L . -- ~. TtX]t)::. '\l)('i - V,X - (~)'(] .I

. ~ ~~) l: =-~V cx,tll--:i?~ ) ~,1 (i) ..\

~~ttA h~ .tf.A, ~~~u ~~~ h-l.t . qt. fl.Lfatt.~ IJtCf-rltOf*P~~~~~~~~~ .

~t.o ~~