Level algebras and Adem-Cartan operads

1. Level algebras and Adem-Cartan operads Muriel LIVERNET Malaga 2003

2. What are level algebras ? Definition, examples The operad Lev Free objects

3. Level algebras Definition: (A,) commutative satisfying (x y) (zt)=(x z) (y t) Ex2: colimits of a graded commutative algebra A Φ: At → A2t x → x2 At∞ := colim{A2ⁿt, Φ} |x|=2ht, |y|=2kt, k>h xy=Φk-h(x)y Ex1: commutative and associative algebras

4. Level algebras Definition: (A,) commutative satisfying (x y) (zt)=(x z) (y t) Ex3: Carlsson unstable module over A2 K=F2 [xk, kZ] ||xk||=(2-k,1) Sq1(xk)=xk+12 K(i)j= F2 xkk  k 2-k=i, k=j  Proposition: K(i) is a level algebra ( xkk)(xll)=  xk+1k xl+1l  satisfies the Cartan formula

5. 3 2 1 0 Planar rooted binary trees summit node root

6. 3 2 1 0 Level trees Equivalence relation: exchanging part above same level = K(1)n=xkk k 2-k=1, k=n  x22x34 Proposition: 1-to-1 correspondance between K(1)nand {n-level trees}.

7. 3 1 3 4 2 2 1 0 n-labelled level trees I=(,{2},{4},{1,3}) Proposition: 1-to-1 correspondance between {n-labelled level trees} and binary ordered partition of {1,…,n}. I=(Ij)j0 {1,…,n}=Ij |Ij |2-j=1

8. Operads Operad P: collection (P(n))n≥1 together with  Σn-action  1  P(1)  compositions P(n) P(i1 ) …  P(in) P(i1 +…+ in) p q1 …  qn p(q1,…,qn) or compositions « i » for 1 i  n i : P(n) P(m) P(n+m-1) Algebras over P: vector space A together with evaluation maps evn : P(n)  An A

9. 4 3 5 4 1 1 2 1 3 3 2 2 2 = 2 The operad Lev Operad Lev:Lev(n)=<n-labelled level trees>  Σn-action on labels  1  Lev(1) is the tree “root”  compositions « i » for 1 i  n

10. 2 1 1 2 = 1 1 2 3 4 4 2 3 = (xy)(zt)= (xt)(yz) Level algebras Theorem: algebras over the operad Lev are level algebras xy=yx Corollary: K(1)= F2 xkk  k 2-k=1  is the free level algebra over 1 generator

11. V vs in degree 1 Φn : V→ S2ⁿ(V) x → x2ⁿ A(*,*)=k0 S*( ΦkV) Theorem: A∞ (V):= colim{A(*,2ⁿ), Φ} is a level algebra ||Φ (x)||=(s,2t) ||x||=(s,t) xy=Φk-h(x)y Free level algebras bidegree (n n,nn2n) x=x0x12…xn2ⁿ… n0 Sn( ΦnV) Φ(x)=1x02x14… n0 Sn-1( ΦnV) It is the free level algebra generated by V

12. Adem-Cartan operads Unstable level algebras over B2 Adem-Cartan operads Secondary cohomological operations Link with E∞ -operads

13. level Unstable algebras over B2 A2 B2 Extended generated by Sqi, iN* Z Steenrod algebra A2 : Adem relations: Sqi Sqj =kkSqi+j-k Sqk Unstable module M: Sqi x=0, i>|x| level level algebra A commutative and associative Unstable algebra A: Cartan formula: Sqi xy= k+l=i Sqk x Sql y Sq|x| x= x2

14. Main theorem Theorem: there exists a family of operads called Adem-Cartan operads such that algebras over an Adem-Cartan operad are unstable level algebras over B2 An Adem-Cartan operad is closely related to a specific operad LevAC

15. Construction of LevAC The standard bar resolution of F2 Definition: LevAC(2)-i=<ei,ei> dei=ei-1+ei-1 ev2 : LevAC(2) A2  A x i y ei x  y  define «Steenrod» operations: Sqi x=x |x|-i x xy:=x 0 y is commutative if dA =0 Algebras over LevAC: A is a dgvs

16. (dp) (a1,…,an)=0 if dA=0 x 0 y =e0(x,y) de1=e0+e0 (de1) (x,y)=xy+yx=0 xy:=x 0 y is commutative P an operad, A an algebra over P dA (p(a1,…,an))=(dp) (a1,…,an)+i p(a1,…,dai,…,an)

17.  LevAC is a dg operad, free as a graded operad  E the standard bar resolution of F2 Free(E) the free operad on E Construction of LevAC V free 4-module generated by Gnm ‘s, degree -n LevAC =Free(E) Ud Free(V)

18.  there is a fibration f: LevAC  Lev such that f(n) is a quasi-isomorphism for n<4 f(n) induces an iso H0(LevAC(n))Lev(n) (xy)(zt)+(zy)(xt)=0 A a graded LevAC -algebra, dG11 (x,y,z,t)=0 (xy)(zt)=(xz)(yt) A is a level algebra Construction of LevAC Example: G11 LevAC(4), dG11 = e0 (e0 ,e0 )(Id+(3214))

19. Theorem: Let A be a graded LevAC -algebra dGn1 (x,x,y,y)=0 gives the Cartan relations dGnm (x,x,x,x)=0 gives the Adem relations Let A be a dg LevAC -algebra Theorem: A is an unstable level algebra over B2 H*(A) is an unstable level algebra over B2 An Adem-Cartan operad is such that O(2)= LevAC (2) and LevAC  O Definition: Same theorems for an algebra over an Adem-Cartan operad Theorems

20. Gp+1m  LevAC(4) Adem relation iSqmiSqni(x)=0 c A [Gp+1m (c,c,c,c)+b] H*(A)/iIm(Sqmi) Secondary cohomological operations A a dg algebra over O an Adem-Cartan operad x H*(A),x iKer(Sqni) Construction 1: Ψm,p: iKer(Sqni)  Hn(A) → H4n-p-1(A) /iIm(Sqmi)

21. Gp+1m  LevAC(4) Adem relation iSqmiSqni(x)=0 Construction 1: Ψm,p: iKer(Sqni) Hn(A) → H4n-p-1(A) /iIm(Sqmi) these two constructions coincide Theorem: Secondary cohomological operations H*(A) is a LevAC -algebra Construction 2: Θm,p: Hn(A) → H4n-p-1(A) x → Gp+1m (x,x,x,x)

22. E ∞(n)is k[n]-projective Definition: E Com Theorem:(Mandell) C*(X,Fp) is endowed with an E-structure which determines the p-adic homotopy type of X By homotopy invariance principle H*(X;Fp) has the same properties Link with E∞-operads

23.  E LevAC Lev Com E is an Adem Cartan operad Theorem: The cohomology of an E -algebra is an unstable algebra over B2 and has secondary cohomological operations In case H*(X;Fp) they coincide with Adams operations Link with E∞-operads The secondary cohomological operations coincide with Adams operations

24. LevAC is endowed with a diagonal Theorem: Cartan relations for secondary cohomological operations Prospects: Higher ordered cohomological operations Work in progress