Fun with Dyer-Lashof operations

Transcription

1 Nordic Topology Meeting, Stockholm (27th-28th August 2015) based on arxiv: last updated 27/08/2015

2 Power operations and coactions Recall the extended power construction for n 1: D n X = EΣ n Σn X (n) = EΣ n Σn X } {{ X }. n For a commutative S-algebra R and an R-module Y, this extends to a construction D R n Y = EΣ n Σn Y n. Suppose that A, B, E are commutative S-algebras (it is enough to assume that E is H ). There are maps D n A A, D n B B and D n E E which give rise to a diagram of A-module morphisms. A D n E I µ n D A n (A E) A E

3 If x : S m A E, then the composition of solid arrows in the commutative diagram e S k Θ e (x) A D n S m id Dnx A D n (A E) I µn A A E A E x defines a power operation Θ e : A m (E) A k (E); Θ e (x) = x e for each element e A k (D n S m ) = π k (A D n S m ). Remark: it is well-known that D 2 S m Σ m RP m = Σ m RP /RP m 1.

4 Generalised coactions Let A, B be two commutative ring spectra (later we will take them to be E ). For any spectrum X, the unit S B induces and so a homomorphism If B (A) is B -flat then A X A S X A B X A (X ) A (B X ) B (X A). A (B) B B (X ) A (B X ) B (X ) B B (A), where denotes bimodule tensor product and denotes left module tensor product. If A = B, this gives the familiar left A (A)-comodule structure ψ : A (X ) A (A) A A (X ) and a right analogue ψ : A (X ) A (X ) A A (A).

5 Power operations and coactions Now assume that A, B, E are E and B (A) is B -flat. The following commutative diagram is the source of formulae showing the interaction between power operations and right coactions. A (S D n(a E)) (id Dnx) A (D n(s A E)) A (S A E) A (S E) A (S D ns m ) S ((A A) E) S (A E) A (B D ns m ) (id Dnx) A (B D n(a E)) A (Dn B (B (A E))) B ((A A) E) B (A E) A (B (A E)) A (B E) B (E) B B (A)

6 Eilenberg-Mac Lane spectra We will give explicit formulae in the case A = B = HF 2. Here A (A) = A = F 2 [ξ s : s 1] = F 2 [ζ s : s 1] is the dual Steenrod algebra which is a Hopf algebra over F 2. The generators have degrees ξ s = ζ s = 2 s 1 and ζ s = χ(ξ s ) and we set ζ 0 = ξ 0 = 1. The comultiplication is given by ψ(ξ s ) = ψ(ζ s ) = 0is ξ 2i s i ξ i, 0is ζ i ζ 2i s i and the right coaction is given by ψ(ξ s ) = ξ i ζs i, 2i 0is ψ(ζs ) = 0is ζ 2i s i ξ i.

7 Here is the formula for the right coaction on a Dyer-Lashof operation applied to an element x H m (E) and s m: [ s (ζ(t) ) ] k ψq s (x) = Q k ( ψ(x)) t k=m t s k. Here we use the Cartan formula to evaluate Dyer-Lashof operations on tensors and the right hand factor involves the generating function ζ(t) = ζ i t 2i. i0 For example, if ψ(x) = i a i x i then ψ(x) = i x i χ(a i ) and ψq m+1 (x) = i x 2 i χ(a i ) 2 ζ 1 + i Q m+1 (x i χ(a i )) = i x 2 i χ(a i ) 2 ζ 1 + i Q m+1 j x i Q j χ(a i )). j

8 Dyer-Lashof operations on A The Dyer-Lashof action on A was determined by Kochman (implicitly) and Steinberger. For example, Q 2s ζ s = ζ s+1. Other useful formula (which seem not to be well-known) are Q 2s ξ s = ξ s+1 + ξ 1 ξs 2, { ζ 4 Q 2s +1 1 if s = 1, ζ s = 0 otherwise, Q 2s +1 ξ s = ξ 2 1ξ 2 s.

9 Use of symmetric functions There is an E orientation MO HF 2 which induces a ring epimorphism H (MO) A. The Thom isomorphism H (BO) H (MO) is also a ring isomorphism. In fact these are both compatible with the Dyer-Lashof actions, but the second is not an A -comodule homomorphism. The ring H (BO) = F 2 [a k : k 1] can be identified with the ring of symmetric functions where a k is the k-th elementary function. The k-th Newton polynomial is defined recursively by N k (a) =a 1 N k 1 (a) a 2 N k 2 (a) + + ( 1) k 2 a k 1 N k 2 (a) + ( 1) k 1 ka k =a 1 N k 1 (a) + a 2 N k 2 (a) + + a k 1 N k 2 (a) + ka k. In general, N 2k (a) = N k (a) 2.

10 Kochman showed that Q r N k (a) = ( ) r 1 N k+r (a). k 1 The elements q k H (MO) corresponding to the N k (a) satisfy simple formulae for the A -coaction, in particular ψ(q 2 s 1) = i q 2i 2 s 1 1 ξ i. Furthermore, under the orientation homomorphism, q 2 s 1 ζ s. If we set N k (ξ) = N k (ξ 1, 0,..., 0, ξ 2, 0,...) where we replace a r by 0 except when r = when we replace it by ξ s, this gives Then we get N k (ξ) = ξ 1 N k 1 (ξ) + ξ 2 N k (ξ) +. Q r ζ s = Q r N 2 s 1(ξ) = ( ) r 1 2 s N 2 2 s 1+r (ξ) which is non-zero precisely when r 0, 1 mod 2 s.

11 Proving a splitting result Consider the factorisation of the bottom cell of BO through an infinite loop map S 1 QS 1 j BO. The associated Thom spectrum Mj is an E ring spectrum which is weakly equivalent to the reduced free spectrum P(S 0 2 e 1 ), hence for any E ring spectrum E with char π 0 (E) = 2 there is an E morphism P(S 0 2 e 1 ) E and we denote this spectrum by S//2. The homology of S//2 is H (S//2) = F 2 [Q I x 1 : I admissible, exc(i ) > 1], where x 1 H 1 (S//2) satisfies Sq 1 x 1 = 1. There is a morphism S//2 HF 2 which induces H (S//2) A so that x 1 ζ 1.

12 Define a sequence of elements x s H 2 s 1(S//2) by x s = Q 2s 1 x s 1 = Q 2s 1 Q 2s 2 Q 22 x 1. These form a regular sequence in the polynomial ring H (S//2) and generate an ideal I H (S//2) whose quotient ring H (S//2)/I is polynomial. The right coaction on the x s is ψ(x s ) = xs i 2i ξ i 0is where x 0 = 1. Induction shows that x s ζ s A 2 2 1, so H (S//2) A is an epimorphism.

13 Theorem The composition H (S//2) H (S//2)/I A ψ H (S//2) A is an isomorphism of A -comodule algebras. Hence S//2 is a wedge of suspensions of HF 2. A similar conclusion holds for any E ring spectrum E with char π 0 (E) = 2; this is a result of Steinberger.