ups that Suppose T em 0.0.1. Lemma h ordc on coproduct The E httakes that r Hence em 0.0.2. Lemma pcrmwihi an is which G ru ihcodnt Spf coordinate with group pcr otectgr fgae bla rusta ae sum takes that groups abelian graded of category the to spectra yinspection. by map Preamble: uoopim of automorphisms otemlilcto map multiplication the to Proof. arXiv:2109.00048v2element [math.AT] 7 Sep 2021 1 E ˝ s : Ñ 3. .Fran For 4. 2. .Teei map a is There 1. h iercoefficient linear The . l lersaegae n commutative. and graded are algebras All E F ev EA from M theory. ssafwsml nvra rpriso eti cohomolog certain of l properties tools universal computational simple on few relying a of uses Instead of derivation group. non-computational additive a gives note two-page This M Ñ f ˝ E Let ˚ p T ,where Σ, EA sra retdwt elorientation real with oriented real is e M Spec : nteiaeof image the in p q s f e e ensasrc uoopimof automorphism strict a defines q to sfltover flat is Let etecnnclgnrtrof generator canonical the be to F otempo oooyidcdby induced homotopy on map the to a eietfidwt oe series power a with identified be may E 2 s E -algebra ˚ g 1 hr emti eiaino h ulSeno algebra Steenrod dual the of derivation geometric short A ups htteei oooycmuaiern spectrum commutative homotopy a is there that Suppose Let . MT Let E ˚ r titioopim ffra ru laws group formal of isomorphisms strict are E r G E 1 sfltover flat is p s H E ˚ R etectgr fpairs of category the be eoe h hf-fgaigfntr morphism A functor. shift-of-grading the denotes M E E p E Ñ q R eara retdhmtp igsetu uhthat such spectrum ring homotopy oriented real a be mdl,hrlsl eoe by denoted harmlessly -module, p R M nue ytemap the by induced eahmtp igsetu and spectrum ring homotopy a be q η Ñ E E hnteei opimSpec morphism a is there Then . E f ˚ BC e Aut let Aut ˚ ˚ If . 0 H E n h map the and , BC sfre ob ycnieigteplbc of pullback the considering by 1 be to forced is 2 Ý Ñ E nuiga smrhs on isomorphism an inducing f G EA ˆ 2 ˚ E G A othat so , n rt Aut write and BC p R ÞÑ q M sachmlg hoy(..if (e.g. theory a is q p 2 R hc sceryntrlin natural clearly is which Ñ η q E etegopi hs bet are objects whose groupoid the be E BC p E f ev A ˚ q p η BC E G 2 e p M E : E Ñp ÞÑ C h oe series power the , ,s F, p “ M ˚ E p G 2 Spec : ^ E 2 R rcmoiino h assignment the of Precomposition . P Let . E stecci ru fodr2. order of group cyclic the is X E q q q E E etme ,2021 8, September M ˚ srpeetdby represented is where f o h ru-audpeha nAlg on presheaf group-valued the for ffre ytepeiu em sijciewhen injective is lemma previous the by afforded ÞÑ ^ X p Ý Ý Ý Ý Ý Ñ 1 id ia Luecke Kiran E e BC E R P q ^ EA Ý Ý Ý Ñp ˚ 1 EA 1 E Abstract ^ Ý Ý Ý Ñ ean be ^ id 2 p F id E R ˚ id E Ñ ^ π n hr sa nes to inverse an is there and , n omlgroup formal and X theories. y ˚ rr k pcrlsqecsadSeno prtos h argume the operations, Steenrod and sequences spectral ike ˚ h ulSeno ler steatmrhsso h formal the of automorphisms the as algebra Steenrod dual the sacnrvratfntrfo h oooyctgr ffinite of category homotopy the from functor contravariant a is T 0 1 x E . “ Hom E E φ : ss E “ E q from R π ^ ^ f oevrb utpiaiiyof multiplicativity by Moreover . ˚ Ý Ý Ñ A opout and products to s E ev ´˚ ^ p agbaadlet and -algebra E n omtswt h oodlsrcueo ohsides both on structure monoidal the with commutes and e MT E ˚ E sfltover flat is q EA E q X E ^ ˚ utb nedmrhs ftefra ru a of law group formal the of endomorphism an be must f p osdrte“onrFod uco Alg functor “Conner-Floyd” the Consider . Aut X ˚ ,E E, b p ^ E F ,s F, Ñ E M » E G nue n“nenl opsto fmorphisms of composition “internal” an induces ˚ F EA. osdrtentrltransformation natural the Consider . ´q E E p Ñ q 2 E to A e agbamaps -algebra p G M ˚ q fmni-audpehae nAlg on presheaves monoid-valued of along E E g . E M ˚ ˚ ˚ uhthat such F F E hni orsod oa(oooyring) (homotopy a to corresponds it then ) sfltover flat is ihfra ru law group formal with T M 1 q s , ˚ S Write . ea lmn fHom of element an be s T X 1 1 q santrlioopimfrom isomorphism natural a is η ã ÞÑ Ñ E santrltransformation natural a is Ý Ñ F ie ysnigatransformation a sending by given ˚ BC f sending M EA p R e E ˚ q 2 f ˚ ˚ Ý Ñ with n noigsaiiyof stability invoking and X f o h induced the for Write . q T R R n auaiyapplied naturality and η n hs morphisms whose and MR E otegopo strict of group the to enstedesired the defines F G MT M E sacohomology a is . o h formal the for p ,ER E, M E E ˚ ˚ ˚ -algebra F nt Ñ . q The . Ñ F MT F T to . 1 structure whose underlying ring is R. Note that MRf is canonically real oriented by the class eMRf “ reM b 1s P ˚ ˚ MRf BC2 “ M BC2 bM ˚ Rf . Then there is a functor

γ : AutGM pRq Ñ MT

which on objects sends f to MRf and on morphisms sends φ to the transformation γpφq : MRf Ñ MRg such that ˚ γpφqpeMRf q“ φpeMRg qP MRg BC2 » RgrreMRg ss as a power series.

Then the evaluation map evH from the previous lemma is an isomorphism. In particular if A2 denotes a corepresenting AutGH then there is an isomorphism of Hopf algebras H˚H » A2.

˚ ˚ Proof. By item 1 F2 is an algebra over M . By item 2 the formal group law of M BC2 has vanishing 2-series r2speM q“ 0 and so it is isomorphic to the additive one. Transporting such an isomorphism φ with γ produces a natural isomorphism

˚ ˚ ˚ γpφq ˚ F ˚ F ˚ M X » M X bM ˚ M ÝÝÝÑ M X bM ˚ 2 bF2 M “ M 2 bF2 M .

Thus MF2 is a summand of a cohomology theory and hence a cohomology theory. The map in item 1 induces a map MF2 Ñ H which by Eilenberg-Steenrod uniqueness must be an isomorphism. Thus, combining item 3 with the pullback along the surjection M Ñ MF2 » H, the map evH must be injective. So to produce an inverse it suffices to produce a section. Now since F2 is a field we have MR » HR for all F2-algebras R. Note that there is an inclusion BAutGH ãÑ AutFH ˚ since FM is sent to the additive formal group law by the map M Ñ F2 in item 1. Then the desired section of evH is given u as follows. Write F2 ÝÑ R for the unit map of an F2-algebra R. Start with an automorphism of AutGH pRq and view this ˚ as an automorphism of the object M Ñ F2 Ñ R in AutGM pRq. Using γ that produces an automorphism of MR » HR, which one then precomposes with the morphism H Ñ HR induced by u. Lemma 0.0.3. A cohomology theory M satisfying the conditions of the previous lemma exists. It is MO.

8 Proof. Let eMO denote the homotopy class of the inclusion RP ãÑ ΣMO. This exhibits pMO,eMOq as the universal real ˚ oriented multiplicative cohomology theory. Hence there is a map MO Ñ H sending eMO to e P H BC2. Thus items 1 1 and 2 are satisfied. Since MO is real oriented, an easy calculation shows that MO˚MO » MO˚ra1,a2, ...s which is free and hence flat over MO˚. Let A be an MO˚-algebra and T : MO Ñ MOA a multiplicative natural transformation. Let ˚ Θn P MO MOpnqq be the universal Thom class. By the splitting principle T pΘ1q determines T pΘnq, which determines all of T by universality. Therefore if MOA is a cohomology theory then evMOpAq is the composition of an isomorphism ˚ (ηMOpAq) and an injection (T ÞÑ T pΘ1q) so item 3 is satisfied. Let A be an F2-algebra receiving two maps f,g : MO Ñ A and let φ P Arrxss be an isomorphism from f˚FMO to g˚FMO . Note that MOAf and MOAg are canonically oriented by eMOAf and eMOAg as defined in item 4. It remains to construct (functorially) an automorphism γpφq : MOAf Ñ MOAg F 2 such that T peMOAf q“ φpeMOAg q. By the calculation MO˚MO » MO˚ bF2 2ra1,a2, ...s, the pair pg, φq corresponds to an F2-algebra map Φ : MO˚MO Ñ A and so using the natural transformation ηMO from the preamble one gets multiplicative transformation ηMOpΦq : MO Ñ MOAg with the property that ηMOpΦqpeMO q “ fpeMOAg q. Indeed the image of eMO 1 id ˚ ^ ˚ ˚ 2 under the map MO BC1 ÝÝÝÑpMO ^ MOq BC2 » MO BC2 bMO˚ MO˚MO is reMO b 1s`reMO b a1s` ... by a simple 3 ˚ calculation using the pairing between homology and cohomology . Note that the map ηMO pΦqpptq : MO Ñ A is not equal ˚ ˚ to g. Now ηMO pΦq induces an A-linear map MOAf X Ñ MOAg X and produces the desired map γpφq : MOAf Ñ MOAg which is what we are really after. Functoriality4 is proved by noting that γpφq is uniquely characterized by the properties of being A-linear, multiplicative, and sending eMOAf to φpeMOAg q. Remark 0.0.4. There is a similar but slightly messier derivation of the dual Steenrod algebra at odd primes which uses a mod p analog of MO constructed by Shaun Bullett [2] using manifolds with singularities. I decided to leave the odd primes to forthcoming work5 since the added technical difficulties (which in this business are usual at odd primes) obscures the simplicity and brevity of the exposition, which is a cardinal virtue of this note. Acknowledgements: Thanks to Tim Campion for extensive comments on previous drafts, and to Robert Burklund for catching a gap in a proof.

1 The real orientation eMO implies that all the differentials in the relevant Atiyah-Hirzebruch sequences are trivial. 2 2 By strictness fpeMOA q“ eMOA ` f1e ` ... and the corresponding map Φ : MO˚ra1,...sÑ A sends ai to fi`1. g g MOAg 3See e.g. Adams [1] Lemma 6.3 page 60 for the complex version. 4This argument for functoriality is due to Quillen in [4]. 5This work will also discuss derivations of algebras of cohomology operations of other spectra as well as a derivation of the Dyer-Lashof algebra.

2 1 References

[1] Adams, F. Stable homotopy and generalised homology. Chicago Lectures in Mathematics. The University of Chicago Press, 1995.

[2] Bullett, S. A Z/p Analogue for Unoriented Bordism. University of Warwick PhD Thesis (1973)

[3] Peterson, E. Formal geometry and bordism operations. Cambridge Studies in Advanced Mathematics, v17 (2019)

[4] Quillen, D. Elementary Proofs of Rome Results of Cobordism Theory Using Steenrod Operations. Advances in Mathematics, 7 29-56 (1971)

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