For complex oriented cohomology theories, p -typicality is atypical Niles Johnson Joint with Justin Noel (IRMA, Strasbourg) Department of Mathematics University of Georgia April 28, 2010 Niles Johnson (UGA) p -typicality is atypical April 28, 2010 1 / 37
Introduction Complex Cobordism MU is a nexus in stable homotopy theory. There is a spectrum MU satisfying: π n MU ∼ = { Complex cobordism classes of n-manifolds } . There is a spectral sequence (ANSS) Ext ∗ , ∗ MU ∗ MU ( MU ∗ , MU ∗ ) = ⇒ π ∗ S . MU serves as a conduit between the theory of formal group laws and stable homotopy theory. This project: use power series calculations to get results about power operations in complex-oriented cohomology theories Niles Johnson (UGA) p -typicality is atypical April 28, 2010 2 / 37
Introduction Goal Conjecture The p -local Brown Peterson spectrum BP admits an E ∞ ring structure Partial Results: (Basterra-Mandell) BP is E 4 . (Richter) BP is 2 ( p 2 + p − 1 ) homotopy-commutative. (Goerss/Lazarev) BP and many of its derivatives are E 1 = A ∞ -spectra under MU (in many ways). (Hu-Kriz-May) There are no H ∞ ring maps BP → MU ( p ) . H ∞ is an “up to homotopy” version of E ∞ Niles Johnson (UGA) p -typicality is atypical April 28, 2010 3 / 37
Introduction Goal Theorem (J. – Noel) Suppose f : MU ( p ) → E is map of H ∞ ring spectra satisfying: f factors through Quillen’s map to BP . 1 f induces a Landweber exact MU ∗ -module structure on E ∗ . 2 Small Prime Condition: p ∈ { 2 , 3 , 5 , 7 , 11 , 13 } . 3 then π ∗ E is a Q -algebra. Application: The standard complex orientations on E n , E ( n ) , BP � n � , and BP do not respect power operations; The corresponding MU -ring structures do not rigidify to commutative MU -algebra structures. Niles Johnson (UGA) p -typicality is atypical April 28, 2010 4 / 37
Introduction Plan Motivate structured ring spectra Describe MU , BP , and the connection to formal group laws Topological question � algebraic question (power series) Display some calculations Niles Johnson (UGA) p -typicality is atypical April 28, 2010 5 / 37
Introduction Background Spectra ↔ Cohomology theories A (pre-)spectrum is a sequence of pointed spaces, E n , with structure maps Σ E n → E n + 1 such that the adjoint is a homotopy equivalence: ≃ E n − → Ω E n + 1 . This yields a reduced cohomology theory on based spaces: E n ( X ) = [ X , E n ] ∼ � = [ X , Ω E n + 1 ] ∼ = � E n + 1 (Σ X ) Niles Johnson (UGA) p -typicality is atypical April 28, 2010 6 / 37
Introduction Background Spectra ↔ Cohomology theories Some motivating examples: Ordinary reduced cohomology is represented by Eilenberg-Mac Lane spaces � H n ( X , R ) = [ X , K ( R , n )] Topological K -theory is represented by BU × Z and U (Bott periodicity): � [ X , BU × Z ] n = even n ( X ) = � KU [ X , U ] n = odd Complex cobordism is represented by MU ( n ) = colim q Ω q TU ( n + q ) n ( X ) = [ X , MU ( n )] � MU etc. Niles Johnson (UGA) p -typicality is atypical April 28, 2010 7 / 37
Introduction Background Spectra ↔ Cohomology theories From a spectrum E we get an unreduced cohomology theory on unbased spaces by adding a disjoint basepoint. For an unbased space X , E ∗ ( X ) = � E n ( X + ) = [ X + , E ∗ ] E ∗ ( − ) takes values in graded abelian groups. When E is a ring spectrum, E ∗ ( − ) takes values in graded commutative rings (with unit). E ∗ denotes the graded ring E ∗ ( pt . ) . Niles Johnson (UGA) p -typicality is atypical April 28, 2010 8 / 37
Introduction Background Spectra ↔ Cohomology theories Brown Representibility Every generalized cohomology theory is represented by a spectrum. Viewed through this lens, it is desirable to express the “commutative ring” property in the category of spectra. Doing so allows us to work with cohomology theories as algebraic objects. Difficulty: organizing higher homotopy data (motivates operads & monads) There are many good categories of spectra, having well-behaved smash products and internal homs. Niles Johnson (UGA) p -typicality is atypical April 28, 2010 9 / 37
� � � Introduction Background Structured Ring Spectra The category of E ∞ ring spectra is one category of structured ring spectra. An E ∞ ring spectrum is equipped with a coherent family of structure maps E ∧ s E � µ � � � � � � � D s which extend over the Borel construction D s E = E Σ s ⋉ Σ s E ∧ s ; a “homotopy-fattened” version of E s coherent: D s D t → D st , D s ∧ D t → D s + t , etc. Niles Johnson (UGA) p -typicality is atypical April 28, 2010 10 / 37
Introduction Background Power Operations and H ∞ Ring Spectra The definition of E ∞ predated applications by about 20 years For many applications, it suffices to have the coherent structure maps defined only in the homotopy category. This defines the notion of an H ∞ ring spectrum. This data corresponds precisely to a well-behaved family of power operations in the associated cohomology theory. For an unbased space X , and π ≤ Σ n → E 0 ( D s X ) δ ∗ µ P π : E 0 ( X ) → E 0 ( B π × X ) . − − µ : H ∞ structure maps δ ∗ : pulling back along diagonal X → X × s Niles Johnson (UGA) p -typicality is atypical April 28, 2010 11 / 37
Introduction Background Power Operations and H ∞ Ring Spectra MU has a natural H ∞ ring structure arising from the group structure on BU . Thom isomorphism for MU ⇒ wider family of even-degree power operations P π : MU 2 i ( X ) → MU 2 in ( B π × X ) π ≤ Σ n take π = C p , X = pt . MU ∗ ( C P ∞ ) = MU ∗ � x � also has a (formal) group structure induced by the multiplication on C P ∞ . This gives us computational access to the MU power operations. Niles Johnson (UGA) p -typicality is atypical April 28, 2010 12 / 37
Formal Group Laws Formal Group Laws A (commutative, 1-dimensional) formal group law over a ring R is determined by a power series F ( x , y ) ∈ R � x , y � which is unital, commutative, and associative, in the following sense: F ( x , 0 ) = x = F ( 0 , x ) . F ( x , y ) = F ( y , x ) . F ( F ( x , y ) , z ) = F ( x , F ( y , z )) . Example ( G a ) : F ( x , y ) = x + y . Example ( G m ) : F ( x , y ) = x + y + xy . Example ( MU ) : C P ∞ × C P ∞ → C P ∞ induces � MU ∗ ( C P ∞ × C P ∞ ) MU ∗ ( C P ∞ ) � MU ∗ � x , y � MU ∗ � x � � x + MU y x � Niles Johnson (UGA) p -typicality is atypical April 28, 2010 13 / 37
Formal Group Laws Formal Group Laws Theorem (Lazard) There is a universal formal group law � a ij x i y j F univ . ( x , y ) = and it is defined over L = Z [ U 1 , U 2 , U 3 , . . . ] Theorem (Quillen) MU ∗ = Z [ U 1 , U 2 , U 3 , . . . ] and x + MU y = F univ . ( x , y ) Niles Johnson (UGA) p -typicality is atypical April 28, 2010 14 / 37
MU ∗ and BP ∗ Formal Group Laws MU ∗ and BP ∗ MU ∗ = Z [ U 1 , U 2 , U 3 , . . . ] MU −∗ ⊗ Q ∼ = H Q ∗ ( MU ) ∼ = Q [ m 1 , m 2 , m 3 , . . . ] [ C P n ] ∈ MU − 2 n Under the Hurewicz map to rational homology [ C P n ] �→ ( n + 1 ) m n . � � ∼ → MU ∗ ⊗ Q = [ C P 1 ] , [ C P 2 ] , [ C P 3 ] , . . . ֒ − Q Niles Johnson (UGA) p -typicality is atypical April 28, 2010 15 / 37
MU ∗ and BP ∗ Formal Group Laws MU ∗ and BP ∗ r ∗ � BP ∗ MU ∗ MU ∗ ⊗ Q � BP ∗ ⊗ Q � if i � = p k − 1 0 m i �→ if i = p k − 1 ℓ k ∼ → BP ∗ ⊗ Q = Q [ ℓ 1 , ℓ 2 , ℓ 3 , . . . ] ֒ − r ∗ [ C P p k − 1 ] = p k ℓ k ∈ BP − 2 ( p k − 1 ) ⊗ Q for n � = p k − 1 r ∗ [ C P n ] = 0 Niles Johnson (UGA) p -typicality is atypical April 28, 2010 16 / 37
MU ∗ and BP ∗ Formal Group Laws MU ∗ and BP ∗ � Σ d BP MU = some d BP ∗ ∼ = Z ( p ) [ v 1 , v 2 , v 3 , . . . ] Hazewinkel generators: p + v 1 + p ℓ 1 = v 1 ℓ 2 = v 2 1 p , p 2 , 2 + v 2 v p 2 + v 1 + p + p 2 p + v 1 v p ℓ 3 = v 3 1 1 , etc. p 2 p 3 Araki generators: v 1 ℓ 1 = p − p p , etc. Niles Johnson (UGA) p -typicality is atypical April 28, 2010 17 / 37
Formal Group Laws log BP , exp BP , and formal sum log BP , exp BP , and formal sum Rationally, every formal group law is isomorphic to the additive formal group x + F y = log − 1 F ( log F ( x ) + log F ( y )) log BP ( t ) = t + ℓ 1 t p + ℓ 2 t p 2 + · · · ( p -typical) exp BP ( t ) = log − 1 BP ( t ) ξ + BP x = exp BP ( log BP ( ξ ) + log BP ( x ) ) = ξ + x + · · · [ i ] ξ = i ξ + · · · = ξ · � i � ξ Niles Johnson (UGA) p -typicality is atypical April 28, 2010 18 / 37
� � � � � � Calculations Topological Question H ∞ ring structure for BP ? Consider P C p : MU 2 i ( pt . ) → MU 2 pi ( BC p ) q ∗ MU ∗ ( BC p ) ∼ = MU ∗ � ξ � / [ p ] ξ → MU ∗ � ξ � / � p � ξ − q ∗ BP ∗ ( BC p ) ∼ = BP ∗ � ξ � / [ p ] ξ → BP ∗ � ξ � / � p � ξ − P Cp q ∗ a 2 n 0 MU 2 p ∗ � ξ � / [ p ] ξ MU 2 p ∗ + 4 n ( p − 1 ) � ξ � / � p � ξ MU 2 ∗ r ∗ r ∗ r ∗ q ∗ a 2 n P Cp � BP 2 p ∗ + 4 n ( p − 1 ) � ξ � / � p � ξ 0 BP 2 p ∗ � ξ � / [ p ] ξ BP 2 ∗ � � � � � Calculate MC n = r ∗ q ∗ a 2 n 0 P C p [ C P n ] for n � = p k − 1 Niles Johnson (UGA) p -typicality is atypical April 28, 2010 19 / 37
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