Obstruction theory for E maps Niles Johnson Joint with Justin Noel - PowerPoint PPT Presentation

Obstruction theory for E ∞ maps Niles Johnson Joint with Justin Noel (Uni Bonn) Department of Mathematics University of Georgia January, 2012 Niles Johnson (UGA) Obstruction Theory January, 2012 1 / 22

Introduction Two main points Obstruction theory for T -algebra maps, where T is a monad on a topological category C . Demonstrations in rational homotopy when T is an E ∞ monad on Spectra . Niles Johnson (UGA) Obstruction Theory January, 2012 2 / 22

Introduction Differential graded algebras A toy question Suppose A and B are commutative d.g. algebras over Q : · · · ← A 2 ← A 1 ← A 0 ← A − 1 ← · · · | xy | = | x | + | y | d ( xy ) = ( dx ) y + ( − 1 ) | x | x ( dy ) xy = ( − 1 ) | x | | y | yx . Then H ∗ A and H ∗ B are (graded-)commutative Q -algebras. Niles Johnson (UGA) Obstruction Theory January, 2012 3 / 22

Introduction Differential graded algebras A toy question f Let A − → B be a map of chain complexes such that f ∗ H ∗ A → H ∗ B − is a map of (graded-)commutative Q -algebras. Question Is f a commutative d.g. algebra map? Is it chain homotopic to one? Answer Not always Niles Johnson (UGA) Obstruction Theory January, 2012 4 / 22

Introduction Differential graded algebras A toy question f Let A − → B be a map of chain complexes such that f ∗ H ∗ A → H ∗ B − is a map of (graded-)commutative Q -algebras. Question Is f a commutative d.g. algebra map? Is it chain homotopic to one? Better Answer Develop an obstruction theory to analyze f . Basic idea: Take P • + 1 A → A a simplicial resolution of A by free commutative d.g. Q -algebras. Consider the cosimplicial set Comm Q - alg ( P • + 1 A , B ) . . . Niles Johnson (UGA) Obstruction Theory January, 2012 4 / 22

Introduction Ring spectra More serious question(s) Let A and B be E ∞ ring spectra, and let f − → B A be a map of underlying spectra. Question(s) Is f H ∞ ? If so, does it rigidify to an E ∞ map? If so, is the rigidification unique? Niles Johnson (UGA) Obstruction Theory January, 2012 5 / 22

Introduction Ring spectra Main demonstration Let X and Y be spaces. There is an E ∞ mapping spectrum H Q X : π ∗ H Q X = H ∗ ( X ; Q ) . (a graded Q -algebra) Consider maps of spectra f H Q X → H Q Y − such that π ∗ f induces a commutative Q -algebra map π ∗ f H ∗ ( X ; Q ) → H ∗ ( Y ; Q ) . − − Question Is f homotopic to an E ∞ map? Note: A map of spaces Y → X induces an E ∞ map H Q X → H Q Y . Niles Johnson (UGA) Obstruction Theory January, 2012 6 / 22

Introduction Ring spectra Main demonstration Let Y = S 2 and let X = N = the Heisenberg nilmanifold: � N � T 2 S 1 � integer  1 ∗ ∗  � Z � π 1 N � Z 2 � ∗ N = 0 1 ∗ ∗   entries   0 0 1 π r N = 0 for r > 1 xy = 0 , � α 2 = β 2 = 0 , H ∗ ( N ; Q ) = Λ( x 1 , y 1 , α 2 , β 2 ) H ∗ ( S 2 ; Q ) = Λ( e 2 ) / e 2 x α = y β = 0 , x β + y α = 0 Consider maps H Q N → H Q S 2 dual to α or β . . . Niles Johnson (UGA) Obstruction Theory January, 2012 7 / 22

� � General framework Monads and homotopy algebra maps Monads Let T be a monad on C : � C T : C � T (unit) Id µ � T (mult.) T 2 σ An object X is a T -algebra if there is a structure map TX → X − compatible with unit and multiplication structure of T . T σ � T 2 X TX µ σ σ � X TX E.g. X is a set, TX is the free group on X σ X is a group if there is a structure map TX − → X such that . . . Niles Johnson (UGA) Obstruction Theory January, 2012 8 / 22

General framework Monads and homotopy algebra maps Example: Monad arising from an E ∞ operad Let O be an E ∞ operad on spectra, and let T = P be the associated monad: ho ( C P ) is the homotopy category of E ∞ ring spectra Let h P be the induced monad on ho C : ( ho C ) h P is the category of H ∞ ring spectra There are forgetful functors ho ( C P ) → ( ho C ) h P → ho C . Rephrased Question(s) Are these functors full? Are these functors faithful? Niles Johnson (UGA) Obstruction Theory January, 2012 9 / 22

General framework Monads and homotopy algebra maps Other examples Monads encoding group action on spaces or spectra Descent monads for commutative ring map k → K Monads arising from A ∞ operads, E n operads Your favorite topological monad! There are forgetful functors ho ( C T ) → ( ho C ) h T → ho C . Rephrased Question(s) Are these functors full? Are these functors faithful? Niles Johnson (UGA) Obstruction Theory January, 2012 10 / 22

General framework The obstruction spectral sequence The simplicial resolution Suppose that A and B are T -algebras in C . Successively applying T yields a simplicial object T • + 1 A . C T ( T s + 1 A , B ) is the space of T -algebra maps May be empty! C T ( T s + 1 A , B ) ∼ = C ( T s A , B ) . C T ( T • + 1 A , B ) ∼ = C ( T • A , B ) is a cosimplicial space � tower of fibrations, Bousfield-Kan spectral sequence under General Assumptions. Niles Johnson (UGA) Obstruction Theory January, 2012 11 / 22

General framework The obstruction spectral sequence General Assumptions C is a topologically enriched model category. T is a topological monad on C . C T has an induced topological model category structure such that all limits and tensors are calculated in C as well as all sifted colimits. A ∈ C T is such that T • + 1 A is Reedy cofibrant in s C T . One of the following: T • + 1 A is Reedy cofibrant in s C . T commutes with geometric realization of simplicial T -algebras. Niles Johnson (UGA) Obstruction Theory January, 2012 12 / 22

� � � General framework The obstruction spectral sequence An observation on E 1 = E 1 ( f ) s π 1 C ( T 2 A , B ) • • π 0 C ( TA , B ) π 1 C ( TA , B ) • π 0 C ( A , B ) π 1 C ( A , B ) t − s -1 0 1 Observation: d 1 is the difference around TA TB in ho C . � B A Niles Johnson (UGA) Obstruction Theory January, 2012 13 / 22

General framework The obstruction spectral sequence Theorem Let T be a monad on C satisfying the General Assumptions and let h T be the induced monad on ho C . For B in C T there is a fringed Bousfield-Kan spectral sequence E s , t r : E 0 , 0 = π 0 C ( A , B ) = ho C ( A , B ) . 1 1 A homotopy class [ f ] ∈ E 0 , 0 survives to E 0 , 0 if and only if [ f ] is 2 1 2 an h T -algebra map, that is, E 0 , 0 = ( ho C ) h T ( A , B ) . 2 When the E 2 = E 2 ( f ) page of the obstruction spectral sequence 3 is defined, we have E s , t = π s π t ( C ( T • A , B ) , f ) . 2 Niles Johnson (UGA) Obstruction Theory January, 2012 14 / 22

General framework The obstruction spectral sequence Theorem Let T be a monad on C satisfying the General Assumptions and let h T be the induced monad on ho C . For B in C T there is a fringed Bousfield-Kan spectral sequence E s , t r : The prospective basepoint f survives to the E ∞ page if and only 4 if f lifts to a T -algebra map. In this case, the spectral sequence conditionally converges to π ∗ ( C T ( A , B ) , f ) . ( E ∞ = E ∞ !) The edge maps 5 → E 0 , 0 → E 0 , 0 π 0 C T ( A , B ) − = ( ho C ) h T ( A , B ) − = π 0 C ( A , B ) . 2 1 are the forgetful functors from T -algebras to h T -algebras to the homotopy category of C , respectively. Niles Johnson (UGA) Obstruction Theory January, 2012 14 / 22

General framework The obstruction spectral sequence Corollaries, T = P Consider forget E ∞ ( A , B ) − − − → H ∞ ( A , B ) . Corollary The forgetful functor from the homotopy category of E ∞ ring spectra to H ∞ ring spectra is faithful if and only if E t , t ∞ = 0 for t > 0. Corollary The forgetful functor from the homotopy category of E ∞ ring spectra to H ∞ ring spectra is full if and only if the differential d r on E 0 , 0 is trivial for all r ≥ 2. r Niles Johnson (UGA) Obstruction Theory January, 2012 15 / 22

Demonstrations Demonstrations For pointed spaces X and Y , recall π ∗ H Q X ∼ = H ∗ ( X ; Q ) H ∞ ( H Q X , H Q Y ) ∼ = Comm Q - alg ( H ∗ ( X ; Q ) , H ∗ ( Y ; Q )) Consider the forgetful functor E ∞ ( H Q X , H Q Y ) − → H ∞ ( H Q X , H Q Y ) . Note: there is a natural base point for the obstruction spectral sequence ε : H Q X → H Q → H Q Y induced by X → ∗ → Y . Niles Johnson (UGA) Obstruction Theory January, 2012 16 / 22

Demonstrations Faithfulness η : S 3 → S 2 The Hopf map The Hopf map induces an E ∞ map H Q S 2 → H Q S 3 . The rational cohomology of S 2 H ∗ ( S 2 ; Q ) = Λ( e 2 ) / e 2 has resolution R = Λ( a 3 , e 2 ) , da = e 2 . The dual map a = η : R → H ∗ ( S 3 ; Q ) is a commutative Q -algebra map. Niles Johnson (UGA) Obstruction Theory January, 2012 17 / 22

� � � Demonstrations Faithfulness The Hopf map, E 2 page s 0 0 0 3 0 0 0 2 Q { a } 0 0 1 ǫ 0 0 t − s -1 0 1 The Hopf map induces a nontrivial E ∞ map which η H Q S 2 H Q S 3 is in the kernel of the forgetful functor. The diagram does not commute in ho E ∞ , H Q but does commute in H ∞ . Niles Johnson (UGA) Obstruction Theory January, 2012 18 / 22