Higher Toda brackets and the Adams spectral sequence Dan - PowerPoint PPT Presentation

Higher Toda brackets and the Adams spectral sequence Dan Christensen University of Western Ontario Joint work with Martin Frankland CT2016, Halifax, Aug 10, 2016 Outline: Triangulated categories and injective classes The Adams spectral sequence 3-fold Toda brackets, and the relation to d 2 Higher Toda brackets, and the relation to d r 1 / 12

Triangulated categories A triangulated category is an additive category T equipped with an equivalence Σ : T → T , and with a specified collection of triangles of the form f g h X − → Y − → Z − → Σ X. (1) These must satisfy the following axioms motivated by (co)fibre sequences in topology. TR0: The triangles are closed under isomorphism. The following is a triangle: 1 − → X − → 0 − → Σ X. X TR1: Every map X → Y is part of a triangle (1). TR2: (1) is a triangle iff (2) is a triangle: g − Σ f h − → Z − → Σ X − → Σ Y. Y (2) 2 / 12

� � � � � Triangulated categories, II T additive, Σ : T → T an equivalence. TR0: Triangles are closed under isomorphism and contain the trivial triangle. TR1: Every map appears in a triangle. TR2: Triangles can be rotated. TR3: Given a solid diagram � Z � Σ X X Y u Σ u � Y ′ � Z ′ � Σ X ′ X ′ in which the rows are triangles, the dotted fill-in exists making the two squares commute. TR4: The octahedral axiom holds. 3 / 12

Examples and consequences Example. The homotopy category of spectra. Example. The derived category of a ring. Example. The stable module category of a group algebra. Example. The homotopy category of any stable Quillen model category. Consequences: (1) For any object A , the sequences · · · − → T ( A, X ) − → T ( A, Y ) − → T ( A, Z ) − → T ( A, Σ X ) − → · · · and · · · ← − T ( X, A ) ← − T ( Y, A ) ← − T ( Z, A ) ← − T (Σ X, A ) ← − · · · are exact sequences of abelian groups. (2) The triangle containing a map X → Y is unique up to (non-unique) isomorphism. 4 / 12

Injective classes Eilenberg and Moore (1965) gave a framework for homological algebra in any pointed category. When the category is triangulated, their axioms are equivalent to the following: Definition. An injective class in T is a pair ( I , N ), where I ⊆ ob T and N ⊆ mor T , such that: (i) I consists of exactly the objects I such that every composite X → Y → I is zero for each X → Y in N , (ii) N consists of exactly the maps X → Y such that every composite X → Y → I is zero for each I in I , (iii) for each Y in T , there is a triangle X → Y → I with I in I and X → Y in N . The first two conditions are easy to satisfy. The third says that there are enough injectives. 5 / 12

Examples of injective classes Example. Let E be an object in any triangulated category T with infinite products. Take I to be all retracts of products of suspensions of E and N to consist of all maps X → Y such that every composite X → Y → I is zero, for I in I . Then ( I , N ) is an injective class. If we write E k ( − ) for the cohomological representable functor T ( − , Σ k E ), then N consists of the maps inducing the zero map under E ∗ ( − ). Example. In the category of spectra, if we take E = H F p , this injective class leads to the classical Adams spectral sequence. We always assume that our injective classes are stable, that is, that they are closed under suspension and desuspension. 6 / 12

� � � � � � � � � � Adams resolutions Definition. An Adams resolution of an object Y in T with respect to an injective class ( I , N ) is a diagram i 0 i 1 i 2 Y = Y 0 Y 1 Y 2 Y 3 · · · p 0 p 1 p 2 δ 0 δ 1 δ 2 I 0 I 1 I 2 · · · where each I s is injective, each map i s is in N , and the triangles are triangles. Axiom (iii) says exactly that you can form such a resolution. Adams resolutions biject with injective resolutions with respect to the injective class. 7 / 12

� � � � � � � � � � The Adams spectral sequence Miller, 1974; C, 1997 Given objects X and Y and an Adams resolution i 0 i 1 i 2 Y = Y 0 Y 1 Y 2 Y 3 · · · p 0 p 1 p 2 δ 0 δ 1 δ 2 � I 1 � I 2 I 0 · · · d 1 d 1 of Y , applying T ( X, − ) leads to an exact couple and therefore a spectral sequence; it is called the Adams spectral sequence. The E 1 term is E s,t = T (Σ t − s X, I s ), and the first differential d 1 is 1 given by composition with d 1 := pδ : I s − → ◦ Y s +1 − → I s +1 . The E 2 term is Ext s I (Σ t X, Y ), essentially by definition. We regard d 1 as a primary operation. 8 / 12

� � � � � � � � � � � � Adams d 2 differential Recall that E 2 is the homology of T ( X, I s ) w.r.t. d 1 . Given a class [ x ] in the E 2 term of an Adams spectral sequence, d 2 [ x ] is defined in the following way: i s +1 i s · · · Y s Y s +1 Y s +2 · · · p s +1 p s +2 p s δ s δ s +1 � I s +1 � I s +2 I s d 1 d 1 � x x d 2 [ x ] X d 2 [ x ] is a subset of T ( X, I s +2 ). We’ll describe this subset using “higher operations”. 9 / 12

� � 3-fold Toda brackets Toda, 1962 f 1 f 2 f 3 Let X 0 − → X 1 − → X 2 − → X 3 be a diagram in T . The Toda bracket � f 3 , f 2 , f 1 � ⊆ T (Σ X 0 , X 3 ) consists of all composites β ◦ Σ α : Σ X 0 → X 3 , where α and β appear in a commutative diagram f 1 � X 1 X 0 α f 2 � X 1 � X 2 � C f 2 Σ − 1 C f 2 β f 3 � X 3 , X 2 where the middle row is a triangle. The indeterminacy can be explicitly described, and there are other equivalent definitions. 10 / 12

Adams d 2 in terms of Toda brackets β Proposition (C-Frankland) . d 2 [ x ] = � d 1 , p s +1 , δ s x � = � d 1 , d 1 , x � . The first equality is an elementary exercise, using the properties of injective classes. The second requires some explanation. Recall that � f 3 , f 2 , f 1 � was defined to consist of certain composites Σ α β Σ X 0 − → C f 2 − → X 3 . β The notation � f 3 , f 2 , f 1 � denotes the subset of the Toda bracket with β held fixed and only α allowed to vary. The choice of β is determined from the Adams resolution and the octahedral axiom. 11 / 12

Adams d r in terms of Toda brackets Following Cohen, Shipley and McKeown, we define r -fold Toda brackets in any triangulated category, and prove basic properties about them. Our main result is: Theorem (C-Frankland) . d r can be expressed in terms of ( r + 1)-fold Toda brackets as: d r [ x ] = � d 1 , d 1 , . . . , d 1 , p s +1 , δ s x � = � d 1 , d 1 , . . . , d 1 , x � fixed The first equality is straightforward, using our results. In the second equality, “fixed” means that you choose a particular “filtered object” derived from the Adams resolution, which fixes all of the choices except the very last α . Details are in arxiv:1510.09216, and these slides are on my website. Thanks for listening! 12 / 12

Overflow slides The remaining slides are just in case I have extra time.

� Higher Toda brackets McKeown, nLab, 2012 f 1 f 2 f 3 Definition. Given X 0 − → X 1 − → X 2 − → X 3 , define the Toda family T( f 3 , f 2 , f 1 ) to consist of all pairs ( β, Σ α ), where α and β appear in a commutative diagram � Σ X 1 Σ X 0 − Σ f 1 Σ α � f 2 � X 2 � C f 2 � Σ X 1 X 1 β f 3 � X 3 , X 2 with middle row a triangle. f 1 f 2 f 3 f n − → X 1 − → X 2 − → · · · − → X n , define the Toda bracket Given X 0 � f n , . . . , f 1 � ⊆ T (Σ n − 2 X 0 , X n ) inductively as follows: If n = 2, it is the set consisting of just the composite f 2 f 1 . If n > 2, it is the union of the sets � β, Σ α, Σ f n − 3 , . . . , Σ f 1 � , where ( β, Σ α ) is in T ( f n , f n − 1 , f n − 2 ).

� � � � � � � 4-fold Toda bracket Example. We have β ′ ,α ′ { β ′ ◦ Σ α ′ } . � f 4 , f 3 , f 2 , f 1 � = � β,α � β, Σ α, Σ f 1 � = � � β,α Σ α ′ � C Σ α Σ 2 X 0 Σ 2 X 1 row = − Σ 2 f 1 β ′ Σ α � C f 3 Σ X 1 Σ X 2 row = − Σ f 2 β f 3 � X 3 � X 4 X 2 f 4 0 The middle column is what is called a filtered object by Cohen, Shipley and Sagave, and so this reproduces their definition.

Self-duality for higher Toda brackets The definition is asymmetrical. What happens in the opposite category? More generally, we can reduce an n -fold Toda bracket to a 2-fold Toda bracket in ( n − 2)! ways, inserting the Toda family operation in any position. Lemma (C-Frankland) . The pair ( β, Σ α ) is in T ( T ( f 4 , f 3 , f 2 ) , Σ f 1 ) iff the pair ( − β, Σ α ) is in T ( f 4 , T ( f 3 , f 2 , f 1 )). This is stronger than saying that the two ways of computing the Toda bracket � f 4 , f 3 , f 2 , f 1 � are negatives, and the stronger statement will be important for us. The proof is a careful application of the octahedral axiom.