Two classical problems Given a spectrum with a homotopy associative - PowerPoint PPT Presentation

MASSEY PRODUCTS AND UNIQUENESS OF A ∞ -ALGEBRA STRUCTURES Operations in Highly Structured Homology Theories, Banff, 22–27 May 2016. Fernando Muro Universidad de Sevilla

Two classical problems Given a spectrum with a homotopy associative multiplication, does it come from an A ∞ -algebra structure? If so, is it unique? 2

Two classical problems Given a spectrum with a homotopy associative multiplication, does it come from an A ∞ -algebra structure? If so, is it unique? Kadesihvili’88 These questions have been considered by many people. Robinson’89 For spectra, chain complexes, Rezk’98 simplicial modules... Tamarkin’98 For many operads: A ∞ , E ∞ , L ∞ , Lazarev’01 G ∞ ... Goerss–Hopkins’04 Using (variations of) Hochschild cohomology. Angeltveit’08 Roitzheim–Whitehouse’11 ... 2

The space of A ∞ -algebras B A ∞ B S A ∞ � category of A ∞ -algebras S � category of spectra B M � classifying space of a model category M � nerve of the category of weak equivalences in M 3

The space of A ∞ -algebras . . . . . . B A ∞ � lim B A n B A n +1 B A n B A 1 � B S A n � category of A n -algebras S � category of spectra B M � classifying space of a model category M � nerve of the category of weak equivalences in M 3

The space of A ∞ -algebras A fixed base point R ∈ B A ∞ allows for the construction of the Bousfield–Kan’72 fringed spectral sequence of the tower, . . . . . . B A ∞ � lim B A n B A n +1 B A n B A 1 � B S A n � category of A n -algebras S � category of spectra B M � classifying space of a model category M � nerve of the category of weak equivalences in M 3

Bousfield–Kan’s fringed spectral sequence E s , t 2 � ⇒ π t − s ( B A ∞ , R ) t e n i l d e g n fi s 4

Bousfield–Kan’s fringed spectral sequence E s , t 2 � ⇒ π t − s ( B A ∞ , R ) t − s t s 4

Bousfield–Kan’s fringed spectral sequence E s , t 2 � ⇒ π t − s ( B A ∞ , R ) t − s t d 5 s 4

Bousfield–Kan’s fringed spectral sequence E s , t 2 � ⇒ π t − s ( B A ∞ , R ) t − s 0 1 t groups abelian groups s t e s d e t n i o p s 4

The fringed line and uniqueness E s , s � weak equivalence classes of A r +1 -algebras which extend r to A r + s -algebras and restrict to the same A r -algebra as R , s ≤ r . t r s 5

The fringed line and uniqueness E s , s � weak equivalence classes of A r +1 -algebras which extend r to A r + s -algebras and restrict to the same A r -algebra as R , s ≤ r . t r s If the green line vanishes, the A r -algebra underlying R extends uniquely to an A n -algebra for all n ≥ r . 5

The fringed line and uniqueness The obstruction to A ∞ -uniqueness is the lim 1 in the Milnor s.e.s. 1 π 1 ( B A n , R ) ֒ → π 0 ( B A ∞ , R ) ։ lim lim n π 0 ( B A n , R ) n 6

The fringed line and uniqueness The obstruction to A ∞ -uniqueness is the lim 1 in the Milnor s.e.s. 1 π 1 ( B A n , R ) ֒ → π 0 ( B A ∞ , R ) ։ lim lim n π 0 ( B A n , R ) n which vanishes provided lim 1 n E s , s +1 � 0 for all s ≥ 0, n 1 t s 6

The fringed line and uniqueness Proposition If E s , s � 0 for all s ≥ r then R is uniquely determined by its r underlying A r -algebra. 7

The fringed line and uniqueness Proposition If E s , s � 0 for all s ≥ r then R is uniquely determined by its r underlying A r -algebra. Over a field k , E s , t 2 � HH s +1 , 1 − t ( π ∗ R ) for t ≥ s ≥ 1 and r � 2. 7

The fringed line and uniqueness Proposition If E s , s � 0 for all s ≥ r then R is uniquely determined by its r underlying A r -algebra. Over a field k , E s , t 2 � HH s +1 , 1 − t ( π ∗ R ) for t ≥ s ≥ 1 and r � 2. Corollary (Kadeishvili’88) If HH n , 2 − n ( π ∗ R ) � 0, n ≥ 3, then R is quasi-isomorphic to π ∗ R . 7

The fringed line and uniqueness Proposition If E s , s � 0 for all s ≥ r then R is uniquely determined by its r underlying A r -algebra. Over a field k , E s , t 2 � HH s +1 , 1 − t ( π ∗ R ) for t ≥ s ≥ 1 and r � 2. Corollary (Kadeishvili’88) If HH n , 2 − n ( π ∗ R ) � 0, n ≥ 3, then R is quasi-isomorphic to π ∗ R . What about existence? We could even be unable to choose a base point in B A ∞ with given algebra π ∗ R . 7

Below the fringed line and existence (Angeltveit’08 and ’11) E s , t HH s , − t ( π ∗ R ) ⇒ HH s − t ( R ) 2 ⇒ π t − s ( B A ∞ , R ) t t s s 8

Below the fringed line and existence (Angeltveit’08 and ’11) E s , t HH s , − t ( π ∗ R ) ⇒ HH s − t ( R ) 2 ⇒ π t − s ( B A ∞ , R ) t t s s E s , t 2 � HH s +1 , 1 − t ( π ∗ R ) , t ≥ s ≥ 1 , π t − s ( B A ∞ , R ) � HH s − t +2 ( R ) , t − s ≥ 3 (Toën’07) . 8

Below the fringed line and existence (Angeltveit’08 and ’11) E s , t HH s , − t ( π ∗ R ) ⇒ HH s − t ( R ) 2 ⇒ π t − s ( B A ∞ , R ) t t s s E s , t 2 � HH s +1 , 1 − t ( π ∗ R ) , t ≥ s ≥ 1 , π t − s ( B A ∞ , R ) � HH s − t +2 ( R ) , t − s ≥ 3 (Toën’07) . 8

Below the fringed line and existence (Angeltveit’08 and ’11) E s , t HH s , − t ( π ∗ R ) ⇒ HH s − t ( R ) 2 ⇒ π t − s ( B A ∞ , R ) t t s s Defined up to E r if R is just an A 2 r − 1 -algebra. E s , t 2 � HH s +1 , 1 − t ( π ∗ R ) , t ≥ s ≥ 1 , π t − s ( B A ∞ , R ) � HH s − t +2 ( R ) , t − s ≥ 3 (Toën’07) . 8

Below the fringed line and existence (Angeltveit’08 and ’11) E s , t HH s , − t ( π ∗ R ) ⇒ HH s − t ( R ) 2 ⇒ π t − s ( B A ∞ , R ) t t s n o i t c u r t s b o s s Defined up to E r if R is just an A 2 r − 1 -algebra. E s , t 2 � HH s +1 , 1 − t ( π ∗ R ) , t ≥ s ≥ 1 , π t − s ( B A ∞ , R ) � HH s − t +2 ( R ) , t − s ≥ 3 (Toën’07) . 8

Below the fringed line and existence (Angeltveit’08 and ’11) E s , t HH s , − t ( π ∗ R ) ⇒ HH s − t ( R ) 2 ⇒ π t − s ( B A ∞ , R ) t t ? n s o n i s o n i t e c t u x r e t s b o s s Defined up to E r if R is just an A 2 r − 1 -algebra. E s , t 2 � HH s +1 , 1 − t ( π ∗ R ) , t ≥ s ≥ 1 , π t − s ( B A ∞ , R ) � HH s − t +2 ( R ) , t − s ≥ 3 (Toën’07) . 8

Extending the fringed spectral sequence Bousfield’89 defined for the tower of the totalization of a cosimplicial space: � an extension of the fringed spectral sequence, given a global base point; � truncated spectral sequences, given an intermediate base point; � obstructions to lifting intermediate base points. 9

Extending the fringed spectral sequence Bousfield’89 defined for the tower of the totalization of a cosimplicial space: � an extension of the fringed spectral sequence, given a global base point; � truncated spectral sequences, given an intermediate base point; � obstructions to lifting intermediate base points. Our tower is not naturally like this. We proceed in a different way, suitable for explicit computations beyond the second page. 9

Extending the fringed spectral sequence S � Hk -module spectra, k a field (in order to stay safe). . . . . . . B A ∞ � lim B A n B A n +1 B A n B A 1 � B S 10

Extending the fringed spectral sequence S � Hk -module spectra, k a field (in order to stay safe). ∗ X . . . . . . B A ∞ � lim B A n B A n +1 B A n B A 1 � B S 10

Extending the fringed spectral sequence S � Hk -module spectra, k a field (in order to stay safe). End A ∞ X � lim End A n . . . End A n +1 End A n . . . ∗ X X X X (Rezk’96) pulling back . . . . . . B A ∞ � lim B A n B A n +1 B A n B A 1 � B S A n � operad for A n -algebras End X � the endomorphism operad of a spectrum X Q P � Map ( P , Q ) � the space of maps P → Q in the category of (non- Σ ) operads 10

Extending the fringed spectral sequence The spectral sequences of these towers substantially overlap. S.s. of { End A n S.s. of { B A n } n ≥ 1 X } n ≥ 2 t t s s 11

Extending the fringed spectral sequence The spectral sequences of these towers substantially overlap. S.s. of { End A n S.s. of { B A n } n ≥ 1 X } n ≥ 2 t t s s We can take advantage of the homotopy theory of A ∞ . From now on, we work with the second one. 11

Where do classical obstructions come from? The operad A ∞ has cells µ n in arity n and dimension n − 2, n ≥ 2. 12

Where do classical obstructions come from? The operad A ∞ has cells µ n in arity n and dimension n − 2, n ≥ 2. F ( Σ − 1 µ n ) A n − 1 A n 12

Where do classical obstructions come from? The operad A ∞ has cells µ n in arity n and dimension n − 2, n ≥ 2. F ( Σ − 1 µ n ) A n − 1 A n R End X 12

Where do classical obstructions come from? The operad A ∞ has cells µ n in arity n and dimension n − 2, n ≥ 2. F ( Σ − 1 µ n ) A n − 1 A n R extension? End X 12

Where do classical obstructions come from? The operad A ∞ has cells µ n in arity n and dimension n − 2, n ≥ 2. F ( Σ − 1 µ n ) A n − 1 A n obstruction! R extension? End X 12