E k -algebras and homological stability Oscar Randal-Williams - PowerPoint PPT Presentation

E k -algebras and homological stability Oscar Randal-Williams

Premise Want to study the homology of things like GL n ( ❦ ) , in particular its behaviour with respect to varying n . Have stabilisation maps � � A 0 : GL n − 1 ( ❦ ) − A �→ → GL n ( ❦ ) 0 1 and homological stability hopes these are homology isomorphisms in a range of degrees going to ∞ with n . Equivalently, it hopes that H d ( GL n ( ❦ ) , GL n − 1 ( ❦ )) = 0 for all d ≤ f ( n ) for some divergent function f . 1

Reformulation The space � R + = BGL n ( ❦ ) n ≥ 0 is a unital E ∞ -algebra in the category of N -graded spaces. Write H n , d ( R + ) := H d ( BGL n ( ❦ )) for homology in this category. For the basepoint σ ∈ H 0 ( BGL 1 ( ❦ )) the stabilisation map can be described in terms of the E ∞ -multiplication as − · σ : H d ( BGL n − 1 ( ❦ )) − → H d ( BGL n ( ❦ )) . Writing R + /σ for the cofibre in graded spaces of − · σ : R + [ 1 ] → R + , H d ( GL n ( ❦ ) , GL n − 1 ( ❦ )) = H n , d ( R + /σ ) . Goal : Exploit the E ∞ -structure on R + to analyse homological stability. Everything is based on joint work with S. Galatius and A. Kupers. 2

Homotopy theory of E k -algebras

Graded objects Let C denote sSet, sSet ∗ , Sp, or (because we are eventually interested in taking ❦ -homology) sMod ❦ . Write ⊗ for the cartesian, smash, or tensor product. We will consider N -graded objects in C, meaning C N := Fun ( N , C ) . This is given the Day convolution monoidal structure: � ( X ⊗ Y )( n ) = X ( a ) ⊗ Y ( b ) . a + b = n Define bigraded homology groups as H n , d ( X ; ❦ ) := H d ( X ( n ); ❦ ) . Define graded spheres in sSet N as � S d if n = m S n , d ( m ) = else ∅ and similarly discs D n , d . Analogously in the other categories. 3

E k -algebras Let C k denote the non-unital ( C k ( 0 ) = ∅ ) little k -cubes operad. e 2 · · · C 2 ( n ) = e 1 e n The categories C N are all tensored over Top: can make sense of the monad � C k ( n ) ⊙ S n X ⊗ n E k ( X ) := n ≥ 1 and so of E k -algebras X in C N . Call the category of these Alg E k ( C N ) . 4

E k -cells Each C N may be given the levelwise model structure, and Alg E k ( C N ) then has the projective model structure, making X �→ E k ( X ) : C N ⇄ Alg E k ( C N ) : X ← � X a Quillen adjunction. Given an E k -algebra X and a map f : S n , d − 1 → X can define the cell f D n , d as the pushout in Alg E k ( C N ) of attachment X ∪ E k f ad E k ( D n , d ) ← − E k ( S n , d − 1 ) → X . − Cellular E k -algebras are those formed by iterated cell attachments. (Every object is equivalent to a cellular one, as usual.) 5

Filtrations Let D := C N and Z ≤ be the poset of integers. A filtered object in D is a functor Z ≤ → D, and D Z ≤ is the category of such. The underlying object of a filtered X is colim Z ≤ X ∈ D. The filtration quotients of a filtered X are the cofibres, i.e. the pointed objects given by the pushouts − X ( n − 1 ) − → X ( n ) . ∗ ← Taking associated graded gives a strong monoidal functor gr : D Z ≤ − → D Z ∗ . If X is cofibrant have a spectral sequence E 1 H n , p + q (gr( X )( q )) ⇒ H n , p + q (colim X ) . n , p , q = � A filtered E k -algebra in D is an E k -algebra in D Z ≤ . A CW-E k -algebra is (roughly) a cellular object in filtered E k -algebras, where the attaching maps of the d -cells have filtration ≤ d − 1. 6

Indecomposables For X ∈ Alg E k ( C N ∗ ) define the E k -indecomposables of X by E k ( X ) = � µ X n ≥ 1 C k ( n ) ⊙ S n X ⊗ n Q E k ( X ) X c where c collapses all factors with n > 1 to the basepoint, and applies the augmentation ε : C k ( 1 ) + → S 0 . Q E k is left adjoint to the inclusion C N ∗ → Alg E k ( C N ∗ ) by imposing the trivial E k -action. Have Q E k ( E k ( X )) = X (the coequaliser is split). If X is a cellular E k -algebra then it follows that Q E k ( X ) is a cellular object with a ( g , d ) -cell for each E k - ( g , d ) -cell of X . If X is not cofibrant we should instead evaluate the derived functor Q E k L ( X ) := Q E k ( cofibrant replacement of X ) . AKA topological Quillen homology (for the operad C k ). 7

E k -homology and minimal cell structures Define E k -homology as H E k n , d ( X ) := H n , d ( Q E k L ( X )) . If ❦ is a field, the discussion so far shows dim ❦ H E 2 n , d ( X ; ❦ ) ≤ number of E 2 - ( n , d ) -cells in any E 2 -cellular approximation of X . Just as in classical homotopy theory, homology can be used to detect minimal cell structures as long as we work in a stable context. The following will suffice for now. Theorem. Let ❦ be a field and C be the category of simplicial ❦ -modules (or H ❦ -module spectra). Then X ∈ Alg E 2 ( C N ) has a cellular → X with dim ❦ H E 2 approximation c X ∼ g , d ( X ) -many E 2 - ( g , d ) -cells. Furthermore c X can be taken to be “CW”, not just “cellular”. 8

Computing E k -homology Q E k L ( X ) may also be computed by a k -fold bar construction. Instances of this have previously been given by Getzler–Jones, Basterra–Mandell, Fresse, Francis. In particular, if X is an E 1 -algebra it can be rectified to a nonunital + . associative algebra X and unitalised to an associative algebra X + → ✶ . Then there is an This unitalisation has an augmentation ε : X equivalence ✶ ∨ Σ Q E 1 + ; ✶ ) L ( X ) ≃ B ( ✶ ; X with the two-sided bar construction. (Something similar can be done for all E k .) From this perspective it is easy to see that vanishing lines for E 1 -homology imply vanishing lines for E 2 -homology, and so on. 9

The mapping class group E 2 -algebra

The mapping class group E 2 -algebra The surface Σ g , 1 = · · · has a mapping class group Γ g , 1 = π 0 ( Diff ∂ (Σ g , 1 )) . The collection � g ≥ 0 Γ g , 1 has the structure of a braided monoidal groupoid, so taking nerves gives a unital E 2 -algebra R + in sSet N with R + ( g ) ≃ B Γ g , 1 . Write R + ❦ ∈ Alg E + k ( sMod N ❦ ) for its ❦ -linearisation. 10

A vanishing line for E 2 -homology The bar construction model for Q E 1 L ( R ) leads us to study the simplicial complex whose p -simplices are ( p + 1 ) arcs on the surface Σ g , 1 , which cut it into ( p + 2 ) components each of which have non-zero genus. • • Σ 3 , 1 • • This is analogous to the Tits building of a vector space. We show that this simplicial complex is ( g − 3 ) -connected, and deduce Theorem (Galatius–Kupers–R-W). H E 2 g , d ( R ) = 0 for d < g − 1. Thus there is an E 2 -cellular approximation C → R only having ∼ ( g , d ) -cells for d ≥ g − 1. 11

Data Many calculations of H d (Γ g , 1 ) available for small g and d through the efforts of many mathematicians: Abhau, Benson, B¨ odigheimer, Boes, F. Cohen, Ehrenfried, Godin, Harer, Hermann, Korkmaz, Looijenga, Meyer, Morita, Mumford, Pitsch, Sakasai, Stipsicz, Tommasi, Wang, ... Z / 2 Z ⊕ Z / 2 2 Z Z Z Z / 10 1 Z 0 Z Z Z Z Z Z Z d / g 3 4 5 0 1 2 6 (Rows eventually constant = homological stability!) However need more refined information than just abstract groups: E 2 -structure, as encoded by multiplication − · − , Browder bracket [ − , − ] , Dyer–Lashof operations Q i ℓ ( − ) for all primes ℓ , ... 12

Refined data Z / 2 Z ⊕ Z / 2 2 Z Z Z Z / 10 Z { τ } 1 Z { σ 2 } Z { σ 3 } Z { σ 4 } Z { σ 5 } Z { σ 6 } Z { σ } 0 Z d / g 3 4 5 0 1 2 6 Here τ is the class of a right-handed Dehn twist. H 2 , 1 ( R + ) = Z / 10 generated by στ . Have [ σ, σ ] = 4 στ , Q 1 Z ( σ ) = 3 στ . (For an integral lift Q 1 Z : H ∗ , 0 ( R + ) → H ∗ , 1 ( R + ) of the F 2 Dyer–Lashof operation Q 1 2 , defined by universal example.) −· σ H 2 , 2 ( R + ) H 3 , 2 ( R + ) H 3 , 2 ( R + /σ ) ∂ H 2 , 1 ( R + ) inj λ �→ 10 µ µ �→ στ Z / 2 Z { λ } ⊕ Z / 2 Z / 10 { στ } Z { µ } 13

E 2 -homology The vanishing line gives the following chart for H E 2 g , d ( R ) . ? ? ? ? ? ? 5 ? ? ? ? ? 4 ? ? ? ? 3 ? ? ? 2 ? ? 1 ? 0 d / g 0 1 2 3 4 5 7 6 14

E 2 -homology Reverse engineering the low-degree E 2 -homology lets us complete the chart for H E 2 g , d ( R ) as follows. ? ? ? ? ? ? 5 ? ? ? ? ? 4 ? ? ? ? 3 ? ⊕ Z { ρ, ρ ′ } Z { ρ ′′ } 2 Z { τ } 1 Z { σ } 0 d / g 0 1 2 3 4 5 7 6 Attaching maps are ∂ρ = 10 στ , ∂ρ ′ = Q 1 Z ( σ ) − 3 στ , ∂ρ ′′ = σ 2 τ . 15

Homological stability

Homological stability Theorem (Harer, Ivanov, Boldsen, R-W). H d (Γ g , 1 , Γ g − 1 , 1 ) = 0 if d < 2 g 3 . The slope in this statement has been steadily improved, from Harer’s original 1 3 to Ivanov’s 1 2 , to the 2 3 obtained by Boldsen and myself. These proofs were similar in spirit to each other, but all very different to what I present here. Proof using E 2 -cells. Need H g , d ( R + /σ ) = 0 for d < 2 g 3 . Enough to show this with ❦ -coefficients for prime fields ❦ . Construct a minimal CW-complex model for R ❦ ∈ Alg E 2 ( sMod N ❦ ) , a filtered object f C with colim f C → R ❦ . Then ∼ � � � S g α , d α , d α gr( f C ) ≃ E 2 . ❦ cells α Can unitalise and form the cofibre f C + /σ in filtered objects, with � � � S g α , d α , d α gr( f C + /σ ) ≃ E + /σ. 2 ❦ 16 cells α