Homological Stability 11 06 2015 Rachael Boyd Table of contents 1 - PowerPoint PPT Presentation

Homological Stability 11 06 2015 Rachael Boyd

Table of contents 1 Definition 2 Examples 3 Why 4 Idea of the proof for symmetric group 5 Current work/people

Homological Stability

Homological Stability Definition

Homological Stability Definition A family of groups and inclusions: G 1 ֒ → G 2 ֒ → . . . ֒ → G n ֒ → . . .

Homological Stability Definition A family of groups and inclusions: G 1 ֒ → G 2 ֒ → . . . ֒ → G n ֒ → . . . is said to satisfy homological stability when the induced maps on homology H i ( G n ) → H i ( G n +1 ) H i ( BG n ) → H i ( BG n +1 ) are isomorphisms for n sufficiently large.

Examples: families of groups Symmetric groups Σ 1 ֒ → Σ 2 ֒ → . . . ֒ → Σ n ֒ → . . .

Examples: families of groups Symmetric groups Σ 1 ֒ → Σ 2 ֒ → . . . ֒ → Σ n ֒ → . . . Mapping class groups of surfaces Γ 1 , 1 ֒ → Γ 2 , 1 ֒ → . . . ֒ → Γ g , 1 ֒ → . . . ֒ → ֒ →

Examples: families of groups Braid groups B 1 ֒ → B 2 ֒ → . . . ֒ → B n ֒ → . . . �→

Examples: families of groups Braid groups B 1 ֒ → B 2 ֒ → . . . ֒ → B n ֒ → . . . �→ General linear groups Gl 1 ( K ) ֒ → Gl 2 ( K ) ֒ → . . . ֒ → Gl n ( K ) ֒ → . . .

Examples: range of stability Symmetric groups (Nakaoka): ∼ = H i (Σ n ; Z ) − → H i (Σ n +1 ; Z ) n ≥ 2 i

Examples: range of stability Symmetric groups (Nakaoka): ∼ = H i (Σ n ; Z ) − → H i (Σ n +1 ; Z ) n ≥ 2 i Mapping class groups of surfaces (Harer): g ≥ 3 ∼ = H i (Γ g , 1 ; Z ) − → H i (Γ g +1 , 1 ; Z ) 2 i + 1

Examples: range of stability Symmetric groups (Nakaoka): ∼ = H i (Σ n ; Z ) − → H i (Σ n +1 ; Z ) n ≥ 2 i Mapping class groups of surfaces (Harer): g ≥ 3 ∼ = H i (Γ g , 1 ; Z ) − → H i (Γ g +1 , 1 ; Z ) 2 i + 1 Braid groups (Arnold): ∼ = H i ( B n ; Z ) − → H i ( B n +1 ; Z ) n ≥ 2 i

Examples: range of stability Symmetric groups (Nakaoka): ∼ = H i (Σ n ; Z ) − → H i (Σ n +1 ; Z ) n ≥ 2 i Mapping class groups of surfaces (Harer): g ≥ 3 ∼ = H i (Γ g , 1 ; Z ) − → H i (Γ g +1 , 1 ; Z ) 2 i + 1 Braid groups (Arnold): ∼ = H i ( B n ; Z ) − → H i ( B n +1 ; Z ) n ≥ 2 i General linear groups (Quillen): ∼ = H i ( Gl n ( K ); Z ) − → H i ( Gl n +1 ( K ); Z ) n ≥ i + 1

Stability in action For symmetric groups stability theorem is: ∼ = H i (Σ n ; Z ) − → H i (Σ n +1 ; Z ) n ≥ 2 i .

Stability in action For symmetric groups stability theorem is: ∼ = H i (Σ n ; Z ) − → H i (Σ n +1 ; Z ) n ≥ 2 i . X Σ 1 Σ 2 Σ 3 Σ 4 Σ 5

Stability in action For symmetric groups stability theorem is: ∼ = H i (Σ n ; Z ) − → H i (Σ n +1 ; Z ) n ≥ 2 i . X Σ 1 Σ 2 Σ 3 Σ 4 Σ 5 H 0 ( X ; Z ) Z Z Z Z Z

Stability in action For symmetric groups stability theorem is: ∼ = H i (Σ n ; Z ) − → H i (Σ n +1 ; Z ) n ≥ 2 i . X Σ 1 Σ 2 Σ 3 Σ 4 Σ 5 H 0 ( X ; Z ) Z Z Z Z Z H 1 ( X ; Z ) 0 Z 2 Z 2 Z 2 Z 2

Stability in action For symmetric groups stability theorem is: ∼ = H i (Σ n ; Z ) − → H i (Σ n +1 ; Z ) n ≥ 2 i . X Σ 1 Σ 2 Σ 3 Σ 4 Σ 5 H 0 ( X ; Z ) Z Z Z Z Z H 1 ( X ; Z ) 0 Z 2 Z 2 Z 2 Z 2 H 2 ( X ; Z ) 0 0 0 Z 2 Z 2

Why? - Infinite homology If we let G ∞ = lim n →∞ G n we can consider H ∗ ( G ∞ ) which we can often compute using completely different methods.

Why? - Infinite homology If we let G ∞ = lim n →∞ G n we can consider H ∗ ( G ∞ ) which we can often compute using completely different methods. This allows us to ‘work backwards’ and compute H ∗ ( G n ) in some range.

Classifying space B Σ n Classifying space R ∞ 2 3 1 4 7 5 6 Conf ( { 1 , . . . , n } , R ∞ )

Classifying space B Σ n Classifying space R ∞ 5 4 3 1 2 6 7 Conf ( { 1 , . . . , n } , R ∞ )

Classifying space B Σ n Classifying space R ∞ B Σ n = Conf ( { 1 , . . . , n } , R ∞ ) / Σ n

Classifying space B Σ n Classifying space Stabilization map R ∞ R ∞ B Σ n = Conf ( { 1 , . . . , n } , R ∞ ) / Σ n B Σ n → B Σ n +1

Idea of proof for symmetric groups For each n there is a space � B Σ • n � such that

Idea of proof for symmetric groups For each n there is a space � B Σ • n � such that Built out of spaces B Σ p n

Idea of proof for symmetric groups For each n there is a space � B Σ • n � such that Built out of spaces B Σ p n There is a homotopy equivalence B Σ p n ≃ B Σ n − p − 1

Idea of proof for symmetric groups For each n there is a space � B Σ • n � such that Built out of spaces B Σ p n There is a homotopy equivalence B Σ p n ≃ B Σ n − p − 1 There is a map � B Σ • n � → B Σ n

Idea of proof for symmetric groups For each n there is a space � B Σ • n � such that Built out of spaces B Σ p n There is a homotopy equivalence B Σ p n ≃ B Σ n − p − 1 There is a map � B Σ • n � → B Σ n This map is highly connected

Idea of proof for symmetric groups For each n there is a space � B Σ • n � such that Built out of spaces B Σ p n There is a homotopy equivalence B Σ p n ≃ B Σ n − p − 1 There is a map � B Σ • n � → B Σ n This map is highly connected After such a space is found, the argument is to compute its homology in a range: we do this using a spectral sequence argument and induction on n .

Defining B Σ p n R ∞ An element of B Σ p n is: 1 a choice of configuration C in B Σ n 2

Defining B Σ p n R ∞ An element of B Σ p n is: 1 a choice of configuration C in B Σ n 2 p + 1 distinct labelled points 0 1 in that configuration. 2

Defining B Σ p n R ∞ An element of B Σ p n is: 1 a choice of configuration C in B Σ n 2 p + 1 distinct labelled points 0 1 in that configuration. 2 Recall we want a homotopy equivalence B Σ p n → B Σ n − p − 1 .

The homotopy equivalence Want to show B Σ p n ≃ B Σ n − p − 1 .

The homotopy equivalence Want to show B Σ p n ≃ B Σ n − p − 1 . Consider the fibration F → B Σ p n → Conf ( { 1 , . . . , p + 1 } , R ∞ ) ≃ ∗

The homotopy equivalence Want to show B Σ p n ≃ B Σ n − p − 1 . Consider the fibration F → B Σ p n → Conf ( { 1 , . . . , p + 1 } , R ∞ ) ≃ ∗ Want fibre F to be B Σ n − p − 1 , so need a contractible space with a free action.

The homotopy equivalence Want to show B Σ p n ≃ B Σ n − p − 1 . Consider the fibration F → B Σ p n → Conf ( { 1 , . . . , p + 1 } , R ∞ ) ≃ ∗ Want fibre F to be B Σ n − p − 1 , so need a contractible space with a free action. R ∞ 0 1 R N 2

Map to classifying space What to show there is a highly connected map � B Σ • n � → B Σ n .

Map to classifying space What to show there is a highly connected map � B Σ • n � → B Σ n . Build by defining a map for each B Σ p n : B Σ p n → B Σ n .

Map to classifying space What to show there is a highly connected map � B Σ • n � → B Σ n . Build by defining a map for each B Σ p n : B Σ p n → B Σ n . This is the obvious map which forgets the distinguished points.

Map to classifying space What to show there is a highly connected map � B Σ • n � → B Σ n . Build by defining a map for each B Σ p n : B Σ p n → B Σ n . This is the obvious map which forgets the distinguished points. We can show that the fibre of the resulting map � B Σ • n � → B Σ n is homotopy equivalent to a wedge of spheres, hence the map is highly connected.

Spectral sequences

Spectral sequences A book where each page is a lattice of abelian groups

Spectral sequences A book where each page is a lattice of abelian groups Differentials move between groups and form chain complexes

Spectral sequences A book where each page is a lattice of abelian groups Differentials move between groups and form chain complexes First quadrant spectral sequences result in ∞ page

Our spectral sequence A spectral sequence can be constructed from the skeletal filtration of � B Σ • n +1 � such that: First quadrant

Our spectral sequence A spectral sequence can be constructed from the skeletal filtration of � B Σ • n +1 � such that: First quadrant First page terms are H t ( B Σ s n +1 ) at ( s , t )

Our spectral sequence A spectral sequence can be constructed from the skeletal filtration of � B Σ • n +1 � such that: First quadrant First page terms are H t ( B Σ s n +1 ) at ( s , t ) First page differentials are zero going from odd to even columns the stabilisation map going from even to odd columns

Our spectral sequence A spectral sequence can be constructed from the skeletal filtration of � B Σ • n +1 � such that: First quadrant First page terms are H t ( B Σ s n +1 ) at ( s , t ) First page differentials are zero going from odd to even columns the stabilisation map going from even to odd columns On the ∞ page groups along a diagonal s + t = k are a ‘filtration quotient’ of H k ( � B Σ • n +1 � )

Case n=5 i ≤ 5 ∼ = H i ( B Σ 5 ) − → H i ( B Σ 6 ) 2 . 1st page: ( s , t ) entry is H t ( B Σ s 6 )