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: G1 ֒ → G2 ֒ → . . . ֒ → Gn ֒ → . . .

Homological Stability

Definition

A family of groups and inclusions: G1 ֒ → G2 ֒ → . . . ֒ → Gn ֒ → . . . is said to satisfy homological stability when the induced maps on homology Hi(Gn) → Hi(Gn+1) Hi(BGn) → Hi(BGn+1) are isomorphisms for n sufficiently large.

Examples: families of groups

Symmetric groups Σ1 ֒ → Σ2 ֒ → . . . ֒ → Σn ֒ → . . .

Examples: families of groups

Symmetric groups Σ1 ֒ → Σ2 ֒ → . . . ֒ → Σn ֒ → . . . Mapping class groups of surfaces Γ1,1 ֒ → Γ2,1 ֒ → . . . ֒ → Γg,1 ֒ → . . .

֒ → ֒ →

Examples: families of groups

Braid groups B1 ֒ → B2 ֒ → . . . ֒ → Bn ֒ → . . .

→

Examples: families of groups

Braid groups B1 ֒ → B2 ֒ → . . . ֒ → Bn ֒ → . . .

→

General linear groups Gl1(K) ֒ → Gl2(K) ֒ → . . . ֒ → Gln(K) ֒ → . . .

Examples: range of stability

Symmetric groups (Nakaoka): Hi(Σn; Z)

∼ =

− → Hi(Σn+1; Z) n ≥ 2i

Examples: range of stability

Symmetric groups (Nakaoka): Hi(Σn; Z)

∼ =

− → Hi(Σn+1; Z) n ≥ 2i Mapping class groups of surfaces (Harer): Hi(Γg,1; Z)

∼ =

− → Hi(Γg+1,1; Z) g ≥ 3 2i + 1

Examples: range of stability

Symmetric groups (Nakaoka): Hi(Σn; Z)

∼ =

− → Hi(Σn+1; Z) n ≥ 2i Mapping class groups of surfaces (Harer): Hi(Γg,1; Z)

∼ =

− → Hi(Γg+1,1; Z) g ≥ 3 2i + 1 Braid groups (Arnold): Hi(Bn; Z)

∼ =

− → Hi(Bn+1; Z) n ≥ 2i

Examples: range of stability

Symmetric groups (Nakaoka): Hi(Σn; Z)

∼ =

− → Hi(Σn+1; Z) n ≥ 2i Mapping class groups of surfaces (Harer): Hi(Γg,1; Z)

∼ =

− → Hi(Γg+1,1; Z) g ≥ 3 2i + 1 Braid groups (Arnold): Hi(Bn; Z)

∼ =

− → Hi(Bn+1; Z) n ≥ 2i General linear groups (Quillen): Hi(Gln(K); Z)

∼ =

− → Hi(Gln+1(K); Z) n ≥ i + 1

Stability in action

For symmetric groups stability theorem is: Hi(Σn; Z)

∼ =

− → Hi(Σn+1; Z) n ≥ 2i.

Stability in action

For symmetric groups stability theorem is: Hi(Σn; Z)

∼ =

− → Hi(Σn+1; Z) n ≥ 2i. X Σ1 Σ2 Σ3 Σ4 Σ5

Stability in action

For symmetric groups stability theorem is: Hi(Σn; Z)

∼ =

− → Hi(Σn+1; Z) n ≥ 2i. X Σ1 Σ2 Σ3 Σ4 Σ5 H0(X; Z) Z Z Z Z Z

Stability in action

For symmetric groups stability theorem is: Hi(Σn; Z)

∼ =

− → Hi(Σn+1; Z) n ≥ 2i. X Σ1 Σ2 Σ3 Σ4 Σ5 H0(X; Z) Z Z Z Z Z H1(X; Z) Z2 Z2 Z2 Z2

Stability in action

For symmetric groups stability theorem is: Hi(Σn; Z)

∼ =

− → Hi(Σn+1; Z) n ≥ 2i. X Σ1 Σ2 Σ3 Σ4 Σ5 H0(X; Z) Z Z Z Z Z H1(X; Z) Z2 Z2 Z2 Z2 H2(X; Z) Z2 Z2

Why? - Infinite homology

If we let G∞ = lim

n→∞ Gn

we can consider H∗(G∞) which we can often compute using completely different methods.

Why? - Infinite homology

If we let G∞ = lim

n→∞ Gn

we can consider H∗(G∞) which we can often compute using completely different methods. This allows us to ‘work backwards’ and compute H∗(Gn) in some range.

Classifying space BΣn

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

Classifying space BΣn

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

Classifying space BΣn

Classifying space R∞ BΣn = Conf ({1, . . . , n}, R∞)/Σn

Classifying space BΣn

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

Idea of proof for symmetric groups

For each n there is a space BΣ•

n such that

Idea of proof for symmetric groups

For each n there is a space BΣ•

n such that

Built out of spaces BΣp

n

Idea of proof for symmetric groups

For each n there is a space BΣ•

n such that

Built out of spaces BΣp

n

There is a homotopy equivalence BΣp

n ≃ BΣn−p−1

Idea of proof for symmetric groups

For each n there is a space BΣ•

n such that

Built out of spaces BΣp

n

There is a homotopy equivalence BΣp

n ≃ BΣn−p−1

There is a map BΣ•

n → BΣn

Idea of proof for symmetric groups

For each n there is a space BΣ•

n such that

Built out of spaces BΣp

n

There is a homotopy equivalence BΣp

n ≃ BΣn−p−1

There is a map BΣ•

n → BΣn

This map is highly connected

Idea of proof for symmetric groups

For each n there is a space BΣ•

n such that

Built out of spaces BΣp

n

There is a homotopy equivalence BΣp

n ≃ BΣn−p−1

There is a map BΣ•

n → BΣn

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 BΣp

n

An element of BΣp

n is: 1 a choice of configuration C

in BΣn

2

R∞

Defining BΣp

n

An element of BΣp

n is: 1 a choice of configuration C

in BΣn

2 p + 1 distinct labelled points

in that configuration. R∞ 1 2

Defining BΣp

n

An element of BΣp

n is: 1 a choice of configuration C

in BΣn

2 p + 1 distinct labelled points

in that configuration. R∞ 1 2 Recall we want a homotopy equivalence BΣp

n → BΣn−p−1.

The homotopy equivalence

Want to show BΣp

n ≃ BΣn−p−1.

The homotopy equivalence

Want to show BΣp

n ≃ BΣn−p−1.

Consider the fibration F → BΣp

n → Conf ({1, . . . , p + 1}, R∞) ≃ ∗

The homotopy equivalence

Want to show BΣp

n ≃ BΣn−p−1.

Consider the fibration F → BΣp

n → Conf ({1, . . . , p + 1}, R∞) ≃ ∗

Want fibre F to be BΣn−p−1, so need a contractible space with a free action.

The homotopy equivalence

Want to show BΣp

n ≃ BΣn−p−1.

Consider the fibration F → BΣp

n → Conf ({1, . . . , p + 1}, R∞) ≃ ∗

Want fibre F to be BΣn−p−1, so need a contractible space with a free action. R∞ 1 2

RN

Map to classifying space

What to show there is a highly connected map BΣ•

n → BΣn.

Map to classifying space

What to show there is a highly connected map BΣ•

n → BΣn.

Build by defining a map for each BΣp

n:

BΣp

n → BΣn.

Map to classifying space

What to show there is a highly connected map BΣ•

n → BΣn.

Build by defining a map for each BΣp

n:

BΣp

n → BΣn.

This is the obvious map which forgets the distinguished points.

Map to classifying space

What to show there is a highly connected map BΣ•

n → BΣn.

Build by defining a map for each BΣp

n:

BΣp

n → BΣn.

This is the obvious map which forgets the distinguished points. We can show that the fibre of the resulting map BΣ•

n → BΣn 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

• f BΣ•

n+1 such that:

First quadrant

Our spectral sequence

A spectral sequence can be constructed from the skeletal filtration

• f BΣ•

n+1 such that:

First quadrant First page terms are Ht(BΣs

n+1) at (s, t)

Our spectral sequence

A spectral sequence can be constructed from the skeletal filtration

• f BΣ•

n+1 such that:

First quadrant First page terms are Ht(BΣs

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

• f BΣ•

n+1 such that:

First quadrant First page terms are Ht(BΣs

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 Hk(BΣ•

n+1)

Case n=5

Hi(BΣ5)

∼ =

− → Hi(BΣ6) i ≤ 5 2. 1st page: (s, t) entry is Ht(BΣs

6)

Case n=5

Hi(BΣ5)

∼ =

− → Hi(BΣ6) i ≤ 5 2. 1st page: (s, t) entry is Ht(BΣs

6)

H2(BΣ0

6)

H3(BΣ0

6)

H0(BΣ0

6)

H1(BΣ0

6)

H0(BΣ1

6)

H1(BΣ1

6)

H2(BΣ1

6)

H3(BΣ1

6)

H0(BΣ2

6)

H1(BΣ2

6)

H2(BΣ2

6)

H3(BΣ2

6)

H0(BΣ3

6)

H1(BΣ3

6)

H2(BΣ3

6)

H3(BΣ3

6)

H0(BΣ4

6)

H1(BΣ4

6)

H2(BΣ4

6)

H3(BΣ4

6)

← − ← − ← − ← − ← ← ← ← ← − ← − ← − ← − ← ← ← ←

Case n=5

Hi(BΣ5)

∼ =

− → Hi(BΣ6) i ≤ 5 2. 1st page: (s, t) entry is Ht(BΣ5−s) H3(BΣ5) H0(BΣ5) H1(BΣ5) H2(BΣ5) H0(BΣ4) H1(BΣ4) H2(BΣ4) H3(BΣ4) H0(BΣ3) H1(BΣ3) H2(BΣ3) H3(BΣ3) H0(BΣ2) H1(BΣ2) H2(BΣ2) H3(BΣ2) H0(BΣ1) H1(BΣ1) H2(BΣ1) H3(BΣ1) ← − ← − ← − ← − ← ← ← ← ← − ← − ← − ← − ← ← ← ←

Case n=5

Hi(BΣ5)

∼ =

− → Hi(BΣ6) i ≤ 5 2. 1st page: (s, t) entry is Ht(BΣ5−s) H3(BΣ5) H0(BΣ5) H1(BΣ5) H2(BΣ5) H0(BΣ4) H1(BΣ4) H2(BΣ4) H3(BΣ4) H0(BΣ3) H1(BΣ3) H2(BΣ3) H3(BΣ3)

∼ =

← −

∼ =

← −

∼ =

← − H0(BΣ2) H1(BΣ2) H2(BΣ2) H3(BΣ2) H0(BΣ1) H1(BΣ1) H2(BΣ1) H3(BΣ1) ← − ← − ← − ← − ← ← ← − ← − ← − ← − ← ← ←

Case n=5

Hi(BΣ5)

∼ =

− → Hi(BΣ6) i ≤ 5 2. 2nd page H3(BΣ5) H0(BΣ5) H1(BΣ5) H2(BΣ5) ? ? ? ? ? ? ? ? ? ?

Case n=5

Hi(BΣ5)

∼ =

− → Hi(BΣ6) i ≤ 5 2. ∞ page ?′ H0(BΣ5) H1(BΣ5) H2(BΣ5) ?′ ?′ ?′ ?′ ?′ ?′ ?′ ?′ ?′ ?′

Case n=5

Hi(BΣ5)

∼ =

− → Hi(BΣ6) i ≤ 5 2. ∞ page ?′ H0(BΣ5) H1(BΣ5) H2(BΣ5) ?′ ?′ ?′ ?′ ?′ ?′ ?′ ?′ ?′ ?′

Case n=5

Hi(BΣ5)

∼ =

− → Hi(BΣ6) i ≤ 5 2. ∞ page ?′ H0(BΣ5) H1(BΣ5) H2(BΣ5) ?′ ?′ ?′ ?′ ?′ ?′ ?′ ?′ ?′ ∼ = H0(BΣ•

6)

∼ = H1(BΣ•

6)

∼ = H2(BΣ•

6)

Case n=5

Hi(BΣ5)

∼ =

− → Hi(BΣ6) i ≤ 5 2. ∞ page ?′ H0(BΣ5) H1(BΣ5) H2(BΣ5) ?′ ?′ ?′ ?′ ?′ ?′ ?′ ?′ ?′ ∼ = H0(BΣ6) ∼ = H1(BΣ6) ∼ = H2(BΣ6)

Current work

Coefficient systems - Djament - Vespa, Wahl ...

Current work

Coefficient systems - Djament - Vespa, Wahl ... Broadening to families - Hepworth

Current work

Coefficient systems - Djament - Vespa, Wahl ... Broadening to families - Hepworth Configurations of manifolds - Palmer

Current work

Coefficient systems - Djament - Vespa, Wahl ... Broadening to families - Hepworth Configurations of manifolds - Palmer Automating parts of proof - Wahl

Current work

Coefficient systems - Djament - Vespa, Wahl ... Broadening to families - Hepworth Configurations of manifolds - Palmer Automating parts of proof - Wahl Representation stability - Church, Ellenberg, Farb ...

Thank you