Finite approximations of p -local compact groups Alex Gonz alez - - PowerPoint PPT Presentation

Finite approximations of p-local compact groups

Alex Gonz´ alez

Kansas State University

Workshop on Group Actions - Classical and Derived The Fields Institute

June 13-17, 2016

Some results for compact Lie groups

Let p be a prime, fixed for the rest of the talk.

Alex Gonz´ alez Finite approximations of p-local compact groups 2/13

Some results for compact Lie groups

Let p be a prime, fixed for the rest of the talk. Stable Elements Theorem for finite groups (Cartan-Eilenberg): Let G be a finite group, and let S ∈ Sylp(G). Then, H∗(BG, Fp) ∼ = lim ← −

FS(G)

H∗(−; Fp) ≤ H∗(BS; Fp).

Alex Gonz´ alez Finite approximations of p-local compact groups 2/13

Some results for compact Lie groups

Let p be a prime, fixed for the rest of the talk. Stable Elements Theorem for finite groups (Cartan-Eilenberg): Let G be a finite group, and let S ∈ Sylp(G). Then, H∗(BG, Fp) ∼ = lim ← −

FS(G)

H∗(−; Fp) ≤ H∗(BS; Fp). FS(G) is the category with Ob(FS(G)) = {P ≤ S} and, for P, Q ≤ S, MorFS(G)(P, Q) = {cx : P → Q | x ∈ G, xPx−1 ≤ Q}.

Alex Gonz´ alez Finite approximations of p-local compact groups 2/13

Some results for compact Lie groups

Question: Is there a version of the Stable Elements Theorem for compact Lie groups?

Alex Gonz´ alez Finite approximations of p-local compact groups 3/13

Some results for compact Lie groups

Question: Is there a version of the Stable Elements Theorem for compact Lie groups? Let G be a compact Lie group. What is FS(G) in this case?

Alex Gonz´ alez Finite approximations of p-local compact groups 3/13

Some results for compact Lie groups

Question: Is there a version of the Stable Elements Theorem for compact Lie groups? Let G be a compact Lie group. What is FS(G) in this case? Let T ≤ G be a maximal torus, and let W be the Weyl group.

• T

NG( T) W

Alex Gonz´ alez Finite approximations of p-local compact groups 3/13

Some results for compact Lie groups

Question: Is there a version of the Stable Elements Theorem for compact Lie groups? Let G be a compact Lie group. What is FS(G) in this case? Let T ≤ G be a maximal torus, and let W be the Weyl group.

• T

NG( T) W Choose π ∈ Sylp(W ):

Alex Gonz´ alez Finite approximations of p-local compact groups 3/13

Some results for compact Lie groups

Question: Is there a version of the Stable Elements Theorem for compact Lie groups? Let G be a compact Lie group. What is FS(G) in this case? Let T ≤ G be a maximal torus, and let W be the Weyl group.

• T

NG( T) W Choose π ∈ Sylp(W ):

• T

NG( T) W

• T

S

• π

incl

• Alex Gonz´

alez Finite approximations of p-local compact groups 3/13

Some results for compact Lie groups

Question: Is there a version of the Stable Elements Theorem for compact Lie groups? Let G be a compact Lie group. What is FS(G) in this case? Let T ≤ G be a maximal torus, and let W be the Weyl group.

• T

NG( T) W Choose π ∈ Sylp(W ):

• T

NG( T) W

• T

S

• π

incl

• S is a Sylow p-subgroup of G, but it is not discrete...

Alex Gonz´ alez Finite approximations of p-local compact groups 3/13

Some results for compact Lie groups

Let T ≤ T be the subgroup of all pn-th roots of 1, for n ≥ 0:

Alex Gonz´ alez Finite approximations of p-local compact groups 4/13

Some results for compact Lie groups

Let T ≤ T be the subgroup of all pn-th roots of 1, for n ≥ 0:

• T

S π T

• S
• π

S is a discrete Sylow p-subgroup of G.

Alex Gonz´ alez Finite approximations of p-local compact groups 4/13

Some results for compact Lie groups

Let T ≤ T be the subgroup of all pn-th roots of 1, for n ≥ 0:

• T

S π T

• S
• π

S is a discrete Sylow p-subgroup of G. Define FS(G) as the category with Ob(FS(G)) = {P ≤ S} and, for P, Q ≤ S, MorFS(G)(P, Q) = {cx : P → Q | x ∈ G, xPx−1 ≤ Q}.

Alex Gonz´ alez Finite approximations of p-local compact groups 4/13

Some results for compact Lie groups

Let T ≤ T be the subgroup of all pn-th roots of 1, for n ≥ 0:

• T

S π T

• S
• π

S is a discrete Sylow p-subgroup of G. Define FS(G) as the category with Ob(FS(G)) = {P ≤ S} and, for P, Q ≤ S, MorFS(G)(P, Q) = {cx : P → Q | x ∈ G, xPx−1 ≤ Q}. There is a natural map H∗(BG; Fp)

ρ

− → lim ← −

FS(G)

H∗(−; Fp) ≤ H∗(BS; Fp).

Alex Gonz´ alez Finite approximations of p-local compact groups 4/13

Some results for compact Lie groups

How can we prove that ρ is an isomorphism (using the finite version of the Stable Elements Theorem)?

Alex Gonz´ alez Finite approximations of p-local compact groups 5/13

Some results for compact Lie groups

How can we prove that ρ is an isomorphism (using the finite version of the Stable Elements Theorem)? A Theorem by Friedlander and Mislin Let l = p be a prime. Then, there exist an algebraic group G(Fl) and a map ϕ: BG(Fl) → BG such that H∗(BG; Fp)

ϕ∗

− → H∗(BG(Fl); Fp) is an isomorphism.

Alex Gonz´ alez Finite approximations of p-local compact groups 5/13

Some results for compact Lie groups

How can we prove that ρ is an isomorphism (using the finite version of the Stable Elements Theorem)? A Theorem by Friedlander and Mislin Let l = p be a prime. Then, there exist an algebraic group G(Fl) and a map ϕ: BG(Fl) → BG such that H∗(BG; Fp)

ϕ∗

− → H∗(BG(Fl); Fp) is an isomorphism. We can choose G(Fl) such that S ∈ Sylp(G(Fl)), and ϕ|BS = Id.

Alex Gonz´ alez Finite approximations of p-local compact groups 5/13

Some results for compact Lie groups

How can we prove that ρ is an isomorphism (using the finite version of the Stable Elements Theorem)? A Theorem by Friedlander and Mislin Let l = p be a prime. Then, there exist an algebraic group G(Fl) and a map ϕ: BG(Fl) → BG such that H∗(BG; Fp)

ϕ∗

− → H∗(BG(Fl); Fp) is an isomorphism. We can choose G(Fl) such that S ∈ Sylp(G(Fl)), and ϕ|BS = Id. Nontrivial Fact FS(G) = FS(G(Fl)).

Alex Gonz´ alez Finite approximations of p-local compact groups 5/13

Some results for compact Lie groups

For each i ≥ 1,

Alex Gonz´ alez Finite approximations of p-local compact groups 6/13

Some results for compact Lie groups

For each i ≥ 1,

1

Set Gi = G(Fli) ≤ G(Fl). Then G(Fl) =

i≥1 Gi.

Alex Gonz´ alez Finite approximations of p-local compact groups 6/13

Some results for compact Lie groups

For each i ≥ 1,

1

Set Gi = G(Fli) ≤ G(Fl). Then G(Fl) =

i≥1 Gi.

2

Choose Si ∈ Sylp(Gi) such that S =

i≥1 Si.

Alex Gonz´ alez Finite approximations of p-local compact groups 6/13

Some results for compact Lie groups

For each i ≥ 1,

1

Set Gi = G(Fli) ≤ G(Fl). Then G(Fl) =

i≥1 Gi.

2

Choose Si ∈ Sylp(Gi) such that S =

i≥1 Si.

3

Form the category FSi(Gi). Then, F0 def =

i≥0 FSi(Gi) ⊆ FS(G).

Alex Gonz´ alez Finite approximations of p-local compact groups 6/13

Some results for compact Lie groups

For each i ≥ 1,

1

Set Gi = G(Fli) ≤ G(Fl). Then G(Fl) =

i≥1 Gi.

2

Choose Si ∈ Sylp(Gi) such that S =

i≥1 Si.

3

Form the category FSi(Gi). Then, F0 def =

i≥0 FSi(Gi) ⊆ FS(G).

Stable Elements Theorem for compact Lie groups H∗(BG; Fp) ∼ = H∗(BG(G(Fl)); Fp) ∼ = lim ← −

i

H∗(BGi; Fp) ∼ =

Alex Gonz´ alez Finite approximations of p-local compact groups 6/13

Some results for compact Lie groups

For each i ≥ 1,

1

Set Gi = G(Fli) ≤ G(Fl). Then G(Fl) =

i≥1 Gi.

2

Choose Si ∈ Sylp(Gi) such that S =

i≥1 Si.

3

Form the category FSi(Gi). Then, F0 def =

i≥0 FSi(Gi) ⊆ FS(G).

Stable Elements Theorem for compact Lie groups H∗(BG; Fp) ∼ = H∗(BG(G(Fl)); Fp) ∼ = lim ← −

i

H∗(BGi; Fp) ∼ = ∼ = lim ← −

i

( lim ← −

FSi (Gi)

H∗(−; Fp)) ∼ =

Alex Gonz´ alez Finite approximations of p-local compact groups 6/13

Some results for compact Lie groups

For each i ≥ 1,

1

Set Gi = G(Fli) ≤ G(Fl). Then G(Fl) =

i≥1 Gi.

2

Choose Si ∈ Sylp(Gi) such that S =

i≥1 Si.

3

Form the category FSi(Gi). Then, F0 def =

i≥0 FSi(Gi) ⊆ FS(G).

Stable Elements Theorem for compact Lie groups H∗(BG; Fp) ∼ = H∗(BG(G(Fl)); Fp) ∼ = lim ← −

i

H∗(BGi; Fp) ∼ = ∼ = lim ← −

i

( lim ← −

FSi (Gi)

H∗(−; Fp)) ∼ = ∼ = lim ← −

F0

H∗(−; Fp) ∼ = lim ← −

FS(G)

H∗(−; Fp) ≤ H∗(BS; Fp).

Alex Gonz´ alez Finite approximations of p-local compact groups 6/13

Getting rid of the groups...

There are other topological spaces that give rise to categories of the form FS(G), but where no such group G is involved.

Alex Gonz´ alez Finite approximations of p-local compact groups 7/13

Getting rid of the groups...

There are other topological spaces that give rise to categories of the form FS(G), but where no such group G is involved. The main example (besides compact Lie groups) is given by p-compact groups: p-complete loop spaces with finite mod p cohomology.

Alex Gonz´ alez Finite approximations of p-local compact groups 7/13

Getting rid of the groups...

There are other topological spaces that give rise to categories of the form FS(G), but where no such group G is involved. The main example (besides compact Lie groups) is given by p-compact groups: p-complete loop spaces with finite mod p cohomology. p-local compact groups A p-local compact group is a triple (S, F, L), where

Alex Gonz´ alez Finite approximations of p-local compact groups 7/13

Getting rid of the groups...

There are other topological spaces that give rise to categories of the form FS(G), but where no such group G is involved. The main example (besides compact Lie groups) is given by p-compact groups: p-complete loop spaces with finite mod p cohomology. p-local compact groups A p-local compact group is a triple (S, F, L), where

1

S fits in a group extension T → S → π where T ∼ = (Z/p∞)×r and π is a finite group.

Alex Gonz´ alez Finite approximations of p-local compact groups 7/13

Getting rid of the groups...

There are other topological spaces that give rise to categories of the form FS(G), but where no such group G is involved. The main example (besides compact Lie groups) is given by p-compact groups: p-complete loop spaces with finite mod p cohomology. p-local compact groups A p-local compact group is a triple (S, F, L), where

1

S fits in a group extension T → S → π where T ∼ = (Z/p∞)×r and π is a finite group.

2

F is a saturated fusion system: a category where Ob(F) = {P ≤ S}, and whose morphisms satisfy a certain set of conditions.

Alex Gonz´ alez Finite approximations of p-local compact groups 7/13

Getting rid of the groups...

There are other topological spaces that give rise to categories of the form FS(G), but where no such group G is involved. The main example (besides compact Lie groups) is given by p-compact groups: p-complete loop spaces with finite mod p cohomology. p-local compact groups A p-local compact group is a triple (S, F, L), where

1

S fits in a group extension T → S → π where T ∼ = (Z/p∞)×r and π is a finite group.

2

F is a saturated fusion system: a category where Ob(F) = {P ≤ S}, and whose morphisms satisfy a certain set of conditions.

3

L is a centric linking system: a category where Ob(L) ⊆ Ob(L), and whose morphisms satisfy another set of conditions.

Alex Gonz´ alez Finite approximations of p-local compact groups 7/13

Getting rid of the groups...

F satisfies the following conditions for all P, Q ≤ S:

1

HomS(P, Q) ⊆ MorF(P, Q) ⊆ Inj(P, Q).

2

if f ∈ HomF(P, Q), then every homomorphism in the triangle is a morphism in F Q P

f

• f ′

∼ =

f (P)

incl

• In addition, there are three more conditions, inspired in the case FS(G).

Alex Gonz´ alez Finite approximations of p-local compact groups 8/13

Getting rid of the groups...

F satisfies the following conditions for all P, Q ≤ S:

1

HomS(P, Q) ⊆ MorF(P, Q) ⊆ Inj(P, Q).

2

if f ∈ HomF(P, Q), then every homomorphism in the triangle is a morphism in F Q P

f

• f ′

∼ =

f (P)

incl

• In addition, there are three more conditions, inspired in the case FS(G).

The category L supplies the absence of an ambient group.

Alex Gonz´ alez Finite approximations of p-local compact groups 8/13

Getting rid of the groups...

F satisfies the following conditions for all P, Q ≤ S:

1

HomS(P, Q) ⊆ MorF(P, Q) ⊆ Inj(P, Q).

2

if f ∈ HomF(P, Q), then every homomorphism in the triangle is a morphism in F Q P

f

• f ′

∼ =

f (P)

incl

• In addition, there are three more conditions, inspired in the case FS(G).

The category L supplies the absence of an ambient group. The classifying space of (S, F, L) is the space |L|∧

p .

Alex Gonz´ alez Finite approximations of p-local compact groups 8/13

Examples

Every finite group determines a p-local finite group.

Alex Gonz´ alez Finite approximations of p-local compact groups 9/13

Examples

Every finite group determines a p-local finite group. Let G be a finite group, and let S ∈ Sylp(G). We have already defined FS(G) above, it remains to define LS(G).

Alex Gonz´ alez Finite approximations of p-local compact groups 9/13

Examples

Every finite group determines a p-local finite group. Let G be a finite group, and let S ∈ Sylp(G). We have already defined FS(G) above, it remains to define LS(G).

1

Ob(LS(G) = {P ≤ S | CG(P) ∼ = Z(P) × C ′

G(P)}, where C ′ G(P) has

• rder prime to p.

Alex Gonz´ alez Finite approximations of p-local compact groups 9/13

Examples

Every finite group determines a p-local finite group. Let G be a finite group, and let S ∈ Sylp(G). We have already defined FS(G) above, it remains to define LS(G).

1

Ob(LS(G) = {P ≤ S | CG(P) ∼ = Z(P) × C ′

G(P)}, where C ′ G(P) has

• rder prime to p.

2

For all P, Q ∈ Ob(LS(G)), set MorLS(G)(P, Q) = NG(P, Q)/C ′

G(P),

where NG(P, Q) = {x ∈ G | xPx−1 ≤ Q}.

Alex Gonz´ alez Finite approximations of p-local compact groups 9/13

Examples

Every finite group determines a p-local finite group. Let G be a finite group, and let S ∈ Sylp(G). We have already defined FS(G) above, it remains to define LS(G).

1

Ob(LS(G) = {P ≤ S | CG(P) ∼ = Z(P) × C ′

G(P)}, where C ′ G(P) has

• rder prime to p.

2

For all P, Q ∈ Ob(LS(G)), set MorLS(G)(P, Q) = NG(P, Q)/C ′

G(P),

where NG(P, Q) = {x ∈ G | xPx−1 ≤ Q}. Theorem (Broto-Levi-Oliver

1

Let G be a compact Lie group, and let S ∈ Sylp(G). Then, there exists (S, FS(G), LS(G)) such that |LS(G)|∧

p ≃ (BG)∧ p .

2

Let (X, BX, e) be a p-compact group, and let S ∈ Sylp(X). Then, there exists (S, FS(X), LS(X)) such that |LS(X)|∧

p ≃ BX.

Alex Gonz´ alez Finite approximations of p-local compact groups 9/13

Some facts about p-local compact groups

Stable Elements Theorem for p-local finite groups (Broto-Levi-Oliver) Let (S, F, L) be a p-local finite group. Then, H∗(|L|∧

p ; Fp) ∼

= lim ← −

F

H∗(−; Fp).

Alex Gonz´ alez Finite approximations of p-local compact groups 10/13

Some facts about p-local compact groups

Stable Elements Theorem for p-local finite groups (Broto-Levi-Oliver) Let (S, F, L) be a p-local finite group. Then, H∗(|L|∧

p ; Fp) ∼

= lim ← −

F

H∗(−; Fp). Let (S, F, L) be a p-local compact group.

Alex Gonz´ alez Finite approximations of p-local compact groups 10/13

Some facts about p-local compact groups

Stable Elements Theorem for p-local finite groups (Broto-Levi-Oliver) Let (S, F, L) be a p-local finite group. Then, H∗(|L|∧

p ; Fp) ∼

= lim ← −

F

H∗(−; Fp). Let (S, F, L) be a p-local compact group.

1

If (S, F, L) comes from a compact Lie group or a p-compact group, then the Stable Elements Theorem holds.

Alex Gonz´ alez Finite approximations of p-local compact groups 10/13

Some facts about p-local compact groups

Stable Elements Theorem for p-local finite groups (Broto-Levi-Oliver) Let (S, F, L) be a p-local finite group. Then, H∗(|L|∧

p ; Fp) ∼

= lim ← −

F

H∗(−; Fp). Let (S, F, L) be a p-local compact group.

1

If (S, F, L) comes from a compact Lie group or a p-compact group, then the Stable Elements Theorem holds.

2

There are p-local compact groups which do not correspond to compact Lie groups or p-compact groups

Alex Gonz´ alez Finite approximations of p-local compact groups 10/13

Some facts about p-local compact groups

Stable Elements Theorem for p-local finite groups (Broto-Levi-Oliver) Let (S, F, L) be a p-local finite group. Then, H∗(|L|∧

p ; Fp) ∼

= lim ← −

F

H∗(−; Fp). Let (S, F, L) be a p-local compact group.

1

If (S, F, L) comes from a compact Lie group or a p-compact group, then the Stable Elements Theorem holds.

2

There are p-local compact groups which do not correspond to compact Lie groups or p-compact groups Question: Is there a general Stable Elements Theorem for p-local compact groups?

Alex Gonz´ alez Finite approximations of p-local compact groups 10/13

Unstable Adams operations

Some obstacles:

Alex Gonz´ alez Finite approximations of p-local compact groups 11/13

Unstable Adams operations

Some obstacles:

1

The proof for p-local finite groups uses certain bisets which are not available in the compact case.

Alex Gonz´ alez Finite approximations of p-local compact groups 11/13

Unstable Adams operations

Some obstacles:

1

The proof for p-local finite groups uses certain bisets which are not available in the compact case.

2

The proof for compact Lie groups or p-compact groups uses transfer arguments which are not available for p-local compact groups.

Alex Gonz´ alez Finite approximations of p-local compact groups 11/13

Unstable Adams operations

Some obstacles:

1

The proof for p-local finite groups uses certain bisets which are not available in the compact case.

2

The proof for compact Lie groups or p-compact groups uses transfer arguments which are not available for p-local compact groups. We need a different strategy.

Alex Gonz´ alez Finite approximations of p-local compact groups 11/13

Unstable Adams operations

Some obstacles:

1

The proof for p-local finite groups uses certain bisets which are not available in the compact case.

2

The proof for compact Lie groups or p-compact groups uses transfer arguments which are not available for p-local compact groups. We need a different strategy. Unstable Adams operations for p-local compact groups Let ζ ∈ (Z∧

p )×. An unstable Adams operation of degree ζ for (S, F, L) is

a self-equivalence Ψ: L → L such that

1

preserves the structure of L;

2

Ψ induces an automorphism of S, Ψ: S → S; and

3

Ψ(t) = tζ for all t ∈ T.

Alex Gonz´ alez Finite approximations of p-local compact groups 11/13

Finite approximations of p-local compact groups

We can use unstable Adams operations to produce approximations of p- local compact groups by p-local finite groups.

Alex Gonz´ alez Finite approximations of p-local compact groups 12/13

Finite approximations of p-local compact groups

We can use unstable Adams operations to produce approximations of p- local compact groups by p-local finite groups. Let (S, F, L) be a p-local compact group, and let Ψ be an unstable Adams operation.

Alex Gonz´ alez Finite approximations of p-local compact groups 12/13

Finite approximations of p-local compact groups

We can use unstable Adams operations to produce approximations of p- local compact groups by p-local finite groups. Let (S, F, L) be a p-local compact group, and let Ψ be an unstable Adams operation. Set

1

Ψ0 = Ψ, and Ψi+1 = Ψp

i for all i ≥ 0.

Alex Gonz´ alez Finite approximations of p-local compact groups 12/13

Finite approximations of p-local compact groups

We can use unstable Adams operations to produce approximations of p- local compact groups by p-local finite groups. Let (S, F, L) be a p-local compact group, and let Ψ be an unstable Adams operation. Set

1

Ψ0 = Ψ, and Ψi+1 = Ψp

i for all i ≥ 0.

2

Si = {x ∈ S | Ψi(x) = x}.

Alex Gonz´ alez Finite approximations of p-local compact groups 12/13

Finite approximations of p-local compact groups

We can use unstable Adams operations to produce approximations of p- local compact groups by p-local finite groups. Let (S, F, L) be a p-local compact group, and let Ψ be an unstable Adams operation. Set

1

Ψ0 = Ψ, and Ψi+1 = Ψp

i for all i ≥ 0.

2

Si = {x ∈ S | Ψi(x) = x}.

3

Li ⊆ L, the subcategory with Ob(Li) = {P ∈ Ob(L) | P ≤ Si} and MorLi(P, Q) = {φ ∈ MorL(P, Q) | Ψi(φ) = φ}.

Alex Gonz´ alez Finite approximations of p-local compact groups 12/13

Finite approximations of p-local compact groups

We can use unstable Adams operations to produce approximations of p- local compact groups by p-local finite groups. Let (S, F, L) be a p-local compact group, and let Ψ be an unstable Adams operation. Set

1

Ψ0 = Ψ, and Ψi+1 = Ψp

i for all i ≥ 0.

2

Si = {x ∈ S | Ψi(x) = x}.

3

Li ⊆ L, the subcategory with Ob(Li) = {P ∈ Ob(L) | P ≤ Si} and MorLi(P, Q) = {φ ∈ MorL(P, Q) | Ψi(φ) = φ}.

4

Fi ⊆ F, the fusion system over Si generated by Li.

Alex Gonz´ alez Finite approximations of p-local compact groups 12/13

Finite approximations of p-local compact groups

Finite Approximation Theorem (G.)

1

There exists M ∈ N such that (Si, Fi, Li) is a p-local finite group for all i ≥ M.

2

|L|∧

p ≃

• hocolim|Li|∧

p

∧

p .

Alex Gonz´ alez Finite approximations of p-local compact groups 13/13

Finite approximations of p-local compact groups

Finite Approximation Theorem (G.)

1

There exists M ∈ N such that (Si, Fi, Li) is a p-local finite group for all i ≥ M.

2

|L|∧

p ≃

• hocolim|Li|∧

p

∧

p .

Stable Elements Theorem for p-local compact groups H∗(|L|∧

p ; Fp) ∼

= lim ← −

i

H∗(|Li|∧

p ; Fp) ∼

= lim ← −

i

(lim ← −

Fi

H∗(−; Fp)) ∼ = ∼ = lim ← −

F0

H∗(−; Fp) ∼ = lim ← −

F

H∗(−; Fp) ≤ H∗(BS; Fp).

Alex Gonz´ alez Finite approximations of p-local compact groups 13/13

Bonus level: mapping spaces

The homotopy type of mapping spaces is difficult to study in general.

Alex Gonz´ alez Finite approximations of p-local compact groups 13/13

Bonus level: mapping spaces

The homotopy type of mapping spaces is difficult to study in general. There is a particular situation where mapping spaces are reasonably easy to describe: mapping spaces between classifying spaces of finite groups.

Alex Gonz´ alez Finite approximations of p-local compact groups 13/13

Bonus level: mapping spaces

The homotopy type of mapping spaces is difficult to study in general. There is a particular situation where mapping spaces are reasonably easy to describe: mapping spaces between classifying spaces of finite groups. Theorem Let f : H → G be a homomorphism between discrete groups. Then Map(BH, BG)Bf ≃ BCG(f (H)).

Alex Gonz´ alez Finite approximations of p-local compact groups 13/13

Bonus level: mapping spaces

The homotopy type of mapping spaces is difficult to study in general. There is a particular situation where mapping spaces are reasonably easy to describe: mapping spaces between classifying spaces of finite groups. Theorem Let f : H → G be a homomorphism between discrete groups. Then Map(BH, BG)Bf ≃ BCG(f (H)). Question Does this Theorem extend (in some sense) to p-local compact groups?

Alex Gonz´ alez Finite approximations of p-local compact groups 13/13

Bonus level: mapping spaces

Let (S, F, L) be a p-local compact group. In general, there is no notion

• f centralizer of a p-local compact subgroup.

Alex Gonz´ alez Finite approximations of p-local compact groups 13/13

Bonus level: mapping spaces

Let (S, F, L) be a p-local compact group. In general, there is no notion

• f centralizer of a p-local compact subgroup.

If P ≤ S, then there is a well-defined centralizer of P, which is again a p-local compact group: (CS(P), CF(P), CL(P)).

Alex Gonz´ alez Finite approximations of p-local compact groups 13/13

Bonus level: mapping spaces

Let (S, F, L) be a p-local compact group. In general, there is no notion

• f centralizer of a p-local compact subgroup.

If P ≤ S, then there is a well-defined centralizer of P, which is again a p-local compact group: (CS(P), CF(P), CL(P)). Mapping Spaces and Centralizers Theorem (B.-L.-O., G.) Let P be a locally finite p-group satisfying the descending chain condition, and let γ : BP → |L|∧

p be a map. Then,

Map(BP, |L|∧

p )γ ≃ |CL(γ(P))|∧ p .

Alex Gonz´ alez Finite approximations of p-local compact groups 13/13