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 ∈ Syl p ( G ). Then, H ∗ ( BG , F p ) ∼ H ∗ ( − ; F p ) ≤ H ∗ ( BS ; F p ) . = lim ← − F S ( G ) 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 ∈ Syl p ( G ). Then, H ∗ ( BG , F p ) ∼ H ∗ ( − ; F p ) ≤ H ∗ ( BS ; F p ) . = lim ← − F S ( G ) F S ( G ) is the category with Ob ( F S ( G )) = { P ≤ S } and, for P , Q ≤ S , Mor F S ( G ) ( P , Q ) = { c x : 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 F S ( 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 F S ( G ) in this case? Let � T ≤ G be a maximal torus, and let W be the Weyl group. � � N G ( � � W T ) T 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 F S ( G ) in this case? Let � T ≤ G be a maximal torus, and let W be the Weyl group. � � N G ( � � W T ) T Choose π ∈ Syl p ( 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 F S ( G ) in this case? Let � T ≤ G be a maximal torus, and let W be the Weyl group. � � N G ( � � W T ) T Choose π ∈ Syl p ( W ): � � N G ( � � W T T ) incl � � � T S π 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 F S ( G ) in this case? Let � T ≤ G be a maximal torus, and let W be the Weyl group. � � N G ( � � W T ) T Choose π ∈ Syl p ( W ): � � N G ( � � W T T ) incl � � � T S π � 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 p n -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 p n -th roots of 1, for n ≥ 0: � � � � π T S 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 p n -th roots of 1, for n ≥ 0: � � � � π T S T S π S is a discrete Sylow p -subgroup of G . Define F S ( G ) as the category with Ob ( F S ( G )) = { P ≤ S } and, for P , Q ≤ S , Mor F S ( G ) ( P , Q ) = { c x : 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 p n -th roots of 1, for n ≥ 0: � � � � π T S T S π S is a discrete Sylow p -subgroup of G . Define F S ( G ) as the category with Ob ( F S ( G )) = { P ≤ S } and, for P , Q ≤ S , Mor F S ( G ) ( P , Q ) = { c x : P → Q | x ∈ G , xPx − 1 ≤ Q } . There is a natural map ρ H ∗ ( BG ; F p ) H ∗ ( − ; F p ) ≤ H ∗ ( BS ; F p ) . − → lim ← − F S ( G ) 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 ( F l ) and a map ϕ : BG ( F l ) → BG such that ϕ ∗ H ∗ ( BG ; F p ) → H ∗ ( BG ( F l ); F p ) − 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 ( F l ) and a map ϕ : BG ( F l ) → BG such that ϕ ∗ H ∗ ( BG ; F p ) → H ∗ ( BG ( F l ); F p ) − is an isomorphism. We can choose G ( F l ) such that S ∈ Syl p ( G ( F l )), 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 ( F l ) and a map ϕ : BG ( F l ) → BG such that ϕ ∗ H ∗ ( BG ; F p ) → H ∗ ( BG ( F l ); F p ) − is an isomorphism. We can choose G ( F l ) such that S ∈ Syl p ( G ( F l )), and ϕ | BS = Id. Nontrivial Fact F S ( G ) = F S ( G ( F l )) . 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, Set G i = G ( F l i ) ≤ G ( F l ). Then G ( F l ) = � i ≥ 1 G i . 1 Alex Gonz´ alez Finite approximations of p -local compact groups 6/13

Some results for compact Lie groups For each i ≥ 1, Set G i = G ( F l i ) ≤ G ( F l ). Then G ( F l ) = � i ≥ 1 G i . 1 Choose S i ∈ Syl p ( G i ) such that S = � i ≥ 1 S i . 2 Alex Gonz´ alez Finite approximations of p -local compact groups 6/13

Some results for compact Lie groups For each i ≥ 1, Set G i = G ( F l i ) ≤ G ( F l ). Then G ( F l ) = � i ≥ 1 G i . 1 Choose S i ∈ Syl p ( G i ) such that S = � i ≥ 1 S i . 2 = � Form the category F S i ( G i ). Then, F 0 def i ≥ 0 F S i ( G i ) ⊆ F S ( G ). 3 Alex Gonz´ alez Finite approximations of p -local compact groups 6/13

Some results for compact Lie groups For each i ≥ 1, Set G i = G ( F l i ) ≤ G ( F l ). Then G ( F l ) = � i ≥ 1 G i . 1 Choose S i ∈ Syl p ( G i ) such that S = � i ≥ 1 S i . 2 = � Form the category F S i ( G i ). Then, F 0 def i ≥ 0 F S i ( G i ) ⊆ F S ( G ). 3 Stable Elements Theorem for compact Lie groups H ∗ ( BG ; F p ) ∼ = H ∗ ( BG ( G ( F l )); F p ) ∼ H ∗ ( BG i ; F p ) ∼ = lim = ← − i Alex Gonz´ alez Finite approximations of p -local compact groups 6/13

Some results for compact Lie groups For each i ≥ 1, Set G i = G ( F l i ) ≤ G ( F l ). Then G ( F l ) = � i ≥ 1 G i . 1 Choose S i ∈ Syl p ( G i ) such that S = � i ≥ 1 S i . 2 = � Form the category F S i ( G i ). Then, F 0 def i ≥ 0 F S i ( G i ) ⊆ F S ( G ). 3 Stable Elements Theorem for compact Lie groups H ∗ ( BG ; F p ) ∼ = H ∗ ( BG ( G ( F l )); F p ) ∼ H ∗ ( BG i ; F p ) ∼ = lim = ← − i ∼ H ∗ ( − ; F p )) ∼ = lim ( lim = ← − ← − i F Si ( G i ) Alex Gonz´ alez Finite approximations of p -local compact groups 6/13

Some results for compact Lie groups For each i ≥ 1, Set G i = G ( F l i ) ≤ G ( F l ). Then G ( F l ) = � i ≥ 1 G i . 1 Choose S i ∈ Syl p ( G i ) such that S = � i ≥ 1 S i . 2 = � Form the category F S i ( G i ). Then, F 0 def i ≥ 0 F S i ( G i ) ⊆ F S ( G ). 3 Stable Elements Theorem for compact Lie groups H ∗ ( BG ; F p ) ∼ = H ∗ ( BG ( G ( F l )); F p ) ∼ H ∗ ( BG i ; F p ) ∼ = lim = ← − i ∼ H ∗ ( − ; F p )) ∼ = lim ( lim = ← − ← − i F Si ( G i ) ∼ H ∗ ( − ; F p ) ∼ H ∗ ( − ; F p ) ≤ H ∗ ( BS ; F p ) . = lim = lim ← − ← − F 0 F S ( G ) Alex Gonz´ alez Finite approximations of p -local compact groups 6/13