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The BZp-homotopy theory of classifying spaces of compact Lie groups

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The BZp-homotopy theory of classifying spaces of compact Lie groups

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									 BZ/p-
homotopy
theory of
   BG

     `
 Natalia
Castellana




             The BZ /p-homotopy theory of classifying spaces of
                           compact Lie groups


                        `                        ´
                     Natalia Castellana (with Ramon J. Flores)



                           Skye Conference, June 2009
 BZ/p-
homotopy
theory of
   BG                                                            A-homotopy
     `
 Natalia
Castellana




             Given a connected pointed space A, the A-homotopy of X is the study of
             [Σi A, X ], or the homotopy type of map∗ (A, X ). This notion of A-homotopy
             theory was introduced by E. Dror Farjoun.
             Definition
             We say that X is A-null if the evaluation map at the base point

                                            map(A, X ) → X

             is a weak homotopy equivalence. If X is connected, it is equivalent to
             map∗ (A, X ) being weakly contractible.
             This is the analogue of weakly contractible spaces in classical homotopy
             theory.
 BZ/p-
homotopy
theory of
   BG                                     Cellularization and nullification
     `
 Natalia
Castellana




             Bousfield and Dror Farjoun described a functorial way of studying spaces X
             through the eyes of A.
               • Nullification functor: PA : Spaces → Spaces and η : Id → PA .
                 It is such that PA (X ) is A-null and η : X → PA (X ) is initial among maps
                 to A-null spaces.
               • Cellularization functor: CWA : Spaces → Spaces and ι : CWA → Id.
                 It is such that CWA (X ) is A-cellular and ι : CWA (X ) → X is terminal
                 among maps from A-cellular spaces.
                 And X and CWA (X ) have the same A-homotopy,

                                     map∗ (A, X )   map∗ (A, CWA (X )).
 BZ/p-
homotopy
theory of
   BG                                             Examples (A = BZ/p)
     `
 Natalia
Castellana




             • The BZ/p-homotopy theory of a finite complex X : map∗ (BZ/p, X )?
               Miller’s proof of the Sullivan conjecture: map∗ (BZ/p, X ) is weakly
               contractible. That is, X is BZ/p-null.
               We have CWBZ/p (X ) ∗ and PBZ/p (X ) X .
             • If X = BZ/p, then CWBZ/p (BZ/p)       BZ/p and PBZ/p (BZ/p)       ∗.
 BZ/p-
homotopy
theory of
   BG
                                      ´
                                 Chacholski’s way of computing CWA
     `
 Natalia
Castellana




                                            W
             Consider the evaluation map [A,X ]∗ A → X where the wedge is taken over
             representatives of all pointed homotopy classes of maps, and let C denote
             its homotopy cofiber.
             Then CWA (X ) is the homotopy fiber of the composite map

                                         X → C → PΣA (C).
 BZ/p-
homotopy
theory of
   BG                                  Classifying spaces of finite groups
     `
 Natalia
Castellana




             Let G be a finite group and let Tp (G) be the smallest normal subgroup
             containing all the p-torsion in G.
             Theorem (R. Flores)
             There is a fibration
                             Y
                                   B(Tp (G))∧ → PBZ/p (BG) → B(G/Tp (G)).
                                            q
                            q=p
 BZ/p-
homotopy
theory of
   BG                                Classifying spaces of finite groups
     `
 Natalia
Castellana




             Theorem (R. Flores, J. Scherer)
             Let G be a finite group. The cellularization CWBZ/p (BG) is either the
             classifying space of a finite p-group generated by elements of order p or it
             has infinitely many non-trivial homotopy groups.

             Proposition (R. Flores, J. Scherer)
             If G is a finite group generated by elements of order p then
             CWBZ/p (BG)∧ CWBZ/p (BGp ).
                            p
                                           ∧
 BZ/p-
homotopy
theory of
   BG                             Reduction on the fundamental group
     `
 Natalia
Castellana
             Definition
                                                                ˜
             We say that g ∈ π1 (X ) lifts to X if there exists g : B g → X such that
                 ˜
             p ◦ g Bi where Bi : B g → Bπ1 (X ) and p : X → Bπ1 (X ).

                                                        u: X
                                                      uu
                                                  ˜
                                                  g
                                                  uuu
                                                uu
                                                              p
                                              uu            
                                          B g        / Bπ1 (X ).

             Let S be the normal subgroup generated by order p elements which lift to X .
             Consider the pullback diagram

                                           E
                                                   f
                                                          /X


                                                            
                                          BS           / Bπ1 (X ).

             Then, f is a BZ/p-equivalence, that is CWBZ/p (E)       CWBZ/p (X ).
 BZ/p-
homotopy
theory of
   BG                            BZ/p-cellularization and p-completion
     `
 Natalia
Castellana




             Proposition
             Let X be a space with π1 (X ) finite and generated by elements of order p
             which lift to X , then
                                                               ∧
                                    CWBZ/p (X ) → CWBZ/p (Xp )
             is a mod p equivalence.
                                                                                  ∧
               • (Miller) X simply connected then CWBZ/p (X )            CWBZ/p (Xp ).
                                        ˜
               • X simply connected and H ∗ (X ; Q) = 0, then CWBZ/p (X ) is p-complete.
               • Let F → E → BW be a fibration where W is a finite group of order
                 prime to p, and F is BZ/p-cellular then

                                                  CWBZ/p (E)    F
                                ∧          ∧
                 but   CWBZ/p (Ep )   →   Ep   is a mod p equivalence.
 BZ/p-
homotopy
theory of
   BG                                                             Examples
     `
 Natalia
Castellana




             • Let X = BS 3 , then CWBZ/2 (BS 3 )   CWBZ/2 ((BS 3 )∧ ), and there is an
                                                                   2
               equivalence map∗ (BZ/2, BZ/2)        map∗ (BZ/2, (BS 3 )∧ ) . Then
                                                                        2
               CWBZ/2 (BS 3 ) BZ/2.
             • If V is an elementary abelian group and W is a finite group of order
               prime to p, then CWBZ/p ((BV )hW ) BV and
               CWBZ/p (((BV )hW )∧ ) → ((BV )hW )∧ is a mod p equivalence.
                                 p               p
             • Let X = BO(2), then CWBZ/2 (BO(2))∧
                                                 2         (BD2∞ )∧ .
                                                                  2
             • Let X = BSO(3), then CWBZ/2 (BSO(3))         CWBZ/2 (BSO(3)∧ ). There
                                                                          2
               is a fibration

                           CWBZ/2 (BSO(3)∧ ) → BSO(3)∧ → (BSO(3)∧ )Q .
                                         2           2          2
 BZ/p-
homotopy
theory of
   BG                                              BZ/p-cellularization
     `
 Natalia
Castellana




             Theorem
             Let G be a compact connected Lie group, p a prime. The CWBZ/p (BG) has
             infinitely many homotopy groups with p-torsion or
             (CWBZ/p (BG) 1 )∧ K (⊕Z/p∞ , 1)∧ K (⊕Z∧ , 2).
                              p                p          p
 BZ/p-
homotopy
theory of
   BG                                                      BZ/p-nullification
     `
 Natalia
Castellana




             Theorem (Dwyer)
             Let G be a compact Lie group such that π0 (G) is a p-group. Then
             PBZ/p (BG) LZ[ 1 ] (BG) where LZ[ 1 ] is the homological localization with
                               p                p
                               1
             respect H∗ (−; Z[ p ]).
 BZ/p-
homotopy
theory of
   BG                                                     BZ/p-nullification
     `
 Natalia
Castellana




             Theorem
             Let G be a compact Lie group, and let Tp (π) be the smallest normal
             subgroup of π0 (G) which contains all p-torsion. Then there is a fibration

                             LZ[ 1 ] (BGp ) → PBZ/p (BG) → B(π0 (G)/Tp (π))
                                p


             where Gp is the subgroup of G whose group of components is Tp (π).

								
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