VIEWS: 5 PAGES: 13 POSTED ON: 3/8/2010
The BZp-homotopy theory of classifying spaces of compact Lie groups
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).