Periodic Localization, Tate Cohomology, and Infinite Loopspaces Talk 1

Periodic Localization, Tate Cohomology, and Infinite Loopspaces Talk 1 Nicholas J. Kuhn University of Virginia University of Georgia, May, 2010 University of Georgia, May, 2010 1 / Three talks Introduction My goal is to introduce a circle of ideas and results involving Localization with respect to periodic homology theories. Infinite loopspace theory. The Tate construction in equivariant stable homotopy theory. The topics fit with the functor calculus theme of the conference. University of Georgia, May, 2010 2 /

Introduction In more detail... Talk 1 Periodicity in stable homotopy. Bousfield localization. associated to Morava K theories. Talk 2 Homotopy orbits and fixed points. The norm map and the Tate spectrum of a G spectrum. Vanishing results for Tate spectra after periodic localization. Talk 3 Applications to splitting Goodwillie towers. Application to computing E n (Ω X ). Open questions and speculation (after Arone-Ching). University of Georgia, May, 2010 3 / The stable and unstable worlds The unstable and stable worlds T = the category of based topological spaces. ho(t ) = its homotopy category: weak equivalences have been inverted. S = the category of spectra. In its basic flavor... Definition A spectrum X is a sequence of based spaces X 0, X 1,..., together with maps ΣX d X d+1. Homotopy groups... π (X ) = colim d π +d (X d ). Homology groups... E (X ) = colim d E +d (X d ). Weak equivalences: f such that π (f ) is an iso. Invert these... ho(s) = the associated homotopy category. University of Georgia, May, 2010 4 /

Why we like spectra The stable and unstable worlds ho(s) is much more algebraic than ho(t )... It is triangulated. Cofibration sequences are equivalent to fibration sequences. Every spectrum is naturally equivalent to an Ω spectrum, X such that each X d ΩX d+1 is a weak equivalence of spaces, and Ω spectra represent cohomology theories. S is home for many of our friends: Thom spectra, K theories, elliptic spectra, topological modular forms,... University of Georgia, May, 2010 5 / The stable and unstable worlds Back and forth: suspension spectra and infinite loopspaces Definition For Z T, Σ Z S has dth space Σ d Z, with identity structure maps. Definition For X S, Ω X T is the 0th space of an Ω spectrum weakly equivalent to X. The pair T Σ S induces adjoint functors on ho(t ). Ω E (Σ Z) E (Z); Σ preserves cofiber sequences, hocolimits,.... π (Ω Z) π (Z) for 0; Ω preserves fib. sequences, holims,.... University of Georgia, May, 2010 6 /

The stable and unstable worlds An unstable/stable hybrid: periodic unstable homotopy Computing homotopy groups π (Z) is hard. Example π (S 3 ) =? Generalize a hard problem... Definition If F is a finite complex, π n (Z; F ) = [Σ n F, Z] Example Computing π (S 3 ; RP 2 ) is still hard. Simplify with localization... a self map v : Σ d F F (almost) makes π (Z; F ) a Z[v] module: given f π n (Z; F ), v f π n+d (Z; F ) is Σ n+d F v Σ n F f Z. Localize: v 1 π (Z; F ) is π of the telescope of Map T (F, Z) v Map T (Σ d F, Z) v Map T (Σ 2d F, Z) v... or Map T (F, Z) v Ω d Map T (F, Z) v Ω 2d Map T (F, Z) v... University of Georgia, May, 2010 7 / Morava K theories Periodicity in stable homotopy Localized at a prime p, there exist p local spectra K(1), K(2), K(3),... K(n) is a complex oriented ring spectrum. K(n) = Z/p[v n, v 1 n ], v n = 2p n 2. K(n) (X Y ) K(n) (X ) K(n) K(n) (Y ). K(1) is (essentially) complex K theory with mod p coefficients. It s handy to define K(0) = HQ. University of Georgia, May, 2010 8 /

Morava E theories Periodicity in stable homotopy Localized at a prime p, there exist p local spectra E 1, E 2, E 3,... E n is a complex oriented commutative S algebra (a ring spectrum with a very nice multiplication). E n = W (F p n)[u 1,..., u n 1 ][u, u 1 ], u i = 0, u = 2. F p n is the field with p n elements, W (F p n) Z n p is its Witt ring. E 1 is complex K theory with p adic coefficients. E n should be viewed as an integral lift of K(n). University of Georgia, May, 2010 9 / Periodicity in stable homotopy The chromatic filtration of finite spectra Subcategories of ho(s)... C = p local finite CW spectra. C n = K(n 1) acyclics in C. Theorem The categories C n are properly nested: C = C 0 C 1 C 2.... Proper: (Mitchell, 1983). Definition An object F C n C n+1 is said to be of type n. University of Georgia, May, 2010 10 /

Periodicity in stable homotopy The Nilpotence Theorem Work of Devanitz, Hopkins, and Jeff Smith (1984 5)... Nilpotence Theorem Given F C, v : Σ d F F is nilpotent K(n) (v) is nilpotent for all n 0. v is nilpotent means that there exists k such that the composite is null. Σ kd F v... v Σ 2d F v Σ d F v F A categorical characterization of the C n s... Thick Subcategory Theorem A nonempty full subcategory of C that is closed under taking cofibers and retracts is C n for some n. University of Georgia, May, 2010 11 / v n self maps Periodicity in stable homotopy Definition Given F C, v : Σ d F F is a v n self map if K(n) (v) is an isomorphism. K(m) (v) is nilpotent for all m n. Example p : S 0 S 0 is a v 0 self map. Example (Adams) There exists a stable map A : Σ 8 RP 2 RP 2 that is multiplication by v 4 1 in K theory. A is a v 1 self map. Remark The cofiber of a v n self map will be in C n+1. University of Georgia, May, 2010 12 /

Periodicity in stable homotopy Every kid wants an ice cream cone Periodicity Theorem F C n F has a v n self map. Given F, G C n with v n self maps u : Σ c F F and v : Σ d G G, and f : F G, there exist i, j such that ic = jd and the diagram Σ ic F F u i Σ ic f f Σ jd G G v j homotopy commutes. University of Georgia, May, 2010 13 / Telescopes Periodicity in stable homotopy For F of type n, let T (F ) be the mapping telescope of a v n self map: T (F ) = hocolim{f v Σ d F v Σ 2d F v... }. Consequences of the Periodicity Theorem... T (F ) is independent of choice of self map. If F and F are both of type n, then T (F ) (W ) = 0 T (F ) (W ) = 0. Definition Let T (n) ambiguously denote T (F ) for any particular type n finite spectrum F. T (n) (W ) = 0 K(n) (W ) = 0. Telescope Conjecture The converse is true. Still open for n > 1. University of Georgia, May, 2010 14 /

Periodicity in stable homotopy Resolutions Another consequence of the Periodicity Theorem... Resolution Theorem There exists a diagram in C, f (1) f (2) F (1) F (2) F (3) S 0... such that each F (k) C n, and hocolim F (k) S 0 induces an T (m) isomorphism for all m n. k University of Georgia, May, 2010 15 / E local spectra Bousfield localization Definitions Fix a spectrum E. f : Y Z is an E iso if E (f ) is an isomorphism. X is E local if every E iso f : Y Z induces a weak equivalence f : Map S (Z, X ) Map S (Y, X ). Remark This condition can be stated in terms of E acyclics... X is E local X (W ) = 0 whenever E (W ) = 0 [W, X ] = 0 whenever E (W ) = 0. Example K(n) (W ) = 0 K(n) (W ) = 0 E n (W ) = 0. Thus E n is K(n) local. University of Georgia, May, 2010 16 /

Localization functors Bousfield localization (Bousfield, 1970 s) Given E S, there exists A functor L E : S S. A natural transformation η X : X L E X satisfying L E X is E local. η X : X L E X is an E iso. A formal consequence... L E is idempotent... η LE X L(η X ) : L E X L E L E X. L E inverts E isos, and kills E acyclic spectra, in a minimal way. University of Georgia, May, 2010 17 / Examples Bousfield localization Let SG = Moore spectrum of type G. Examples L SZ(p) = localization at p. L SZ/p = completion at p. L E = L F when E (W ) = 0 F (W ) = 0. Examples L SZ/p = L SZ/p 2. L K = L KO. More generally, E (W ) = 0 F (W ) = 0 implies that L F X L F L E X. Example L K(n) = L K(n) L T (n). Remark Telescope Conjecture asks if L T (n) = L K(n). University of Georgia, May, 2010 18 /

Bousfield localization More examples Constructions yielding E local objects... X Y Z a fib. seq. with 2 out of 3 E local the 3rd is E local. X is E local Map S (Y, X ) is E local. X i, i I, are E local X i is E local. i I G acts on an E local X X hg is E local. R a ring spectrum R module spectra (including R!) are R local. Example Completed at 2, L K(1) S is the fiber of KO Ψ3 1 KO. Similar for p odd. Note: KO = K hz/2. There are now nice descriptions of L K(2) S in terms of spectra of the form E2 hg with G finite. These are computationally useful! University of Georgia, May, 2010 19 / The Periodicity Theorem well packaged for infinite loopspace theory... Theorem (Bousfield n = 1, K. all n, 1980 s) For each n 1 (and each p), there is a functor Φ n : T S satisfying the following properties. For all spaces Z, Φ n (Z) is T (n) local. There is a natural isomorphism v 1 π (Z; F ) [F, Φ n (Z)] for all unstable v n maps v : Σ d F F, and spaces Z. For all spectra X, there is a natural weak equivalence Φ n (Ω X ) L T (n) X. Remark The first two properties (almost) characterize Φ n. University of Georgia, May, 2010 20 /

Applications to spectra Different spectra can have homotopy equivalent 0th spaces. However... Proposition Ω X Ω Y L T (n) X L T (n) Y for all n. Proof: Apply Φ n to the equivalence Ω X Ω Y. Example If R is a commutative S algebra, Ω 1 R has a delooping classically denoted by R, and more recently by gl 1 (R). So L T (n) gl 1 (R) L T (n) R. (Bousfield, 1982): With n = 1, recover the Adams Priddy Thm that BO BO, suitably completed. (Rezk, 2006): Let R = E n, then identify, in terms of formal groups, the resulting logarithm l n,p : E 0 n (Z) E 0 n (Z). University of Georgia, May, 2010 21 / Applications to spectra (cont.) Proposition After T (n) localization, the evaluation map ɛ : Σ Ω X X has a natural section η n : L T (n) X L T (n) Σ Ω X. Proof: Apply Φ n to the inclusion (Ω X ) Ω Σ (Ω X ). Corollary For all X, ɛ : E n (X ) E n (Ω X ) is split monic. Remark By contrast, the kernel of ɛ : H (X ; Z/2) H (Ω X ; Z/2) contains all elements of the form Sq i x, i > x. Remark η n will make an appearance in the other talks. University of Georgia, May, 2010 22 /

Applications to spectra (cont.) One more example, with p = 2... there is a cofibration sequence S 1 i RP 1 p RP 0 t S 0. i is the inclusion of the bottom cell. t is the transfer map. Kahn Priddy: Ω t has a section. (Not quite true, but close enough.) Thus L T (n) t has a section. We deduce that, localized at 2, L K(n) i is null for all n. L K(n) RP 0 L K(n) (RP 1 S 0 ) for all n. Contrast with speculation by Hopkins, Hovey... Conjecture X,Y finite and L K(n) X L K(n) Y for all n X Y. University of Georgia, May, 2010 23 / Applications to spaces Roughly put, the spectrum Φ n (Z) determines the unstable v n periodic homotopy groups of a space Z. Problem For ones favorite Z, identify Φ n (Z) in familiar terms. More useful properties of Φ n : Φ n takes homotopy pullbacks in T to homotopy pullbacks in S. Φ n (Map T (A, Z)) Map S (A, Φ n (Z)) for all A, Z T. Φ n (Z d ) Φ n (Z) for all Z T and all d. University of Georgia, May, 2010 24 /

Applications to spaces (cont.) A strategy... resolve the space Z by infinite loopspaces, and apply Φ n. Mahowald (1980): At 2, the James Hopf map Ω 2m+1 S 2m+1 Ω Σ RP 2m induces an isomorphism on v 1 periodic homotopy. Apply Φ 1 and deduce Theorem Localized at 2, Φ 1 (S 2m+1 ) L K(1) Σ 2m+1 RP 2m. R. Thompson: the odd primary analogue. University of Georgia, May, 2010 25 / Applications to spaces (cont.) A conceptual proof, and generalization to higher n... Apply Φ n to the resolution of S 2m+1 by infinite loopspaces arising from the Goodwillie Weiss tower of the identity, as analyzed by Arone Mahowald. (Help from B.Johnson and Dwyer.) Theorem Φ n (S 2m+1 ) has a finite resolution with fibers the T (n) localization of known suspension spectra. Example Localized at 2, there is a cofibration sequence Φ 2 (S 3 ) L T (2) Σ 3 RP 2 L T (2) Σ 3 B, with B both K(1) acyclic and fitting into a cofibration sequence ΣRP /RP 2 B Σ 2 RP /RP 4. University of Georgia, May, 2010 26 /

Construction of Φ n Step 1 An unstable map v : Σ d F F induces natural maps Map T (F, Z) v Ω d Map T (F, Z) v Ω 2d Map T (F, Z)... Not interesting if v is nilpotent. But if v is v n periodic, define Φ F : T S by letting Φ F (Z) have rdth space Map T (F, Z), and structure maps Φ F (Z) rd v Ω d Φ F (Z) (r+1)d. By periodicity, this extends to stable F, and is natural in F. Φ F satisfies versions of the properties of Φ n : Φ F (Z) is T (n) local. π (Φ F (Z)) = v 1 π (Z; F ). Φ F (Ω X ) Map S (F, L T (n) X ). University of Georgia, May, 2010 27 / Step 2 Recall that the Resolution Theorem says that one has f (1) f (2) F (1) F (2) F (3) S 0... such that each F (k) is of type n, and hocolim F (k) S 0 is a T (n) iso. k Definition Φ n (Z) = holim Φ F (k) (Z). k Then... Φ n (Ω X ) = holim k Φ F (k) (Ω X ) holim Map S (F (k), L T (n) X ) k = Map S (hocolim F (k), L T (n) X ) k Map S (S 0, L T (n) X ) = L T (n) X. University of Georgia, May, 2010 28 /

Some References A. K. Bousfield, Uniqueness of infinite deloopings for K-theoretic spaces, Pac.J.M. 1 (1987), 1 31. A. K. Bousfield, On the telescopic homotopy theory of spaces, T.A.M.S. 353 (2001), 2391 2426. N. J. Kuhn, Morava K-theories and infinite loop spaces, Algebraic Topology (Arcata, CA, 1986), SLNM 1370 (1989), 243 257. N. J. Kuhn, A guide to telescopic functors, Algebraic Topology (Baltimore, 2007), H.H.A. 10 (2008), 1 319. C. Rezk, The units of a ring spectrum and a logarithmic cohomology operation, J.A.M.S. 19 (2006), 969 1014 University of Georgia, May, 2010 /