Kathryn Hess. International Conference on K-theory and Homotopy Theory Santiago de Compostela, 17 September 2008

Transcription

1 Institute of Geometry, Ecole Polytechnique Fédérale de Lausanne International Conference on K-theory and Homotopy Theory Santiago de Compostela, 17 September 2008

2 Joint work with John Rognes.

3 Outline 1 2 Power maps in algebra and rational homotopy 3 4 5

4 Conventions R is a PID Ch R is the category of chain es of R-modules that are R-free as graded R-modules. Alg R is the category of augmented, associative dg over R. Coalg R is the category of coaugmented, connected, coassociative dg co over R. Ω : Coalg R Alg R is the (reduced) cobar construction. B : Alg R Coalg R is the (reduced) bar construction.

5 Hopf Let (H,, µ) be a (dg) Hopf algebra, and let f, g End R (H). The convolution product f g of f and g is the composite H H H f g H H µ H. The r th -power map on H is the R-linear endomorphism λ r = Id r H : H H.

6 Philosophy of this lecture: algebra case When A is an algebra such that BA is actually a Hopf algebra and therefore admits power maps, construct a lift of the power maps to of A. This lift was known ([BFG], [V]) to be topologically meaningful when A = A PL (X; Q): The algebraic power maps model the topological power maps on LX, the free loop space on X, over Q.

7 Philosophy of this lecture: coalgebra case Similarly, when C is a coalgebra such that ΩC is actually a Hopf algebra, construct a lift of the power maps to the cohochschild of C. We show that this lift is topologically meaningful if C = C X, where X is a double suspension, or C = C (X; Z/2Z), where X = Σ j J RPn j k j : The algebraic power maps model the topological power maps on LX, over Z, respectively Z/2Z.

8 Why do we care? The algebraic power map is essential to the construction of our algebraic model for the spectrum homology of the topological cyclic homology TC(X; p) of a space X. Possible purely algebraic applications, e.g., Hodge decompositions?

9 The classical Hochschild is a functor H : Alg R Ch R such that for all A, there is a twisted tensor extension of chain es A H A BA. In particular, BA = (TsA, d B ) and H A = (TsA A, d H ), where TV := R V V 2.

10 Multiplicative structure in the case If A is a algebra, then BA is naturally a Hopf algebra, with multiplication given by BA BA E Z B(A A) Bm A BA. Moreover, the multiplication on BA lifts to H A, so that A H A BA becomes a sequence of dg algebra maps.

11 Power maps in the case [Loday,...] If A is a algebra, then the r th -power map λ r : BA BA lifts to an r th -power map such that commutes. A λ r : H A H A H A BA = bλ r λ r A H A BA

12 Free loop spaces Let X be a topological space. The free loop space on X LX := Map(S 1, X). Evaluation at the basepoint gives a fiber sequence ΩX LX X.

13 loop spaces The r th -power maps on ΩX and on LX l r : ΩX ΩX and l r : LX LX are defined by l r (α)(z) = α(z r ) for all z S 1 and l r = l r ΩX, so that ΩX l r LX bl r X = ΩX LX X commutes. Note that l r = µ (r 1) (r 1) : LX LX X LX LX. X The power map is given by iterated loop concatenation.

14 Parallels between algebra and topology Compare and C X C LX C ΩX = C b l r C l r C X C LX C ΩX A H A BA = bλ r λ r A H A BA.

15 : the rational case [BFG, V] If X is a simplicial, then there is a commuting diagram HH ( APL (X) ) H b λr a = HH ( APL (X) ) a = H (LX; Q) H b l r H (LX; Q), where A PL (X) is the dg algebra of piecewise-linear Q-valued forms on X; HH (A) denotes the reduced homology of H A.

16 Goals of this lecture Generalize construction to a larger category. Provide conditions under which the generalized Hochschild construction admits power maps. Explain the topological of the algebraic power maps.

17 Twisting cochains Let (C,, d) Coalg R and (A, m, d) Alg R. A linear map t : C A of degree 1 such that dt + td = m(t t) is a. {s t : C A} = {dg alg maps α t : ΩC A} = {dg coalg maps β t : C BA}

18 Important examples I The universal t Ω : C ΩC: For every t : C A, the diagram C t Ω ΩC t A α t commutes.

19 Important examples II The couniversal t B : BA A: For every t : C A, the diagram BA t B A β t t C commutes.

20 Important examples III Let K be a reduced simplicial set, and let C K denote the normalized chains on K. Let GK denote the Kan loop group on K (simplicial analog of the based loop space). Szczarba s is natural in K and t K : C K C GK α K := α tk : ΩC K C GK is a quasi-isomorphism of dg for all K.

21 Let t : C A be a. of t, denoted H (t), is the chain with underlying graded R-module C A and with differential d t, defined on c a C A by d t (c a) = dc a ± c da ± c j t(c j ) a ± c j a t(c j ), where (c) = c j c j. There is a twisted extension of chain es A H (t) C.

22 Important examples H (t B ) is the usual Hochschild H A on A. H (t Ω ) is the cohochschild Ĥ C on C [HPS].

23 Naturality of construction I Given s t : C A and t : C A and a diagram in Alg R ΩC ϕ α t ΩC α t A g A there is a commuting diagram of chain maps ΩC α t A H (t) C ϕ ΩC α t g H (ϕ,g) f A H (t ) C, where f is the linear part of ϕ.

24 Naturality of construction II Given s t : C A and t : C A and a diagram in Coalg R C β t f C β t BA γ BA there is a commuting diagram of chain maps A H (t) C β t BA g H (f,γ) f A H (t ) C β t γ BA where g is the linear part of γ.

25 Special case of naturality I If f : C C is a coalgebra map and g : A A is an algebra map, then each of the three diagrams below commutes if and only if the other two do. ΩC Ωf α t ΩC α t A g C f t A g C f β t BA A C t A C β t BA Bg

26 Special case of naturality II Thus, if C t A f g C t A commutes, then there is a commuting diagram of chain maps ΩC α t A H (t) C β t BA Ωf ΩC α t g H (f,g) f Bg A H (t ) C β t BA.

27 Hirsch co: definition A Hirsch coalgebra is a dg coalgebra C, together with a dg algebra map ψ : ΩC ΩC ΩC such that (ΩC, ψ) is a dg Hopf algebra. A Hirsch coalgebra (C, ψ) is balanced if ψ is co.

28 Hirsch co: examples I Let C be a dg coalgebra with diagonal : C C C. If is co, and therefore a coalgebra map, then (C, Ω ) is a balanced Hirsch coalgebra.

29 Hirsch co: examples II [Baues, HPST, HPS] If K is a reduced simplicial set, then there is a natural comultiplication with respect to which ψ K : ΩC K ΩC K ΩC K α K : ΩC K C GK is strongly homotopy comultiplicative. Moreover, if C K is co (e.g., if K is simplicial suspension), then (C EK, ψ EK ) is a balanced Hirsch coalgebra (E = simplicial suspension functor).

30 Hirsch co: examples III [HR] If H is a dg Hopf algebra, then there is natural comultiplication ψ H : ΩBH ΩBH ΩBH with respect to which the natural algebra map is a map of Hopf. ε H : ΩBH H

31 The existence theorem: hypotheses Given a Hirsch coalgebra (C, ψ), a dg Hopf algebra (H, ), and a t : C H such that α t : (ΩC, ψ) (H, ) is a map of Hopf, and τ t = t : C H H. (τ : H H = H H is the symmetry isomorphism.)

32 The existence theorem: conclusion Then, for all r 1, there exists a natural chain map such that commutes. λ r : H (t) H (t) H λ r H (t) bλ r C = H H (t) C In particular, if (ΩC, ψ) is primitively generated, then λ r = Id C λ r.

33 Special case: balanced Hirsch co If (C, ψ) is a balanced Hirsch coalgebra, then for all r 1, there exists a natural chain map such that commutes. ΩC λ r : Ĥ C Ĥ C Ĥ C λ r bλ r ΩC Ĥ C C = C

34 Special case: co Hopf If (H, ) is a co Hopf algebra, then for all r 1, there exists such that commutes. H λ r λ r : H H H H H H bλ r BH = H H H BH

35 A small, explicit model for free loop spaces [HPS] If X is a topological space such that X K for some reduced simplicial set K, then there is a quasi-isomorphism ζ : Ĥ C K S LX. Here, S denotes integral singular chains.

36 Compatability with power maps: integral case [HR] If K = E 2 L, where L is any simplicial set, and X K, then for all r 1, Ĥ C K bλ r Ĥ C K ζ ζ commutes up to chain homotopy. S LX S b lr S LX Remark The condition K = E 2 L implies that (C K, ψ K ) is a balanced Hirsch coalgebra and therefore that the power map λ r exists.

37 Compatability with power maps: mod 2 case [HR] Let X = Σ j J RPn j k j. There exists a finite-type simplicial set K such that for all r 1, Ĥ C (K ; F 2 ) bλ r Ĥ C (K ; F 2 ) ζ ζ commutes up to chain homotopy. S (LX; F 2 ) S b lr S (LX; F 2 ) Remark Any such X admits a simplicial model K such that ( C (K ; F 2 ), ψ K ) is a balanced Hirsch coalgebra, which implies that the power map λ r exists.

38 Sketch of the proof of compatibility I [BHM] Let G be a topological group, ZG its cyclic nerve and BG its usual nerve. Then there exists a homotopy equivalence and an operator such that commutes. h : ZG L BG l cyc r ZG l cyc r ZG : ZG ZG h h L BG bl r L BG

39 Sketch of the proof of compatibility II Let G be a topological group, ZG its cyclic nerve and BG its usual nerve. There is a simplicial map l simp r : ZG ZG such that l simp r l cyc r : ZG ZG.

40 Sketch of the proof of compatibility III For any simplicial group G, there is a twisted cartesian product (Kan fibration) G HG WG, where W is Kan s classifying space functor and HG is the simplicial Hochschild construction, admitting a power map λ hoch r. For any reduced simplicial set K, there is a twisted cartesian product (Kan fibration) GK ĤK K, where ĤG is the simplicial cohochschild construction, admitting a power map λ cohoch r.

41 Sketch of the proof of compatibility IV Let K be a reduced simplicial set, and let G = GK. Then there is a commuting diagram C GK C ĤK C GK S CG C HGK S ZG C K C WGK S BG S G S L BG S BG, compatible at every stage with power maps.

42 Sketch of the proof of compatibility V (With contribution from [HPS]) If K = E 2 L, where L is any simplicial set, then there is a natural quasi-isomorphism such that θ K : Ĥ C K C ĤK Ĥ C K θ K C ĤK bλ r Ĥ C K θ K C ĤK commutes up to chain homotopy. C λ cohoch r