Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Algebraic K -Theory of Strict Ring Spectra
John Rognes
University of Oslo, Norway
Seoul ICM 2014
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Outline
1
Algebraic K -Theory and Automorphisms of Manifolds
2
Topological Cyclic Homology and p-Complete Calculations
3
Logarithmic Ring Spectra and Localization Sequences
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Outline
1
Algebraic K -Theory and Automorphisms of Manifolds
2
Topological Cyclic Homology and p-Complete Calculations
3
Logarithmic Ring Spectra and Localization Sequences
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Symmetric Spectra (Smith)
A spectrum is a sequence of based spaces
X0 , X1 , X2 , . . .
and maps σ : Xn ∧ S 1 → Xn+1 , for n ≥ 0.
A symmetric spectrum is a spectrum equipped with a
Σn -action on each Xn , such that
σ k : Xn ∧ S k → Xn+k
is Σn × Σk -equivariant for each n, k ≥ 0.
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Symmetric Ring Spectra
The category SpΣ of symmetric spectra is closed
symmetric monoidal, with unit the sphere spectrum S and
monoidal pairing the smash product X ∧ Y .
Its localization Ho(SpΣ ) with respect to the stable
equivalences is Boardman’s stable homotopy category.
A symmetric ring spectrum is a symmetric spectrum A with
associative and unital structure maps µ : A ∧ A → A and
η : S → A.
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Algebraic K -Theory of Symmetric Ring Spectra
Mandell defined K (A) as the algebraic K -theory of a
category CA of finite cell A-modules.
The algebraic K -theory spectrum K (A) exhibits a group
completion
|hCA | → Ω∞ K (A)
of the left hand classifying space, turning cofiber
sequences into sums.
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Algebraic K -Theory of Spaces
Let X ' BG be a space, with loop group G ' ΩX .
Let S[G] be the spherical group ring spectrum.
Waldhausen first defined
A(X ) = K (S[G])
as the algebraic K -theory of an unstable model for the
category of finite cell S[G]-modules, the category of
retractive spaces over X .
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
h-Cobordism Spaces
If X is a compact smooth manifold, let H(X ) be the space
of h-cobordisms (W ; X , Y ) with X at one end:
∂W = X ∪ Y ,
'
'
X →W ←Y
Let H (X ) = colimk H(X × [0, 1]k ) be the stable
h-cobordism space.
Theorem (Igusa)
H(X ) → H (X ) is about n/3-connected, for n = dim X .
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
The Stable Parametrized h-Cobordism Theorem
A(X ) = K (S[G]) splits as
A(X ) ' S[X ] ∨ Wh(X ) ,
defining the Whitehead spectrum.
Let ΩWh(X ) = Ω∞+1 Wh(X ) be the looped Whitehead
space.
Theorem (Waldhausen–Jahren–R.)
There is a natural homotopy equivalence H (X ) ' ΩWh(X ).
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Diffeomorphism Groups: Rational
When X is contractible, A(∗) = K (S) ' S ∨ Wh(∗).
Theorem (Borel)
(
Q for i = 0 or 4k + 1 6= 1,
Ki (S) ⊗ Q ∼
= Ki (Z) ⊗ Q ∼
=
0 otherwise.
Example (Farrell–Hsiang)
(
Q for i = 4k − 1, n odd,
πi Diff (D rel ∂D ) ⊗ Q ∼
=
0 otherwise,
n
n
for i up to about n/3.
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Outline
1
Algebraic K -Theory and Automorphisms of Manifolds
2
Topological Cyclic Homology and p-Complete Calculations
3
Logarithmic Ring Spectra and Localization Sequences
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Topological Cyclic Homology
Bökstedt–Hsiang–Madsen constructed a natural
cyclotomic trace map
K (A) → TC(A; p)
to the topological cyclic homology of A.
It is a homotopy limit
TC(A; p) = holim THH(A)Cpn
n,R,F
of cyclic fixed points of the topological Hochschild
homology of A.
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Nilpotent extensions
∧
An integral version satisfies TC(A)∧
p ' TC(A; p)p .
Theorem (Dundas–Goodwillie–McCarthy)
Let A → B be a map of connective symmetric ring spectra, with
π0 (A) → π0 (B) surjective with nilpotent kernel. The square
K (A)
/ K (B)
/ TC(B)
TC(A)
is homotopy Cartesian.
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
The Sphere Spectrum and the Integers
Example
Homotopy Cartesian square
K (S)∧
p
/ K (Z)∧
p
/ TC(Z; p)∧ .
TC(S; p)∧
p
p
R. used this to calculate H∗ and π∗ of
∧
∧
K (S)∧
p ' Sp ∨ Wh(∗)p
for regular primes p.
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
K -Theory of the Sphere Spectrum: Cohomology
Let A be the mod p Steenrod algebra.
For p = 2 let C ⊂ A be generated by admissible Sq I
where I = (i1 , . . . , in ) with n ≥ 2 or I = (i) with i odd.
Theorem (R.)
The mod 2 cohomology of Wh(∗) is the nontrivial extension
Σ−2 C/A (Sq 1 , Sq 3 ) → H ∗ Wh(∗) → Σ3 A /A (Sq 1 , Sq 2 )
of A -modules.
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
K -Theory of the Sphere Spectrum: Homotopy
Example (R.)
The homotopy groups of Wh(∗), modulo p-torsion for irregular
primes p, begin:
i
πi Wh(∗)
i
πi Wh(∗)
i
πi Wh(∗)
0
0
1
0
2
0
3
Z/2
10
Z/8 ⊕ (Z/2)2
15
(Z/2)2
4
0
11
Z/6
16
Z/24 ⊕ Z/2
John Rognes
5
Z
6
0
12
Z/4
7
Z/2
13
Z
17
Z ⊕ (Z/2)2
8
0
9
Z ⊕ Z/2
14
Z/36 ⊕ Z/3
18
Z/480 ⊕ (Z/2)3
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Localization and Descent for Algebraic K -Theory
Seek a conceptual understanding of these calculational
results on K (A)p for A = S.
Can we recover K (A)p from K (B)p for suitably local
symmetric ring spectra B?
Can we descend to K (B)p from K (C)p for appropriate
extensions B → C?
Is there a simple description of K (Ω)p for sufficiently large
such extensions B → Ω?
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Algebraic K -Theory of Topological K -Theory
Adams summand A = `p of kup , with π∗ `p = Zp [v1 ].
Localization B = Lp , with π∗ Lp = Zp [v1±1 ].
/ KUp
O
Lp
O
Sp
/ `p
φ
/ kup
/ HZp
Theorem (Blumberg-Mandell)
Homotopy cofiber sequence
K (`p ) → K (Lp ) → ΣK (Zp ) .
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Chromatic Redshift
For p ≥ 5, the type 2 Smith–Toda complex
V (1) = S ∪p e1 ∪α1 e2p−1 ∪p e2p
is a ring spectrum up to homotopy, with v2 ∈ π2p2 −2 V (1).
Theorem (Ausoni–R.)
V (1)∗ K (`p ) and
V (1)∗ K (Lp )
are finitely generated free Fp [v2 ]-modules, each on 4p + 4
generators, up to small error terms.
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Lichtenbaum–Quillen Conjecture
Suggests that K (Ω)p is a connective form of the Lubin–Tate
spectrum E2 , with π∗ E2 = WFp2 [[u1 ]][u ±1 ] and
V (1)∗ E2 = Fp2 [u ±1 ].
Conjecture (R.)
For purely v1 -periodic commutative symmetric ring spectra B
there is a spectral sequence
−s
2
Es,t
= Hmot
(B; Fp2 (t/2)) =⇒ V (1)s+t K (B) .
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
E 2 -Term for V (1)∗ K (Lp ), p = 5
−3
•
−2
• • •
−1
0
•
s/t
0
•
•
•
•
•
•
•
•
• • •
•
•
•
•
•
2p
|
|
|
John Rognes
Algebraic K -Theory of Strict Ring Spectra
•
•
•
2p2
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Beilinson–Lichtenbaum Conjecture
Set
r
r
(Lp ; Fp2 (∗)) .
Het
(Lp ; Fp2 (∗)) = v2−1 Hmot
Observe motivic truncation:
(
r (L ; F (m)) for 0 ≤ r ≤ m,
Het
p
p2
r
Hmot
(Lp ; Fp2 (m)) ∼
=
0
otherwise.
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Tate–Poitou Duality
Symmetry about (s, t) = (−3/2, p + 1) similar to arithmetic
duality.
Conjecture (R.)
For finite extensions B of Lp there is a perfect pairing
3−r
r
Het
(B; Fp2 (m)) ⊗ Het
(B; Fp2 (p+1−m))
∪
3
→ Het
(B; Fp2 (p+1)) ∼
= Z/p
for each r and m.
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Outline
1
Algebraic K -Theory and Automorphisms of Manifolds
2
Topological Cyclic Homology and p-Complete Calculations
3
Logarithmic Ring Spectra and Localization Sequences
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Logarithmic Geometry
Seek to realize more of motivic cohomology as Galois
cohomology.
Difficult to classify/construct ramified extensions B → C by
obstruction theory.
Tamely ramified extensions behave as unramified when
rigidified by logarithmic structures.
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Logarithmic Rings (Fontaine–Illusie, Kato)
A pre-log ring consists of
a commutative ring R;
a commutative monoid M;
a monoid homomorphism α : M → (R, ·).
Log ring if α−1 GL1 (R) → GL1 (R) is an isomorphism.
Trivial log structure on R has M = GL1 (R).
Localization R → R[M −1 ] factors in log rings as
R → (R, M) → R[M −1 ] .
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
J -spaces (R., Sagave, Schlichtkrull)
The “underlying graded space” of a symmetric spectrum A
is a J -shaped diagram of spaces
ΩJ (A) : (n1 , n2 ) 7→ Ωn2 An1
Indexing category J is isomorphic to Quillen’s construction
Σ−1 Σ, with BJ ' QS 0 .
Homotopy type of a J -space X : J → S is detected by
XhJ = hocolimJ X . Positive projective model structure.
Convolution product X Y maps to smash product under
S J [−] : S J → SpΣ , Quillen adjoint to ΩJ (−) : SpΣ → S J .
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Topological logarithmic structures
Definition
A pre-log ring spectrum consists of
a commutative symmetric ring spectrum A;
a commutative J -space monoid M;
a commutative J -space monoid map α : M → ΩJ (A).
J
Log ring spectrum if α−1 GLJ
1 (A) → GL1 (A) is
J -equivalence.
J
Trivial log structure on A has M = GLJ
1 (A) ⊂ Ω (A).
Localization A → A[M −1 ] = A ∧S J [M] S J [M gp ] factors as
A → (A, M) → A[M −1 ] .
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
The replete bar construction
The group completion η : M → M gp makes (M gp )hJ a
group completion of the E∞ space MhJ .
The cyclic bar construction B cy (M) is the usual simplicial
object [q] 7→ M M · · · M.
The replete bar construction is a homotopy pullback
B cy (M)
M
ρ
/ B rep (M)
/ B cy (M gp )
/M
/ M gp
=
Repletion in topology plays the role of working with fine
and saturated log structures in algebra.
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Logarithmic Topological Hochschild Homology
Definition
Log THH of a pre-log ring spectrum (A, M, α) is the pushout
S J [B cy (M)]
THH(A)
ρ
ρ
/ S J [B rep (M)]
/ THH(A, M)
of cyclic commutative symmetric ring spectra.
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Log Étale Extensions
(A, M) → (B, N) is formally log étale if
B ∧A THH(A, M) ' THH(B, N).
The direct image log structure of (B, N) along j : A → B is
j∗ N = ΩJ (A) ×ΩJ (B) N.
Theorem (R.–Sagave–Schlichtkrull)
J
φ : (`p , j∗ GLJ
1 (Lp )) → (kup , j∗ GL1 (KUp ))
is log étale.
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Localization Sequences
Theorem (R.–Sagave–Schlichtkrull)
Let E be a d-periodic commutative symmetric ring spectrum,
with connective cover j : e → E. Homotopy cofiber sequence
ρ
∂
THH(e) → THH(e, j∗ GLJ
1 (E)) → ΣTHH(e[0, d))
where e[0, d) is the (d − 1)-th Postnikov section of e.
Example
Homotopy cofiber sequence
THH(`p ) → THH(`p , j∗ GLJ
1 (Lp )) → ΣTHH(Zp ) .
John Rognes
Algebraic K -Theory of Strict Ring Spectra
Algebraic K -Theory and Automorphisms of Manifolds
Topological Cyclic Homology and p-Complete Calculations
Logarithmic Ring Spectra and Localization Sequences
Future Work
Develop log TC, with a cyclotomic trace map from log
K -theory, related to K (A[M −1 ]).
Develop log obstruction theory to realize tamely ramified
extensions A → B as part of log étale extensions
(A, M) → (B, N).
John Rognes
Algebraic K -Theory of Strict Ring Spectra