Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Topological Hochschild homology of
topological modular forms
Bob Bruner1
John Rognes2
1 Department
of Mathematics
Wayne State University
2 Department
of Mathematics
University of Oslo
Nordic Topology Meeting, NTNU, November 2008
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Outline
1
Hyperelliptic Cohomology
2
K (n)-Local THH(tmf )
3
Homology of THH(tmf )
4
Homotopy of THH(tmf )
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Red-Shift
Trace Invariants
Circle Action
Results
Outline
1
Hyperelliptic Cohomology
2
K (n)-Local THH(tmf )
3
Homology of THH(tmf )
4
Homotopy of THH(tmf )
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Red-Shift
Trace Invariants
Circle Action
Results
Chromatic Red-Shift
Algebraic K -theory often increases chromatic complexity by
one.
Algebraic K -theory of a finite field is a form of integral
cohomology.
Algebraic K -theory of the integers is a form of topological
K -theory.
Algebraic K -theory of topological K -theory is a form of
elliptic cohomology.
We study algebraic K -theory of elliptic cohomology, K (tmf ),
expecting to find a form of a v3 -periodic cohomology theory,
tentatively called hyperelliptic cohomology.
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Red-Shift
Trace Invariants
Circle Action
Results
Periodic Families
With increasing chromatic complexity, more of the stable
homotopy groups of spheres is detected.
Rational cohomology detects the 0-stem π0 (S).
Topological K -theory detects the image-of-J summand in
π∗ (S). This includes all classes in dimensions ∗ ≤ 5.
Elliptic cohomology detects the v2 -periodic families in
π∗ (S). For p = 2, this includes all classes in dimensions
∗ ≤ 30.
With K (tmf ) we may hope to show that ηθ4 in the 31-stem, or
certain classes in the 39- to 41-stems, are part of v3 -periodic
families. No such periodic family is presently known for p = 2.
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Red-Shift
Trace Invariants
Circle Action
Results
Trace Invariants of Algebraic K -Theory
We study the algebraic K -theory of an S-algebra B by the
Bökstedt–Hsiang–Madsen trace maps
tr : K (B)
trc
// TC(B; p)
// THH(B) .
The right hand map factors through the S 1 -homotopy fixed
points
1
1
∞
THH(B)hS = F (S+
, THH(B))S
and the approximate S 1 -homotopy fixed points
1
1
3
THH(B)aS = F (S+
, THH(B))S .
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Red-Shift
Trace Invariants
Circle Action
Results
σ-operator
The cyclic structure on THH(B) gives a circle action
1
S+
∧ THH(B) → THH(B) .
The σ-operator
σ : H∗ THH(B) → H∗+1 THH(B)
is induced by circle action and the fundamental class in
1 ).
H1 (S+
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Red-Shift
Trace Invariants
Circle Action
Results
Summary of Results
Let p = 2 and B = tmf , the topological modular forms
spectrum. We can:
Compute the Morava K (n)-localizations LK (n) THH(tmf ) for
0 ≤ n ≤ 2.
Describe H∗ THH(tmf ) as an A∗ -comodule algebra.
Give a (quite complete) calculation of π∗ THH(tmf ).
To pass from homological to homotopical calculations, we use
the Adams spectral sequence.
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Red-Shift
Trace Invariants
Circle Action
Results
Plans for Further Work
Jointly with Sverre Lunøe-Nielsen we plan to:
1
Determine H∗ THH(tmf )aS as an A∗ -comodule algebra
1
Compute π∗ THH(tmf )aS (in a range)
Use this to detect potential v3 -periodic classes in π∗ (S)
1
Similar results for THH(tmf )hS or the S 1 -Tate construction
1
THH(tmf )tS would establish v3 -periodicity.
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
K (0)-Local THH(tmf )
K (1)-Local THH(tmf )
K (2)-Local THH(tmf )
Chromatic Assembly
Outline
1
Hyperelliptic Cohomology
2
K (n)-Local THH(tmf )
3
Homology of THH(tmf )
4
Homotopy of THH(tmf )
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
K (0)-Local THH(tmf )
K (1)-Local THH(tmf )
K (2)-Local THH(tmf )
Chromatic Assembly
Rational THH(tmf )
In rational (= K (0)-local) homotopy
π∗ (tmf ) ⊗ Q = Q[c4 , c6 ]
equals elliptic modular forms, with |ci | = 2i.
Theorem
π∗ THH(tmf ) ⊗ Q is an exterior algebra over π∗ (tmf ) ⊗ Q on two
algebra generators σc4 and σc6 in dimensions 9 and 13.
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
K (0)-Local THH(tmf )
K (1)-Local THH(tmf )
K (2)-Local THH(tmf )
Chromatic Assembly
K (1)-Local THH(tmf )
By Hopkins and Laures, the KO∗ -algebra unit map for tmf
factors
// KO∗ [x] f // KO∗ tmf
KO∗
where f is étale.
Theorem
π∗ LK (1) THH(tmf ) is an exterior algebra over
π∗ LK (1) tmf = KO∗ [j −1 ] on one generator σf in dimension 1.
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
K (0)-Local THH(tmf )
K (1)-Local THH(tmf )
K (2)-Local THH(tmf )
Chromatic Assembly
K (2)-Local THH(tmf )
By the Morava change-of-rings theorem the Hopkins–Miller
spectrum LK (2) tmf = EO2 is a pro-étale extension of LK (2) S.
Theorem
π∗ LK (2) THH(tmf ) is isomorphic to π∗ LK (2) tmf = π∗ EO2 .
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
K (0)-Local THH(tmf )
K (1)-Local THH(tmf )
K (2)-Local THH(tmf )
Chromatic Assembly
Chromatic Assembly Problem
THH(tmf ) is
K (0)-locally like four = 22 copies of tmf ,
K (1)-locally like two = 21 copies of tmf , and
K (2)-locally like one = 20 copy of tmf .
What is the global picture?
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Homology Calculation
A∗ -Comodule Decomposition
The Layer Comodules
The tmf -Module Filtration
Outline
1
Hyperelliptic Cohomology
2
K (n)-Local THH(tmf )
3
Homology of THH(tmf )
4
Homotopy of THH(tmf )
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Homology Calculation
A∗ -Comodule Decomposition
The Layer Comodules
The tmf -Module Filtration
The Steenrod Algebra
Let A = hSq i | i ≥ 1i be the mod 2 Steenrod algebra and
let
A∗ = P(ξ¯k | k ≥ 1)
be the dual Steenrod algebra, with |ξ¯k | = 2k − 1.
The coproduct on A∗ is given by
X
i
ξ¯i ⊗ ξ¯j2 .
ψ(ξ¯k ) =
i+j=k
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Homology Calculation
A∗ -Comodule Decomposition
The Layer Comodules
The tmf -Module Filtration
Homology of tmf
In cohomology
H ∗ (tmf ) = A ⊗A(2) F2
where A(2) = hSq 1 , Sq 2 , Sq 4 i.
In homology
H∗ (tmf ) = P(ξ¯18 , ξ¯24 , ξ¯32 , ξ¯k | k ≥ 4)
is an A∗ -comodule subalgebra of A∗ .
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Homology Calculation
A∗ -Comodule Decomposition
The Layer Comodules
The tmf -Module Filtration
The Bökstedt Spectral Sequence
The Bökstedt spectral sequence
2
E∗∗
= HH∗ (H∗ (tmf )) =⇒ H∗ (THH(tmf ))
collapses at
2
= H∗ (tmf ) ⊗ E(σ ξ¯18 , σ ξ¯24 , σ ξ¯32 , σ ξ¯k | k ≥ 4)
E∗∗
since the algebra generators are in filtration ≤ 1.
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Homology Calculation
A∗ -Comodule Decomposition
The Layer Comodules
The tmf -Module Filtration
The Homology of THH(tmf )
The multiplicative extensions are determined by the
Dyer–Lashof operations.
Theorem
H∗ THH(tmf ) = H∗ (tmf ) ⊗ E 3 P∗
as an A∗ -comodule algebra, where
E 3 P∗ = E(σ ξ¯18 , σ ξ¯24 , σ ξ¯32 ) ⊗ P(σ ξ¯4 ) .
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Homology Calculation
A∗ -Comodule Decomposition
The Layer Comodules
The tmf -Module Filtration
The Adams Spectral Sequence for π∗ THH(tmf ), I
The E2 -term of the Adams spectral sequence
∗
∧
E2s,t = Exts,t
A (H THH(tmf ), F2 ) =⇒ πt−s THH(tmf )2
can, by change-of-rings, be rewritten as
∗∗
3
3 ∗
E2∗∗ = Ext∗∗
A(2) (E P , F2 ) = ExtA(2)∗ (F2 , E P∗ ) .
We must understand E 3 P∗ as an A(2)∗ -comodule.
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Homology Calculation
A∗ -Comodule Decomposition
The Layer Comodules
The tmf -Module Filtration
A∗ -Coaction
The A∗ -coaction is generated by
σ ξ¯18 →
7 1 ⊗ σ ξ¯18
σ ξ¯24 7→ 1 ⊗ σ ξ¯24 + ξ¯14 ⊗ σ ξ¯18
σ ξ¯2 7→ 1 ⊗ σ ξ¯2 + ξ¯2 ⊗ σ ξ¯4 + ξ¯2 ⊗ σ ξ¯8
3
3
1
2
2
1
σ ξ¯4 7→ 1 ⊗ σ ξ¯4 + ξ¯1 ⊗ σ ξ¯32 + ξ¯2 ⊗ σ ξ¯24 + ξ¯3 ⊗ σ ξ¯18
so the square (σ ξ¯4 )2 in dimension 32 is A∗ -comodule primitive.
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Homology Calculation
A∗ -Comodule Decomposition
The Layer Comodules
The tmf -Module Filtration
A∗ -Comodule Decomposition of E 3 P∗
Definition
L[1]∗ = F2 {σ ξ¯18 , σ ξ¯24 , σ ξ¯32 , σ ξ¯4 }
with exterior powers the layer comodules
L[j]∗ = Λj L[1]∗
for 0 ≤ j ≤ 4.
Lemma
E 3 P∗ = (L[0]∗ ⊕ · · · ⊕ L[4]∗ ) ⊗ P((σ ξ¯4 )2 )
is the direct sum of the terms Σ32i L[j]∗ for i ≥ 0, 0 ≤ j ≤ 4.
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Homology Calculation
A∗ -Comodule Decomposition
The Layer Comodules
The tmf -Module Filtration
The A∗ -Comodules L[j]∗ , I
The bottom and top exterior powers
L[0]∗ = F2 {1}
0
L[4]∗ = F2 {σ ξ¯18 σ ξ¯24 σ ξ¯32 σ ξ¯4 }
53
are concentrated in dimensions 0 and 53.
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Homology Calculation
A∗ -Comodule Decomposition
The Layer Comodules
The tmf -Module Filtration
The A∗ -Comodules L[j]∗ , II
The generating comodule
Sq 4
L[1]∗
Sq 2
¹¹
9
!!
13
15
Sq 1
// 16
is dual to the third exterior power
Sq 4
Sq 2
L[3]∗
37
Sq 1
// 38
Bob Bruner, John Rognes
!!
40
¹¹
44
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Homology Calculation
A∗ -Comodule Decomposition
The Layer Comodules
The tmf -Module Filtration
The A∗ -Comodules L[j]∗ , III
The middle exterior power L[2]∗
Sq 4
== 24
22
Sq
Sq 2
ºº
// 25
1
Sq 2
28
Sq 1
// 29
GG
!!
31
Sq 4
is self-dual.
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Homology Calculation
A∗ -Comodule Decomposition
The Layer Comodules
The tmf -Module Filtration
A Realization Lemma
Lemma
For each 0 ≤ j ≤ 4 there exists a finite CW spectrum L[j] with
H∗ L[j] = L[j]∗
as A∗ -comodules. This determines L[j] uniquely up to 2-adic
equivalence.
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Homology Calculation
A∗ -Comodule Decomposition
The Layer Comodules
The tmf -Module Filtration
A Linear Ordering
Each A∗ -comodule Σ32i L[j]∗ in the sum decomposition of
E 3 P∗ = E(σ ξ¯18 , σ ξ¯24 , σ ξ¯32 ) ⊗ P(σ ξ¯4 )
has a unique A∗ -comodule primitive.
We linearly order the summands according to the dimension of
this primitive:
L[0]∗ , L[1]∗ , L[2]∗ , Σ32 L[0]∗ , L[3]∗ ,
Σ32 L[1]∗ , L[4]∗ , Σ32 L[2]∗ , Σ64 L[0]∗ , . . .
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Homology Calculation
A∗ -Comodule Decomposition
The Layer Comodules
The tmf -Module Filtration
A tmf -Module Filtration
Lemma
There is a filtration of tmf -module spectra
tmf = T 0 → · · · → T k −1 → T k → · · · → THH(tmf )
with homotopy cofiber sequences
T k −1 → T k → tmf ∧ Σ32i L[j]
such that
H∗ T k =
M
H∗ (tmf ) ⊗ Σ32i L[j]∗
is the sum of terms 0 through k in the linear ordering.
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Homology Calculation
A∗ -Comodule Decomposition
The Layer Comodules
The tmf -Module Filtration
Comment on Proof
This is approximately the tmf -module filtration generated by a
skeleton filtration.
When the E 3 P∗ -summands overlap, as for L[3]∗ and Σ32 L[1]∗ ,
the proof is incomplete, due to a possible attachment of a cell
of the “lower” piece to a cell of the “higher” piece by η 2 . This
first plays a role in dimension 44, and can probably be resolved
by the K (1)-local calculation.
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Outline
1
Hyperelliptic Cohomology
2
K (n)-Local THH(tmf )
3
Homology of THH(tmf )
4
Homotopy of THH(tmf )
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
The Adams Spectral Sequence for π∗ THH(tmf ), II
The Adams spectral sequence E2 -term
∗
E2∗∗ = Ext∗∗
A (H THH(tmf ), F2 )
3 ∗
∧
= Ext∗∗
A(2) (E P , F2 ) =⇒ π∗ THH(tmf )2
is machine computable using Bruner’s ext-program.
It gets crowded after the 30-stem.
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Adams Chart for π∗ THH(tmf ), 0 ≤ ∗ ≤ 44
THHtmf/A2 from s,n=0,0
THHtmf/A2 from s,n=0,22
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Adams Chart for π∗ THH(tmf ), 44 ≤ ∗ ≤ 88
THHtmf/A2 from s,n=0,44
THHtmf/A2 from s,n=0,66
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Plan for the Calculation of π∗ THH(tmf )
To clarify we use the tmf -module filtration:
tmf
// . . .
// T k −1
// T k
// . . .
// THH(tmf )
²²
tmf ∧ Σ32i L[j]
First calculate homotopy tmf∗ (Σ32 L[j]) of the filtration
quotients, for 0 ≤ j ≤ 4.
Then assemble homotopy π∗ (T k ) of filtration stages, for
k ≥ 0.
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Adams Spectral Sequence for Filtration Layers
The Adams spectral sequence for the (i, j)-th layer
∗
32i
E2s,t = Exts,t
A (H (tmf ∧ Σ L[j]), F2 )
32i
∗
32i
= Exts,t
A(2) (Σ L[j] , F2 ) =⇒ (tmf )t−s (Σ L[j])
is practically independent of i.
Reduces to the five cases 0 ≤ j ≤ 4.
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Adams E2 -Term for π∗ tmf
For j = 0, L[0] = S and we are computing π∗ tmf .
The Adams E2 -term
∧
E2s,t = Ext∗∗
A(2) (F2 , F2 ) =⇒ πt−s (tmf )2
was computed by Iwai–Shimada. It has algebra generators:
h 0 , h1 , h 2
α0 = v28 , α1 , α2 , . . . , α6 , α7
ω0 = v14 , ω1
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Chart for π∗ tmf , 0 ≤ ∗ ≤ 24
s
12
11
10
9
8
7
6
5
4
α4
ω0
α5
ω1
3
α1
α2
α3
2
1
h1
h0
h2
0
t−s
0
1
2
3
4
5
6
7
8
9
10
11
Bob Bruner, John Rognes
12
13
14
15
16
17
18
19
20
21
22
23
24
Topological Hochschild homology of topological modular forms
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Chart for π∗ tmf , 24 ≤ ∗ ≤ 48
s
20
19
18
17
16
15
14
13
12
11
10
9
8
α0
7
ω1 α3
α7
6
5
α6
t−s
24
25
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27
28
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30
31
32
33
34
35
36
37
38
Bob Bruner, John Rognes
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Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Adams Differentials for π∗ tmf
Hopkins–Mahowald computed these Adams differentials.
Permanent cycles are black. Dead classes are white.
To describe the differentials, write the E2 -term as the sum of
two pieces:
The Bott periodic part: free over P(ω0 , α0 ) = P(v14 , v28 ).
The Mahowald–Tangora wedge: free of rank one over
P(v1 , w, α0 ) on ω1 α3 in dimension 35.
The first piece comes in several stages: Infantile, Puerile,
Juvenile, Virile, Senile.
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Chart for tmf — The Bott Periodic Part
s
16
15
ω03 α1
14
13
12
ω02 α4
ω03
ω02 α2
ω02 ω1
ω02 α5
11
ω0 α2 α4
ω02 α3
10
ω0 α22
9
α4 α5
ω1 α2
7
ω0 α32
ω0 α2 α3
ω0 α6
8
ω0 α3 α4
α23
ω1 α4
α7 + ω1 α2
t−s
24
26
28
30
32
Bob Bruner, John Rognes
34
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Topological Hochschild homology of topological modular forms
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Chart for tmf — The Wedge Part
s
20
19
18
17
16
15
14
13
ω02 ω1
12
ω1 α4 α5
11
ω1 α2 α4
10
9
α4 α6
8
7
ω1 α4
α7 + ω1 α2
ω1 α22
α2 α3 α4
ω0 α2 α3
α5 α6
ω12 α2
ω1 α3 α4
ω1 α32
ω1 α2 α3
ω1 α6
ω12
ω1 α5
ω1 α3
t−s
32
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40
42
Bob Bruner, John Rognes
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52
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Adams Spectral Sequence for tmf — Summary
The Adams E2 -term for tmf is completely known, including
cup and Massey products, by machine computation.
The Adams differentials are completely known, using E∞
structure and/or the Adams–Novikov spectral sequence.
The additive extensions of π∗ (tmf ) are completely known,
using Massey products and Moss’ theorem.
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Adams E2 -Term for tmf∗ (L[1])
For j = 1, L[1] = S 9 ∪ν e13 ∪η e15 ∪2 e16 .
The Adams E2 -term
∗
∧
E2s,t = Ext∗∗
A(2) (L[1] , F2 ) =⇒ tmf∗ (L[1])2
was computed by Davis–Mahowald.
This spectral sequence is a module over the tmf spectral
sequence.
We write gn (or xn ) for a module generator in dimension
n = t − s.
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Adams Chart for tmf∗ (L[1]), 9 ≤ ∗ ≤ 33
s
12
ω03 g9
11
ω02 g13
10
9
8
ω02 g9
7
ω0 g25
ω0 α2 g9
ω0 g13
6
α22 g9
α2 g13
5
ω0 g18
4
3
x33
α3 g18
g25
ω0 g9
α2 g9
g13
α3 g9
2
1
0
g18
g9
9
t−s
11
13
15
17
19
Bob Bruner, John Rognes
21
23
25
27
29
31
33
Topological Hochschild homology of topological modular forms
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Chart for tmf∗ (L[1]), 33 ≤ ∗ ≤ 57
s
16
15
ω03 g13
14
13
12
ω03 g9
11
ω02 α2 g9
10
9
α22 g13
8
x49
ω0 g25
7
6
α22 g9
5
α32 g9
α6 g9
α3 ω1 g18
α0 g9
α32 g18
α2 x33
x37
α6 g18
ω1 g18
x33
4
α3 g18
0
t−s
33
35
37
39
41
43
Bob Bruner, John Rognes
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51
53
55
57
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Adams Differentials for tmf∗ (L[1]
This spectral sequence is quite sparse.
The first nonzero differential is
d3 (α02 g18 ) = ω14 α3 g18
landing in dimension t − s = 113.
This is well beyond the initial range of interest.
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Adams E2 -Term for tmf∗ (L[2])
For j = 2, L[2] is a self-dual 6-cell CW spectrum.
The Adams E2 -term
∗
∧
E2s,t = Ext∗∗
A(2) (L[2] , F2 ) =⇒ tmf∗ (L[2])2
is machine computable.
This spectral sequence is a module over the tmf spectral
sequence.
We write gn for a module generator in dimension n = t − s.
The two generators in dimension 34 are called g34,1 and
g34,2 .
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Chart for tmf∗ (L[2]), 22 ≤ ∗ ≤ 46
s
12
11
10
9
8
7
6
5
g45
4
3
g33
2
g27
1
0
g36
g39
g34,2
g31
g34,1
g22
22
t−s
24
26
28
30
32
Bob Bruner, John Rognes
34
36
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40
42
44
46
Topological Hochschild homology of topological modular forms
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Chart for tmf∗ (L[2]), 46 ≤ ∗ ≤ 70
s
16
15
14
13
12
11
10
g67
9
8
7
6
g51
5
4
t−s
46
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52
54
56
Bob Bruner, John Rognes
58
60
62
64
66
68
70
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Adams Differentials for tmf∗ (L[2])
We have computed the Adams differentials for tmf∗ (L[2]).
To describe the differentials, write the E2 -term as the sum of
two pieces:
A Bott periodic part, which is free over P(ω0 , α0 ).
A double Mahowald–Tangora wedge, which is free of rank
two over P(v1 , w, α0 ) on ω1 g39 and α32 g34,1 in
dimensions 59 and 64.
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Adams Chart for tmf∗ (L[2]) — The Bott Periodic Part
s
16
15
14
13
12
11
10
9
8
7
6
5
t−s
46
48
50
52
54
56
Bob Bruner, John Rognes
58
60
62
64
66
68
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Adams Chart for tmf∗ (L[2]) — The Wedge Part
s
14
13
12
11
10
9
8
7
α32 g34,1
ω1 g39 = α6 g34,2
6
5
t−s
58
60
62
64
66
Bob Bruner, John Rognes
68
70
72
74
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Adams Spectral Sequence for tmf∗ (L[2]) — Summary
The Adams E2 -term for tmf∗ (L[2]) is completely known,
including cup and Massey products.
The Adams differentials are completely known, using
rational information and the tmf -module structure.
The additive extensions of tmf∗ (L[2]) are (almost)
completely known.
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Adams spectral sequences for tmf∗ (L[3]), tmf∗ (L[4])
For j = 3, with
L[3] = S 37 ∪2 e38 ∪η e40 ∪ν e44
the Adams spectral sequence for tmf∗ (L[3]) is sparse like
the one for tmf∗ (L[1]).
For j = 4, with L[4] = S 53 the Adams spectral sequence for
tmf∗ (L[4]) is a shifted copy of the one for π∗ (tmf ).
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Assembling the Layers
The zeroth layer T 0 = tmf splits off from THH(tmf ).
The second layer tmf ∧ L[2] is nontrivially attached to the
first layer tmf ∧ L[1]:
Theorem
There is a differential
d2 (g22 ) = h2 g18
in the Adams spectral sequence for π∗ THH(tmf ).
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
The String Orientation
The proof uses the string orientation
M String = MOh8i → tmf
and the induced map
tmf ∧ BBOh8i+ = THH(MOh8i, tmf ) → THH(tmf )
to prove that g92 = g18 in π∗ THH(tmf ).
From h2 g9 = 0 it follows that h2 g18 is a boundary.
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
The NeXT Step
The third layer tmf ∧ S 32 is nontrivially attached to the second
layer:
Lemma
There is a nonzero differential
dk +2 (g32 ) = h0k g31
for some k ≥ 0.
May need Pontryagin power operations in tmf -homology to
determine k .
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms
Hyperelliptic Cohomology
K (n)-Local THH(tmf )
Homology of THH(tmf )
Homotopy of THH(tmf )
Adams Sp. Seq. for THH(tmf )
Adams Sp. Seq. for tmf
Adams Sp. Seq. for tmf ∧ L[1] and tmf ∧ L[2]
Remaining Steps
Christian Nassau’s Big Ext Chart
140
135
130
125
120
115
110
105
100
95
90
85
80
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200 205 210
Bob Bruner, John Rognes
Topological Hochschild homology of topological modular forms