Lecture notes of Matilde Marcolli - Vanderbilt University

Noncommutative geometry, the
field with one element, and
3-manifolds
Matilde Marcolli
2009

The field with one element
Finite geometries (q = p k , p prime)
#P n−1 (F q ) = #(An (F q ) {0})
#G m (F q )
= qn − 1
q − 1 = [n] q
#Gr(n, j)(F q ) = #{P j (F q ) ⊂ P n (F q )}
=
[n] q !
[j] q ![n − j] q ! = ( n)
j q
[n] q ! = [n] q [n − 1] q · · ·[1] q , [0] q ! = 1
J. Tits: take q = 1
P n−1 (F 1 ) := finite set of cardinality n
Gr(n, j)(F 1 ) := set of subsets of cardinality j
Algebraic geometry over F 1 ?
1

Extensions F 1 n (Kapranov-Smirnov)
Monoid {0} ∪ µ n (n-th roots of unity)
- Vector space over F 1 n: pointed set (V, v) with
free action of µ n on V {v}
- Linear maps: permutations compatible with
the action
F 1 n ⊗ F1 Z := Z[t, t −1 ]/(t n − 1)
Zeta functions (Manin) as variety over F 1 :
Z(T n , s) = s − n
2π
with T n = G n m
⇒ F 1 -geometry and infinite primes (Arakelov)
Γ C (s) −1 = ((2π) −s Γ(s)) −1 = ∏ s + n
2π
n≥0
like a “dual” infinite projective space ⊕ ∞ n=0 T−n over F 1
Note: Spec(Z) not a (fin. type) scheme over
F 1 : what is Z ⊗ F1 Z??
More recently: Soulé, Haran, Dourov, Toen–
Vaquie, Connes–Consani, Borger
2

Focus on two approaches:
Descent data for rings from Z to F 1 :
• by cyclotomic points (Soulé)
• by Λ-ring structure (Borger)
Gadgets over F 1
C. Soulé, Les variétés sur le corps à un élément, Mosc.
Math. J. Vol.4 (2004) N.1, 217–244.
(X, A X , e x,σ )
• X : R → Sets covariant functor,
R finitely generated flat rings
• A X complex algebra
• evaluation maps: for all x ∈ X(R), σ : R → C
⇒ e x,σ : A X → C algebra homomorphism
e f(y),σ = e y,σ◦f
for f : R ′ → R ring homomorphism
Affine varieties V Z ⇒ gadget X = G(V Z ) with
X(R) = Hom(O(V ), R) and A X = O(V ) ⊗ C
3

Affine variety over F 1 (Soulé)
Gadget with X(R) finite; variety X Z and morphism
of gadgets
X → G(X Z )
such that all X → G(V Z ) come from X Z → V Z
Should think: R = Z[Z/nZ] ⇒ X(R) cyclotomic points
Λ-rings, endomotives, and F 1
J. Borger, Λ-rings and the field with one element, preprint
2009
- Grothendieck: characteristic classes, Riemann–
Roch
- Λ-ring structure: “descent data” for a ring
from Z to F 1
Torsion free R with action of semigroup N by
endomorphisms lifting Frobenius
s p (x) − x p ∈ pR,
Morphisms: f ◦ s k = s k ◦ f
∀x ∈ R
Q-algebra A ⇒ Λ-ring
iff action of Gal(¯Q/Q) × N on X = Hom(A, ¯Q) factors
through an action of Ẑ
4

NCG of Q-lattices
Alain Connes, M.M., Quantum statistical mechanics of
Q-lattices in “Frontiers in number theory, physics, and
geometry. I”, 269–347, Springer, Berlin, 2006.
Definition: (Λ, φ) Q-lattice in R n
lattice Λ ⊂ R n + labels of torsion points
φ : Q n /Z n −→ QΛ/Λ
group homomorphism (invertible Q-lat if isom)
φ
general case
invertible case
φ
Commensurability (Λ 1 , φ 1 ) ∼ (Λ 2 , φ 2 )
iff QΛ 1 = QΛ 2 and φ 1 = φ 2 mod Λ 1 + Λ 2
Q-lattices / Commensurability ⇒ NC space
5

1-dimensional case
(Λ, φ) = (λ Z, λ ρ) λ > 0
ρ ∈ Hom(Q/Z, Q/Z) = lim ←−n
Z/nZ = Ẑ
Up to scaling λ: algebra C(Ẑ)
Commensurability Action of N = Z >0
α n (f)(ρ) = f(n −1 ρ)
zero otherwise
(partially defined action of Q ∗ + )
1-dimensional Q-lattices up to scale / Commens.
⇒ NC space C(Ẑ) ⋊ N
Crossed product algebra
f 1 ∗ f 2 (r, ρ) =
f ∗ (r, ρ) = f(r −1 , rρ)
∑
s∈Q ∗ + ,sρ∈Ẑ
f 1 (rs −1 , sρ)f 2 (s, ρ)
6

Bost–Connes algebra
J.B. Bost, A. Connes, Hecke algebras, type III factors
and phase transitions with spontaneous symmetry
breaking in number theory. Selecta Math. (N.S.) Vol.1
(1995) N.3, 411–457.
Algebra A Q,BC = Q[Q/Z] ⋊ N generators and
relations
µ n µ m = µ nm
µ n µ ∗ m = µ∗ m µ n when (n, m) = 1
µ ∗ nµ n = 1
e(r + s) = e(r)e(s), e(0) = 1
ρ n (e(r)) = µ n e(r)µ ∗ n = 1 n
∑
ns=r
e(s)
C ∗ -algebra C ∗ (Q/Z) ⋊ N = C(Ẑ) ⋊ N
Time evolution
σ t (e(r)) = e(r),
σ t (µ n ) = n it µ n
Hamiltonian Tr(e −βH ) = ζ(β)
7

Quantum statistical mechanics
(A, σ t ) C ∗ -algebra and time evolution
State: ϕ : A → C linear ϕ(1) = 1, ϕ(a ∗ a) ≥ 0
Time evolution σ t ∈ Aut(A)
rep. on Hilbert space H ⇒ Hamiltonian H =
d
dt σ t| t=0
Equilibrium states (inverse temperature β = 1/kT)
1
Z(β) Tr ( a e −βH)
Classical points of NC space
Z(β) = Tr ( e −βH)
KMS states ϕ ∈ KMS β (0 < β < ∞)
∀a, b ∈ A ∃ holom function F a,b (z) on strip: ∀t ∈ R
F a,b (t) = ϕ(aσ t (b))
F a,b (t + iβ) = ϕ(σ t (b)a)
8

More on Bost–Connes
• Representations π ρ on l 2 (N):
µ n ǫ m = ǫ nm , π ρ (e(r))ǫ m = ζ m r ǫ m
ζ r = ρ(e(r)) root of 1, for ρ ∈ Ẑ ∗
• Low temperature extremal KMS (β > 1)
ϕ β,ρ (a) = Tr(π ρ(a)e −βH )
Tr(e −βH , ρ ∈ Ẑ ∗
)
High temperature: unique KMS state
• Zero temperature: evaluations ϕ ∞,ρ (e(r)) = ζ r
ϕ ∞,ρ (a) = 〈ǫ 1 , π ρ (a)ǫ 1 〉
Intertwining: a ∈ A Q,BC , γ ∈ Ẑ ∗
ϕ ∞,ρ (γa) = θ γ (ϕ ∞,ρ (a))
θ : Ẑ ∗ ≃ → Gal(Q
ab /Q)
Class field theory isomorphism
9

Bost–Connes endomotive
A.Connes, C.Consani, M.M., Noncommutative geometry
and motives: the thermodynamics of endomotives,
Adv. in Math. 214 (2) (2007), 761–831
A = lim −→n
A n with A n = Q[Z/nZ]
abelian semigroup action S = N on A = Q[Q/Z]
Endomotives (A, S) from self maps of algebraic
varieties s : Y → Y , s(y 0 ) = y 0 unbranched,
X s = s −1 (y 0 ), X = lim ←−
X s = Spec(A)
ξ s,s ′ : X s ′ → X s , ξ s,s ′(y) = r(y), s ′ = rs ∈ S
Bost–Connes endomotive:
G m with self maps u ↦→ u k
s k : P(t, t −1 ) ↦→ P(t k , t −k ), k ∈ N, P ∈ Q[t, t −1 ]
ξ k,l (u(l)) = u(l) k/l , u(l) = t mod t l − 1
X k = Spec(Q[t, t −1 ]/(t k − 1)) = s −1
k
(1) and
X = lim ←−k
X k
u(l) ↦→ e(1/l) ∈ Q[Q/Z],
C(X(¯Q)) = C(Ẑ)
10

Integer model of the Bost–Connes algebra
A.Connes, C.Consani, M.M., Fun with F 1 , to appear in
J.Number Theory, arXiv:0806.2401
A Z,BC generated by Z[Q/Z] and µ ∗ n , ˜µ n
˜µ n˜µ m = ˜µ nm
µ ∗ nµ ∗ m = µ ∗ nm
µ ∗ n˜µ n = n
˜µ n µ ∗ m = µ∗ m˜µ n (n, m) = 1.
µ ∗ nx = σ n (x)µ ∗ n and x˜µ n = ˜µ n σ n (x)
where σ n (e(r)) = e(nr) for r ∈ Q/Z
Note: ρ n (x) = µ n xµ ∗ n ring homomorphism but
not ˜ρ n (x) = ˜µ n xµ ∗ n (correspondences “crossed
product” A Z,BC = Z[Q/Z] ⋊˜ρ N)
11

NCG and Soulé’s F 1 -geometry:
Roots of unity as varieties over F 1 :
µ (k) (R) = {x ∈ R | x k = 1} = Hom Z (A k , R)
A k = Z[t, t −1 ]/(t k − 1)
• Inductive system G m over F 1 :
µ (n) (R) ⊂ µ (m) (R), n|m, A m ։ A n
A X = C(S 1 )
• Projective system (BC): ξ m,n : X n ։ X m
ξ m,n : µ (n) (R) ։ µ (m) (R),
n|m
µ ∞ (R) = Hom Z (Z[Q/Z], R)
⇒ projective system of affine varieties over F 1
ξ m,n : F 1 n ⊗ F1 Z → F 1 m ⊗ F1 Z
A X = C[Q/Z]
12

Bost–Connes and F 1
A.Connes, C.Consani, M.M., Fun with F 1 , arXiv:0806.2401
Affine varieties µ (n) over F 1 defined by gadgets
G(Spec(Q[Z/nZ])); projective system
Endomorphisms σ n (of varieties over Z, of gadgets,
of F 1 -varieties)
Extensions F 1 n: free actions of roots of 1
(Kapranov–Smirnov)
ζ ↦→ ζ n , n ∈ N
and ζ ↦→ ζ α ↔ e(α(r)), α ∈ Ẑ
Frobenius action on F 1 ∞
In reductions mod p of integral Bost–Connes
endomotive ⇒ Frobenius
Bost–Connes = extensions F 1 n plus Frobenius
13

Characteristic p versions of the BC endomotive
Q/Z = Q p /Z p × (Q/Z) (p)
denom = power of p; denom = prime to p
K[Q p /Z p ] ⋊ p Z+
endomorphisms σ n for n = p l , l ∈ Z +
ϕ Fp (x) = x p Frobenius of K char p
f ∈ K[Q/Z]
(σ p l ⊗ ϕ l F p
)(f) = f pl
(σ p l ⊗ϕ l F p
)(e(r)⊗x) = e(p l r)⊗x pl = (e(r)⊗x) pl
⇒ BC endomorphisms restrict to Frobenius on
mod p reductions: σ p l Frobenius correspondence
on pro-variety µ ∞ ⊗ Z K
14

NCG and Borger’s F 1 -geometry
M.M., Cyclotomy and endomotives, preprint arXiv:0901.3167
Multivariable Bost–Connes endomotives
Variety T n = (G m ) n endomorphisms α ∈ M n (Z) +
X α = {t = (t 1 , . . . , t n ) ∈ T n | s α (t) = t 0 }
ξ α,β : X β → X α , t ↦→ t γ , α = βγ ∈ M n (Z) +
t ↦→ t γ = σ γ (t) = (t γ 11
1 tγ 12
2 · · · tγ 1n
n , . . . , tγ n1
1 tγ n2
2 · · · tγ nn
n )
X = lim ←−α
X α with semigroup action
C(X(¯Q)) ∼ = Q[Q/Z] ⊗n generators e(r 1 ) ⊗ · · · ⊗ e(r n )
A n = Q[Q/Z] ⊗n ⋊ ρ M n (Z) +
generated by e(r) and µ α , µ ∗ α
ρ α (e(r)) = µ α e(r)µ ∗ α = 1
det α
∑
α(s)=r
σ α (e(r)) = µ ∗ α e(r)µ α = e(α(r))
Endomorphisms:
σ α (e(r)) = µ ∗ α e(r)µ α
e(s)
15

The Bost–Connes endomotive is a direct limit
of Λ-rings
R n = Z[t, t −1 ]/(t n −1) s k (P)(t, t −1 ) = P(t k , t −k )
Action of Ẑ:
α ∈ Ẑ : (ζ : x ↦→ ζx) ↦→ (ζ : x ↦→ ζ α x)
Ẑ = Hom(Q/Z, Q/Z)
⇒ Frobenius over F 1 ∞ (Haran)
Embeddings of Λ-rings (Borger–de Smit)
Every torsion free finite rank Λ-ring embeds
in Z[Q/Z] ⊗n with action of N compatible with
S n,diag ⊂ M n (Z) +
BC systems as universal Λ-rings
⇒ multivariable Bost–Connes endomotives as
universal Λ-rings
16

Quantum statistical mechanics
of multivariable Bost–Connes endomotives
Representations on l 2 (M n (Z) + ): ζ α i
= ∏ j ζα ij
i
ζ = (ζ 1 , . . . , ζ n ) = ρ(r),
ρ ∈ GL n (Ẑ)
µ β ǫ α = ǫ βα
π ρ (e(r))ǫ α = ∏ i
ζ ατ
i ǫ α
Time evolution:
σ t (e(r)) = e(r),
σ t (µ α ) = det(α) it µ α
Hamiltonian:
Hǫ α = logdet(α) ǫ α
Problem: Infinite multiplicities in the spectrum:
SL n (Z)-symmetry
17

Groupoid and convolution algebra
Similar to: A.Connes, M.M. Quantum statistical mechanics
of Q-lattices, in “Frontiers in Number Theory,
Physics, and Geometry, I” pp.269–350, Springer, 2006.
U Γ = {(α, ρ) ∈ Γ\GL n (Q) + × Γ Ẑ n | α(ρ) ∈ Ẑ n }
Quotient U Γ by SL n (Z) × SL n (Z)
(γ 1 , γ 2 ) : (α, ρ) ↦→ (γ 1 αγ2 −1 , γ 2(ρ))
Convolution algebra
(f 1 ⋆ f 2 )(α, ρ) =
∑
(α,ρ)=(α 1 ,ρ 1 )◦(α 2 ,ρ 2 )∈U Γ
f 1 (α 1 , ρ 1 )f 2 (α 2 , ρ 2 )
f ∗ (α, ρ) = f(α −1 , α(ρ)) and σ t (f)(α, ρ) = det(α) it f(α, ρ)
∑
(π ρ (f)ξ)(α) =
f(αβ −1 , β(ρ))ξ(β)
β∈Γ\GL n (Q) + : βρ∈Ẑ ∗
On l 2 (Γ\G ρ ). If ρ ∈ (Ẑ ∗ ) n :
G ρ = {α ∈ GL n (Q) + | α(ρ) ∈ Ẑ n } = M n (Z) +
Z(β) =
∑
m∈Γ\M n (Z) + det(m) −β 18

Why QSM?
A.Connes, C.Consani, M.M., The Weil proof and the geometry
of the adeles class space, arXiv:math/0703392
For original BC system ⇒ dual system
 = A ⋊ σ R. Restriction map:
δ : Â ♮ β → S♮ (Ω β )
Cyclic module: cokernel HC 0 (D(A, ϕ))
⇒ Spectral realization of zeros of Riemann
zeta function and trace formula
Multivariable case:
Manin (1994): problem of Z ⊗ F1 Z =???
cf. Weil’s proof for function fields C × Fq C
Zeta function viewpoint: Kurokawa
tensor product zeta function: summing over zeros
∏
λ∈Ξ(s − λ) m λ
⊗ ∏ µ∈Θ
∏
(s − µ) n µ
:= (s − λ − µ) m λ+n µ
(λ,µ)
as zeta-regularized infinite products
Question: Kurokawa zeta functions from multivariable
dual systems?
19

Thermodynamics of endomotives and RH
A.Connes, C.Consani, M.M., Noncommutative geometry
and motives: the thermodynamics of endomotives,
Adv. in Math. 214 (2) (2007), 761–831
• Dual system: Â = A BC ⋊ σ R
• Scaling action: θ λ ( ∫ x(t) U t dt) = ∫ λ it x(t) U t dt
• Classical points: ˜Ω β ≃ Ω β × R ∗ +
• Restriction map: (morphism of cyclic modules)
δ : Â β
low T KMS states
π
−→ C(˜Ω β , L 1 ) −→ Tr C(˜Ω β )
• Cokernel (abelian category) D(A, ϕ) = Coker(δ)
• Cyclic homology (with scaling action) HC 0 (D(A, ϕ))
• Galois action+ scaling C Q = Ẑ ∗ × R ∗ +
Weil explicit formula as trace formula
on H 1 = HC 0 (D(A, ϕ)):
Tr(ϑ(f)| H 1) = ˆf(0)+ˆf(1)−∆•∆f(1)− ∑ v
ϑ(f) =
∫
∫ ′
C Q
f(g)ϑ g d ∗ g f ∈ S(C Q )
(K ∗ v,e Kv )
Self inters of diagonal ∆ • ∆ = log |a| = −log |D|
f(u −1 )
|1 − u| d∗ u
(D = discriminant for #-field, Euler char χ(C) for F q (C))
20

Note:
• Connes’ 1997 RH paper: Tr(R Λ U(f)):
- only critical line zeros
- Trace formula (global) ⇔ RH
• Tr(ϑ(f)| H 1):
- all zeros involved
- RH ⇔ positivity
Tr ( ϑ(f ⋆ f ♯ )| H 1)
≥ 0 ∀f ∈ S(CQ )
where
(f 1 ⋆ f 2 )(g) =
∫
f 1 (k)f 2 (k −1 g)d ∗ g
multiplicative Haar measure d ∗ g and adjoint
f ♯ (g) = |g| −1 f(g −1 )
⇒ Better for comparing with Weil’s proof for
function fields
21

Weil’s proof in a nutshell
K = F q (C) function field, Σ K = places deg n v = # orbit
of Fr on fiber C(¯F q ) → Σ K
ζ K (s) = ∏ Σ K
(1 − q −n vs ) −1 =
P(q −s )
(1 − q −s )(1 − q 1−s )
P(T) = ∏ (1−λ n T) char polynomial of Fr ∗ on H 1 et (¯C, Q l )
C(¯F q ) ⊃ Fix(Fr j ) = ∑ k
(−1) k Tr(Fr ∗j |H k et(¯C, Q l ))
RH ⇔ eigenvalues λ n with |λ j | = q 1/2
Correspondences: divisors Z ⊂ C × C; degree, codegree,
trace:
d(Z) = Z • (P × C) d ′ (Z) = Z • (C × P)
Tr(Z) = d(Z) + d ′ (Z) − Z • ∆
RH ⇔ Weil positivity Tr(Z ⋆ Z ′ ) > 0
⇒ Dictionary between NCG and Alg Geom:
C × Fq C ⇔ NCG version of Spec(Z)× F1 Spec(Z)
22

A possible intermediate step: Witt vectors
Affine ring scheme of big Witt vectors
Z[λ 1 , λ 2 , . . . , λ n , . . .]
symmetric functions λ i , ghost coordinates
ψ n = ∑ d|n
d u n/d
d
, with ψ n = x n 1 + xn 2 + · · ·
W(R) = ∏ n≥1 R addition/multipl of ghost coords
Truncated Witt schemes with Z[u 1 , . . . , u N ] ⇒
gadgets W (N)
Z
over F 1 (Manin)
Also Λ-ring structure given by the ψ n
Borger’s suggestion: Witt schemes related to
Spec(Z) × F1 · · · × F1 Spec(Z)
⇒ Realize inside multivariable BC endomotives
23

Other aspects of F 1 -geometry:
Analytic geometry
Yu.I. Manin, Cyclotomy and analytic geometry over F 1 ,
arXiv:0809.1564.
The Habiro ring
K. Habiro, Cyclotomic completions of polynomial rings,
Publ. RIMS Kyoto Univ. (2004) Vol.40, 1127–1146.
Ẑ[q] = lim ←−n
Z[q]/((q) n )
(q) n = (1 − q)(1 − q 2 ) · · ·(1 − q n )
Z[q]/((q) n ) ։ Z[q]/((q) k ) for k ≤ n since (q) k |(q) n
Evaluation maps at roots of 1: surj ring homom
ev ζ :
Ẑ[q] → Z[ζ]
give an injective homomorphism:
ev : Ẑ[q] → ∏
Z[ζ]
Taylor series expansions
ζ∈Z
τ ζ :
Ẑ[q] → Z[ζ][[q − ζ]]
injective ring homomorphism
24

Ring of “analytic functions on roots of unity”
⇒ Another model for the NCG of the cyclotomic
tower, replacing Q[Q/Z] with Ẑ[q]
Multivariable Habiro rings (Manin)
̂ Z[q 1 , . . . , q n ] = lim ←−N
Z[q 1 , . . . , q n ]/I n,N
where I n,N is the ideal
( (q1 − 1)(q 2 1 − 1) · · ·(qN 1 − 1), . . . ,(q n − 1)(q 2 n − 1) · · ·(qN n − 1))
Evaluations at roots of 1
ev (ζ1 ,...,ζ n ) :
Taylor expansions
T Z :
̂ Z[q1 , . . . , q n ] → Z[ζ 1 , . . . , ζ n ]
̂ Z[q1 , . . . , q n ] → Z[ζ 1 , . . . , ζ n ][[q 1 −ζ 1 , . . . , q n −ζ n ]]
Z = (ζ 1 , . . . , ζ n ) in Z n 25

Endomotives: Habiro ring version
M.M., Cyclotomy and endomotives, preprint arXiv:0901.3167
One variable:
σ n (f)(q) = f(q n )
lifts P(ζ) ↦→ P(ζ n ) in Z[ζ] through ev ζ
Action of N ⇒ group crossed product
A Z,q = Ẑ[q] ∞ ⋊ Q ∗ +
A Z,q = generated by Ẑ[q] and µ n, µ ∗ n
µ n σ n (f) = fµ n , µ ∗ nf = σ n (f)µ ∗ n
Ẑ[q] ∞ = ∪ N A N with A N gen by µ N fµ ∗ N
Ẑ[q] ∞ = lim −→n
(σ n :
Ẑ[q] → Ẑ[q])
injective ⇒ automorphisms σ n : Ẑ[q] ∞ → Ẑ[q] ∞
26

P Z = polyn in Q powers q r
ˆP Z = lim ←−N
P Z /J N
J N = ideal gen by (q r ) N = (1 − q r ) · · ·(1 − q rN ), r ∈ Q ∗ +
ρ r (f)(q) = f(q r )
Multivariable:
Ẑ[q] ∞ ≃ ˆP Z , µ n fµ ∗ n ↦→ f(q 1/n )
̂ Z[q 1 , . . . , q n ] = lim ←−N
Z[q 1 , . . . , q n , q −1
1 , . . . , q−1 n ]/J n,N
J n,N ideal generated by the (q i − 1) · · ·(q N i
− 1), for i =
1, . . . , n and the (q −1
i
− 1) · · ·(q −N
i
− 1)
Torus T n = (G m ) n , algebra Q[t i , t −1
i
]
t α = (t α i ) i=1,...,n with t α i = ∏ j
t α ij
j
Semigroup action α ∈ M n (Z) + :
q ↦→ σ α (q) = σ α (q 1 , . . . , q n ) =
(q α 11
1 qα 12
2 · · · qα 1n
n , . . . , qα n1
1 qα n2
2 · · · qα nn
n ) = qα 27

Question: 3-manifolds and F 1 -geometry?
Universal Witten–Reshetikhin–Turaev invariant
K.Habiro, A unified Witten–Reshetikhin–Turaev invariant
for integral homology spheres, Invent. Math. 171
(2008) 1–81
- Chern–Simons path integral (Witten)
- quantum groups at roots of 1 (Reshetikhin–Turaev)
- Ohtsuki series
τ(M) : Z → C,
τ O (M) = 1 +
τ ζ (M)
∞∑
λ n (M)(q − 1) n
n=1
Unified view (Habiro): J M (q) = J L (q)
J M (q) ∈ Ẑ[q]
function in the (one-variable) Habiro ring:
ev ζ (J M (q)) = τ ζ (M)
τ 1 (J M (q)) = τ O (M)
Using: 3-dimensional integral homology sphere M
surgery presentation M = SL 3 , algebraically split link
L = L 1 ∪ · · · ∪ L l in S 3 framing ±1
SL 3 = ∼ SL 3 ⇔ L ∼ ′ L′ Fenn–Rourke moves
28

Integral homology 3-spheres
Zhs = free ab group generated by orientationpreserving
homeomorphism classes of integral
homology 3-spheres
Ring with product M 1 #M 2 connected sum
J M1 #M 2
(q) = J M1 (q)J M2 (q), J S 3(q) = 1
J −M (q) = J M (q −1 )
⇒ WRT ring homomorphism
Ohtsuki filtration
J : Zhs → Ẑ[q]
Zhs = F 0 ⊃ F 1 ⊃ · · · F k ⊃ · · ·
F k Z-submodule spanned by
[M, L 1 , . . . , L k ] =
∑
L ′ ⊂{L 1 ,...,L k }
L i = alg split links ±1-framed
Habiro conjecture
J :
Ẑhs → Ẑ[q]
(−1) |L′ | ML ′
Ẑhs = lim ←−d
Zhs/F d with d : N → Z ≥0
d(n)-components of link have framing ±n
29

Integral homology spheres and F 1 ?
X Zhs (R) := {φ : Zhs → R | ∃˜φ : Ẑ[q] → R, φ = ˜φ◦J}
J : Zhs → Ẑ[q] WRT invariant;
X Zhs (R) = set of “coarser” R-valued invariants
A X := Zhs⊗C WRT evaluations as cyclotomic
points: σ ◦ φ : Zhs → C factors through some
evaluation ev ζ
⇒ X Zhs gadget over F 1
Question: Using Habiro conjecture XẐhs inductive
limit of affine varieties over F 1 ? (finiteness)
30

Question: 3-manifolds and endomotives?
Semigroup action? Is it possible to construct
M = S(L,m) 3 ↦→ σ n(M) = S(L 3 n ,m n )
(inv. under Fenn–Rourke moves)
σ n (M 1 #M 2 ) = σ n (M 1 )σ n (M 2 )
Question: ∃ ? Υ M ∈ Ẑ[q]
Υ σn (M) = σ n(Υ M )
Is it possible to use endomotives (eg multivariable
BC) to construct 3-manifold invariants ?
Zhs ⋊ N → A = A ⋊ N
Quantum statistical mechanics?
31

Categorical approach:
- Reidemeister moves
(link diagrams: same ambient isotopy class)
- Hoste moves (admissible links)
(same Zhs: trivial linking #s and frames 1/m)
component 1/m-framing replaced by −m full
twists
1/m
−m full twists
32

2-category:
- Objects: (D, r) admissible framed link diagrams
- 1-morphisms: (D, r) H → (D ′ , r ′ ) Hoste moves
- 2-morphisms: R = (R, R ′ ) Reidemeister moves
(D 1 , r 1 ) H 1
(D
1 ′ , r′ 1 )
R
R ′
(D 2 , r 2 ) H 2
(D ′ 2 , r′ 2 )
Convolution algebra: A CL (2) functions with finite
support on 2-Mor
- vertical convolution product
(f 1 ◦ f 2 )(R) = ∑
R=R 1 ◦R 2
f 1 (R 1 )f 2 (R 2 )
- horizontal convolution product
(f 1 • f 2 )(R) = ∑
R=R 1 •R 2
f 1 (R 1 )f 2 (R 2 )
33

For compositions of 2-morphisms:
- horizontal R 1 • R 2 with R 1 = (R, R ′ ) and R 2 = (R ′ , R ′′ )
(D 1 , r 1 ) H 1
(D
1 ′ , r′ 1 )
R
R ′
H ′ 1
(D ′′
1 , r′′ 1 )
R ′′
(D 2 , r 2 ) H 2
(D
2 ′ , r′ 2 ) H′ 2
(D
2 ′′,
r′′ 2 )
- vertical R 1 ◦ R 2 with R 1 = (R 1 , R ′ 1 ) and R 2 = (R 2 , R ′ 2 )
(D 1 , r 1 ) H 1
(D
1 ′ , r′ 1 )
R 1
R ′ 1
(D 2 , r 2 ) H 2
(D 2 ′ , r′ 2 )
R 2
R ′ 2
(D 3 , r 3 ) H 3
(D ′ 3 , r′ 3 ) 34

Involution: f ∗ (R) = f(R −1 )
(D ′ 2 , r′ 2 ) H −1
2
R ′−1
(D 2 , r 2 )
R −1
(D
1 ′ , r′ 1 ) H −1
1
(D 1 , r 1 )
- Horizontal time evolution: σ f (f 1 ◦ f 2 ) = σ t (f 1 ) ◦
σ t (f 2 )
σ t (f)(H) = e imt f(H), with m = l − l ′
Hoste move between L = L 1 ∪· · ·∪L l and L ′ = L ′ 1 ∪· · ·∪L′ l ′
- Vertical time evolution: σ f (f 1 •f 2 ) = σ t (f 1 )•σ t (f 2 )
self-linking number with respect to the blackboard
framing of the link diagram
(changed by Reidemeister moves)
35

Variant: only oriented Hoste moves (semigroupoid)
Cabling: L n = c n (L) with n = (n, . . . , n)
(L, r) admissible and n|m i : if oriented Hoste
move
(L, r) H → (L ′ , r ′ )
then n|m i also for L ′ and cabled oriented Hoste
move
(L n , nr) H n
→ (L ′ n , nr′ )
Action:
ρ n (f)(H) =
{
f(Hn ) n|m i , ∀i,
0 otherwise,
endomorphisms of A CL (1)
(after collapsing 2-morphisms to equiv rel)
36

Similar to case of
M.M., A.Zainy al-Yasry, Coverings, correspondences, and
noncommutative geometry, J.Geom.Phys. Vol.58 (2008)
N.12, 1639–1661
2-category with Obj=embedded graphs,
1-Mor=3-manifolds as branched coverings,
2-Mor=4-dim cobordisms
Horizontal and vertical time evolutions
(multiplicity of coverings, index theorems, Hartle–Hawking
gravity)
Further questions: Relation of the Habiro
rings and (semi)group actions to Zagier’s quantum
modular forms ?
37