Comparison of secondary invariants of algebraic K-theory

0.38 MB·English

by  Jens Kaad

COMPARISON OF HIGHER INVARIANTS FOR SPECTRAL TRIPLES
8
J. KAAD
0
0
2 Department of Mathemati
al S
ien
es, University of Copenhagen

r Universitetsparken 5, DK-2100 Copenhagen, Denmark
p
A
Abstra
t
0
1
In this paper we prove that the multipli
ative
hara
ter of A. Connes and M. Karoubi and the

] determinant invariant of L. G. Brown, J. W. Helton and R. E. Howe agree up to a
anoni
al
T
homomorphism.
K
.
h Contents
t
a
m 1. Introdu
tion 2
[
2. The Determinant Invariant 5

1
2.1. Preliminaries on Relative Group Homology 5
v
9
2.2. The Universal Determinant Invariant 7
8
5
2.3. Comparing De(cid:28)nitions of The Determinant Invariant 9
1
.
4 3. The Multipli
ative Chara
ter 12
0
8 3.1. Normalized Singular Homology 12
0
3.2. Continuous Cy
li
Homology and The Relative Chern Chara
ter 17
:
v
i 3.3. The Universal Multipli
ative Chara
ter 20
X
r 4. Comparing The Determinant Invariant and The Multipli
ative Chara
ter 21
a
4.1. Fa
torization through Relative Cy
li
Homology 21
4.2. Determinants and The Relative Logarithm 25
4.3. Proof of The Main Result 28
5. Se
ondary Invariants for Spe
tral Triples 31
5.1. The Multipli
ative Chara
ter of a Spe
tral Triple 31
5.2. The Determinant Invariant of a Spe
tral Triple 32
6. Cal
ulations of the Multipli
ative Chara
ter 34
Referen
es 39
email: kaadmath.ku.dk.
1
2 J. KAAD
1. Introdu
tion
In their paper, [10℄, A. Connes and M. Karoubi de(cid:28)ne a multipli
ative
hara
ter on algebrai
K
-theory M : K (A) → C/(2πi)pZ
F 2p
(F,H)
for ea
h odd Fredholm module satisfying the additional growth
ondition
[F,a] ∈ L2p(H) a ∈ A
(1) for all
By the work of A. L. Carey, J. Phillips and F. A. Suko
hev, [11℄ and [12℄, ea
h odd unital
(A,H,D)
spe
tral triple with resolvent
(λ−D)−1 ∈ Lq(H) λ ∈ C−R
for ea
h
q 1 ≤ q < 2p 2p
th
in the S
hatten ideal for some , gives rise to an odd -summable Fredholm
F ∈ L(H)
module in a
anoni
al way. The Fredholm operator being the sign operator
x x ∈ 1 (D)
F(x) = for [0,∞)
(cid:26) −x x ∈ 1 (D)
(−∞,0)
for
We
an therefore asso
iate a multipli
ative
hara
ter
M : K (A) → C/(2πi)pZ
D 2p
q (A,H,D) 1 ≤ q < 2p
to ea
h odd -dimensional spe
tral triple with .
In a di(cid:27)erent dire
tion, L.G. Brown has de(cid:28)ned a determinant invariant
: K (B) → C∗
X 2
Det
C
for ea
h exa
t sequen
e of -algebras
X : 0 −−−→ L1(H) −−−i→ E −−−s→ B −−−→ 0
ι : E → L(H)
equipped with an inje
tive homomorphism su
h that
(ι◦i)(T) = T T ∈ L1(H)
(2) for all
See the papers [4℄ and [5℄.
B
In the
ase where is
ommutative this invariant is related to the work of J. W. Helton and
R. E. Howe on tra
es of
ommutators, [15℄, via the identity
{s(S),s(T)} = exp( [S,T])
X
Det Tr
{s(S),s(T)} ∈ K (B)
2
Here denotes the Steinberg symbol.
q (A,H,D) 1 ≤ q < 2
Given an odd -dimensional spe
tral triple with we
onstru
t an exa
t
sequen
e
X : 0 −−−→ L1(H) −−−i→ E −−−s→ B −−−→ 0
(3) D
ι : E → L(H)
an inje
tive homomorphism satisfying (2) and a surje
tive homomorphism
R : A → B
(A,H,D)
In parti
ular the spe
tral triple furnishes both a multipli
ative
hara
ter
M : K (A) → C/(2πi)Z
D 2
COMPARISON OF HIGHER INVARIANTS FOR SPECTRAL TRIPLES 3
a determinant invariant
: K (B) → C∗
DetXD 2
and a
anoni
al link between them
R : K (A) → K (B)
∗ 2 2
The main result of this paper
an then be expressed as the
ommutativity of the diagram
K (A) R-∗ K (B)
2 2
(4) MD ? ?DetXD
C/(2πi)Z - C∗
exp
The multipli
ative
hara
ter and the determinant invariant thus agree upto the homomorphism
R : K (A) → K (B)
∗ 2 2
.
This should be seen in relation with a result of A. Connes and M. Karoubi des
ribing the
multipli
ative
hara
ter in terms of a
entral extension
1 −−−→ C∗ −−−→ Γ −−−→ E(M1) −−−→ 1
5
Indeed some of the main ideas for our proof are taken from [10, Paragraphe ℄.
One important advantage of
omparing the multipli
ative
hara
ter and the determinant in-
vCar∞ia(nSt1)is,Lt2h(aSt1)t,h−eiladtter admits
al
ulation. aA:sS1an→eCxample we look at the spe
tral triple
dt . For any smooth loop we prove that
(cid:0) (cid:1)
1 2π
M {z,ea} = a(θ)dθ ∈ C/(2πi)Z
D
2π Z
0
K C∞(S1) 0
2 th
The algebrai
K-group thus
ontains information about the Fourier
oe(cid:30)
ient
a (cid:0) (cid:1) E C∞(S1)
of C. ∗Inthis
ontextweshouldalsomentiontheworkin[28℄ona
entralextensionof (cid:0) (cid:1)
by .
The paper is organized as follows.
In the (cid:28)rst se
tion we give an alternative des
ription of the universal determinant invariant
: K L(H)/L1(H) → C∗
U 2
Det
(cid:0) (cid:1)
The main ingredient is a di(cid:27)erent
onstru
tion of the homomorphism
f : H (H) → (ϕ)/[G, (ϕ)]
ϕ 2
Ker Ker
ϕ : G → H
indu
ed by a surje
tive group homomorphism . This approa
h uses a
okernel

omplex as introdu
ed by M. Levine in [19℄. What is gained at (cid:28)rst sight is a homomorphism
whi
h is more suitable for
al
ulations sin
e it is de(cid:28)ned without the use of set theoreti
al
se
tions. On a deeper level, the alternative de(cid:28)nition of the determinant invariant whi
h arises
seems to be
ru
ial for our main proof to work.
3
In se
tion we review the de(cid:28)nition of the universal odd multipli
ative
hara
ter
M : K (M1) → C/(2πi)pZ
U 2p
4 J. KAAD
introdu
ingthe relevant
on
epts as we need them. In parti
ular we des
ribe the relative Chern

hara
ter
: K (A) → HC (A)

hrnel nrel n
o−n1t
H A (∆) C2
n 0
in details. To this end we introdu
e a homology theory, , of maps
σ : ∆n → GL(A) σ(0)(cid:0)= 1 (cid:1)
as well as the logarithm
γ : H A (∆) → HC (A)
n 0 n
o−n1t
(cid:0) (cid:1)
given by
(−1)n ∂σ ∂σ
γ(σ) = (s) ·σ−1 ⊗...⊗ ·σ−1dt ∧...∧dt
1 n
n! X sgn Z∆n ∂ts(1) ∂ts(n)
s∈Σn
C2
We remark that for ea
h map
σ : ∆1 → G σ(0) = 1
G
where denotes the operators of determinant
lass we have
(exp◦ ◦γ)(σ) = σ(1)
Tr det
(cid:0) (cid:1)
giving a hint of the
onne
tion between determinants and the multipli
ative
hara
ter. Another
4
important observation, whi
h we present in se
tion , is the fa
torizationof the homomorphism
τ : HC (M1) → C
1 1
ont
HC (T 1,M1)
indu
edbythe
y
li

o
y
leofA.Connesthroughtherelativehomologygroup 0
ont
HC (M1) -∂ HC (T 1,M1)
1
ont 0
ont
(cid:8)
(cid:8)
(cid:8)
τ1 (cid:8)
(cid:8)
? (cid:8) T
(cid:8)(cid:25)
C
Together with
onsiderations on the relative normalized singular homology group
H T 1(∆),M1(∆)
1 0 0
(cid:0) (cid:1)
and its relations to relative
ontinuous
y
li
homologythese observations lead to a proof of the
main theorem: The universal multipli
ative
hara
ter agrees with the universal determinant
invariant up to a
anoni
al homomorphism
K (M1) R-∗ K L(H)/L1(H)
2 2
(cid:0) (cid:1)
MU U
? ?Det
C/(2πi)Z - C∗
exp
5
In se
tion we review the transition from spe
tral triples to Fredholm modules and
onstru
t
the asso
iated exa
t sequen
e (3). The
ommutativity of the diagram (4) is then dedu
ed as a

orollary to the main theorem.
The paper
on
ludes with a
al
ulation of the multipli
ative
hara
ter in a
on
rete example.
COMPARISON OF HIGHER INVARIANTS FOR SPECTRAL TRIPLES 5
A
knowledgements We would like to thank Ryszard Nest for his support and helpful
om-
ments.
2. The Determinant Invariant
2.1. Preliminaries on Relative Group Homology. In this se
tion we will review some
standard homologi
al
onstru
tions. This is needed in order to give pre
ise des
riptions of the
isomorphisms in Theorem 2.1 and Theorem 2.2 whi
h we will need throughout the paper.
G
TZh[eGgnr]o,udp homologywith integer
oe(cid:30)
ients of a group isthe homologyof the
hain
omplex
where the boundary
(cid:0) (cid:1)
d : Z[Gn] → Z[Gn−1] d = n (−1)id
i=0 i
P
is given by the fa
e operators
(g ,...,g ) i = 0
2 n
for
d (g ,...,g ) =  (g ,...,g g ,...,g ) i ∈ {1,2,...,n−1}
i 1 n 1 i i+1 n
(5)  (g ,...,g ) for i = n
1 n−1
for

n ∈ {1,2,...} Z (G) B (G)
n n
In degree we denote the
y
les by , the boundaries by and the
H (G)
n
homology by .
ϕ : G → H ϕ :
∗
Ea
h group homomorphism indu
es a group homomorphism on homology
H (G) → H (H)
n n
whi
h on
hains are extended linearly from
ϕ (g ,...,g ) = ϕ(g ),...,ϕ(g )
∗ 1 n 1 n
(cid:0) (cid:1)
Explaining the fun
toriality of group homology.
H (G) H (H)
n n
The homology groups and (cid:28)t in a long exa
t sequen
e
··· ϕ∗ - H (H) ∂- H (G,H) p- H (G)
n+1 n n
ϕ∗
(6) ?
··· (cid:27) H (G) (cid:27) H (G,H) (cid:27) H (H)
ϕ∗ n−1 p n−1 ∂ n
H (G,H) H(G,H)
0
whi
h terminates at . The relative group homology, , is the homology of the
( (ϕ)[−1]),d(ϕ)
n
totalized shifted
one
omplex Tot Cone where the boundary
(cid:0) (cid:1)
d(ϕ) : Z[Hn+1]⊕Z[Gn] → Z[Hn]⊕Z[Gn−1]
is given by
d(ϕ)(y,x) = dy +ϕ (x),−dx (y,x) ∈ Z[Hn+1]⊕Z[Gn]
∗
(cid:0) (cid:1)
p : H (G,H) → H (G) p(y,x) = x ∂ :
n n
The map is indu
ed by the proje
tion . The map
H (H) → H (G,H) ∂(y) = (y,0)
n n−1
is indu
ed by the in
lusion .
6 J. KAAD
Ea
h
ommutative diagram
ϕ-
G H
α β
(7) ? ?
G′ - H′
ϕ′
(β,α) : H (G,H) → H (G′,H′)
∗ n n
of group homomorphisms indu
es a group homomorphism
whi
h on
hains is given by
(β,α) (y,x) = β (y),α (x) (y,x) ∈ Z[Hn+1]⊕Z[Gn]
∗ ∗ ∗
(cid:0) (cid:1)
Explaining the fun
toriality of the relative group homology.
n ∈ {1,2,...} Gn×HnGn (g1,g2) ∈ Gn×Gn
For ea
h let denote the set of pairs whi
h
oin
ide
Hn
in
ϕ (g1) = ϕ (g2) ∈ Hn
∗ ∗
n(δϕ) Z[Gn ×Hn Gn]
Let Coker denote the abelian group modulo the relation
(g1,g2)+(g2,g3) ∼ (g1,g3) (g1,g2),(g2,g3) ∈ Gn ×Hn Gn
(δ ),d
n ϕ
We get a
hain
omplex Coker with boundary
(cid:0) (cid:1)
d : (δ ) → (δ )
n ϕ n−1 ϕ
Coker Coker
extending linearly from
(g1,g2) 7→ ni=0(−1)i di(g1),di(g2) (g1,g2) ∈ Gn ×Hn Gn
P (cid:0) (cid:1)
H (δ )
n ϕ
The homology groups Coker are related to the relative homology groups by the map
(cid:0) (cid:1)
i◦ε : (δ ) → (ϕ)[−1]
n ϕ n
Coker Tot Cone
(cid:0) (cid:1)
extending linearly from
(i◦ε)(g1,g2) = (0,g1−g2) (g1,g2) ∈ Gn ×Hn Gn
Ea
h
ommutative diagram of the form (7) indu
es a group homomorphism
α∗ : Hn (δϕ) → Hn (δϕ′)
Coker Coker
(cid:0) (cid:1) (cid:0) (cid:1)
whi
h on
hains extends linearly from
α (g1,g2) = α (g1),α (g2)
∗ ∗ ∗
(cid:0) (cid:1)
We then get the
ommutative diagram
H (H) ∂ - H (G,H) (cid:27)i◦ε H (δ )
n+1 n n ϕ
Coker
(cid:0) (cid:1)
β∗ (β,α)∗ α∗
(8) ? ? ?
Hn+1(H′) - Hn(G′,H′) (cid:27) Hn (δϕ′)
∂ i◦ε Coker
(cid:0) (cid:1)
We
ite the following theorems from [19℄.
COMPARISON OF HIGHER INVARIANTS FOR SPECTRAL TRIPLES 7
ϕ : G → H
Theorem 2.1. Suppose that is surje
tive. Then the map
i◦ε : (δ ) → (ϕ)[−1]
∗ ϕ ∗
Coker Tot Cone
(cid:0) (cid:1)
indu
es an isomorphism on homology
i◦ε : H (δ ) → H (G,H)
n ϕ n
Coker
(cid:0) (cid:1)
K ϕ : G → H Γ G
ϕ ϕ
Let denote the kernel of and let denote the smallest normal subgroup of

ontaining all the
ommutators
[g,k] = gkg−1k−1 g ∈ G k ∈ K
and
Theorem 2.2. The homomorphism
ψ : Z[G× G] → K /Γ n z (x ,y ) 7→ n (x y−1)zi
H ϕ ϕ i=1 i i i i=1 i i
P Q
des
ends to an isomorphism
ψ : H (δ ) → K /Γ
1 ϕ ϕ ϕ
Coker
(cid:0) (cid:1)
2.2. The Universal Determinant Invariant. In this se
tion we will apply the results of
Theorem 2.1 and Theorem 2.2 to de(cid:28)ne a universal determinant invariant
: K L(H)/L1(H) → C∗
U 2
Det
(cid:0) (cid:1)
This will allows us to asso
iate a determinant invariant
: K (B) → C∗
X 2
Det
C
to ea
h exa
t sequen
e of -algebras
X : 0 −−−→ L1(H) −−−i→ E −−−s→ B −−−→ 0
ι : E → L(H)
equipped with an inje
tive homomorphism satisfying
(ι◦i)(T) = T T ∈ L1(H)
(9) for all
We
onsider the group homomorphism
GL(q) : GL L(H) → GL L(H)/L1(H)
(cid:0) (cid:1) (cid:0) (cid:1)
indu
ed by the quotient map
q : L(H) → L(H)/L1(H)
I GL(q)
GL(q)
Let denote the image of . Theorem 2.1 and Theorem 2.2 then furnish us with an
isomorphism
ψ ◦(i◦ε)−1 : H GL(L(H)),I → K /Γ
1 GL(q) GL(q) GL(q)
(cid:0) (cid:1)
k ∈ K GL(q)(k) = 1
GL(q)
Sin
e ea
h element satis(cid:28)es the Fredholm determinant yields a
homomorphism : K → C∗
GL(q)
det
It des
ends to the quotient : K /Γ → C∗
GL(q) GL(q)
det
g,l ∈ GL L(H) k ∈ K
GL(q)
sin
e for every and we have
(cid:0) (cid:1)
(lgkg−1k−1l−1) = (gkg−1k−1) = (gkg−1) (k−1) = 1
det det det det
8 J. KAAD
For details on the Fredholm determinant we refer to [29℄ or [31℄.
ψ ◦(i◦ ε)−1
In the following we might suppress parts of the isomorphism . We hope that this
will not
ause any
onfusion.
De(cid:28)nition 2.3. The universal determinant invariant
: K L(H)/L1(H) → C∗
U 2
Det
(cid:0) (cid:1)
is de(cid:28)ned as the
omposition
= ◦∂ ◦i ◦h
U ∗ 2
Det det
of the Hurewi
z homomorphism
h : K (L(H)/L1(H)) → H (I )
2 2 2 GL(q)
the boundary map on group homology
∂ : H (I ) → H GL(L(H)),I
2 GL(q) 1 GL(q)
(cid:0) (cid:1)
and the Fredholm determinant
: H GL(L(H)),I → C∗
1 GL(q)
det
(cid:0) (cid:1)
K L(H)
Remark 2.4. By [23℄ the (cid:28)rst algebrai
-group of vanishes. We therefore have the
GL L(H) = E L(H)
equality and by
onsequen
e
(cid:0) (cid:1) (cid:0) (cid:1) I = E L(H)/L1(H)
GL(q)
(cid:0) (cid:1)
This explains the un
onventional
hoi
e of range of the Hurewi
z homomorphism in the above
de(cid:28)nition.
C
Suppose that we have an exa
t sequen
e of -algebras
X : 0 −−−→ L1(H) −−−i→ E −−−s→ B −−−→ 1
ι : E → L(H)
equipped with an inje
tive homomorphism satisfying (9). We then have an
indu
ed homomorphism
ι : B → L(H)/L1(H)
K
whi
h by fun
toriality of algebrai
-theory yields a homomorphism
ι : K (B) → K L(H)/L1(H)
∗ 2 2
(cid:0) (cid:1)
De(cid:28)nition 2.5. The determinant invariant
: K (B) → C∗
X 2
Det
asso
iated with the exa
t sequen
e
X : 0 −−−→ L1(H) −−−i→ E −−π−→ B −−−→ 0
is de(cid:28)ned as the
omposition
= ◦ι
X U ∗
Det Det
of the homomorphism
ι : K (B) → K L(H)/L1(H)
∗ 2 2
(cid:0) (cid:1)
and the universal determinant invariant
: K (L(H)/L1(H)) → C∗
U 2
Det
COMPARISON OF HIGHER INVARIANTS FOR SPECTRAL TRIPLES 9
2.3. Comparing De(cid:28)nitions of The Determinant Invariant. In this subse
tion we relate
the determinant invariant of De(cid:28)nition 2.5 to the determinant invariant of L. G. Brown, J.W.
Helton, R. E. Howe studied and introdu
ed in the papers [4, 5, 15℄.
Consider the exa
t sequen
e of groups
i GL(q)
1 −−−→ K −−−→ GL L(H) −−−→ I −−−→ 1
GL(q) GL(q)
(cid:0) (cid:1)
Γ GL L(H)
GL(q)
We let denote the smallest subgroup of
ontaining the
ommutators
(cid:0) (cid:1)
gkg−1k−1 ∈ Γ g,k ∈ GL L(H) GL(q)(k) = 1
GL(q)
for all with
(cid:0) (cid:1)
Passing to the quotient we get a
entral extension
i GL(q)
1 −−−→ K /Γ −−−→ GL L(H) /Γ −−−→ I −−−→ 1
(10) GL(q) GL(q) GL(q) GL(q)
(cid:0) (cid:1)
and by
onsequen
e a map
f : H (I ) → K /Γ
GL(q) 2 GL(q) GL(q) GL(q)
See [27℄ for example. Composing with the Fredholm determinant
: K /Γ → C∗
GL(q) GL(q)
det
we get a map ◦f : H (I ) → C∗
GL(q) 2 GL(q)
det
This is the main ingredient of the determinant invariant in the de(cid:28)nition of [5℄.
Relating the two alternative de(cid:28)nitions is now a question of showing that the homomorphism
f : H (I ) → K /Γ
GL(q) 2 GL(q) GL(q) GL(q)

oin
ides with the
omposition
ψ ◦(i◦ε)−1 ◦∂ : H (I ) → K /Γ
2 GL(q) GL(q) GL(q)
See Se
tion 2.2.
We undertake a slightly more general task.
ϕ : G → H
Theorem 2.6. Let be a surje
tive homomorphism. The homomorphism
f : H (H) → K /Γ
ϕ 2 ϕ ϕ
indu
ed by the
entral extension
i ϕ
1 −−−→ K /Γ −−−→ G/Γ −−−→ H −−−→ 1
ϕ ϕ ϕ

oin
ides with the
omposition
ψ ◦(i◦ε)−1 ◦∂ : H (H) → K /Γ
2 ϕ ϕ
of the group homologi
al boundary map
∂ : H (H) → H (G,H)
2 1
and the isomorphism
ψ ◦(i◦ε)−1 : H (G,H) → K /Γ
1 ϕ ϕ
of Theorem 2.1 and Theorem 2.2.
10 J. KAAD
Proof. Let there be given a
y
le
2n
x = (−1)i(x ,y ) ∈ Z (H)
i i 2
(11)
Xi=1
x ∈ H (H)
2
representing a
lass . Choose a set theoreti
al se
tion
t : H → G ϕ◦t =
H
Id
ϕ : G → H
of the surje
tive group homomorphism . Re
all that the homomorphism
f : H (H) → K /Γ
ϕ 2 ϕ ϕ
is given by
n n
−1
f (x) = t(x )t(y ) t(x y ) −1 · t(x )t(y ) t(x y ) −1
ϕ 2k 2k 2k 2k 2k−1 2k−1 2k−1 2k−1
(12) (cid:16)kY=1 (cid:0) (cid:1) (cid:17) (cid:16)kY=1 (cid:0) (cid:1) (cid:17)
k ∈ {1,...,n}
For ea
h we set
z := x z := y z := x y
k 2k n+k 2k 2n+k 2k−1 2k−1
l ∈ {1,...,n}
Likewise for ea
h we set
w := x w := y w = x y
l 2l−1 n+l 2l−1 2n+l 2l 2l
Note that we then have
z = w w w = z z
2n+k k n+k 2n+l l n+l
and
Sin
e the boundary
2n n n
d(x) = (−1)i(x +y −x y ) = (z +z +z )− (w +w +w )
i i i i k n+k 2n+k l n+l 2n+l
Xi=1 Xk=1 Xl=1
Z[H]
vanishes in we
an (cid:28)nd a bije
tion
σ : {1,...,3n} → {1,...,3n}
su
h that
z = w j ∈ {1,...,3n}
j σ(j)
for all
The boundary map
∂ : H (H) → H (G,H)
2 1
x ∈ H (H)
2
when applied to is represented by the
y
le
0,−d t(x) ∈ Z (ϕ)[−1]
1
Tot Cone
(cid:16) (cid:17) (cid:16) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1)
t(x) ∈ C (G)
2
Where denotes the
hain
2n
t(x) = (−1)i t(x ),t(y )
i i
Xi=1 (cid:0) (cid:1)