The Noncommutative Geometry Generalization of Fundamental Group

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06 The Noncommutative Geometry Generalization
of Fundamental Group
Petr R. Ivankov and Nickolay P. Ivankov
April 24, 2006
Abstract
A notion of fundamental group of spectral triples [1] has been intro-
duced. The notion uses a noncommutative analogue of unramified cov-
erings. It was shown that in commutative case this fundamental group
is a profinite completion of fundamental group of corresponding Riemann
manifold.
1 Introduction
In algebraic topology the fundamental group may be defined using closed paths
or automorphisms of universal covering. However, noncommutative geometry
spectral triple has no paths and even points in general case. Similary in algebraic
geometry we can not always find a set of paths that provides a good definition
of fundamental group. But it is possible to define unramified coverings. These
coverings enable us to define analogue of fundamental group in algeraic geome-
try. The algebraic geometry fundamental group of complex algebraic manifold
is the profinite completion of algebraic topology fundamental group [2]. The
notion of ramified covering in noncommutative geometry was introduced in [3].
Using very similar approach we have introduced the notion of unramified finite
covering in noncommutative geometry. The spectral triple coverings enable us
to define the noncommutative version of fundamental group. If a spectral triple
is commutative then its fundamental group is the profinite completion of the
fundamental group of Riemann manifold that is associated to the triple.
2 Preliminaries
Recall definition of real spectral triple[1]. A real spectral triple consits of a set
of four (five) objects (A,H, D, J(,Γ)), of following types:
(1) A is a pre− C∗algebra;
(2) H is a Hilbert space carrying faithful representation π of algebra A;
(3) D is a selfadjoint operator on H with compact resolvent;
1

The Noncommutative Geometry Generalization of Fundamental Group 2
(4) J is a antilinear isometry of H onto itself;
(5) Γ is a selfadjoint unitary operator on H so that Γ2 = 1.
If Γ is present we say that triple is even, otherwise it is odd. The reader can
find complete description of spectral triples, and those axioms at [1]. We will use
following notation of spectral triples A = (A,H, D, J,Γ) or A = (A,H, D, J)
or A = (A,H, D).
In the following text D, J ,(Γ) will be called as “operators”.
Recall the notion of unitary equivalence of geometries [1]. Unitary equiva-
lence of spectral triple A = (A,H, D) is associated with such unitary operator
U on H that Uπ(A)U−1 = π(A).
Let G(A) or G(A,H, D) be the group of unitary equivalences of the triple
A = (A,H, D). Every unitary equivalence U defines * - automoprphism σ of
A that satisfies following condition Uπ(a)U−1 = π(σ(a)). Hence G(A) acts on
HA and A.
The orientability axiom [1] assume the existence of fundamental Hochschild
cycle cA ∈ Zn(A ⊗ A0). Every homomorhism f : A → B of algebras defines
natural transformation f∗ of Hochschild cycles. Thus we have f∗(cA) ∈ Zn(B⊗
B0).
3 Finite coverings. Fundamental group
In this section we shall define noncommutative analogue of covevings using
sructures of spectral triple only. We will show below that in commutative case
there is one to one correspondence between these coverings and finitely sheeted
coverings of corresponding Riemann manifold. These coverings enable to define
fundamental group.
3.1 Elements of finite covering
Let G ⊂ G(A) be a finite subgroup of G(A). Elements of G act on A and H.
Thefore group G defines projector P on A and H by following formula:
PG =
1
|G|
∑
g∈G
g (1)
The image of P is a subalgebra (subspace) AG (HG) of A (H).
Definition 3.1. Finite covering of spectral triples p : (A,HA, DA) → (B,HB, DB)
consists of such pair (p1, p2) of injective *-homomorphism p1 : B → A and ho-
momorphism p2 : HB → HA that:
(i) There exists a finite subgroup G(A,B) ⊂ G(A,H, D) that image of p1
(p2) is AG(A,B) (HG(A,B));
(ii) Homomorpisms p1 and p2 satisfy to the condition
p2(π(b)h) = π(p1(b))p2(h) for any b ∈ B and h ∈ HB;
(iii) Spectral triplesA = (A,HA, DA) , B = (B,HB, DB) and groupG(A,B)
satisfy following axioms 1 - 7.

The Noncommutative Geometry Generalization of Fundamental Group 3
3.2 Axioms of finite covering
Axiom 1. A is a finitely generated projective B− module.
Axiom 2. There is the natural isomorphism of A− modules HA ≈ A⊗B HB
Axiom 3. There exists such surjective homomorphism φAB : G(A) → G(B)
that the following diagram is commutative
HA HB
HA HB
w
PG(A,B)
u
g
u
φAB(g)
w
PG(A,B)
where P is the projection defined by equation (1).
Axiom 4. Every element g ∈ G(A) that is identity on p2(HB) belongs to
G(A,B).
Axiom 5. Every g ∈ G(A,B) commute with DA, JA (, ΓA) and DB, JB (,
ΓB) are restrictions of DA, JA (, ΓA) on HB .
Axiom 6. The dimension of the triple A = (A,HA, DA) equals to the dimen-
sion of the triple B = (B,HB, DB). Definition of a triple dimension is contained
in [1].
Axiom 7. If cA ∈ Zn(A⊗A0) is fundamental cycle on A and cB ∈ Zn(B⊗B0)
is fundamental cycle on B then cA = f∗(cB).
In the following text we consider B as subalgebra of A and HB as subspace
of HA.
3.3 Composition of finite coverings
Lemma 3.1. Let
A = (A,HA, DA) B = (B,HB, DB) C = (C,HC , DC)w
f
w
g
be a diagram with finite coverings of spectral triples, and f = (f1, f2), g =
(g1, g2). Then the pair (f1g1, f2g2) defines finite covering gf : A → C
Proof. Let us check (i) - (iii) (i) Let G(A,C) ⊂ G(A) be the subroup of those
automorphisms that are identities on p1(C) and p2(HC) According to axiom 4
we have a surjective homomorphism G(A,C) → G(B,C) and even an exact
sequence
{e} → G(A,B) → G(A,C) → G(B,C) → {e} (2)
Since G(A,B) and G(B,C) are finite then G(A,C) is finite. (ii) Follows
from simple direct calculation and omitted here. (iii)

The Noncommutative Geometry Generalization of Fundamental Group 4
Axiom 1. The A is the direct summand of Bm, and the B is direct summand
of Cn. Hence A is the direct summand of Cmn.
Axiom 2. This axiom follows from next equivalences:
HB ≈ B ⊗C HC
HA ≈ A⊗B HB ≈ A⊗B B ⊗C HC ≈ A⊗C HC
Axiom 3. Follows from the next commutative diagram:
g
HA HB HC
HA HB HC
u
w
PG(A,B)
u
φAB(g)
w
PG(B,C)
u
φAC(g)
w
PG(A,B)
w
PG(B,C)
.
Axiom 4. Follows from the exact sequence (2) of finite groups.
Axiom 5. Let g ∈ G(A,C), H1 is ortogonal supplement of HC in HB, and
H2 is ortogonal supplement of HB in HA. Then HA = HC ⊕ H1 ⊕ H2 and
HB = HC ⊕H1
The φAB(g) is represented by operator on HB that looks like:
(
IdHC 0
0 U1
)
, (3)
where U1 is an unitary operator on H1. if g is represented by operator U then
U


IdHC 0 0
0 U−11 0
0 0 IdH2

 =


IdHC 0 0
0 IdH1 0
0 0 U2

 , (4)
where U2 is an unitary operator on H2. We have
U =


IdHC 0 0
0 U1 0
0 0 U2

 . (5)
Operator DA|H2 commute with U2. Operator DA|H1 = DB|H1 commute
with U1. Hence DA commute with U . We can say the same about JA, ΓA.
Axiom 6. Dimension of A equals to dimension of B and dimension of B
equals to dimension of C. Hence dimension of A equals to dimension of C.
Axiom 7. Let cA, cB, cC are fundamental cycles on A, B, C. Then we have
cA = f∗1 (cB) and cB = g∗1(cC). Hence cA = (f1g1)∗(cC).
3.4 Fundamental group
Let A be a spectral triple. Consider the following category. Object of this
category is a finite covering fi : Ai → A. Every object of this category defines

The Noncommutative Geometry Generalization of Fundamental Group 5
a group G(Ai,A). Morphism from fi : Ai → A to fj : Aj → A is a such finite
covering fij : Ai → Aj that the diagram:
Ai Aj
A
[
[℄
fi
w
fij
�
��
fj
is commutative.
Every morphism of this category naturally defines a surjective group homor-
phism G(Ai,A) → G(Aj,A).
Hence for every spectral triple we have a commutative diagram of groups
and surjective homomorphisms.
Definition 3.2. Fundamental group of spectral triple A = (A,H, D) is an
inverse limit of described above diagram of groups.
We shall use following notation π1(A) or π1(A,H, D) for fundamental group
of A = (A,H, D).
4 Fundamental group of commutative spectral
triple
Let us recall some facts of noncommutative and differential geometry and topol-
ogy. In [1] and [4] it was shown that every commutative spectral triple B =
(B,HB, DB, JB(,ΓB)) defines compact Riemann manifold N , and spinor SN
bundle on it. In this caseB ≈ C∞(N) andHB ≈ L2(SN ). Moreover,DB, JB(,ΓB)
correspond to local “operators” on smooth sections of the bundle SN . Accorg-
ing to [1] the fundametal cycle on B corresponds to the volume form ΩN of
the Riemann manifold N . Inversely, if we have a Riemann manifold N with
spinor bundle SN and local “operators” on the bundle smooth sections which
satisfy to the set of conditions, then we can build corresponding spectral triple.
A completion of a pre−C∗ algebra A is the C∗ algebra A, and latter defines a
topological space. Every *- homomorphism of commutative C∗ algebras defines
continous map of topological spaces [7]. In this case every element of pre− C∗
algebra corresponds to a smooth complex function. Hence if a *- homomor-
phism is an extention of a *-homomorphism of pre−C∗ algebras then the map
is smooth. If f : M → N is a connected finitely sheeted covering of compact
Riemann manifold N then M has the natural structure of Riemann manifold
that is compact. If a local diffeomorphism f : M → N is not surjective then
C∞(M) is not a finitely generated C∞(N) module (we consider the natural
C∞(N) module structure).
These facts will be used below to show that the fundametal group of com-
mutative spectral triple B = (C∞(N), L2(SN ), DB, JB(,ΓB)) is the profinite
completion of π1(N).