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arXiv:math/0108107v2 [math.KT] 13 Mar 2005
Index Defects in the Theory of Non-local
Boundary Value Problems and the
η-Invariant
A. Yu. Savin and B. Yu. Sternin
February 1, 2008
Abstract
The paper deals with elliptic theory of boundary value problems on manifolds
whose boundary is represented as a covering space. We compute the index for a class
of non-local boundary value problems on such manifolds. For a non-trivial covering,
the index defect of the Atiyah–Patodi–Singer boundary value problem is computed.
Poincar´ e duality in K-theory of the corresponding manifolds with singularities is
obtained.
Introduction
This paper deals with boundary value problems for elliptic operators on a manifold whose
boundary is the total space of a finite-sheeted covering. On such manifolds, we consider
boundary value problems for operators that do not satisfy the Atiyah–Bott condition (i.e.,
have no well-posed classical boundary value problems). Recall that this condition does
not hold, in particular, for the Hirzebruch and Dirac operators as well as some other
related geometric operators.
We consider the following two classes of boundary value problems.
1. Non-local boundary value problems. Let M be a smooth manifold such that
the boundary ∂M is a finite-sheeted covering with projection π : ∂M −→ X. Then there
is an isomorphism
C∞(∂M)≃ C∞(X,π!1)
between the space C∞(∂M) of smooth functions on ∂M and the space of sections of the
vector bundle π!1 ∈ Vect(X) on the base of the covering. Here π!1 is the direct image of
the trivial line bundle.
For a scalar elliptic operator D on M, the simplest non-local boundary value problem
of the type considered in this paper is
?Du = f,
β
Bβ u|∂M= g.
(0.1)
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Here u and f are functions on M, g is a function on X, and the operator B of boundary
conditions acts also on X. In terms of the original manifold M, the boundary conditions
in (0.1) are non-local, since they relate the values of functions at distinct points of M.
We prove a finiteness theorem and in the case of regular coverings obtain an index
formula for this class of non-local boundary value problems. Without going into detail at
the moment, let us mention two essential features of the theory.
First, in the proof of the index theorem we embed our manifolds in the classifying space
of a finite group, while in the classical index theorem it is suffices to use embeddings in
RN.
Second, the analogue of the Atiyah–Singer difference element for a non-local boundary
value problem is an element of the K-group of a non-commutative C∗-algebra associated
with the cotangent bundle and the covering. Recall that in the classical index theorem it
suffices to use topological K-theory.
The index formula of this paper is given in a form resembling the K-theoretic statement
of the Atiyah–Singer theorem. Local index formulae will appear elsewhere.
2. Spectral problems on manifolds with a covering. The first generalization
of classical boundary value problems that is free of the Atiyah–Bott obstruction is due
to Atiyah, Patodi, and Singer [1]. For a class of first-order elliptic operators, one has
so-called spectral boundary value problems denoted by (D,Π+). Spectral boundary value
problems enjoy the Fredholm property. However, their index is not determined by the
principal symbol of D.
Interesting invariants arise if the boundary has the structure of a covering. Here we
consider a class of elliptic operators that are lifted from the base of the covering in a
neighbourhood of the boundary. In this case, the principal symbol of an elliptic operator
D defines an element
[σ (D)] ∈ K0?T∗M
in the K-group of the singular space T∗M
we identify all points in each fiber of the covering (for details, see Section 5). The element
[σ(D)] has a topological index
π?
πobtained from the cotangent bundle T∗M if
indt[σ(D)] ∈ Q/nZ,
where n is the number of sheets.
coincide only for trivial coverings. For a general covering, we obtain the index defect
formula
modn-ind(D,Π+) − indt[σ(D)] = η(D|X⊗ 1n−π!1) ∈ Q/nZ.
However, the analytical and the topological index
(0.2)
The index defect (the difference between the analytical index modulo n and the topological
index) is equal to the relative Atiyah–Patodi–Singer η-invariant of the restriction of D to
the boundary with coefficients in the flat bundle π!1. For a trivial covering, the relative
η-invariant is zero, and the index defect formula becomes the index formula
modn-ind(D,Π+) = indt[σ (D)]
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due to Melrose and Freed [2] (see also [3, 4, 5, 6]). However, our proof is new even in this
case. It is interesting to note that the main step in the proof is to realize the fractional
analytic invariant
modn-ind(D,Π+) − η (D|X⊗ 1n−π!1) ∈ Q/nZ
(0.3)
as the index of some non-local boundary problem of the form (0.1) (in a suitable elliptic
theory with coefficients).
There is also a deeper relation between the two elliptic theories described in Subsec. 1
and 2.
3. Poincar´ e isomorphism and duality. We establish Poincar´ e isomorphisms on
the singular spaces T∗M
isomorphisms are just the well-known isomorphisms (e.g., see [7, 8, 9])
πand M
π. For the identity covering π = Id,X = ∂M, these
K0(T∗M) ≃ K0(M,∂M),K0(T∗(M \ ∂M)) ≃ K0(M). (0.4)
(For non-compact spaces, we use K-theory with compact supports.) In contrast to the
smooth case, the Poincar´ e isomorphisms for singular spaces relate the K-groups of a
commutative algebra of functions to those of a dual non-commutative algebra. They are
defined on the elements as quantizations, i.e., take symbols to operators. More precisely,
the analogue of the first isomorphism in (0.4) is defined in terms of the operators described
in Subsec. 2, while in the second case one uses non-local problems introduced in Subsec. 1.
Let us outline the contents of the paper. The first section contains the definition of the
class of non-local boundary value problems on manifolds with a covering on the boundary
and a proof of the Fredholm property. The index formula is obtained in Sec. 2. By way
of example, we define a non-local boundary value problem for the Hirzebruch operator
on a manifold with reflecting boundary. In Sec. 3, we give the homotopy classification
of non-local problems. The index defect formula (0.2) is proved in Sec. 5. This is one
of the central results of the paper. Section 6 contains applications to the computation
of the fractional part of the η-invariant. It is also shown that the invariant (0.3) can be
computed by the Lefschetz formula. Poincar´ e isomorphisms in K-theory of the singular
spaces corresponding to manifolds whose boundary bears the structure of a covering are
constructed in the last two sections.
There are other interesting classes of non-local boundary value problems arising if the
projection has singularities (e.g., the projection on the quotient by a non-free action of
a finite group). Index theory of such boundary value problems is apparently related to
index theory on orbifolds ([10, 11]). Our approach is advantageous in that if the base of
the covering is smooth, then there are no additional analytic and topological difficulties
related to the singularities of the covering. More general classes of non-local boundary
value problems (e.g., see [12]) are beyond the scope of this paper.
Acknowledgements. The results were announced at the conferences “Spring School
2001” in Potsdam, Germany, “Topology, analysis, and related topics” in Moscow, 2001,
and at the International Congress of Mathematicians in Beijing, 2002. We are grateful
to V.E. Nazaikinskii and V. Nistor for helpful discussions. The work was supported in
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part by RFBR grants Nos. 03-02-16336, 02-01-00118, and 02-01-00928. We are grateful
to A.S. Mishchenko, for finding an error in the original version of the paper and to the
referee for helpful remarks.
1 Non-local boundary value problems
1. Coverings and non-local operators. Let Y be a finite covering over a manifold X
with projection π : Y −→ X. The projection defines the direct image mapping
π!: Vect(Y ) −→ Vect(X)
that takes each vector bundle E ∈ Vect(Y ) to the bundle
π!E ∈ Vect(X),(π!E)x= C∞?π−1(x),E?,x ∈ X.
This clearly gives an isomorphism βE: C∞(Y,E)
Y and X, while permits one to identify operators defined on the total space and on the
base. More precisely, the direct image
≃
−→ C∞(X,π!E) of section spaces on
π!D = βEDβ−1
E: C∞(X,π!E) −→ C∞(X,π!E)
of a differential operator
D : C∞(Y,E) −→ C∞(Y,E)
on Y is a differential operator. However, the following example shows that the inverse
image
π!D′= β−1
ED′βE: C∞(Y,E) −→ C∞(Y,E) (1.1)
of a differential operator D′on X may well be a non-local operator. (It is not even
pseudolocal.)
Example 1.1. For the trivial covering
Y =X ⊔ X ⊔ ... ⊔ X
?
???
n copies
−→ X
and the trivial bundle E = C, we have π!E = Cn. The direct image
π!D = diag?D|X1,..., D|Xn
of a differential operator on Y is always a diagonal operator, and hence the inverse image
of a non-diagonal operator can not be a differential operator. The off-diagonal entries pro-
duce non-local operators on Y , since they interchange the values of functions on different
leaves of the covering.
?: C∞(X,Cn) −→ C∞(X,Cn)
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2. Non-local boundary value problems. Let M be a smooth compact manifold
with boundary ∂M. Suppose that the boundary is a covering space over a smooth closed
manifold X with projection
π : ∂M −→ X.
We fix a collar neighbourhood ∂M × [0,1) of the boundary. The normal coordinate will
be denoted by t.
For a smooth function u ∈ C∞(M), let
?
∂M
jm−1
∂Mu =u|∂M, −i∂
∂tu
????
,...,
?
−i∂
∂t
?m−1
u
?????
∂M
?
be the restriction of its (m−1)st jet in the normal direction to the boundary. The operator
jm−1
∂M
is continuous in the Sobolev spaces
jm−1
∂M: Hs(M) −→
m−1
?
k=0
Hs−1/2−k(∂M), s > m − 1/2.
Throughout the paper we assume that for vector bundles E on manifolds with boundary
there are given isomorphisms p∗(E|∂M) ≃ E|∂M×[0,1]in the collar neighbourhood of the
boundary, where p : ∂M ×[0,1] → ∂M is the natural projection. In this case, the normal
jet of a section of E is also well defined.
Definition 1.1. A non-local boundary value problem for a differential operator
D : C∞(M,E) −→ C∞(M,F)
of order m is a system of equations
?
BβEjm−1
Du = f,u ∈ Hs(M,E),f ∈ Hs−m(M,F),
∂Mu = g,g ∈ Hδ(X,G),
(1.2)
where the boundary condition is defined by a pseudodifferential operator
B :
m−1
?
k=0
Hs−1/2−k(X,π!E|∂M) −→ Hδ(X,G)
on X. We assume that the component
Bk: Hs−1/2−k(X,π!E|∂M) −→ Hδ(X,G)
of B has the order s − 1/2 − k − δ.
Remark 1.1. One can also consider problems similar to (1.2) in which the components
of the vector function g belong to Sobolev spaces of different orders. The case in which
all components have the same order δ is more convenient and can always be achieved by
order reduction.
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Note that for the identity covering π = Id, X = ∂M, problem (1.2) is just a classical
boundary value problem (e.g., see [13]).
3. Relation to classical boundary value problems. Finiteness theorem. Note
that ∂M × [0,1) is also a covering with projection
π × 1 : ∂M × [0,1) −→ X × [0,1).
The induced isomorphism of function spaces will be denoted by
β′
E: C∞(∂M × [0,1),E) −→ C∞(X × [0,1),π!E).
The non-local problem (D,B) can be represented in a neighbourhood of the boundary as
the inverse image of the classical boundary value problem
?
More specifically, this is the boundary value problem
?
Bjm−1
X
β′
0
F
0
1
?
◦
?
D
BβEjm−1
∂M
?
◦ (β′
E)−1=
?
β′
FD(β′
Bjm−1
E)−1
X
?
.
β′
FD (β′
E)−1
?
: C∞(X × [0,1),π!E) −→
C∞(X × [0,1),π!F)
⊕
C∞(X,G),
(1.3)
for the differential operator (π × 1)!D = β′
coordinates, this operator is represented by a diagonal matrix with elements acting on
different leaves of the covering. Problem (1.3) will be denoted by ((π × 1)!D,B) for short.
We point out that the classical boundary value problem(1.3) is defined only in a neigh-
bourhood of the boundary, since the covering is defined only near the boundary.
FD(β′
E)−1on the cylinder X × [0,1). In local
Definition 1.2. Problem (D,B) is said to be elliptic if D is elliptic and ((π × 1)!D,B)
is elliptic, i.e., satisfies the Shapiro–Lopatinskii condition (e.g., see [13]).
The proof of the following finiteness theorem is standard.
Theorem 1.1. An elliptic boundary value problem D = (D,B) defines a Fredholm oper-
ator.
Proof. Let D−1be the parametrix of D in the interior of the manifold. Similarly,
the parametrix of the classical boundary value problem on X × [0,1) will be denoted by
(L,K). They can be pasted together globally on M by the formula
?
Here
ϕ1+ ϕ2= 1,
ψ1= 0 near the boundary ,ψ2= 0 far from the boundary.
Furthermore, ψ2 is assumed to be constant in the fiber of π × 1. Obviously, D−1is a
two-sided parametrix of D. The proof is complete.
D−1=ψ1D−1ϕ1+ ψ2(π × 1)!Lϕ2,ψ2K
?
.
ψjϕj= ϕj
(1.4)
?
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2 The index of non-local problems
In the previous section, non-local boundary value problems were represented near the
boundary in terms of equivalent classical boundary value problems. Therefore, we can
apply well-known topological methods (e.g., see [14]) to compute the index of non-local
boundary value problems.
1. Reduction to zero-order operators. We introduce a class of operators that are
non-local in a neighbourhood of the boundary. A linear operator
D : C∞(M,E) −→ C∞(M,F)
will be called an admissible operator of order m if it can be represented modulo operators
with smooth kernels as
D = ψ1D′ϕ1+ ψ2(π × 1)!D′′ϕ2
(2.1)
for cutoff functions ϕ1,2,ψ1,2as in the proof of Theorem 1.1, a pseudodifferential operator
D′: C∞(M,E) → C∞(M,F), and an operator
D′′: C∞(X × [0,1),π!E) −→ C∞(X × [0,1),π!F)
that is a sum of a pseudodifferential operator with compactly supported kernel on X ×
(0,1) and a differential operator
m
?
k=0
Dk(t)
?
−i∂
∂t
?m−k
(2.2)
with respect to the normal variable t.
Here the Dk(t) are smooth families of pseudodifferential operators on X of order k
and D0(t) is induced by a vector bundle isomorphism.
To this class of operators, one can extend the notion of ellipticity, the statement of non-
local boundary value problems, and the finiteness theorem (cf. a similar generalization in
[13] for the classical case). In particular, the symbol of an admissible operator is a pair
(σM,σX), where σM : p∗E → p∗F is defined over M \ (∂M × [0,ε)) (p : S∗M → M is
the natural projection) and σX: p∗
defined over X × [0,1]. Moreover, the symbols are smooth and satisfy the compatibility
condition
(π0)!σM|∂M×(ε,1)= σX,
where the direct image is induced by the natural projection π0: T∗M|∂M×(ε,1)→ T∗(X ×
(ε,1)).
0(π!E) → p∗
0(π!F) (p0: S∗(X × [0,1]) → X × [0,1]) is
Example 2.1. Let E ∈ Vect(M) be a vector bundle. Suppose that its direct image over
U∂Mis decomposed as a sum of two subbundles
π!E|U∂M= E+⊕ E−,E±∈ Vect(X × [0,ε)).
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Consider the operator D±: C∞(M,E) → C∞(M,E) given by
??
(Here Λ stands for first-order pseudodifferential operators with principal symbol |ξ| on
the corresponding manifolds, and the cutoff functions are chosen as before.) This formula
defines an admissible elliptic operator. We equip it with the Dirichlet boundary condition
D±= ψ2(π × 1)!
−i∂
∂t+ iΛX,E+
?
⊕
?
i∂
∂t+ iΛX,E−
??
ϕ2+ ψ1iΛMϕ1. (2.3)
PE−βEu|∂M= g ∈ C∞(X,E−),
where PE−: π!E|∂M→ π!E|∂Mis a projection onto the subbundle E−. Denote this
boundary value problemby D±. By analogy with the classical case (e.g., see [13]), one
proves that the index of this boundary value problem is zero.
For example, let E+= π!E|∂Mand E−= 0. Then the operator (2.3), which will be
denoted by D+, is Fredholm without any boundary condition.
Remark 2.1. Just as in the classical elliptic theory on a closed manifold (see [15]), there
are two equivalent definitions of homotopy of non-local elliptic problems. First, one can
say that two problems are homotopic if they can be connected by a family of non-local
elliptic problems continuous in the operator norm (in some given pair of Sobolev spaces).
Second, two problems are said to be homotopic if there exists a continuous homotopy
of their principal symbols (preserving ellipticity). The equivalence of the two definitions
is based on the smoothing of continuous homotopies and the standard norm estimates
modulo compact operators, e.g., see [16].
Let1Ellm(M,π), m ≥ 1, be the Grothendieck group of the semigroup of homotopy
classes of elliptic boundary value problems for admissible operators of order m modulo
boundary value problems of the form D±◦ Dm−1
The group of stable homotopy classes of zero-order admissible elliptic operators is
denoted by Ell0(M,π). Recall that stabilization is taken modulo trivial operators. In
this case, by trivial operators we mean operators induced by vector bundle isomorphisms.
It should be noted that elliptic operators of order zero do not require boundary conditions,
since near the boundary they are induced by vector bundle isomorphisms.
Just as in the classical theory (see [14] or [13]), the order of a non-local boundary
value problem can be reduced to zero by stable homotopies. More precisely, the following
theorem holds.
+
.
Theorem 2.1 (order reduction). The composition with the operator D+(with coefficients
in vector bundles) induces an isomorphism
×Dm
+: Ell0(M,π) −→ Ellm(M,π),
[D]?→
?D ◦ Dm
+
?.
1Later on, by Ellm(M,π) we denote also the corresponding Grothendieck groups for closed manifolds
and for manifolds with boundary with projection π defined, possibly, on an open subset. Which group is
meant is always clear from the context.
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The proof of this result is a straightforward generalization of the corresponding proof
in the classical case (see [14]) and hence is omitted.
Remark 2.2. Let us explicitly describe order reduction, i.e., the mapping?×Dm
?Du = f,
for a first-order admissible operator D : C∞(M,E) → C∞(M,F) that admits a decom-
position
?D|U∂M
in a neighbourhood of the boundary, where A(t) is a smooth operator family on X and
Γ : π!E|∂M→ π!F|∂Mis a vector bundle isomorphism. The boundary condition is defined
by the projection P in the bundle π!(E|∂M). We assume for simplicity that the symbol
a(x,ξ) of A(0) is symmetric and additionally satisfies a∗a = |ξ|2. We assume that P is
also symmetric.
Let L+(A(0)) ∈ Vect(S∗X) be the Calder´ on bundle. For our first-order operator, this
is the bundle over S∗X generated by eigenvectors of a(x,ξ) with positive eigenvalues.
The ellipticity condition for (D,P) requires that P define an isomorphism
?
+
?−1, in
the important special case of boundary value problems
PβE(u|∂M) = g,g ∈ C∞(X,ImP),
(π × 1)!
?= Γ
?∂
∂t+ A(t)
?
L+(A(0))
P
−→ p∗
0ImP,p0: S∗X → X,
of subbundles. Consider the principal symbol of our operator on the boundary:
?∂
(here τ is dual to t). The linear homotopy
σ
∂t+ A(0)
?
= iτ + a(x,ξ)
(1 − ε)(iτ + a(x,ξ)) + ε(2P(x) − 1),ε ∈ [0,1],
is a homotopy of elliptic symbols for τ2+ξ2= 1 provided that the ellipticity condition for
(D,P) is satisfied. Furthermore, at the end of the homotopy (for ε = 1) the symbol does
not depend on the cotangent variables. Let us treat the homotopy of elliptic symbols on
X as an elliptic symbol on X × [0,1]. Then the symbol of D and the homotopy taken
together define the symbol of an admissible elliptic operator of order zero on the manifold
M with [0,1] × ∂M attached.
This zero-order symbol can be transferred to M by an obvious diffeomorphism M ≃
M ∪∂M([0,1] × ∂M) that is equal to identity far from the boundary. One can show (cf.
[14]) that the element defined by this symbol (operator) is precisely the image of the
problem (D,P) under the order reduction mapping (×D+)−1of Theorem 2.1.
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2. Admissible operators on closed manifolds. Let U be a codimension zero
submanifold of some closed manifold M. We assume that U is a covering space
U
π
−→ Y
with smooth base Y . Let U and Y be the corresponding sets of interior points (we allow U
to have a boundary). Then scalar admissible operators on M are by definition operators
of the form
D = D′+ ψ?π!D′′?ϕ,
where D′is a pseudodifferential operator on M, D′′is a pseudodifferential operator on Y
acting on sections of π!1 ∈ Vect(Y ), and the cutoff functions ϕ and ψ are supported in
U.
In the non-scalar case, we consider operators acting in the spaces slightly more general
than section spaces of vector bundles.
Namely, consider triples (E,E0,α) defined by vector bundles
E ∈ Vect(V ), E0∈ Vect(Y )
(here we fix a neighbourhood V ⊂ M of M\U such that if a point lies in U ∩ V then the
entire fiber containing this point also lies in U ∩ V ) and a vector bundle isomorphism
π!E|U∩V
α
≃ E0|π(U∩V )
on π (U ∩ V ).
Let Vect(M,π) be the set of isomorphism classes of such triples. Here two triples
(E,E0,α),(F,F0,γ) are isomorphic if the vector bundles are pairwise isomorphic, E
F,E0
The linear space of sections corresponding to the triple E = (E,E0,α) is defined as
?
For the identity covering, E defines a vector bundle on M obtained by clutching E with
E0by the transition function α, and C∞(M,E) is just the space of sections of E.
The space C∞(M,E) is generated by the subspaces
a≃
b≃ F0, and the isomorphisms are compatible: γ(π!a) = bα.
C∞(M,E) =(u,v)
????
u ∈ C∞(V,E),v ∈ C∞(Y,E0),
αβE(u|U∩V) = v|π(U∩V )
?
⊂ C∞(V,E) ⊕ C∞(Y,E0).
C∞
0(V,E),C∞
0(Y,E0) ⊂ C∞(M,E)
of compactly supported sections. More precisely, the first embedding takes u to the pair
(u,
?
βEu|U∩V), where the tilde stands for the extension of a function by zero at the points
where the function was not originally defined. Similarly, the second embedding takes v to
?
β−1
be defined by analogy with the scalar case. Namely, an admissible operator of order m is
an operator
D : C∞(M,E) −→ C∞(M,F)
the pair (
Ev|π(U∩V ),v). Now non-local operators acting in spaces C∞(M,E) can readily
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that is equal, modulo operators with smooth kernel, to
D = D1ϕ1+ D2ϕ2, (2.4)
where
D1: C∞
0(V,E) → C∞
0(V,F),D2: C∞
0(Y,E0) → C∞
0(Y,F0)
are mth-order pseudodifferential operators with compactly supported kernels. Here we
assume that the cutoff function ϕ1is zero in some neighbourhood of M\V and ϕ2is zero
in a neighbourhood of M\U.
The symbol of an admissible operator is a pair (σM,σY) of usual elliptic symbols
σM: p∗
ME|M\U−→ p∗
MF|M\U,σY : p∗
YE0|Y−→ p∗
YF0|Y,
where pM: S∗M −→ M and pY : S∗Y −→ Y, are compatible in the sense that
γ((π0)!σM|∂U)α−1= σY|∂Y.
Let Ellk(M,π) be the group of stable homotopy classes of admissible elliptic operators
of order k on M, modulo elliptic operators with principal symbols independent of the
cotangent variables.
Remark 2.3. On manifolds with boundary, one can also consider a similar class of el-
liptic operators and boundary value problems. More precisely, let M be a manifold with
boundary, with a projection π defined on a closed subset U ⊂ M as above. We assume
that U is a codimension zero submanifold in the interior M \ ∂M and is the Cartesian
product [0,ε) × U0in some collar neighbourhood of the boundary for some codimension
zero submanifold U0in ∂M. Then on M we consider operators similar to (2.4), where
both D1and D2are of order m and are differential operators with respect to the normal
variables in neighbourhoods of the boundaries of the corresponding manifolds (see (2.2)).
Such operators are considered in the spaces C∞(M,E). One considers boundary value
problems of the form
(D,Bj) : C∞(M,E) −→ C∞(M,F) ⊕ C∞(∂M,G),
where E,F ∈ Vect(M,π),G ∈ Vect(∂M,π|∂M), j is the jet operator of order m, j :
C∞(M,E) → C∞(∂M,Em|∂M), and the boundary conditions are defined by an admissible
operator B on the boundary. One can readily extend all results of this section, including
the definition of trivial problems D±, the group of stable homotopy classes of boundary
value problems, and order reduction, to this class of boundary value problems.
Remark 2.4. Let M be a manifold with covering π on the boundary. In Subsec. 1,
we defined the group Ellm(M,π) generated by elliptic non-local problems for the usual
operators. At the same time, the projection π ×1 : ∂M ×[0,1) → X ×[0,1) is defined in
a collar neighbourhood of the boundary, and one can consider the corresponding group
Ellm(M,π × 1) generated by non-local problems for admissible operators in the sense of
Remark 2.3. It turns out that these two groups are isomorphic under the natural mapping
Ellm(M,π) −→ Ellm(M,π × 1).
This essentially follows from the isomorphism Vect(M) ≃ Vect(M,π × 1).
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3. Reduction to a closed manifold. We return to the problem of computing the
index of non-local operators on a manifold M with a covering π defined on ∂M. Consider
an embedding f : M → M′in a closed manifold of the same dimension as M (for example,
M′can be the double 2M = M ∪∂MM). Just as in the classical case [15], f induces the
direct image mapping
f!: Ell0(M,π) −→ Ell0(M′,π × 1),
where π × 1 is the extension of π to ∂M × [−1,1] ⊂ M′. This mapping takes the symbol
σ(D) = (σM,σX) of an elliptic operator2
D : C∞(M,E) −→ C∞?M,Ck?
to the symbol on M′that coincides on M with the original symbol and is the identity
id : Ck→ Ckon the complement M′\M. The extended symbol is defined on the bundle
obtained by clutching E with Ckusing the isomorphism σX|Xand maps this bundle to
the bundle Ckover the ambient closed manifold M′.
Lemma 2.1. The mapping f! : Ell0(M,π) −→ Ell0(M′,π × 1) is well defined and is
index preserving.
Proof. This is a restatement of the well-known excision property of the index. The proof
is standard, and hence we omit it altogether.
?
4. Embedding in a universal space. In the index theorems of the present paper,
we assume that the following condition is satisfied.
Assumption 2.1. The covering π is regular and there is a free action of a finite group
G on the submanifold U such that π is the projection onto the quotient.
Let (M,π) and (M′,π′) be two pairs (both manifolds are assumed to be closed) and
let U and U′be the domains of π and π′, respectively.
Definition 2.1. We say that f is an embedding
embedding f : M → M′, f?U?⊂ U
Denote by πN: EGN−→ BGNthe N-universal bundle for G. We assume that EGN
and BGN are closed manifolds. There is an explicit construction for such a model (e.g.,
see [17]). For example, consider the embedding
of (M,π) in (M′,π′) if there is an
′, that is equivariant on the domain of π.
G ⊂ S|G|⊂ U(|G|)
in the unitary group. (Here |G| is the order of G.) Consider the bundle Vk,|G|→ Vk,|G|/G,
where Vk,nis the Stiefel manifold of n-frames in Ck. For sufficiently large k, this bundle
is N-universal.
2An arbitrary operator D′: C∞(M,E) → C∞(M,F) is reduced to this form by adding the identity
operator in the sections of the complementary bundle to F.
12
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Proposition 2.1. For (M,π) satisfying Assumption 2.1, there exists an embedding in
(EGN,πN) provided that N is sufficiently large.
Proof. By N-universality of πN, there exists an equivariant mapping U → EGN. We
can assume that this mapping is a smooth embedding. This can be achieved by a small
deformation provided that the dimension of EGNis sufficiently large.
This embedding can be extended to a smooth mapping M → EGN owing to the N-
connectedness of EGN. Finally, a small deformation outside a neighbourhood of U makes
it a global embedding.
?
5. The Euler operator on the disc. Consider the Neumann problem
?
for the Euler operator in the unit disc Dn⊂ Rnwith the Euclidean metric. Here g ∈
Λe+n(Sn−1). This boundary value problem is elliptic, and Hodge theory shows that the
cokernel is trivial and the one-dimensional kernel consists of constant functions.
The same is true for the homogeneous boundary value problem, which we rewrite in
the operator form
DdR= d + δ : Λe
(d + δ)u = f,
(∗u)|Sn−1 = g,
u ∈ Λe(Dn),f ∈ Λo(Dn),
0(Dn) −→ Λo(Dn).
(Here Λe
operator is O(n)-equivariant with respect to the natural action of the orthogonal group
on Dn.
6. Embeddings and the index of elliptic operators. Let f : (M,π) → (M′,π′)
be an embedding of positive codimension. We choose a Riemannian metric on M′that is
G-invariant over U
neighbourhood W of M in M′is diffeomorphic to the unit ball subbundle DM ⊂ NM.
Additionally, we can assume this diffeomorphism to be G-equivariant over U ⊂ U
Consider an admissible elliptic operator
0(Dn) is the space of forms satisfying the homogeneous boundary condition.) This
′⊂ M′. Denote the normal bundle to M by NM. Then a closed tubular
′.
D : C∞(M,E) −→ C∞(M,F).
We define a boundary value problem on DM as the exterior tensor product of D by
a family of boundary value problems for the Euler operator in the fibers. The definition
of this product is the same as in [15].
More precisely, the exterior tensor product gives the operator
?
D =
?D ⊗ 1Λe
−1F⊗?D∗
dR
1E⊗?DdR
?D∗⊗ 1Λo
?
,
where the pullback of D to the bundle DM with coefficients in even forms on the fibers
is denoted by
?D ⊗ 1Λe : C∞(DM,p∗E ⊗ Λe
0(DM)) −→ C∞(DM,p∗F ⊗ Λe
0(DM)).
13
Page 14
Here?D∗⊗1Λo is the pullback of the adjoint operator with coefficients in odd forms. The
E = (E,E0,α) is denoted by
family of Neumann problems for the Euler operator?DM
1E⊗?DdR: C∞
The off-diagonal entries of D commute with entries on the diagonal by construction. As
in ordinary Atiyah–Singer theory, this leads to the following result.
dRwith coefficients in the triple
α⊗1(DM,p∗E ⊗ Λe
0(DM)) −→ C∞
α⊗1(DM,p∗E ⊗ Λo(DM)).
Lemma 2.2. One has indD = indD.
The proof is similar to [15].
?
Thus an elliptic operator on the submanifold M ⊂ M′induces an elliptic boundary
value problem with the same index on the tubular neighbourhood DM ≃ W ⊂ M′.
Further, we can apply the order reduction procedure to this problem (see Remark 2.2)
and extend the resulting zero-order operator from W to the entire manifold M′as in
Subsec. 2.
Summarizing, we see that the embedding f of (M,π) in (M′,π′) induces the direct
image mapping
f!: Ell1(M,π) −→ Ell0(M′,π′),
which preserves the index.
Remark 2.5. A straightforward computation shows that the linear homotopy of order
reduction for boundary value problems (defined in Remark 2.2) which extends the symbol
σ(d + δ) from T∗Dnto T∗Rnas an invertible element outside a compact set defines an
element of the equivariant K-group equal to the element
j!(1) ∈ KO(n)(T∗Rn),j : pt −→ Rn,
which is used in the standard proof of the Atiyah–Singer theorem.
7. The Index theorem. Let f be an embedding of (M,π) in the universal space
defined in Proposition 2.1. For the universal space EGN, the projection πN is defined
globally. Therefore, the direct image of a non-local operator can be treated as a usual
elliptic operator on the base BGN; i.e., we have a natural mapping
(πN)!: Ell(EGN,πN) −→ Ell(BGN) ≃ K (T∗BGN).
Theorem 2.2. For a pair (M,π) satisfying Assumption 2.1, the diagram
Ell1(M,π)
ind ↓
f!
−→ Ell0(EGN,πN)
↓ (πN)!
indt
←−K (T∗BGN),
Z
commutes. Here indtis the usual topological index on a closed manifold.
14
Page 15
Proof. Indeed, we have
indD = indf![D] = ind(πN)!f![D] = indt((πN)!f![D]).
The first equality here follows from the invariance of the index for embeddings, the second
from the fact that (πN)!does not change the operator, and the last equality is just the
Atiyah–Singer formula on BGN.
?
8. Example. Manifolds with reflecting boundary [18]. Let M be a 4k-dimensional
compact oriented Riemannian manifold with boundary ∂M. Suppose that ∂M is equipped
with an orientation-reversing smooth involution G without fixed points. The involution
defines a free action of the group Z2and the corresponding double covering π : ∂M −→
∂M/Z2. Consider the Hirzebruch operator [19]
d + d∗: Λ+(M) −→ Λ−(M).
In a neighbourhood of the boundary, let us take a metric lifted from [0,1]×∂M/Z2. Then
the Hirzebruch operator can be decomposed near the boundary as (see [1])
∂
∂t+ A
(up to a bundle isomorphism), where A is an elliptic self-adjoint operator on the boundary
and is given by the formula
A : Λ∗(∂M) −→ Λ∗(∂M), Aω = (−1)k+p(d ∗ −ε ∗ d)ω;
here for an even degree form ω ∈ Λ2p(∂M) we set ε = 1, and ε = −1 otherwise. Since G
reverses the orientation, it follows that A and G∗anticommute:
G∗A = −AG∗.
It is known that the Hirzebruch operator has no well-posed classical boundary conditions.
However, it admits the non-local boundary value problem
?(d + d∗)ω = f,
2
ω|∂M= g,g ∈ Λ∗(∂M)Z2≃ Λ∗(∂M/Z2)
(1+G∗)
(2.5)
on the manifold with reflecting boundary. Here Λ∗(∂M)Z2is the subspace of G-invariant
forms on the boundary.
Proposition 2.2. The non-local boundary value problem (2.5) is elliptic.
Proof. Consider an arbitrary point x ∈ ∂M/Z2. An explicit computation shows that
near this point the equivalent classical boundary value problem is
? ?∂
ω1|∂M/Z2+ ω2|∂M/Z2= g.
∂t+ A?ω1= f1,
?∂
∂t− A?ω2= f2,
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It is elliptic (satisfies the Shapiro–Lopatinskii condition), since the symbol of the operator
of boundary conditions defines an isomorphism
Imσ(Π+)(x,ξ) ⊕ Imσ (Π−)(x,ξ) ≃ Λ∗(∂M)x
at an arbitrary point (x,ξ) ∈ S∗(∂M/Z2), where
Π+=A + |A|
2|A|
is the non-negative spectral projection of A and Π−= 1 − Π+is the negative projection.
The ellipticity of the classical boundary value problem proves the desired statement.
?
Proposition 2.3. One has
ind(d + d∗,(1 + G∗)) = signM,
where signM is the signature of M.
Proof. The symbol of (2.5) coincides with that of the composition of the spectral
Atiyah–Patodi–Singer boundary value problem
?(d + d∗)ω = f
and the Fredholm operator
Π+ω|∂M= ω′,ω′∈ ImΠ+⊂ Λ∗(∂M),
(1 + G∗) : ImΠ+−→ Λ∗(∂M)Z2. (2.6)
Let us compute both indices.
1) For the index of the spectral boundary value problem, one has [1]
ind(d + d∗,Π+) = signM −dimkerA
2
.
In addition, by the Hodge–de Rham theory we obtain dimkerA = dimH∗(∂M).
2) On the other hand, one can readily verify that the operator in Eq. (2.6) is surjective
and its kernel coincides with the space of G-antiinvariant harmonic forms. The Hodge
operator ∗ interchanges the antiinvariant and invariant subspaces. Thus we obtain
dimker (1 + G∗)|ImΠ+(A)=dimkerA
2
.
Adding the index of the spectral problem to the index of (1 + G∗), we obtain
ind(d + d∗,(1 + G∗)) = signM −dimkerA
2
+dimkerA
2
= signM.
The proof of the theorem is complete.
?
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3 The homotopy classification of non-local operators
Let us cut M into two parts
M′= M\{∂M × [0,1)} ≃ Mand∂M × [0,1].
Then the symbol σ (D) of an admissible elliptic operator D of order zero is naturally
represented as a pair of usual symbols
σ (D)|M′ and (π × 1)!σ(D)|∂M×[0,1]. (3.1)
Both symbols define difference elements
[σM] ∈ K (T∗M′),[σX] ∈ K (T∗(X × (0,1])).
Here and in what follows, we use K-groups with compact supports. In the latter case,
the elliptic symbol σXof order zero is invertible over X ×{0} (this follows from ellipticity
and the decomposition in Eq. (2.2)) and hence defines element in the above-mentioned
K-group with compact supports.
However, it is impossible to define an element of a single topological K-group; indeed,
the manifolds T∗M′and T∗(X × (0,1]) can not be glued together, for their boundaries
are not diffeomorphic. Nonetheless, we can glue the algebras of functions on these spaces
instead of the original manifolds.
1. The C∗-algebra of a manifold with a covering on the boundary. To each
space, we assign an algebra of continuous functions vanishing at infinity:
C0(T∗M′),C0(T∗(X × (0,1]),Endp∗π!1).
More precisely, on the space T∗(X × (0,1]) we consider functions ranging in the set of
endomorphisms of the bundle π!1 ∈ Vect(X), where p : T∗(X × (0,1]) → X is the natural
projection. In the direct sum of these algebras, consider the subalgebra determined by
the compatibility condition
?
AT∗M,π=
(u,v)
????
u ∈ C0(T∗M′),v ∈ C0(T∗(X × (0,1]),Endp∗π!1)
β u|∂M′β−1= v|t=1
?
. (3.2)
Here t is the coordinate on (0,1].
For the trivial covering ∂M → ∂M = X, this algebra is just the commutative algebra
of continuous functions on T∗(M\∂M) vanishing at infinity. Let us also mention that
this algebra can be also viewed as the groupoid C∗-algebra [20] of the equivalence relation
defined by π (x ∼ y if either x = y or x,y ∈ ∂M and π(x) = π(y)). For a trivial covering,
this algebra was used in [6].
2. The difference construction. Let us define the difference construction for
non-local operators. This will be a mapping
χ : Ell0(M,π) −→ K0(AT∗M,π) (3.3)
17
Page 18
into the K0group of the C∗-algebra AT∗M,π. To this end, we take an elliptic operator
D : C∞(M,E) −→ C∞?M,Ck?,
fix some embeddings of E and Ckin trivial bundles of sufficiently large dimension, and
denote by PEand PCk the projections that define the corresponding subbundles:
E ≃ ImPE⊂ CN⊕ 0,
Ck≃ ImPCk ⊂ 0 ⊕ CL.
We denote the direct images of these projections near the boundary by Pπ!Eand Pπ!Ck.
The difference element of D is, by definition, the difference
χ[D] = [P1⊕ P2] −?PCk ⊕ Pπ!Ck?∈ K0(AT∗M,π),
where the projection P1over M′is given by
?
PCk,
PEcos2|ξ| + PCk sin2|ξ| +?σ−1
M(x,ξ)PCk + σM(x,ξ)PE
?sin|ξ|cos|ξ|, |ξ| ≤ π/2,
|ξ| > π/2.
(3.4)
(We assume that the principal symbol is zero-order homogeneous in ξ.) The projection
P2over X × [0,1] is defined by the formula
where the first case is used for x′∈ X × [1/2,1],|ξ| ≤ π/2, the second for x′∈ X ×
[0,1/2],|ξ| < πt, and the third otherwise. Here we write
P2=
Pπ!Ecos2|ξ| + Pπ!Ck sin2|ξ| + 1/2?? σ−1(x′,ξ)Pπ!Ck + ? σ (x′,ξ)Pπ!E
?sin2|ξ|,
Pπ!Ecos2ϕ + Pπ!Ck sin2ϕ + 1/2?? σ−1(x′,0)Pπ!Ck + ? σ (x′,0)Pπ!E
?sin2ϕ,
Pπ!Ck,
ϕ = |ξ| + π/2(1 − 2t),? σ (x′,ξ) = σX(x′,ξ)
for brevity. Geometrically, these projections define a subbundle that coincides with E ⊂
CN+Lover the zero section (for ξ=0); coincides with the orthogonal bundle Ck⊂ CN+L
for |ξ| ≥ π/2; and is obtained by the rotation of the first bundle towards the second
bundle with the use of σ (D) at the intermediate points. (The symbol is treated as an
isomorphism of the two bundles.) By construction, P1and PCk coincide outside a compact
set in T∗M′, and P2and Pπ!Ck coincide outside a compact set in T∗(X × (0,1]). Therefore,
the difference [P1⊕ P2] −?PCk ⊕ Pπ!Ck?is indeed in K0(AT∗M,π).
Remark 3.1. This element of the K-group can be equivalently defined by different ex-
pressions (cf. [21]).
Theorem 3.1. The difference construction (3.3) is a well-defined group isomorphism.
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Proof. The mapping χ preserves the equivalence relations in Ell0(M,π) and K0(AT∗M,π).
Indeed, under an operator homotopy the symbols vary continuously. Therefore, the cor-
responding projections P1,2are joined by a continuous homotopy. Furthermore, χ[D] is
independent of the choice of an embedding in a trivial bundle, since all such embeddings
are homotopic, and for a trivial D (i.e., one induced by a vector bundle isomorphism)
χ[D] is equal to zero. This shows that χ is well defined. The proof that this mapping is
one-to-one presents no essential difficulties and is left to the reader.
?
3. Index theorem for families. Later on in Section 5, we use a families index
formula. Let us briefly state the corresponding results.
Let P be a compact space. Denote by EllP(M,π) the group of stable homotopy classes
of elliptic families on M parametrized by P.
Theorem 3.2 (the index of families of non-local operators). Let (M,π) satisfy Assump-
tion 2.1. Then for an embedding f : M → EGN the direct image mapping f!for families
is well defined and the following diagram commutes:
EllP(M,π)
ind
??
f!
??EllP(EGN,πN)
(πN)!
??
K0(P)K0(P × T∗BGN)
indt
??
EllP(BGN).
The proof is similar to that of Theorem 2.2 in the previous section (cf. [22]) and
therefore is omitted.
Let us finally note that the difference construction can also be defined in this case as
a mapping
χP: EllP(M,π) −→ K0(C (P,AT∗M,π)),
where C (P,AT∗M,π) is the algebra of continuous functions on P ranging in the C∗-algebra
AT∗M,π.
4 A homotopy invariant for manifolds with covering
on the boundary
1. The class of operators. On a manifold M with covering π on the boundary, we
consider elliptic differential operators
D : C∞(M,E) −→ C∞(M,F)
that are lifted from the base of the covering in a neighbourhood of the boundary . Tech-
nically, we suppose that the following condition is satisfied.
19
Page 20
Assumption 4.1. The restrictions of the bundles E and F to the boundary are lifted
from the base of the covering; moreover, we fix some isomorphisms
E|∂M≃ π∗E0,F|∂M≃ π∗F0,E0,F0∈ Vect(X),
and for some operator D0: C∞(X × [0,1),E0) −→ C∞(X × [0,1),F0) on the cylinder
with base X the direct image of D in a collar neighbourhood of the boundary satisfies
the commutative diagram
C∞(X × [0,1),π!E)
≃
??
(π×1)!D
??C∞(X × [0,1),π!F)
≃
??
C∞(X × [0,1),E0⊗ π!1)
D0⊗1??C∞(X × [0,1),F0⊗ π!1).
(4.1)
Here D0⊗ 1 stands for the operator D0with coefficients in the flat bundle π!1 (e.g., see
[23]).
We also suppose that D is first-order operator and the following assumption is satisfied.
Assumption 4.2. In the neighbourhood X ×[0,ε) of the boundary, the operator has the
form
D0|X×[0,ε)= Γ
?∂
∂t+ A0
?
for a bundle isomorphism Γ, where A0is an elliptic self-adjoint first-order operator on X.
This operator is called the tangential operator of D0.
If D satisfies Assumptions 4.1 and 4.2, then near the boundary it has the form
D =∂
∂t+ π!(A0⊗ 1)
up to a vector bundle isomorphism. For brevity, the self-adjoint operator π!(A0⊗ 1) will
be denoted by A.
2. The homotopy invariant. For an operator D satisfying Assumptions 4.1 and
4.2, consider the spectral Atiyah–Patodi–Singer boundary value problem [1]
?Du
where Π+ = (A + |A|)/2|A| is the non-negative spectral projection of the self-adjoint
operator A. (If A is not invertible, then in this formula one should replace A by A + ε
for some ε less then the absolute value of the greatest negative eigenvalue of A.) The
spectral problem is always Fredholm. However, its index ind(D,Π+) is not invariant
under homotopies of D and is not determined by its principal symbol. Here by definition
a continuous homotopy of D is a continuous homotopy in the interior of M that can be
covered in a neighbourhood of the boundary by a homotopy of the diagram (4.1) and a
continuous homotopy of tangential operators.
= f,
= g,Π+u|∂M
g ∈ ImΠ+⊂ Hs−1/2(∂M,E),
20
Page 21
Proposition 4.1. The sum
?
indD
def
= modn-(ind(D,Π+) + η(A) − nη (A0)) ∈ R/nZ, (4.2)
is a homotopy invariant of D. Here n is the number of sheets of the covering and η(A)
and η(A0) are the spectral Atiyah–Patodi–Singer η-invariants of the tangential operators
A and A0.
Proof. Consider the non-reduced invariant
ind(D,Π+) + η (A) − nη (A0).(4.3)
The results of [23] imply that for a smooth operator family Dtthis expression is a piecewise
smooth function of the parameter t (the corresponding families of tangential operators
are denoted by Atand A0,t).
1) We claim that (4.3) is a piecewise constant function. Indeed, the derivative of the
η-invariant
d
dtη(At)
with respect to t is local, i.e., is equal to an integral over the manifold of an expression
determined by the complete symbol of the tangential family At. However, the complete
symbols of Atand A0,tcoincide locally by Assumption 4.1. Thus we have
d
dtη (At) = nd
dtη(A0,t).
Therefore, (4.3) is a piecewise constant function.
2) Let us show that the jumps of this function are multiples of the number of sheets
of the covering. Indeed, for a homotopy Dtthe index, as well as the η-invariants, changes
by the spectral flow of the corresponding families of tangential operators. Hence
[ind(Dt,Π+,t) + η (At) − nη (A0.t)]|t=0,1= (−1 + 1)sf (At)t∈[0,1]− nsf (A0,t)t∈[0,1]∈ nZ,
as desired.
?
Remark 4.1. For a trivial covering, our invariant is none other than the modn-index
modn-ind(D,Π+) ∈ Zn⊂ R/nZ
of Freed-Melrose [2]. On the other hand, the fractional part of the invariant (4.2) is the
so-called relative Atiyah–Patodi–Singer η-invariant [24, 23]
{η(A0⊗ 1π!1) − nη (A0)} ∈ R/Z
of A0with coefficients in the flat bundle π!1 ∈ Vect(X).
21
Page 22
The invariant?
the boundary
ind has an interesting interpretation as an obstruction. Namely, suppose
that M is the total space of a covering ? π with base Y that induces the covering π over
∂M⊂
π ↓
X⊂
M
↓ ? π
Y.
Proposition 4.2. If a differential operator D : C∞(M,E) → C∞(M,F) is the pullback
of an elliptic operator D0on Y, then
?
indD = 0.
Proof. According to the Atiyah–Patodi–Singer formula (see [1]), the sum
ind(D,Π+) + η(A)
is equal to the integral over the manifold of a local expression defined by the complete
symbol of D. Since D and D0coincide locally, one has
ind(D,Π+) + η(A) = n(ind(D0,Π+,0) + η (A0)).
We obtain the desired formula by transposing the term nη (A0) to the left-hand side. ?
5 The index defect formula
The aim of this section is to find a topological formula for the invariant?
M
ind.
1. The difference construction. The pair (M,π) defines the singular space
π= M /{x ∼ x′, if x,x′∈ ∂M and π (x) = π (x′)} ,
obtained by identification of points in the fibers of π (see Figure 1 in the case of a trivial
covering). Likewise, the boundary of the non-compact manifold T∗M is a covering over
the product T∗X × R, and the corresponding singular space will be denoted by T∗M
Consider an elliptic operator D satisfying Assumption 4.1. The diagram (4.1) implies
that the principal symbol defines a K-theory element
[σ (D)] ∈ K?T∗M
Thus we have a homomorphism
χ : Ell?M
Here Ell?M
The topological formula for the invariant?
22
π.
π?.
π?−→ K?T∗M
π?,χ[D] = [σ (D)].
π?is the Grothendieck group of homotopy classes of elliptic operators D on
ind uses the Poincar´ e pairing on the manifold
πwith singularities. Let us define this pairing.
M that satisfy Assumptions 4.1 and 4.2.
T∗M
Page 23
Figure 1: Singular space
2. A pairing in K-theory of a singular manifold. By analogy with the algebra
AT∗M,πof the cotangent bundle, one can define an algebra for M itself:
?
AM,π=(u,v)
????
u ∈ C0(M′),v ∈ C0(X × (0,1],Endπ!1)
β (u|∂M′)β−1= v|t=1
?
.
Lemma 5.1. The group K0(AM,π) is isomorphic to the group of stable homotopy classes
of triples
(E,F,σ), E,F ∈ Vect(M), σ : π!E|∂M−→ π!F|∂M.
Here σ is a bundle isomorphism, and trivial triples are those with σ induced by an iso-
morphism over M.
Proof. Note that this lemma is similar to Theorem 3.1, which can be also considered
as giving a realization of the group K0(AT∗M,π) in topological terms. Along the same
lines, a triple?E,Ck,σ?defines the element
[PE⊕ P2] −?PCk ⊕ Pπ!Ck?∈ K0(AM,π),
where the projection P2over X × [0,1] is
P2= Pπ!Ecos2ϕ + Pπ!Ck sin2ϕ + Pπ!Ckσ (x)Pπ!Esin2ϕ,ϕ =π
2(1 − t).
Here PE,PCk are projections on subbundles isomorphic to E and F. We also suppose that
the subbundles are orthogonal to each other.
23
Page 24
The proof of the fact that this mapping induces an isomorphism with the group
K0(AM,π) is similar to the previous proof and is omitted.
?
This realization permits one to define a product
K0?T∗M
using the construction of a symbol with coefficients in a vector bundle. More precisely,
for elements
[σ] ∈ K?T∗M
consider the symbol
σ ⊗ 1E⊕ σ−1⊗ 1F
π?× K0(AM,π) −→ K0(AT∗M,π)
π?,[E,F,σ′] ∈ K0(AM,π),
(5.1)
on M. The direct image of its restriction to the boundary can be written as
π!(σ ⊗ 1E⊕ σ−1⊗ 1F|∂M) = π!σ ⊗ 1π!E|∂M⊕ π!σ−1⊗ 1π!F|∂M
≃ (π!σ ⊕ π!σ−1) ⊗ 1π!E|∂M.
The latter isomorphism is induced by a vector bundle isomorphism
π!E|∂M
σ′
≃ π!F|∂M.
Now (π!σ ⊕ π!σ−1) ⊗ 1π!E|∂Mis trivially homotopic to the identity. The homotopy is
?
Thus we have extended the symbol (5.1) to a non-local elliptic symbol on M. Now the
desired product is defined as the difference construction of the latter symbol, which we
denote by
[σ] × [E,F,σ′] ∈ K0(AT∗M,π).
π!σ 0
01
??
cosτ
−sinτ
sinτ
cosτ
??
1
0 π!σ−1
0
??
cosτ
sinτ
−sinτ
cosτ
?
,τ ∈ [0,π/2].
Using the homotopy classification K0(AT∗M,π) ≃ Ell(M,π), we can apply the index map-
ping to this product and define the pairing of groups as the composition
?,? : K0?T∗M
This pairing is an analogue of Poincar´ e duality on the singular manifold T∗M
Section 8). Let us note that for a regular covering the index can be computed topologically
(by the index theorem) and hence the pairing is also topologically computable.
3. The element of K-theory with coefficients defined by a manifold with a
covering on the boundary. We start from a universal example. Denote the half-infinite
cylinder EGN× [0,+∞) by MN, and the projective limit of the groups K0(AMN,πN) as
N → ∞ by K0(AM∞). Let us show that the universal bundle γ = π!1 ∈ Vect(BG),
where 1 ∈ Vect(EG), defines an element
π?× K0(AM,π) −→ K0(AT∗M,π)
ind
−→ Z.
π(see
[? γ] ∈ K0(AM∞,Q/nZ),n = |G|. (5.2)
24
Page 25
To this end, note that the universal bundle gives the element3
[γ] − n ∈ K0(BG).
We use the following lemma to show that this difference defines the desired element.
Lemma 5.2. There is an isomorphism? K0(BG) ≃ K1(BG,Q/nZ), which is defined as
→ K1(BGN) ⊗ Q −→ K1(BGN,Q/nZ)
the coboundary mapping ∂ in the exact sequence
∂
−→ K0(BGN)
×n
−→ K0(BGN) ⊗ Q
induced by the inclusion of the coefficient groups nZ ⊂ Q.
Proof. Let us rewrite the sequence
→ K1(BGN) ⊗ Q −→ K1(BGN,Q/nZ) −→ K0(BGN)
×n
−→ K0(BGN) ⊗ Q
as the short exact sequence
0 → K1(BGN) ⊗ Q/nZ −→ K1(BGN,Q/nZ) −→ TorK0(BGN) → 0.
Then we obtain the following sequence of projective limits as N → ∞ (this sequence may
not be exact):
0 →lim
←−K1(BGN) ⊗ Q/nZ −→ K1(BG,Q/nZ) −→lim
←−TorK0(BGN) → 0. (5.3)
As N increases, the sequence? K∗(BGN) has the following property (e.g., see [24, 25]).
? K∗(BGN+L) −→? K∗(BGN)
is in the torsion subgroup. Using this, we obtain the following expressions for the limits:
For an arbitrary N, there exists an L > 0 such that the range of the mapping
(5.4)
lim
←−K1(BGN) ⊗ Q/nZ = 0,lim
←−TorK0(BGN) =lim
←−
? K0(BGN) =? K0(BG).
One can also use (5.4) to prove the exactness of (5.3). The proof is based on commutative
diagrams of the form
0 → K1(BGN) ⊗ Q/nZ −→ K1(BGN,Q/nZ) −→ TorK0(BGN) → 0
0 ↑
0 →K1(BGN+L) ⊗ Q/nZ −→ K1(BGN+L,Q/nZ) −→ TorK0(BGN+L)→ 0,
↑↑
(5.5)
where L is chosen as in (5.4).
3Here and below, the K-groups of classifying spaces are defined as the projective limits over their
finite-dimensional approximations.
25
Page 26
Thus we obtain the desired isomorphism
lim
←−K1(BGN,Q/nZ) ≃lim
←−
?K0(BGN).
?
Finally, the desired element (5.2) is obtained from the isomorphism
K∗+1(BGN,Q/nZ) ≃ K∗(AMN,πN,Q/nZ)
induced by the inclusion of the ideal C0(BGN× (0,1),End(πN)!1). Thus
K1(BG,Q/nZ) ≃ K0(AM∞,Q/nZ),
and [? γ] can be viewed as an element of both groups.
we assume that π is a principal G-covering for a finite group G. There exists a mapping
f : M → MNthat takes the boundary to the base EGN× {0} and is equivariant on the
boundary (see Proposition 2.1). The inverse image of [? γ] is denoted by
[?
This element does not depend on the choice of f, since a map into the universal space is
unique up to homotopy. Let us obtain a geometric realization of this element. We do this
in two steps.
4. A geometric realization of K0(AM,π,Q/nZ). The group Q/nZ is the direct
limit of the finite groups
ZnN⊂ Q/nZ,
Suppose that we are now given a pair (M,π). In the remaining part of the section,
π!1]
def
= f∗[? γ] ∈ K0(AM,π,Q/nZ). (5.6)
x ?→ x/N.
Therefore, the K-group with coefficients in Q/nZ is defined as the direct limit
K0(AM,π,Q/nZ) =lim
−→K0(AM,π,ZnN). (5.7)
Further, the elements of the groups with finite coefficients can be constructed by using
the following proposition (cf. [26]).
Proposition 5.1. A triple (E,F,σ), where
E ∈ Vect(M), F ∈ Vect(X),π!(E|∂M)
σ≃ kF,(5.8)
and σ is an isomorphism on X, defines an element in K0(AM,π,Zk).
Proof. By analogy with the topological case (e.g., see [24]), the theory with coefficients
in Zkis defined in terms of the Moore space Mkof this group by the formula
K0(AM,π,Zk) = K0
??C0(Mk,AM,π)
26
?
, (5.9)
Page 27
where?C0(Mk,AM,π) is the algebra of AM,π-valued functions on the Moore space vanishing
One can readily generalize Lemma 5.1 to the case of families. More precisely, the
same method shows that the group K0
stable homotopy classes of triples (E′,F′,σ′), where E′,F′∈ Vect(M × Mk) and the
isomorphism
σ′: π!(E′|∂M) −→ π!(F′|∂M)
is defined over X × Mk.
Let ε be the line bundle over the Moore space representing the generator [ε] − 1 ∈
? K (Mk) ≃ Zk. (Further information about the Moore spaces can be found, e.g., in [24, 26].)
To the triple (E,F,σ) in (5.8), we assign the element
at the fixed point.
??C0(Mk,AM,π)
?
is isomorphic to the group of
Let us also fix a trivialization ρ : kε → Ck.
[E ⊗ ε,E,σ′] ∈ K0
??C0(Mk,AM,π)
?
,
where the isomorphism σ′is defined as the composition (see [26])
π!(E|∂M) ⊗ ε
σ⊗1
→ kF ⊗ ε ≃ F ⊗ kε
1⊗ρ
→ F ⊗ Ck≃ kF
σ−1⊗1
→ π!(E|∂M). (5.10)
?
5. A geometric realization of [?
π!1] (see (5.6)). For sufficiently large N, consider
a triple (N,1,α), where
N ∈ Vect(M),1 ∈ Vect(X),
are trivial vector bundles of the corresponding dimensions and π!N
alization. By Proposition 5.1, this triple defines an element
α
≃ CnNis some trivi-
[N,1,α] ∈ K0(AM,π,Q/nZ).
Now, using the diagram (5.5), the reader can verify that this element coincides with [?
Suppose that the range of the classifying mapping
π!1]
if the number N and the trivialization are chosen as follows.
f : M −→ M∞
is contained in the skeleton MN′, then for N′there exists an L′such that property (5.4)
is valid. Now we can choose an N such that the restriction of the direct sum Nγ of the
universal bundle to BGN′+L′ is trivial with some trivialization
Nγ
α′
≃ CnN.
Finally, over M we choose the induced trivialization
α = f∗α′.
6. The index defect theorem.
27
Page 28
Theorem 5.1. Let (M,π) be a manifold with a covering on the boundary corresponding
to a free action of a finite group G. Then the diagram
Ell?M
?
R/nZ,
π?
ind
??
χ??K?T∗M
?·,[?
π?
π!1]?
?????????????
commutes. Here ?,? is the Poincar´ e pairing with coefficients,
?,? : K?T∗M
Remark 5.1. Theorem 5.1 expresses?
using the index theorem for families (Theorem 3.2).
π?× K0(AM,π,Q/nZ) −→ K0(AT∗M,π,Q/nZ)
indD in topological terms via the principal symbol.
Indeed, by (5.7) and (5.9), the index mapping in (5.11) can be expressed topologically
ind
→ Q/nZ. (5.11)
Proof. The proof of the theorem is essentially analytic in nature. The main idea is to
reduce the analytic invariant?
1. First, we define elliptic theory Ell(M,π,Q/nZ) with coefficients Q/nZ. The defi-
nition can be given using the direct limit
ind to the index with values in Q/nZ. Thus, we start by
defining the corresponding operators.
Ell(M,π,Q/nZ) =lim
−→Ell(M,π,ZnN),
ZnN⊂ ZnNM⊂ Q/nZ,
of theories with finite coefficients. More precisely, elliptic theory with coefficients in Zkis
defined by families of non-local elliptic operators of order one parametrized by the Moore
space Mkof Zk:
Ell(M,π,Zk) = EllMk(M,π).
For elliptic theory with coefficients, we refer the reader to [26].
2. Consider the mapping
Ell?M
π?
Φ
−→ Ell(M,π,Q/nZ)
(5.12)
that takes an operator D to the family
D∗⊕ (D ⊗ 1ε) : C∞(M,F ⊕ E ⊗ ε) −→ C∞(M,E ⊕ F ⊗ ε)
of first-order elliptic operators on M parametrized by MnN(the number N will be chosen
below). Here D∗is the adjoint operator, and the operator family obtained by twisting D
with the bundle ε is denoted by D⊗1ε. Consider the direct sum of N copies of this family.
It turns out that if N is sufficiently large, then this family admits an elliptic boundary
condition. Indeed, for N sufficiently large there exists a trivialization
Nπ!1
α
≃ CnN, (5.13)
28
Page 29
since π!1 is flat. Hence on the base of the covering we have the vector bundle isomorphism
π!(N E|∂M) ≃ π!N ⊗ E0
α⊗1
−→ CnN⊗ E0
and the similar isomorphism
π!(N E|∂M) ⊗ ε ≃ π!N ⊗ E0⊗ ε
α⊗1
−→ CnN⊗ E0⊗ ε ≃ nNε ⊗ E0
ρ⊗1
−→ CnN⊗ E0.
We denote the induced isomorphisms on sections by
B1= α ⊗ 1 : C∞(X,π!(N E|∂M)) −→ C∞?X,CnN⊗ E0
B2= ρ ⊗ 1(α ⊗ 1) : C∞(X,π!(N E|∂M) ⊗ ε) −→ C∞?X,CnN⊗ E0
Now we can define the family of non-local boundary value problems (in the sense of
Section 1)
?
B1βEu|∂M+ B2βEv|∂M= g,
?,
?.
ND∗u = f1,N (D ⊗ 1ε)v = f2,
g ∈ C∞?X,CnN⊗ E0
?,
(5.14)
which consists of elliptic elements. We define the mapping (5.12) as follows: it takes D
to the family of non-local problems (5.14). Note that Φ depends on the choice of the
trivialization (5.13).
3. There is a natural index mapping
ind : Ell(M,π,Q/nZ) −→ Q/nZ
that takes an element [D] represented by a family D of elliptic operators parametrized by
MnNto the (reduced) index of the family
ind[D]
def
= indD ∈? K (MnN) ≃ ZnN⊂ Q/nZ.
Ell?M
?
Lemma 5.3. The diagram
π?
Φ
??
ind
???
?
?
?
?
?
?
?
?
?
?
Ell(M,π,Q/nZ)
???????????????
R/nZ,
ind
where N and the trivialization α in (5.13) are chosen as in Subsec. 5, commutes.
Proof of the lemma. The boundary value problem (5.14) is linearly homotopic to the
problem
?
B1βEΠ−u|∂M+ B2βEΠ+v|∂M= g,
ND∗u = f1,N (D ⊗ 1ε)v = f2,
g ∈ C∞?X,CnN⊗ E0
?,
29
Page 30
within the class of elliptic problems. The last formula shows that the index of Φ[D] is
equal to the sum of the index of the family of spectral problems for ND∗and N (D ⊗ 1ε)
and the index of the operator family
NImΠ−(A) ⊕ NImΠ+(A) ⊗ ε
B1+B2
−→ C∞?X,CnN⊗ E0
?
(5.15)
on the boundary. Note that we specify the self-adjoint operators in the notation of spectral
projections. Let us compute the index of the former family on X.
1) There is a decomposition
C∞?X,CnN⊗ E0
of the target space for the family (5.15). This decomposition is defined as
?≃ nNImΠ−(A0) ⊕ nNε ⊗ ImΠ+(A0)
−→ C∞?X,CnN⊗ E0
nNImΠ−(A0) ⊕ nNε ⊗ ImΠ+(A0)
1+(ρ⊗1)
?.
Using this isomorphism, we represent the index of (5.15) in the form
?
Finally, we rewrite the index by pushing forward the space ImΠ+(A) to the base of the
covering:
?
The index of the elliptic operator (not a family!) in the last formula can be expressed by
the Atiyah–Patodi–Singer formula [23]
?
where the brackets ?,? denote the pairing
= indNImΠ+(A)
Π+(A0)βE
−→nNImΠ+(A0)
?
([ε] − 1) ∈? K (MnN).
?
= indNImΠ+(π!A)
Π+(A0)
−→ nNImΠ+(A0) ([ε] − 1).
indNImΠ+(π!A)
Π+(A0)
−→ nNImΠ+(A0)
?
= Nη (A) − nNη (A0) + ?[σ (A0)],[π!1]?,
(5.16)
?,? : K1(T∗X) × K1(X,Q) −→ Q
(5.17)
of the difference element [σ (A0)] ∈ K1(T∗X) of an elliptic self-adjoint operator A0with
the element [π!1] ∈ K1(X,Q) defined by the trivialized flat bundle Nπ!1 (more about
this formula can be found in the book [27]).
2) It turns out that for our choice of the trivialization (5.13) the last term in (5.16) is
equal to zero. Indeed, consider the classifying mapping f : X → BGN′. We can evaluate
(5.17) on the classifying space:
?[σ (A0)],[π!1]? = ?f![σ (A0)],[γ]?,[π!1] = f∗[γ] ∈ K1(X,Q), (5.18)
where [γ] ∈ K1(BGN′)⊗ Q is the element defined by the trivialized flat bundle Nγ. The
inclusion BGN′ ⊂ BGN′+L′ induces the commutative diagram
K1(T∗BGN′)
↓
K1(T∗BGN′+L′) × K1(BGN′+L′) ⊗ Q −→ Q
×K1(BGN′) ⊗ Q
↑
−→ Q
?
30
Page 31
Using this diagram and (5.4), one can prove the triviality of the pairing (5.18) by a
diagram chase argument.
Thus we have reduced indΦ[D] to the desired form
indΦ[D] =?
ind[D].
?
4. To complete the proof of the theorem, it suffices to show that the value of the
Poincar´ e pairing ?[σ(D)],[?
ϕ : K?T∗M
Lemma 5.4. Under the assumptions of Lemma 5.3, the diagram
Ell?M
K?T∗M
where χ′is induced by the difference constructions for families (see Subsec. 3.3).
π!1]? coincides with the index of Φ(D).
π!1] by ϕ:
π?−→ K0(AT∗M,π,Q/nZ).
We denote the product by [?
π?
Φ
−→ Ell(M,π,Q/nZ)
↓ χ′
−→ K (AT∗M,π,Q/nZ),
χ ↓
π?
ϕ
(5.19)
Proof. Substituting the definitions of [σ (D)] and [?
determined by the family of elliptic symbols that are equal to
π!1] (according to Subsecs. 4.4 and 4.5)
into Eq. (5.1), defining the product, one can show that the desired product ϕ[σ(D)] is
Nσ (D) ⊗ 1ε⊕ Nσ (D)−1⊗ 1(5.20)
far from the boundary. The direct image of the restriction of this symbol to the boundary
is equal to
?σ(D) ⊗ 1ε⊕ σ(D)−1⊗ 1?
≃
Nπ!
=
?σ(D0) ⊗ 1Nε⊗π!1⊕ σ (D0)−1⊗ 1Nπ!1
?≃(5.21)
?σ(D0) ⊕ σ (D0)−1?⊗ 1CnN.
≃ CnN,In the last equality, we use the isomorphisms Nπ!1
is extended to a neighbourhood of the boundary using the homotopy of the direct sum
σ(D0) ⊕ σ (D0)−1to the identity.
It remains to prove that the difference elements for the principal symbol of the family
of boundary value problems (5.14) and the symbol defined by (5.20), (5.21) coincide.
Indeed, this equality is obvious far from the boundary, since the only difference here is
in the components σ(D∗) and σ (D)−1. These components are joined by the standard
homotopy
σ (D∗)[σ(D)σ(D∗)]−s,
α
CnNε
ρ
≃ CnN. The symbol
s ∈ [0,1].
The reader can also prove the equality near the boundary using the formulae for order
reduction given in Remark 2.2.
?
By combining Lemmata 5.4 and 5.3, we complete the proof of the theorem.
?
31
Page 32
6 Applications
1. Theorem 5.1 enables one to express the fractional part of the η-invariant in the following
situation.
Let M be an even-dimensional spin manifold with boundary represented as the total
space of a covering such that the spin structure on the boundary is the pullback of a spin
structure on the base. Let us also fix an E ∈ Vect(M) that is also pulled back from
the base near the boundary: E|∂M≃ π∗E0. We choose a metric on M that is a product
metric induced by a metric on the base near the boundary. Finally, we choose a similar
connection in E.
Proposition 6.1. The Dirac operator DM on M with coefficients in E satisfies the as-
sumptions of Theorem 5.1, and the fractional part of the η-invariant is equal to
M
{η(DX)} =1
n
?
? A(M)chE −
?
[σ (DM)],[?
π!1]
?
∈ R/Z,
where DXis the self-adjoint Dirac operator on X with coefficients in E0.
Proof. The formula follows from Theorem 5.1 if we decompose the index of the spectral
problem using the Atiyah–Patodi–Singer formula
?
M
ind(DM,Π+) =
? A(M)chE − η (D∂M).
?
2. The invariant?
G-invariant elliptic differential operator of order one on M. For g ∈ G, let L(D,g) ∈ C
be the usual contribution to the Lefschetz formula (see [28]) of the fixed point set of the
diffeomorphism g : M → M.
ind can be effectively computed via Lefschetz theory. Suppose that
π is regular, i.e., the boundary is a principal G-bundle for a finite group G. Let D be a
Proposition 6.2. One has
?
indD ≡ −
?
g?=e
L(D,g) (modn). (6.1)
Proof. Consider the equivariant index indg(D,Π+) of the Atiyah–Patodi–Singer prob-
lem and the equivariant η-function (see [28]) of the tangential operator A on the boundary.
Denote by (D,Π+)Gand AGthe restrictions of the corresponding operators to the
subspaces of G-invariant sections. Clearly, AGis isomorphic to A0on X. On the other
hand, one can express the usual invariants in terms of their equivariant counterparts:
?
ind(D,Π+)G=
1
|G|
g∈G
indg(D,Π+)η?AG?=
1
|G|
?
g∈G
η (A,g).
32
Page 33
These expression follow from elementary character theory. Using them, we write
?
indD = inde(D,Π+) −
?
g?=e
η (A,g).
Let us substitute the expression for the η-invariant given by the equivariant Atiyah–
Patodi–Singer formula (see [28])
−η (A,g) = indg(D,Π+) − L(D,g)
into this formula. This gives the desired congruence (6.1):
?
indD = |G|ind(D,Π+)G−
?
g?=e
L(D,g).
?
7Poincar´ e isomorphisms
1. A closed smooth manifold. It is well known (see [29, 8] or the monograph [30]) that
elliptic operators of order zero on a compact closed manifold define elements in K-theory:
[σ (D)] ∈ K∗(T∗M),[D] ∈ K∗(C (M)) ≡ K∗(M).
The latter group is the analytic K-homology group, and the grading is odd for self-adjoint
operators and even otherwise. The first element is the difference element of the operator.
To define the second element, we recall that an elliptic operator D of order zero is a
Fredholm operator
D : L2(M,E) −→ L2(M,F),
where both L2-spaces are modules over C (M) (the module structure is given by the
pointwise product of functions). In addition, D commutes with the module structure
up to compact operators. Thus, for a self-adjoint D (of course in this case the bundles
coincide) the pair (L2(M,E),D) is an element
[D] ∈ K1(C (M)).
For a nonself-adjoint D, we consider a self-adjoint matrix operator
T =
?
0
D
D∗
0
?
(7.1)
in the naturally Z2-graded C (M)-module L2(M,E) ⊕L2(M,F). The operator T is odd
with respect to the grading. Hence it defines a K-theory element, denoted by
[D] ∈ K0(C (M)).
33
Page 34
2. Manifold with boundary. On the other hand, elliptic operators of order one on
a manifold with non-empty boundary define similar elements
[σ (D)] ∈ Ki(T∗M),[D] ∈ Ki(M\∂M).
The former is the Atiyah–Singer difference element, and the latter is defined as follows.
Consider an embedding
M ⊂?
an arbitrary extension of D to?
?F =
We define the restriction of this operator to M as the bounded operator
M
of M in some closed manifold?
M of the same dimension (e.g., the double 2M). Let?D be
?
M. On?
M, we consider the zero-order operator
1 +?D∗?D
?−1/2?D.
F = i∗?Fi∗: L2(M,E) −→ L2(M,F),
M) is the extension by zero and i∗: L2(?
For a symmetric D, we find that F satisfies
(7.2)
where i∗: L2(M) → L2(?
M) → L2(M) is the
restriction operator.
F − F∗∈ K,f (F2− 1) ∈ K,[F,f] ∈ K
for functions f ∈ C0(M\∂M) vanishing on the boundary. Here K is the ideal of compact
operators. These relations show that F defines an element of K1(C0(M\∂M)) (see [31]).
If D is nonself-adjoint, then one considers the matrix as in Eq. (7.1). This defines an
element in K0(C0(M\∂M)).
The mappings
K∗(T∗(M\∂M)) −→
K∗(T∗M)
K∗(M),
−→ K∗(M\∂M),
[σ (D)]?→[D],
which take symbols to operators, define Poincar` e isomorphisms on a smooth manifold M
with boundary (e.g., see [8]). The top mapping is defined in terms of elliptic operators of
order zero on M that are induced by vector bundle isomorphisms near the boundary.
3.Manifolds with singularities. An elliptic non-local zero-order operator D
defines elements
[σ (D)] ∈ K∗(AT∗M,π),[D] ∈ K∗?C?M
The first is the difference element defined in Section 2. To define the second element,
we note that a non-local elliptic operator D of order zero does not almost commute with
the entire algebra C (M) but only with the functions pulled back from the quotient space
M
π??≃ K∗
?M
π?.
π. This leads to a smaller algebra.
34
Page 35
On the other hand, the operators of Sections 4 and 5 define similar elements
[σ (D)] ∈ K∗?T∗M
In this case, the corresponding operators (7.2), on the contrary, almost commute with
functions C0(M\∂M) as well as with the elements of the algebra AM,π.
π?,[D] ∈ K∗(AM,π).
Theorem 7.1. For an arbitrary manifold with a covering on the boundary (M,π), the
following Poincar´ e isomorphisms are valid:
K∗(AT∗M,π) −→
K∗?T∗M
[σ (D)]
K∗
?M
π?,
π?
−→ K∗(AM,π),
?→[D].
Proof. 1) Let us prove the latter isomorphism. Consider the ideal
I = C0(T∗(X × (0,1)),Endp∗π!1) ⊂ AT∗M,π
with the quotient AT∗M,π/I ≃ C0(T∗M). The long exact sequence of the pair can be
written as
→ K (T∗X)
α
→ K0(AT∗M,π) → K (T∗M) → K1(T∗X) → ... (7.3)
Here we have taken into account the isomorphism K∗(C0(Y,EndG)) ≃ K∗(C0(Y )) ≃
K∗(Y ) for a vector bundle G ∈ Vect(Y ).
Consider the commutative diagram
→ K0(T∗X) → K0(AT∗M,π) →
↓
→K0(X)→
K0(T∗M)
↓
→ K1(T∗X) ...
↓
K1(X)
↓
K0
?M
π?
→ K0(M,∂M) → ...
Here the lower sequence is the exact sequence of the pair X ⊂ M
The vertical mappings of the diagram (except for the second one) are isomorphisms (see
[32, 8]). Thus, using the 5-lemma, we find that the middle mapping
πin K-homology.
K∗(AT∗M,π) −→ K∗
?M
π?
is also an isomorphism.
2) In the second case, the proof follows the same scheme, but one uses the diagram
← K1(T∗X) ← K0?T∗M
←K1(X)← K0(AM,π) ←
π?
← K0(T∗(M\∂M)) ← K0(T∗X) ...
↓
K0(M)
↓↓↓
←K0(X) ...
The upper row corresponds to the pair R × T∗X ⊂ T∗M
The proof of the theorem is complete.
π.
?
35
Page 36
8
An analogue of the pairing for the groups K0?T∗M
Theorem 8.1. On a manifold M with covering π on the boundary, the pairings
Ki?T∗M
are non-degenerate on the free parts of the groups.
Poincar´ e duality
π?and K0(AM,π) in Section 5 is also
valid for the odd groups. The definition is left to the reader.
π?× Ki(AM,π) −→ Z,i = 1,2, (8.1)
Proof. Fixing the first argument of the pairing, we obtain a mapping
Ki?T∗M
where for brevity we write G′= Hom(G,Q). This mapping is part of the commutative
diagram
K1(T∗X) ⊗ Q ← K0?T∗M
K1′(X)←K′
←
π?⊗ Q −→ K′
i(AM,π),
π?⊗ Q ← K0(T∗(M\∂M)) ⊗ Q ← K0(T∗X) ⊗ Q
0(AM,π)K0′(M)
↓↓↓↓
←K0′(X).
Here the vertical mappings, except for the second one, are isomorphisms (by virtue of
Poincar´ e duality on a closed manifold and on a manifold with boundary). Thus, by the 5-
lemma, the second mapping is an isomorphism. Hence the pairing (8.1) is non-degenerate
in the second variable.
The non-degeneracy with respect to the first argument can be proved in a similar way.
?
By way of example, consider M with a spinc-structure that on the boundary is induced
by a spinc-structure on the base X of the covering π. Then the group K∗?T∗M
take the difference construction
[σ (D)] ∈ Kn?T∗M
of the principal symbol of the Dirac operator on M (this can be proved by analogy with the
usual case of closed manifolds; e.g., see [33]). Consequently, one can define the Poincar´ e
duality pairing
K∗+n?M
pairing is non-degenerate on the free parts of the groups.
π?is a
free K∗+n?M
π?-module with one generator (where n = dimM); as a generator one can
π?
π?× K∗(AM,π) −→ Z
as the composition with K∗+n?M
π?→ K?T∗M
π?. The above theorem shows that this
36
Page 37
References
[1] M. Atiyah, V. Patodi, and I. Singer. Spectral asymmetry and Riemannian geometry
I. Math. Proc. Cambridge Philos. Soc., 77, 1975, 43–69.
[2] D. Freed and R. Melrose. A mod k index theorem. Invent. Math., 107, No. 2, 1992,
283–299.
[3] N. Higson. An approach to Z/k-index theory. Int. J. Math., 1, No. 2, 1990, 189–210.
[4] W. Zhang. On the modk index theorem of Freed and Melrose. J. Differ. Geom., 43,
No. 1, 1996, 198–206.
[5] B. Botvinnik. Manifolds with singularities accepting a metric of positive scalar cur-
vature. Geom. Topol., 5, 2001, 683–718.
[6] J. Rosenberg. Groupoid C∗-algebras and index theory on manifolds with singularities.
Geom. Dedicata, 100, 2003, 65–84.
[7] A. Connes. Noncommutative geometry. Academic Press Inc., San Diego, CA, 1994.
[8] G. Kasparov. Equivariant KK-theory and the Novikov conjecture. Inv. Math., 91,
No. 1, 1988, 147–201.
[9] R. Melrose and P. Piazza. Analytic K-theory on manifolds with corners. Adv. in
Math., 92, No. 1, 1992, 1–26.
[10] T. Kawasaki. The index of elliptic operators over V -manifolds. Nagoya Math. J., 84,
1981, 135–157.
[11] C. Farsi. K-theoretical index theorems for orbifolds. Quart. J. Math. Oxford, 43,
1992, 183–200.
[12] A. B. Antonevich. Lineinye funktsionalnye uravneniya. Operatornyi podkhod. “Uni-
versitetskoe”, Minsk, 1988.
[13] L. H¨ ormander. The Analysis of Linear Partial Differential Operators. III. Springer–
Verlag, Berlin Heidelberg New York Tokyo, 1985.
[14] A. Savin, B.-W. Schulze, and B. Sternin. The Homotopy Classification and the Index
of Boundary Value Problems for General Elliptic Operators. Univ. Potsdam, Institut
f¨ ur Mathematik, Oktober 1999. Preprint N 99/20, arXiv: math/9911055.
[15] M. F. Atiyah and I. M. Singer. The index of elliptic operators I. Ann. of Math., 87,
1968, 484–530.
[16] J. Kohn and L. Nirenberg. An algebra of pseudo-differential operators. Comm. Pure
Appl. Math., 18, 1965, 269–305.
37
Page 38
[17] G. Luke and A. S. Mishchenko. Vector bundles and their applications, volume 447 of
Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht, 1998.
[18] Ch.-Ch. Hsiung. The signature and G-signature of manifolds with boundary. J. Diff.
Geometry, 6, 1972, 595–598.
[19] R. S. Palais. Seminar on the Atiyah–Singer index theorem. Princeton Univ. Press,
Princeton, NJ, 1965.
[20] J.-L. Brylinski and V. Nistor. Cyclic cohomology of etale groupoids. K-theory, 8,
1994, 341–365.
[21] N. Higson. On the K-theory proof of the index theorem. In Index Theory and
Operator Algebras (Boulder, CO, 1991), volume 148 of Contemp. Math., 1993, pages
67–86, Providence, RI. AMS.
[22] M. F. Atiyah and I. M. Singer. The index of elliptic operators IV. Ann. Math., 93,
1971, 119–138.
[23] M. Atiyah, V. Patodi, and I. Singer. Spectral asymmetry and Riemannian geometry
III. Math. Proc. Cambridge Philos. Soc., 79, 1976, 71–99.
[24] M. Atiyah, V. Patodi, and I. Singer. Spectral asymmetry and Riemannian geometry
II. Math. Proc. Cambridge Philos. Soc., 78, 1976, 405–432.
[25] M. F. Atiyah. Characters and cohomology of finite groups. Publ. Math. IHES, 9,
1961, 23–64.
[26] A. Savin, B.-W. Schulze, and B. Sternin. Elliptic Operators in Subspaces and the
Eta Invariant. K-theory, 27, No. 3, 2002, 253–272.
[27] P. B. Gilkey.
theorem. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, second
edition, 1995.
Invariance theory, the heat equation, and the Atiyah-Singer index
[28] H. Donnelly. Eta-invariants for G-spaces. Indiana Univ. Math. J., 27, 1978, 889–918.
[29] M. F. Atiyah. Global theory of elliptic operators. In Proc. of the Int. Symposium on
Functional Analysis, 1969, pages 21–30, Tokyo. University of Tokyo Press.
[30] N. Higson and J. Roe. Analytic K-homology. Oxford University Press, Oxford, 2000.
[31] Yu. P. Solovyov and E. V. Troitsky. C∗-algebras and elliptic operators in differen-
tial topology, volume 192 of Translations of Mathematical Monographs. American
Mathematical Society, Providence, RI, 2001.
[32] P. Baum and R. G. Douglas. K-homology and index theory. In R. Kadison, editor,
Operator Algebras and Applications, number 38 in Proc. Symp. Pure Math, 1982,
pages 117–173. American Mathematical Society.
38
Page 39
[33] H. B. Lawson and M. L. Michelsohn. Spin geometry. Princeton Univ. Press, Prince-
ton, 1989.
39
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