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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.
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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)).
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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.
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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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