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arXiv:math/0503694v1 [math.OA] 29 Mar 2005
On the Homotopy Classification of Elliptic
Operators on Manifolds with Edges
V.E. Nazaikinskii, A. Yu. Savin, B.-W. Schulze, B. Yu. Sternin
Abstract
We obtain a stable homotopy classification of elliptic operators on manifolds
with edges.
Contents
Introduction1
1Operators on Manifolds with Edges3
2 An Element in the K-Homology of the Singular Space9
3 The Homotopy Classification 11
4 Computations on the Edge 14
5 Comparison of the Boundary Mappings 16
6 Some Remarks 19
7Appendix 21
Introduction
The paper deals with the classification of elliptic operators on manifolds with edges, i.e.,
a description of the set of elliptic operators up to stable homotopy.
For the first time, such a classification for the case of smooth manifolds was given
by Atiyah and Singer [1] in terms of the topological K-functor. Later, an approach to
the homotopy classification on manifolds with singularities in terms of the analytic K-
homology of the manifold was suggested in [2]. Namely, an elliptic operator on a manifold
M with singularities is represented as an abstract (elliptic) operator in the sense of Atiyah
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[3] and hence defines a cycle in the analytic K-homology of the manifold M viewed as a
compact topological space. In other words, there is a well-defined homomorphism
Ell0(M) −→ K0(M), (0.1)
where Ell0(M) is the group of stable homotopy classes of elliptic operators. For the case
of isolated singularities, the fact that (0.1) is an isomorphism was proved in [2]. (See
also [4], where the case of one singular point was considered.)
The classification (0.1) has many corollaries and applications, including a formula for
the obstruction, similar to the Atiyah–Bott obstruction [5], to the existence of Fredholm
problems for elliptic equations, the fact that the group Ell0(M) is equal modulo torsion
to the homology Hev(M), a generalization of Poincar´ e duality in K-theory to manifolds
with singularities, etc.
In the present paper, we prove that the mapping (0.1) is an isomorphism for elliptic
operators on manifolds with edges in the sense of [6].
The idea underlying the proof is simple. A manifold M with edges is a stratified
manifold with two strata, the singularity stratum X and the open stratum M \ X; both
strata are smooth. The isomorphism (0.1) on smooth manifolds is known [7, 8], and
hence it is natural to extend the assertion about the isomorphism to the union of these
two strata using the sequences
Ell1(M \ X) → Ell0(X) → Ell0(M) → Ell0(M \ X) → Ell1(X)
↓↓↓
K1(M \ X)→K0(X)→K0(M)
↓↓
→K0(M \ X)→K1(X)
(0.2)
of the pair X ⊂ M in K-homology and Ell-theory. Once this commutative diagram is
defined, the desired isomorphism follows from the five lemma.
Actually, we construct Ell-theory (Ell-functor) for the manifold M with edges and
then establish the isomorphism (0.1) by mimicking the proof of the uniqueness theorem
for extraordinary cohomology theory. We however point out the following important facts.
1. Exact sequences in elliptic theory are not known in general. To construct these exact
sequences, we represent the group Ell0(M) as the K-group of some C∗-algebra and
then use the exact sequence of K-theory of algebras.
2. The main difficulty is the boundary map in the upper row (i.e., the boundary map
in K-theory of C∗-algebras). We use semiclassical quantization (e.g., see [9]), which
permits us to replace the algebra of edge symbols by a simpler algebra of families of
parameter-dependent operators in the computation of the boundary map. For the
latter algebra, the boundary map is given by the index theorem in [10].
Let us briefly outline the contents of the paper. We recall the main notions of elliptic
theory on manifolds with edges in Sec. 1. Section 2 describes a construction that assigns
a cycle in K-homology to each elliptic operator. Then (Sec. 3) we state a homotopy clas-
sification theorem and present its proof except for two especially lengthy computations,
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which are given separately in Secs. 4 and 5. The last section contains some additional
remarks (the classification of edge morphisms and the topological obstruction to the exis-
tence of elliptic edge problems). The desired properties of semiclassical quantization are
established in the Appendix.
The research was supported by the Deutsche Forschungsgemeinschaft and by RFBR
grants Nos. 02-01-00118, 02-01-00928, and 03-02-16336.
1 Operators on Manifolds with Edges
First, we recall some facts of the theory of elliptic operators on manifolds with edges. We
systematically use the results of the paper [6].1
1. Manifolds with edges.
equipped with the structure of a smooth locally trivial bundle π : ∂M → X with base
X and fiber Ω. A manifold M with edge X ⊂ M is the space obtained from M by
identifying the points lying in the same fiber:
M = M?∼,
The complement M◦= M\X is an open smooth manifold, and an arbitrary point of the
edge X has a neighborhood homeomorphic to the model wedge
Let M be a smooth compact manifold with boundary ∂M
x ∼ y ⇐⇒ x = y, or (x,y ∈ ∂M and π(x) = π(y)).
W = Rn× KΩ, (1.1)
where n is the dimension of X and KΩ= Ω × R+
this local model, the points of the edge form the subset Rn× {0} ⊂ Rn× KΩ.
We often use local coordinates (x,ω,r). For the model wedge (1.1), these coordinates
have the following form: x ∈ Rn, r ∈ R+, and ω is a coordinate on Ω.
?Ω × {0} is the cone with base Ω. In
2. Differential operators and function spaces.
with smooth coefficients on M◦having the form
Consider a differential expression
D =
?
|α|+|β|+j+l≤m
aαβjl(r,ω,x)
?
−i∂
∂x
?α?
−i
r
∂
∂ω
?β?
−i∂
∂r
?j?1
r
?l
(1.2)
in a neighborhood of the edge, where m is the order of the expression and the coefficients
aαβjl(r,ω,x) are smooth functions up to r = 0. Such differential expressions can be
realized as bounded operators
D : Ws,γ(M) −→ Ws−m,γ−m(M) (1.3)
1The paper [6] contains a version of the theory of elliptic operators on manifolds with edges [11, 12]
especially suited for the study of topological aspects of the theory. In particular, one deals only with
smooth (or continuous) symbols and does not impose any analyticity requirements.
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in the edge Sobolev spaces Ws,γobtained by gluing of the standard Sobolev space Hson
the smooth part of the manifold and the function space on the infinite wedge (1.1) with
the norm
??
where [ξ] =
?u?s,γ=[ξ]2s?κ−1
[ξ]? u?Ks,γdξ
?1/2
,s,γ ∈ R,
?1 + ξ2, ? u is the Fourier transform of u with respect to the variable x,
κλu(r) = λ(k+1)/2u(λr)
in a unitary action of the group R+ in the space L2(KΩ,dvol) with the volume form
dvol = rkdrdω corresponding to the cone metric dr2+ r2dω2(here k = dimΩ), and the
norm ?·?Ks,γ for functions on the cone KΩis determined by the formula
?u?Ks,γ = ?(1 + r−2+ ∆KΩ)s/2ρs−γu?L2(KΩ,dvol),
in which ∆KΩis the Beltrami–Laplace operator with respect to the conical metric and ρ
is a smooth weight function equal to r in a neighborhood of r = 0 and equal to unity for
large r.
3. Pseudodifferential operators and symbols.
of pseudodifferential operators on manifolds with edges extending the calculus of edge-
degenerate differential operators.A pseudodifferential operator D of order m in the
spaces (1.3) has a well-defined interior symbol, which is a function σ(D) homogeneous
of degree m on the cotangent bundle T∗
section (the definition of T∗M ∈ Vect(M) can be found in the cited paper) and a well-
defined edge symbol, which is an operator-valued function
The paper [6] describes a calculus
0M of the manifold with edges without the zero
σ∧(D)(x,ξ) : Ks,γ(KΩ) −→ Ks−m,γ−m(KΩ),(x,ξ) ∈ T∗
0X,(1.4)
in the function space on the infinite wedge. The edge symbol possesses the twisted homo-
geneity property
σ∧(D)(x,λξ) = λmκλσ∧(D)(x,ξ)κ−1
λ,λ ∈ R+. (1.5)
In particular, the edge symbol of the operator (1.2) is equal to
D =
?
|α|+|β|+j+l=m
aαβjl(0,ω,x)ξα
?
−i
r
∂
∂ω
?β?
−i∂
∂r
?j?1
r
?l
.
The notion of interior and edge symbols is important in that the following assertion holds.
Proposition 1.1. A pseudodifferential operator D of order m in the spaces (1.3) is com-
pact if and only if σ(D) = 0 and σ∧(D) = 0.
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4. The composition theorem and ellipticity.
of pseudodifferential operators on manifolds with edges is expressed by the composition
theorem.
The main property of the calculus
Theorem 1.2. The composition of edge-degenerate pseudodifferential operators corre-
sponds to the composition of their symbols (interior and edge):
σ(D1D2) = σ(D1)σ(D2),σ∧(D1D2) = σ∧(D1)σ∧(D2).
In conjunction with the compactness criterion given by proposition 1.1, the composi-
tion formula results in the following finiteness theorem.
Theorem 1.3. If a pseudodifferential operator D of order m acting in the space (1.3) is
elliptic (i.e., its interior symbol is invertible everywhere outside the zero section on T∗M
and the edge symbol is invertible in the spaces (1.4) everywhere outside the zero section
on T∗X), then it is Fredholm.
Naturally, all the preceding is also valid for operators acting in spaces of sections of
vector bundles on M.
5. Order reduction.
pseudodifferential operators (1.3) modulo stable homotopies. (The precise definition of
stable homotopy will be given below.) Let us show that this problem can actually be
reduced to the special case of zero-order operators in the spaces Ws,γfor s = γ = 0.
Indeed, an operator D is elliptic for given s (and fixed γ) if and only if it is elliptic for
any s (with the same γ); see, e.g., [6]. Consequently, a homotopy in the class of elliptic
operators for some s is valid for all s. Hence without loss of generality we can assume
that s = γ, i.e., consider the operators
In this paper, we are interested in the classification of elliptic
D : Wγ,γ(M) −→ Wγ−m,γ−m(M). (1.6)
Next, there exist elliptic operators
V : W0,0(M) −→ Wγ,γ(M),
?V : Wγ−m,γ−m(M) −→ W0,0(M)
D ?−→?V DV
(1.7)
of index zero. Then the mapping
reduces elliptic operators (and homotopies) in the spaces (1.6) to those in the space
W0,0(M). The inverse mapping (modulo compact operators, which plays no role if ho-
motopies in the class of elliptic operators are considered) naturally has the form
D ?−→?V−1DV−1,
where?V−1and V−1are almost inverses of?V and V , respectively.
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6. Pseudodifferential operators of order zero.
what follows we are mainly interested in pseudodifferential operators of order zero in
the space W0,0(M). Hence we give a more detailed description of their construction
and properties, mainly following [6] with some simplifications (related to the fact that
the paper [6] deals with operators of arbitrary order and not only compactness, but
also smoothing properties of remainders in composition formulas are taken into account).
For simplicity, we present all facts for operators acting in function spaces on M. The
generalization to operators acting in spaces of sections of vector bundles on M is trivial.
By virtue of order reduction, in
Edge symbols. First, we describe the class of edge symbols used here.
Definition 1.4. An edge symbol is a family D(x,ξ), (x,ξ) ∈ T∗
spaces on the cones KΩwith the following properties.
0X, of operators in function
1. For any multi-indices α,β, |α| + |β| = 0,1,2,... , the derivatives
D(α,β)(x,ξ) : K0,0(KΩ) −→ K0,0(KΩ)
are continuous operators.
2. The twisted homogeneity condition
D(x,λξ) = κλD(x,ξ)κ−1
λ,λ ∈ R+,
holds.
3. Modulo compact operators, one has the representation
D(x,ξ) = d
?
x,ξ
2r,i
1
r∂
∂r
?
, (1.8)
where d(x,η,p) is a classical pseudodifferential operator with parameters (η,p) ∈
T∗
depending on the additional parameter x ∈ X.
xX ×L−(k+1)/2in the sense of Agranovich–Vishik [13] of order zero on Ω smoothly
Here L−(k+1)/2 = {Imp = −(k + 1)/2} is the weight line, and the function of the
operator ir∂/∂r in (1.8) is defined with the help of the Mellin transform on this weight
line.
Definition 1.5. The interior symbol of the edge symbol D(x,ξ) is the principal symbol
σ(D) = σ(d) in the sense of Agranovich–Vishik of the corresponding pseudodifferential
operator d(x,η,p).The conormal symbol of D(x,ξ) is the operator family σc(D) =
d(x,0,p) in the space L2(Ω).
The main properties of edge symbols are expressed by the following theorem.
Theorem 1.6. The following assertions hold.
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1. Definition 1.4 is consistent; i.e., the operator (1.8) is always bounded in K0,0(KΩ).
2. The operator (1.8) is compact if and only if its interior and conormal symbol are
zero. (In particular, it follows that the interior and conormal symbol of an edge
symbol are well defined.)
3. Edge symbols form a local C∗-algebra, and, modulo compact edge symbols, the prod-
uct of pseudodifferential operators d(x,η,p) corresponds to the product of the respec-
tive edge symbols D(x,ξ) and the adjoint operator corresponds to the adjoint edge
symbol.
4. (Corollary.) The mapping that takes each edge symbol to its interior and conormal
symbols is linear and multiplicative and commutes with the passage to the adjoint
operator.
5. (Norm estimates modulo compact operators.) For an edge symbol D(x,ξ) of order
zero, one has2
inf
K∈C(S∗X,K)max
S∗X?D + K?B(K0,0(KΩ))= max
?
max
∂S∗M|σ(D)|, sup
X×R?σc(D)?B(L2(Ω))
?
(1.9)
.
6. The commutator [D(x,ξ),ϕ] is compact for any continuous function ϕ(r) on R+
equal to zero for sufficiently large r.
7. The product ϕ(r)D(x,ξ) is a symbol of order zero and has a compact fiber variation
[14] on T∗
0X if ϕ(r) is the same as in item 6.
8. For any pair (interior symbol, conormal symbol) satisfying the compatibility condi-
tion (σ(d) for η = 0 is equal to σ(σc)) one can construct an edge symbol.
The algebra of zero-order edge symbols will be denoted by Ψ∧(X).
Pseudodifferential operators.
dodifferential operators.
Let A be the algebra of classical zero-order pseudodifferential operator A on the open
manifold M◦with the following properties:
Now we can describe the class of zero-order pseu-
1) the principal symbol σ(A) is a smooth function on T∗
0M up to the boundary;
2) the operator A is continuous in the space W0,0(M);
3) A compactly commutes with C(M).
2Here and in the following, K is either the ideal of compact operators or (where there is a bundle) a
bundle of algebras of compact operators acting in function spaces on the fibers. Likewise, B corresponds
to the algebra of bounded operators.
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Definition 1.7. A zero-order pseudodifferential operator on M is a continuous operator
B : W0,0(M) −→ W0,0(M) (1.10)
representable modulo compact operators in the form
B = (ϕ(r)D)
?
2x,−i
1
∂
∂x
?
+ rA,(1.11)
where D(x,ξ) ∈ Ψ∧(X) is an edge symbol, ϕ(r) is a smooth function on M equal to 1 on
∂M and zero outside a sufficiently small neighborhood of the boundary, and A ∈ A is a
classical pseudodifferential operator on M◦. (Here r is the distance to ∂M.)
Definition 1.8. The edge symbol of the operator (1.10) is the operator family
σ∧(B) = D(x,ξ). (1.12)
The interior symbol of the operator (1.10) is the function
σ(B) = ϕ(r)σ(D) + rσ(A). (1.13)
The set of zero-order pseudodifferential operator on M will be denoted by Ψ(M). The
main properties of pseudodifferential operators are expressed by the following theorem.
Theorem 1.9. The following assertions hold.
1. The definition is consistent; i.e., the operator (1.11) is always bounded in W0,0(M).
2. The interior and edge symbols of the operator (1.11) are well defined. It is compact
if and only if both symbols are zero.
3. The interior and edge symbols satisfy the compatibility condition
σ(B)|∂T∗
0M= σ(σ∧(B)). (1.14)
For any pair (interior symbol, edge symbol) satisfying the compatibility condition,
one can construct the corresponding pseudodifferential operator.
4. Pseudodifferential operators form a local C∗-algebra, and the mapping taking each
pseudodifferential operator to its interior and edge symbols is a ∗-homomorphism.
5. (Norm estimates modulo compact operators.) For a zero-order edge-degenerate pseu-
dodifferential operator D one has
?
6. If B ∈ Ψ(M), then the commutator [B,ϕ] is compact for any function ϕ ∈ C(M).
inf
K∈K?D + K?B(W0,0(M))= maxmax
S∗M|σ(D)|,max
S∗X?σΛ(D)?B(K0,0(KΩ))
?
(1.15)
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The norm closure.
lowing description of the Calkin algebras of the closures Ψ(M) and ΨΛ(X).
The norm estimates modulo compact operators imply the fol-
Corollary 1.10. The interior and edge symbol homomorphisms (for operators on M)
Ψ(M)
σ→ C∞(S∗M), Ψ(M)
σΛ
→ ΨΛ(X),
induce the isomorphism
Ψ(M)/K ≃
?
(a,aΛ)
???a ∈ C(S∗M),aΛ∈ ΨΛ(X) :
a|∂S∗M= σ(aΛ)
?
.
The interior and conormal symbol homomorphisms (for edge symbols)
ΨΛ(X)
σ→ C∞(∂S∗M), ΨΛ(X)
σc
→ Ψc(X),
where Ψcis the algebra of conormal symbols, induce the isomorphism
?
ΨΛ(X)/C(S∗X,K) ≃(a,ac)
???a ∈ C(∂S∗M),ac∈ Ψc(X) :
a|S(T∗X⊕1)= σ(ac)
?
.
2 An Element in the K-Homology of the Singular
Space
In this section, we show how an elliptic operator on a manifold with edges gives rise to
an element in the analytic K-homology of the space M. (A detailed exposition of the
theory of analytic K-homology can be found in [15], [16], and [8]. An introduction to the
theory can be found in [17].) To the best of the authors’ knowledge, this correspondence
was used for the first time in [18]. In accordance with the preceding, we consider only
zero-order operators.
Let
D : W0,0(M,E) −→ W0,0(M,F)
be an elliptic operator of order zero in sections of vector bundles E,F ∈ Vect(M). The
commutator [D,f] with a function f ∈ C∞(M) is compact if the restrictions of the
function to the fibers of π are constant functions. (This follows from the composition
formula.) Thus D is a generalized elliptic operator on M in the sense of Atiyah [3] and
hence defines a class in the analytic K-homology K0(M) of the singular space M. Let
us give a precise construction of the corresponding cycle.
If D is self-adjoint (and E = F), then we consider the normalization
D = (PkerD+ D2)−1/2D : W0,0(M,E) −→ W0,0(M,E), (2.1)
where PkerDis the orthogonal projection on the kernel.
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In the general case, we consider the self-adjoint operator
?
D =
0D(PkerD+ D∗D)−1/2
(PkerD+ D∗D)−1/2D∗
0
?
: W0,0(M,E⊕F) → W0,0(M,E⊕F),
(2.2)
which is odd with respect to the Z2-grading of the space W0,0(M,E)⊕ W0,0(M,F). By
C(M) we denote the algebra of continuous functions on M.
Proposition 2.1. The operators (2.1) and (2.2) are zero-order elliptic pseudodifferential
operators and define elements in K-homology; these elements will be denoted by
[D] ∈ K∗(M),
where ∗ = 1 for self-adjoint operators and ∗ = 0 in the general case.
Proof. 1. The operator D∗D is pseudodifferential, and the same is true for PkerD, since the
latter operator is finite rank and hence compact. Thus D∗D+PkerDis a pseudodifferential
operator, and since it is invertible, it follows that the inverse is also a pseudodifferential
operator (recall that Ψ(M) is a C∗-algebra). To prove that (D∗D + PkerD)−1/2is a
pseudodifferential operator, it remains to use the formula
A−1/2=1
π
∞
?
0
λ−1/2(A + λ)−1dλ
for a self-adjoint strongly positive operator A (e.g., see [16, p. 165]).
2. The operators D in (2.1) and (2.2) are self-adjoint operators acting in a ∗-module
over the C∗-algebra C(M) and have the properties
[D,f] ∈ K,(D2− 1)f ∈ K.
Thus we have the Fredholm modules
[D] =
?
[D,W0,0(M,E)] ∈ K1(M)
[D,W0,0(M,E ⊕ F)] ∈ K0(M)
if D = D∗
if D ?= D∗.
(2.3)
Remark 2.2. If only the interior symbol of D is elliptic, then there is a well-defined
element in the K-homology of the open smooth part of M:
[D] ∈ K∗(M \ X). (2.4)
Here in the definition of the elements (2.1) and (2.2) one should replace the expression
(PkerD+D∗D)−1/2by an arbitrary self-adjoint edge-degenerate pseudodifferential operator
with interior symbol (σ(D)∗σ(D))−1/2.
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3 The Homotopy Classification
We recall the standard equivalence relation on the set of pseudodifferential operator acting
in sections of vector bundles, namely, stable homotopy.
Definition 3.1. Two operators
D : W0,0(M,E) → W0,0(M,F),D′: W0,0(M,E′) → W0,0(M,F′)
are said to be stably homotopic if they are homotopic modulo stabilization by vector
bundle isomorphism, i.e., if there exists an continuous homotopy of elliptic operators
D ⊕ 1E0∼ f∗?D′⊕ 1F0
where E0,F0∈ Vect(M) are vector bundles and
?e∗,
e : E ⊕ E0−→ E′⊕ F0,f : F′⊕ F0−→ F ⊕ E0
are vector bundle isomorphisms.
Stable homotopy is an equivalence relation on the set of all elliptic edge-degenerate
pseudodifferential operators acting in sections of vector bundles. By Ell0(M) we denote
the set of elliptic operators modulo stable homotopies. This set is a group with respect to
the direct sum of elliptic operators. The inverse element corresponds to an almost inverse
(i.e., an inverse modulo compact operators), and the unit is the equivalence class of trivial
operators.
In a similar way, one defines odd elliptic theory Ell1(M) as the group of stable homo-
topy equivalence classes of elliptic self-adjoint operators. Here the class of trivial operators
consists of Hermitian isomorphisms of vector bundles.
The homotopy classification problem for elliptic operators on the manifold M is the
problem of computing the groups Ell∗(M).
The following theorem solves the classification problem for manifolds with edges and
is the main result of the paper.
Theorem 3.2. There is an isomorphism
Ell∗(M)
ϕ
≃ K∗(M),
which takes each elliptic operator D to the element defined in Proposition 2.1.
Corollary 3.3. Two elliptic operators D1and D2are stably rationally homotopic if and
only if they have the same indices with coefficients in an arbitrary bundle on M:
ind(1 ⊗ p)(D1⊗ 1N)(1 ⊗ p) = ind(1 ⊗ p)(D2⊗ 1N)(1 ⊗ p), (3.1)
where p ∈ Mat(N × N,C(M)) is a matrix projection.
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This follows from the nondegeneracy (on the free parts of the groups) of the natural
pairing K0(M) × K0(M) −→ Z, which is just defined by the formula (3.1).
Proof of the theorem. The mapping is well defined, since homotopies of elliptic operators
give rise to homotopies of the corresponding Fredholm modules, i.e., result in the same
element in K-homology. Bundle isomorphisms give degenerate modules.
Let us now prove that the mapping is an isomorphism. We split the proof into three
stages.
1. Reduction of Ell-groups to K-groups of C∗-algebras (see [2]).
edge-degenerate operators in sections of vector bundles as operator generated by the pair
of algebras
C∞(M) ⊂ Ψ(M)
We interpret
of scalar operators. The embedding corresponds to the conventional action of functions
as multiplication operators. Namely, an arbitrary edge-degenerate pseudodifferential op-
erator of order zero can be represented in the form
D′: ImP −→ ImQ,
where P = P2and Q = Q2are matrix projections with coefficients in the function algebra
C∞(M) and D′is a matrix operator whose entries belong to the operator algebra Ψ(M).
We obtain a group isomorphic to Ell(M) if, instead of operators with smooth symbols,
we consider operators whose symbols are only continuous, i.e., pass to the closure Ψ(M)
of the algebra of pseudodifferential operators with respect to the operator norm. (The
fact that these groups are isomorphic follows from Theorem 1.9). By Σ
denote the algebra of continuous symbols.
The results of [2] give the isomorphisms3
def
= Ψ(M)/K we
Ell∗(M)
χ
≃ K∗(Con(C(M) → Σ)).
Here
Con(A
f
−→ B) =
?
(a,b(t)) ∈ A ⊕ C0([0,1),B) | f(a) = b(0)
?
is the cone of the algebra homomorphism f : A → B. In the odd case, one can rewrite
the K-group in the form
K1(Con(C(M) → Σ)) ≃ K0(Σ)/K0(M).
The composition of the last isomorphism with χ is a generalization of the Atiyah–Patodi–
Singer isomorphism [19]; i.e., self-adjoint elliptic operators modulo stable homotopy are
isomorphic to symbols-projections modulo projections determining sections of bundles.
3Note that if the edges are absent, then the isomorphism is just the Atiyah–Singer difference construc-
tion [1].
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2.
commutative diagram
A diagram relating K-theory of algebras and K-homology. Consider the
0 → Σ0 −→
↑
0 →0
Σ
↑
σ
−→ C(S∗M) → 0
↑
=C(M) −→ C(M)→ 0
(3.2)
with exact rows. Here σ is the interior symbol and Σ0= kerσ is the corresponding ideal.
The diagram induces the exact sequence
0 → SΣ0−→ Con(C(M) → Σ) −→ C0(T∗M) → 0(3.3)
of the cones of the vertical homomorphisms. Here S stands for the suspension: SΣ0=
C0((0,1),Σ0).
For brevity, we set A = Con(C(M) → Σ).
The key point of the proof is the construction of the commutative diagram
K∗+1
c
(T∗M)
↓
∂→
A
∂→
K∗(SΣ0)
↓
K∗(X)
→
B
→
K∗(A)
↓
K∗(M)
→
C
→
K∗
c(T∗M)
↓
K∗(M \ ∂M)
∂→
D
→
K∗+1(SΣ0)
↓
K∗+1(X)K∗+1(M \ ∂M)
(3.4)
relating the exact sequence induced by (3.3) in K-theory of algebras to the exact sequence
of the pair X ⊂ M in K-homology.
The vertical arrows in (3.4) are induced by quantizations. Namely, the elements of
K-groups in the upper row correspond to some symbols, and the vertical mappings take
these symbols to the corresponding operators. More precisely,
• the mappings K∗
quantizations (see the preceding section);
c(T∗M) → K∗(M \ ∂M) and K∗(A) → K∗(M) are determined by
• the mappings K∗(SΣ0) −→ K∗(X) are induced by quantization of edge symbols.
In more detail, the mapping K1(Σ0) −→ K0(X) takes an edge symbol 1 + u(x,ξ),
u ∈ Σ0, invertible on S∗X to the operator
?
in the space L2(X,K0,0(KΩ)) on the infinite wedge. The quantization is well defined,
since u(x,ξ) has a compact fiber variation (the interior symbol of the edge symbol is
zero). The mapping K0(Σ0) −→ K1(X) takes a self-adjoint edge symbol-projection
p(x,ξ) to the operator
?
1 + ux,−i∂
∂x
?
(3.5)
2p x,−i∂
∂x
?
− 1. (3.6)
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3. An application of the five lemma.
mutes, it follows from the five lemma that the middle vertical arrow is an isomorphism,
since the quantizations K∗
manifold M with boundary and are isomorphisms (e.g., see [8]). The isomorphism of edge
quantizations K∗(SΣ0) → K∗(X) will be established below in Sec. 4.
The fact that the diagram commutes can be established as follows:
Once we prove that the diagram (3.4) com-
c(T∗M) → K∗(M\∂M) are defined on the interior of the smooth
• The square C in (3.4) commutes automatically, since the horizontal arrows are
forgetful mappings;
• The fact that the square B commutes will be proved in Sec. 4.
• The fact that the square A , including the boundary maps, commutes will be verified
in Sec. 5.
This completes the proof of the theorem up to the above-mentioned computations.
4 Computations on the Edge
1. The isomorphism K∗(Σ0) ≃ K∗+1(X). Consider the diagram
→ K∗(S∗X) →
?
→ K∗(S∗X) → K∗+1
K∗(Σ0)
↓ L
(T∗X) → K∗
→ K∗
c(X × R)
?
c(X × R) →,
→
c
(4.1)
which compares the K-theory sequence corresponding to the short exact sequence
0 → C(S∗X,K) −→ Σ0
σc
−→ C0(X × R,K) → 0(4.2)
(where σcis the conormal symbol) with the sequence of K-groups of the pair S∗X ⊂ B∗X
formed by the unit sphere and ball bundles in T∗X. The mapping L is the difference
constructions for pseudodifferential operators (3.5), (3.6) with operator-valued symbols
in the sense of Luke (see [14] and [20]).
Lemma 4.1. The diagram (4.1) commutes.
Proof. 1. The commutativity of the squares
K∗(S∗X) −→
?
K∗(S∗X) −→ K∗+1(T∗X)
K∗(Σ0)
↓ L
follows from the fact that for finite-dimensional symbols the difference constriction coin-
cides with the Atiyah–Singer difference constriction.
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2. The commutativity of the squares
K∗(Σ0)
↓ L
K∗+1
c
σc
−→ K∗
c(X × R)
?
K∗+1(X),(T∗X)
j∗
−→
j : X → T∗X,
follows from the index formula [21]
β indDy= indσc(Dy) ∈ K1
c(Y × R) (4.3)
for a family of elliptic operators Dy, y ∈ Y , with unit interior symbol on the infinite cone.
Here Y is a compact parameter space and β is the periodicity isomorphism K(Y ) ≃
K1
The formula (4.3) applies directly to the group K1(Σ0), and for the K0-group one uses
the suspension (cf. (3.6)).
3. The commutativity of the squares
c(Y × R).
K∗+1
c
(X × R)
?
K∗(X)
∂
−→ K∗(S∗X)
?
p∗
−→ K∗(S∗X)
,p : S∗X → X,
also follows from the above-mentioned index formula, since the boundary mapping in
K-theory of algebras is an index mapping. We leave details to the reader.
By applying the five lemma, we arrive at the desired corollary.
Corollary 4.2. The quantization K∗(Σ0) → K∗+1(X) is an isomorphism: K∗(Σ0) ≃
K∗+1
c
(T∗X) ≃ K∗+1(X).
2. Commutativity of the square B .
odd case can be treated in a similar way.
The image of the composite mapping K0(SΣ0) → K0(A) → K0(M) corresponds to
elliptic operators on M of the form 1 + G, where G is an operator with zero interior
symbol. We must show that the element
To be definite, we consider the even case. The
[1 + G : W0,0(M) → W0,0(M)] ∈ K0(M)
coincides with the element
?
1 + g
?
x,−i∂
∂x
?
: W0,0(W) −→ W0,0(W)
?
∈ K0(M)
determined by the operator on the infinite wedge W with symbol g(x,ξ) = σΛ(G). For
the latter operator, the module structure on the spaces is defined as follows: a function
f ∈ C(M) acts as the multiplication by its restriction to the edge.
The equality of these two elements can be established in two steps:
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