By definition ofind(A)for a gap continuous familyA:X → CF0(H), it is clear that the results from section 5.3 on the dimension of the support are verbatim true also in this case.
Hence we can state without further work the following theorem which is an immediate conse-quence of proposition 5.3.1.
6.3.1 Proposition. LetXbe a compact orientable manifold of dimensionnsuch thatH2k−1(X;Z) is free for some1≤k≤ n−12 . IfA:X→ CF0(H)is gap continuous and
ck(ind(A))6= 0∈H2k(X;Z),
then the dimension of the singular set Σ(A)is at least n−2kand it is not contractible to a point insideX.
Moreover, we obtain from proposition 5.3.4 and theorem 1.1.8:
6.3.2 Proposition. LetX be a connected and orientable manifold having aCW-structure. Let k∈ Nbe a natural number such that k≤ n−12 , Hj(X) = 0, 1≤j ≤2k−1, and π2k(X) = 0.
Finally, letA:X → CF0(H)be gap continuous having a compact singular set and A⊂X be a closed subspace such thatA∩Σ(A) =∅. Ifπ1(X\Σ(A))is abelian and there existsf ∈Ω2k(X, A) such that
f∗indA 6= 0∈K(I2k, ∂I2k)∼=Z, thendim Σ≥n−2kandΣ(A)is not contractible.
6.3.3 Remark. As in section 5.3 we want to point out that our estimates can be improved when working on real Hilbert spaces and defining the index bundle inKO-theory.
Chapter 7
s-ind and Spectral Flow
In this chapter we introduce a variant of the index bundle defined for families of generally unbounded selfadjoint Fredholm operators which corresponds to ind in the same way as the classical Atiyah-Jänich bundle toindswhich we introduced in section 4.2.
In the first section we give the definitions and prove basic properties. The second section is concerned with estimates of the dimension of the singular set of a given family of selfadjoint Fredholm operators. Here we can apply proposition 3.3.6 for the first time in order to conclude results on the dimension by homotopy classes of paths. In the final third section we show at first that the restriction of our selfadjoint index bundle to bounded families can be identified with inds canonically. Afterwards we prove the important result that the first Chern number of our selfadjoint index bundle of paths of Fredholm operators is the spectral flow [BLP05] as introduced in section 4.3.
7.1 The Selfadjoint Index Bundle
Let X be a compact topological space and A : X → CFsa(H) a gap continuous family of selfadjoint Fredholm operators. Thenind(A) = 0by lemma 6.2.4 and our aim is to modify the construction of the index bundle in order to make it accessible for selfadjoint operators as well.
We denote byΣ(A)⊂X the singular set ofAand define
A(x,s)u=Axu+isu, D(A(x,s)) =D(Ax), (x, s)∈X×R.
ThenAis a gap continuous family of closed operators on H according to theorem 2.2.1.
7.1.1 Lemma. Ais a family of Fredholm operators of index 0 whose singular set is
Σ(A) = Σ(A)× {0} ⊂X×R.
Proof. Let(x, s)∈X×R. Ifs= 0, thenA(x,0)=Axis a selfadjoint Fredholm operator and so in particular of index0. Moreover,A(x,0)does not have a bounded inverse if and only ifx∈Σ(A).
Ifs6= 0, thenA(x,s)u=Axu+isuhas a bounded inverse because otherwise06=−is∈σ(Ax)in contradiction to the selfadjointness ofAx.
7.1.2 Definition. Let A : X → CFsa(H) be a gap continuous family of selfadjoint operators parametrised by a compact spaceX andA⊂X a closed subspace such that A∩Σ(A) =∅. The selfadjoint index bundleofA is defined by
s-ind(A) = ind(A)∈K(X×R, A×R) =K−1(X, A).
Note that we do not give the most general definition here but restrict to compact spacesX for the sake of simplicity. Accordingly the singular setΣ(A)is automatically compact.
We assume throughout thatA:X → CFsa(H)is a gap continuous family andA⊂X a closed subspace of the compact spaceX such thatA∩supp(A) =∅.
7.1.3 Lemma (Normalisation). If A(X)⊂GCsa(H), thens-ind(A) = 0∈K−1(X, A).
Proof. SinceΣ(A) =∅this is an immediate consequence of lemma 6.2.1.
7.1.4 Lemma. Let Y be compact, B⊂Y closed and f : (Y, B)→(X, A)continuous. Then
s-ind(f∗A) =f∗s-ind(A)∈K−1(Y, B), where
f :Y ×R→X×R, f(y, s) = (f(y), s) Proof. At first observe that
(f∗A)(y,s)=Af(y)+is·I= (f∗A)(y,s), (y, s)∈Y ×R. Now we obtain
s-ind(f∗A) = ind(f∗A) = ind(f∗A) =f∗ind(A) =f∗s-ind(A), where the third equality follows by the corresponding property 6.2.2 ofind.
7.1.5 Lemma. If H :I×X → CFsa(H) is gap continuous andA∩Σ(Hλ) = ∅ for all λ∈I, then
s-ind(H0) = s-ind(H1)∈K−1(X, A).
Proof. Define
H :I×X×R→ C(H), H(t, x, s) =H(t, x) +is·I
which is a gap continuous family of closed operators by theorem 2.2.1. Moreover, we infer as in the proof of lemma 7.1.1 that H(t, x, s) ∈ CF0(H) for all (t, x, s) ∈ I ×X ×R and Σ(H) = Σ(H)× {0}. Now we finally obtain by lemma 6.2.3
s-ind(H0) = ind(H0) = ind(H1) = s-ind(H1).
By the corresponding discussion in section 6.2 we obtain the following invariance under com-pact perturbations.
7.1.6 Corollary. If K : X → K(H) is a family of compact selfadjoint operators such that A∩Σ(A+λ·K) =∅ for any0≤λ≤1, then
s-ind(A+K) = s-ind(A)∈K−1(X, A).
7.1.7 Lemma. Let A1,A2 : X → CFsa(H) be two gap continuous families such that their singular sets are disjoint fromA. Then
s-ind(A1⊕ A2) = s-ind(A1) + s-ind(A2)∈K−1(X, A).
Proof. SinceA1⊕ A2=A1⊕ A2, the assertion follows from 6.2.6.
Note that the logarithmic property makes no sense here, because the product of two selfadjoint operators is in general not even symmetric. Nevertheless we have the following result which is an immediate consequence of lemma 6.2.7.
7.1.8 Lemma. If U : X → U(H) is a continuous family of unitary operators, then U∗AU is gap continuous and
s-ind(U∗AU) = s-ind(A)∈K−1(X, A).
Now we discuss properties that hold exclusively in the selfadjoint case.
7.1.9 Lemma. IfA:X → BFsa(H)is a family of bounded selfadjoint Fredholm operators, then
s-ind(−A) =−s-ind(A)∈K−1(X, A).
Proof. We compute forz= (x, s)∈X×R
−AzAz= (−Ax+isI)(Ax+isI) =−A2x−s2I.
By using the homotopy
H :I×X×R→ CF(H), H(λ, x, s) =−A2x−(s2+λ)I and the homotopy invariance 6.2.3 we infer thatind(−A A) = 0. Since
D(−A A) =D(−A) =D(A) =X×H (7.1) according to corollary 6.1.14, we now can use the logarithmic property 5.2.7 and finally obtain
s-ind(−A) = ind(−A) =−ind(A) =−s-indA ∈K−1(X, A).
7.1.10 Remark. i) Since (7.1)is essential in the proof of lemma 7.1.9, we have not found a proof of the corresponding property for general gap continuous familiesA:X→ CFsa(H) yet.
ii) Note that in contrast to lemma 7.1.9 we have ind(−A) = ind(A) for all A:X → CF(H) which follows from the homotopy
H :I×X → CF(H), H(λ, x) =eiπλAx.
Before we can go on, we need an auxiliary result about selfadjoint operators.
7.1.11 Lemma. Let Abe a selfadjoint operator acting on the Hilbert spaceH.
i) IfB∈GL(H) is a bounded, invertible and selfadjoint operator, thenBAB is selfadjoint.
ii) If B ∈GL(H)is bounded, selfadjoint and either positive or negative, then A+iB has a bounded inverse.
Proof. i) Note at first thatBAB is densely defined becauseD(BAB) =B−1D(A)and hence its adjoint is defined.
Ifu, v∈ D(BAB), then
hBABu, vi=hABu, Bvi=hBu,ABvi=hu, BABvi,
where we use in the second equality thatBv ∈ D(A). Hence BAB is symmetric, which means
B−1D(A) =D(BAB)⊂ D((BAB)∗).
Now, letv∈ D((BAB)∗)be given. By definition this means that
D(BAB)3u7→ hBABu, vi ∈C is bounded and hence there exists a constantc >0such that
|hBABu, vi| ≤ckuk, u∈ D(BAB).
SinceB is invertible andD(BAB) =B−1D(A)we obtain for anyu∈ D(A)
|hAu, Bvi|=|hBAu, vi|=|hBABB−1u, vi| ≤ckB−1uk ≤ckB−1kkuk.
HenceBv∈ D(A∗) =D(A)implyingv∈B−1D(A) =D(BAB).
To sum up,BAB is symmetric and D((BAB)∗) =D(BAB)which implies that BAB is selfadjoint.
ii) First of all note that we can assume without loss of generality thatB is positive because otherwise we consider−(−A+i(−B)).
If we assumeB to be positive we can build its square rootB12 ∈ L(H)which is selfadjoint and invertible (cf. [Ka76, V.3.11]). We obtain
A+iB=A+iB12B12 =B12(B−12AB−12 +i·I)B12 (7.2) and since the operatorB−12AB−12 is selfadjoint on the domainB12D(A) by the first part of this lemma, we infer thatB−12AB−12 +i·Ihas a bounded inverse.
Now we consider the bounded operator
B−12(B−12AB−12 +i·I)−1B−12 ∈ L(H)
onH which mapsH ontoD(A). By using (7.2), we obtain
(A+iB)B−12(B−12AB−12 +i·I)−1B−12
=B12(B−12AB−12 +i·I)(B−12AB−12 +i·I)−1B−12 =IH
and
B−12(B−12AB−12 +i·I)−1B−12(A+iB)
=B−12(B−12AB−12 +i·I)−1(B−12AB−12 +i·I)B12 =ID(A). HenceA+iB has a bounded inverse.
The following result will be of high importance in the sequel. It shows that we actually have more freedom in the definition ofs-indwhich can be very useful in computations as we will see in the proof of the first part of theorem 9.1.1 below.
As before, we assumeA:X→ CFsa(H)to be gap continuous andA⊂Xto be closed such that A∩Σ(A) =∅.
7.1.12 Proposition. Let B : X → L(H) be a continuous family of selfadjoint non negative operators such thatAx+isBx has a bounded inverse for alls6= 0. Then
s-ind(A) = ind(A+isB)∈K−1(X, A).
Proof. Consider the homotopy
H(λ, x, s) =Ax+is(λI+ (1−λ)Bx), D(H(λ, x, s)) =D(Ax), (λ, x, s)∈I×X×R, which is gap continuous by theorem 2.2.1. SinceλI+ (1−λ)Bxis selfadjoint for eachλ∈I, x∈X, and
σ(λI+ (1−λ)Bx) =λ+ (1−λ)σ(Bx), λ∈I, x∈X,
these operators are positive if λ 6= 0. We obtain from lemma 7.1.11 that H(λ, x, s) has a bounded inverse wheneverλ6= 0ands6= 0. Since H(0, x, s) =Ax+isBx has by assumption a bounded inverse for alls6= 0we conclude thatΣ(H) =I×Σ(A)× {0}. Hence the homotopyH has a constant compact singular set inI×X×Rand we obtain from the homotopy invariance property 6.2.3
ind(A+isB) = ind(H(0)) = ind(H(1)) = ind(A+is·I) = s-ind(A)∈K−1(X, A).
7.1.13 Corollary. If B : X → L(H) is a continuous family of selfadjoint positive operators, then
s-ind(A) = ind(A+isB)∈K−1(X, A).
Proof. We only have to show that anyAx+isBxhas a bounded inverse ifs6= 0which follows from lemma 7.1.11.
If we assume for the moment thatA:X→ CFsa(H)is continuous with respect to the Riesz topology, then we can build s-ind(A) because every Riesz continuous family is gap continuous as well by lemma 4.3.1. On the other hand, we can also proceed as usual in the case of Riesz continuous families and considers-indof the bounded transformA(I+A2)−12 :X→ BFsa(H).
The next lemma ensures that we obtain the same elements inK-theory in both cases.
7.1.14 Lemma. Let A:X → CFsa(H) be Riesz continuous. Then
s-ind(A) = s-ind(A(I+A2)−12).
Proof. First of all, by definition of the Riesz topology and (4.4) we know that the maps
X 3x7→ Ax(I+A2x)−12 ∈ L(H) X 3x7→(I+A2x)−12 ∈ L(H)
(7.3) are continuous. Next we want to show that the map
M :X×R×H →D(A), (x, s, u)7→(I+A2x)−12u
is a Banach bundle isomorphism. We note at first that each Mz, z = (x, s) ∈ X ×R, is bijective and by lemma A.2.6 continuous with respect to the topology inD(A)zwhich is induced by the normkukH+k(Ax+isI)ukH. Moreover, ifτz0 denotes the trivialisation ofD(A)at some z0∈X×Ras defined in (6.2), we obtain for allz∈Uz0
(τz0◦M)z=Pgraph(A
z0)((I+A2x)−12·,Ax(I+A2x)−12 ·+is(I+A2x)−12·)∈ L(H,graph(Az0))
which is continuous in z due to the continuity of the maps (7.3). Now lemma 1.2.2 shows thatM is indeed a Banach bundle isomorphism.
Now we conclude from lemma 5.2.7 that
s-ind(A) = ind(A) = ind(A) + ind(M) = ind(A ◦M)∈K−1(X, A), where
(A ◦M)z=Ax(I+A2x)−12 +is(I+A2x)−12
and each (I+A2x)−12 is bounded and non negative. Since(A ◦M)z has for each s6= 0 the inverse(I+A2x)12(Ax+is)−1 which is bounded by corollary A.2.6, we obtain from proposition 7.1.12
ind(A ◦M) = s-ind(A(I+A2x)−12)∈K−1(X, A).
7.2 On the Dimension of the Singular Set
We now study which information we can gain from the selfadjoint index about the dimension of the singular set ofA.
We have for any compact topological spaceX an isomorphismK−1(X)∼= ˜K(Σ(X+))by lemma B.2.1, where, according to our notation,X+ denotes the union ofX and a disjoint point.
7.2.1 Definition. For a compact spaceX we define the oddk-th Chern class ˆck,k∈N, as the composition of the following maps
ˆ
ck :K−1(X) =K(X×R)−∼=→K(Σ(X˜ +))−→ck H2k(Σ(X+))−∼=→H2k−1(X).
We obtain immediately from the definition that ˆck is indeed a characteristic class in the ordinary sense:
7.2.2 Lemma. If X,Y are compact spaces andf :X →Y continuous, then
ˆ
ck(f∗η) =f∗ˆck(η), η ∈K−1(Y).
Proof. This follows from the corresponding property of ck, the naturality of the suspension isomorphismHn+1(ΣX)−∼=→Hn(X),n∈N, and lemma B.2.1.
7.2.3 Proposition. Let X be a compact orientable manifold of dimension nand 1≤k≤ n2 a natural number such that H2k−2(X) is free. LetA:X → CFsa(H) be a gap continuous family such that
ˆ
ck(s-ind(A))6= 0∈H2k−1(X).
Thendim Σ≥n−2k+ 1 andΣis not contractible to a point inX.
Proof. We define for any compact subspaceX0⊂X
σk(X0) = ˆck(s-ind(A ◦ιX0))∈H2k−1(X0),
whereιX0 : X0 ,→X denotes the inclusion. It is clear by lemma 7.1.3, 7.1.4 and 7.2.2 that σk satisfies the properties required in theorem 3.2.1.
LetA: (X, A)→(CFsa(H),GCsa(H))be a gap continuous family and letf ∈Ω2k−1(X, A) be given, whereA is a closed subspace of the compact spaceX. ThenΓ2k−1(f) := s-ind(f∗A) is an element inK−1(I2k−1, ∂I2k−1)∼=Z. In order to study the dimension of its singular set by this construction, we need as in section 5.3 the following technical result.
7.2.4 Lemma. Let f1, f2∈Ωk(X, Y) be such thatf1∗f2 exists. Moreover, assume that D(A) is a trivial bundle. Then
s-ind((f1∗f2)∗A) = s-ind(f1∗A) + s-ind(f2∗A).
Proof. We observe at first that by lemma 7.1.4 and the definition ofs-ind
s-ind((f1∗f2)∗A) =f1∗f2
∗s-ind(A) =f1∗f2
∗ind(A).
Hence as in the proof of lemma 5.3.3 we can use the normalisation and logarithmic property 5.2.1 and 5.2.8 in order to assume without loss of generality thatD(A)is a productX×H0 for some Hilbert spaceH0.
Now the rest of the proof is verbatim as the proof of lemma 5.3.3 if we add a further component s∈Rto each of the mapsf1, f2, f1∗f2, g, g1andg2which is, moreover, not even affected by the homotopies.
Note thatD(A)is in particular trivial ifX is a CW-complex by theorem 1.1.8. Because the remaining assumptions of proposition 3.3.6 and 3.3.7 are clear for Γ2k−1 by lemma 7.1.3 and lemma B.1.2, we obtain immediately the following two results.
7.2.5 Proposition. Let X be a simply connected manifold of dimension n ≥ 2 possessing a CW-complex structure andA: (X, A)→(CFsa(H), GCsa(H))a gap continuous family. If there existsf ∈Ω1(X, A)such thats-ind(f∗A)6= 0∈Z thendim Σ≥n−1 andΣis not contractible.
7.2.6 Proposition. Let X be a connected orientable manifold of dimensionnpossessing aCW structure and A : (X, A) → (CFsa(H), GCsa(H)) be gap continuous. Assume that for some k ∈ N, 2 ≤ k ≤ n2, Hj(X) = 0 for all 1 ≤ j ≤ 2k−2, π2k−1(X) = 0 and that there exists f ∈Ω2k−1(X, A)such thats-ind(f∗A)6= 0∈Z. Ifπ1(X\Σ)is abelian, thendim Σ≥n−2k+ 1 andΣis not contractible.