COARSE GEOMETRY

G. Yu

23. 5. 2002

Sommerschule Göttingen

Let Γ be a finitely generated (just for convenience, countable would still be ok) discrete group, let S be a finite symmetric generating set (g ∈ S iff g^{−1} ∈ S). We can define a word length ℓ on Γ by:

ℓ(γ) = \{ \begin{matrix}0 & if γ = id \\ min\{ n | γ = g_{i_{1}} ⋯ g_{i_{n}}, g_{i_{k}} ∈ S \} & otherwise. \end{matrix}

Define a word metric d on Γ by

d(γ_{1}, γ_{2}) = ℓ(γ_{1}γ − 2^{−1}) .

From now on, let Γ be a locally finite metric space, i.e., #B(γ, r) < ∞ for all γ ∈ Γ, r > 0.

0.1. Example. Γ a finitely generated group with a word metric.

0.2. Example. M a non-compact Riemannian manifold, Γ ⊂ M is a net, i.e., Γ is locally finite and there exists c ≥ 0 such that Γ is c-dense in M, so d(x, Γ) ≤ c for all x ∈ M.

1. A geometric algebra associated to Γ (John Roe).

Let H be a separable and infinite dimensional Hilbert space.

ℓ^{2}(Γ) = { f: Γ ⟶ ℂ | ∑_{γ∈Γ} |f(γ)| < ∞ } .

Consider ℓ^{2}(Γ) ⊗ H, let T: ℓ^{2}(Γ) ⊗ H → ℓ^{2}(Γ) ⊗ H be a bounded linear operator. We can write:

T = (T_{γγ′})_{γγ′∈Γ}

with respect to ℓ^{2}(Γ) ⊗ H = ⨁_{γ∈Γ}(δ_{γ} ⊗ H).

1.1. Definition.

(1) T is called locally compact if T_{γγ′} is compact for all γ, γ′ ∈ Γ.

(2) The propagation of T is defined to be the smallest r such that T_{γγ′} = 0 if d(γ, γ′) > r.

Typeset by AMS-TEX

These notes were taken during the lecture. They are provided “as is”, with no guarantee for correctness whatsoever.

If the propagation of T is finite, then T is said to have finite propagation.

Let

ℂ |Γ| = { T: ℓ^{2}(Γ) ⊗ H ⟶ ℓ^{2}(Γ) ⊗ H | T is locally compact and has finite propagation } .

1.2. Example. Let Γ = ℕ, let T be an infinite matrix which is zero except at a few off-diagonals.

1.3. Example. Let Γ be a finite metric space, then ℂ |Γ| = K(ℓ^{2} ⊗ H) ≅ ℂ |pt| = K(H).

The coarse C^{∗}-algebra.

C^{∗} |Γ| = \overline{ℂ |Γ|} ,

completed with respect to the operator norm.

Let Γ_{1}, Γ_{2} be locally finite metric spaces, a map f: Γ_{1} → Γ_{2} is called a coarse map if

(1) f is proper, i.e., the preimage of a finite set is finite,

(2) for every r > 0 there exists R ≥ 0 such that if d(x, y) ≤ r, then d(f(x), f(y)) ≤ R.

1.4. Definition. Γ_{1} and Γ_{2} are said to be coarse equivalent if there exist coarse maps f: Γ_{1} → Γ_{2}, g: Γ_{2} → Γ_{1} such that there exists C ≥ 0 such that d(g ∘ f(x), x) ≤ C, d(f ∘ g(y), y) ≤ C for all x ∈ Γ − 1, y ∈ Γ − 2.

1.5. Example. Let Γ_{1} be a finite metric space, then Γ_{1} is coarse equivalent to a point.

1.6. Proposition If Γ_{1} and Γ−2 are coarse equivalent, then ℂ |Γ_{1}| ≅ ℂ |Γ_{2}| and C^{∗} |Γ_{1}| ≅ C^{∗} |Γ_{2}|.

Proof. If f: Γ_{1} → Γ − 2 is a coarse map, then there exists an injective isometry V_{f}: ℓ^{2}(Γ_{1}) ⊗ H → ℓ_{2}(Γ_{2}) ⊗ H satisfying

V_{f}(δ_{x} ⊗ H) ⊂ δ_{f(x)} ⊗ H .

This is not uniquely defined.

Define a homomorphism

Ad(V_{f}): C^{∗} |Γ_{1}| ⟶ C^{∗} |Γ_{2}| , T ⟼ V_{f}T V_{f}^{∗} .

From this, construct isomorphisms …

1.7. Question. What are the K-groups K_{∗}(C^{∗} |Γ|)?

1.8. Example. Let Γ = ℕ, then K_{∗}(C^{∗}ℕ) = 0 (Eilenberg’s swindle). There is a shift map f: ℕ → ℕ, n ↦ n + 1, so there is a homomorphism W = V_{f}: ℓ^{2}(ℕ) ⊗ H → ℓ^{2}(ℕ) ⊗ H. Let H^{∞} = H ⊕ H ⊕ … (infinitely many copies). Embed

i: C^{∗}(ℕ, ℓ^{2}(ℕ) ⊗ H) ⟶ C^{∗}(ℕ, ℓ^{2}(ℕ) ⊗ H^{∞}) , T ⟼ T ⊕ 0 ⊕ 0 ⊕ …

Claim: i_{∗}: K_{∗}(C^{∗}(ℕ, ℓ^{2}(ℕ) ⊗ H)) → K_{∗}(C^{∗}(ℕ, ℓ^{2}(ℕ) ⊗ H^{∞})) is an isomorphism.

For all [p] ∈ K_{0}(C^{∗}ℕ), p a projection in C^{∗}ℕ of finite propagation, it is enough to show that i_{∗}[p] = 0, i.e., [p ⊕ 0 ⊕ …] = 0 in K_{∗}(C^{∗}(ℕ, ℓ^{2}(ℕ) ⊗ H^{∞})). Regard

p ⊕ W pW^{∗} ⊕ W^{2}pW^{2∗} ⊕ ⋯ ∈ C^{∗}(ℕ, ℓ^{2}(ℕ) ⊗ H) ,

this operator still has the same propagation as p, and is still locally compact:

W pW^{∗}(δ_{1} ⊗ H) = 0 ,  (W^{2}pW^{2∗})(δ_{x} ⊗ H) = 0  if k ≤ 2,

so at any x ∈ N, you see at most finitely many of the compact operators. In K_{∗}(C^{∗}ℕ),

[p ⊕ W pW^{∗} ⊕ W^{2}pW^{2∗} ⊕ …]

∼ ( \begin{matrix}W \\ & 1 \end{matrix} ) ( \begin{matrix}p \\ & 0 \end{matrix} ) ( \begin{matrix}W^{∗} \\ & 1 \end{matrix} ) ⊕ ( \begin{matrix}W \\ & 1 \end{matrix} ) ( \begin{matrix}W pW^{∗} \\ & 0 \end{matrix} ) ( \begin{matrix}W^{∗} \\ & 1 \end{matrix} ) ⊕ …

∼ ( \begin{matrix}1 \\ & W \end{matrix} ) ( \begin{matrix}p \\ & 0 \end{matrix} ) ( \begin{matrix}1 \\ & W^{∗} \end{matrix} ) ⊕ ( \begin{matrix}1 \\ & W \end{matrix} ) ( \begin{matrix}W pW^{∗} \\ & 0 \end{matrix} ) ( \begin{matrix}1 \\ & W^{∗} \end{matrix} ) ⊕ …

∼ [W pW^{∗} ⊕ W^{2}pW^{2∗} ⊕ …]

Subtracting [W pW^{∗} ⊕ W^{2}pW^{2∗} ⊕ …] on both sides proves that p ⊕ 0 ∼ 0.

2. Positive scalar curvature

Let M be a Riemannian spin manifold, let D = e_{i} · ∇_{e_{i}} be the Dirac operator, then

D^{2} = ∇^{∗}∇ + \frac{k}{4} .

If k ≥ c > 0, then D is invertible. Construct a higher index ind(D) ∈ K_{∗}(C^{∗}Γ), where Γ is a net in M. This index then is an obstruction to invertibility of D, and hence to uniformly positive scalar curvature.

2.1. Example. X = T^{n} cannot carry a positive scalar curvature metric. Regard M = X̃ = ℝ^{n}; it turns out that ind(D) ≠ 0 in K_{∗}(C^{∗} |ℤ^{n}|). The K-theory and the indices only depend on the coarse geometry of M, which is independent of the Riemannian metric.

Let S → M be the spinor bundle, then D acts on L^{2}(M, S). Define

F = \frac{D}{\sqrt{1 + D^{2}}} ,

which is a bounded operator. Then (L^{2}(M, S), F, φ) is a K-homology class in KK(C_{0}(M), ℂ), where φ: C_{0}(M) → L(L^{2}(M, S)) is given by

f ⟼ M_{f} , (M_{f}ξ)(x) = f(x) ξ(x)

for all ξ ∈ L^{2}(M, S). Then (1) F^{∗} = F;

(2) F φ(f) − φ(f)F is compact (because diffential operators are local …);

(3) (F^{2} − 1)φ(f) compact.

Let {U_{γ}}_{γ} be a locally finite open cover of M such that there exists R > 0 satisfying diam(U_{γ}) ≤ R for all γ ∈ Γ and γ ∈ U_{γ}. Let {ψ_{γ}}_{γ} be a partition of unity associated to {U_{γ}}_{γ}. Define a perturbation

F′ = ∑_{γ} ψγ^{\frac{1}{2}} F ψγ^{\frac{1}{2}} .

Then (L^{2}(M, S), F′, φ) ∼ (L^{2}(M, S), F, phy) as K-homology classes, but F′ has finite propagation speed.

Let {V_{γ}}_{γ} be a disjoint collection of Borel subsets of M satisfying

(1) M = ⋃_{γ} V_{γ};

(2) γ ∈ V_{γ} for all γ;

(3) there exists R ≥ 0 such that diam V_{γ} ≤ R for all γ.

(4) The volume of V_{γ} is nonzero for all γ.

Then

L^{2}(M, S) = ⨁_{γ}(V_{γ}, S) ≅ ℓ^{2}(Γ) ⊗ H ,

so F′ has finite propagation.

Let L_{fp}(L^{2}(M, S)) be the C^{∗}-algebra generated by finite perturbation operators on L^{2}(M, S), with respect to the operator norm. We have a short exact sequence

0 ⟶ C^{∗} |Γ| ⟶ L_{fp}(L^{2}(M, S)) ⟶ L_{fp}(L^{2}(M, S)) / C^{∗} |Γ| ⟶ 0 .

In particular, F′ ∈ L_{fp}(L^{2}(M, S)), and F′ is invertible in F′ ∈ L_{fp}(L^{2}(M, S))/C^{∗} |Γ|, so it defines a class in K_{1}(L_{fp}(L^{2}(M, S)) / C^{∗} |Γ|), which gives a K_{0}-index by the six-term exact sequence (sign!) The algebra C^{∗} |Γ| plays the role of the compact operators.

We define ind D = ind F′ ∈ K_{∗}(C^{∗} |Γ|). More generally, for any K-homology class (H, F, φ), we can define ind(H, F, φ) ∈ K_{∗}(C^{∗} |Γ|). Thus, we have defined an assembly map

μ: KK(C_{0}(M), ℂ) ⟶ K_{∗}(C^{∗} |Γ|)

(geometric Baum-Connes map). This index is in fact independent of all the choices made above. Note that KK(C_{0}(M), ℂ) is not invariant under coarse equivalence.

2.2. Example. Let M = ℝ^{n}, Γ = ℤ^{n}, so

μ: KK(C_{0}(ℝ^{n}), ℂ) ⟶ K_{∗}(C^{∗} |ℤ^{n}|) .

We claim that this map is an isomorphism, and the Dirac operator gives us a generator [D] ∈ KK(C_{0}(ℝ^{n}), ℂ) ≅ ℤ, in particular, [D] ≠ 0. If indeed μ is an isomorphism, then ind(D) ≠ 0, so T^{n} cannot carry a positive scalar curvature metric.

Mayer-Vietoris sequence for K_{∗}(C^{∗} |Γ|). We say that Γ = Γ_{1} ∪ Γ_{2} is ω-excisive if for all r ≥ 0, there exists R ≥ 0 such that

N(Γ_{1}, r) ∩ N(Γ_{2}, r) ⊂ N(Γ_{1} ∩ Γ_{2}, R) ,

where N(Γ, r) is the r-neighbourhood of Γ.

2.3. Proposition (Higson-Roe-Yu). If Γ = Γ_{1} ∪ Γ_{2} is ω-excisive, then the following sequence is exact,

⋯ ⟶ K_{0}(C^{∗} |Γ_{1} ∩ Γ_{2}|) ⟶ K_{0}(C^{∗} |Γ_{1}|) ⊕ K_{0}(C^{∗} |Γ_{2}|) ⟶
⟶ K_{0}(C^{∗} |Γ|) ⟶ K_{1}(C^{∗} |Γ_{1} ∩ Γ_{2}|) ⟶ …

Proof of Example 2.1, 2.2. Assume n = 1. Let ℤ_{±} = { n ∈ ℤ | ±n ≥ 0}, then ℤ = ℤ_{+} ∪ ℤ_{−} is ω-excisive. We have Mayer-Vietoris sequences

… ⟶ K^{0}(C_{0}({0})) ⟶ K^{0}(C_{0}(ℝ_{+})) ⊕ K^{0}(C_{0}(ℝ_{+})) ⟶ K^{0}(C_{0}(ℝ)) ⟶ …
≅↓^{μ}  ≅↓^{μ}  ↓
… ⟶ K_{0}(C^{∗} |{0}|) ⟶ K_{0}(C^{∗} |ℤ_{+}|) ⊕ K_{0}(C^{∗} |ℤ_{−}|) ⟶ K_{0}(C^{∗} |ℤ|) ⟶ …

The KK-group KK_{0}(C_{0}(ℝ_{+}), ℂ) = K^{0}(C_{0}(ℝ_{+})) vanishes by an exact sequence involving the closed and open half-line, or just by the long exact sequence in the top row. The claim follows from the five-lemma.

For higher n, the K-theory associated with half spaces and half lattices vanishes by the same shifting argument as before. □

3. Rips’ complex

Let Γ be a locally finite metric space, let C^{∗} |Γ| be the coarse (Roe) algebra. If X is a proper metric space (every closed ball is compact, i.e., the functions d_{x} are proper for x ∈ X), and Γ is a net in X, then there is a Baum-Connes map (index map)

μ: K_{∗}(X) = KK(C_{0}(X), ℂ) ⟶ K_{∗}(C^{∗} |Γ|) .

The left hand side is easier to compute because the Mayer-Vietoris sequence for the left hand side is better behaved than the one for the right. In general, μ will not be an isomorphism. In order to obtain all elements on the right hand side, one should look at all possible spaces X containing Γ as a net, or build a universal example X.

For all d ≥ 0, we define P_{d}(Γ) to be the simplicial complex with

(1) set of vertices V = Γ,

(2) simplices {γ_{0}, …, γ_{n}} ⊂ Γ whenever d(γ_{i}, γ_{j}) ≤ d for all 1 ≤ i, j ≤ n. The Rips’ complex carries a natural simplicial metric, and Γ is a net in P_{d}(Γ).

For example, P_{1}(ℤ) = ℝ. In general, the homotopy type of P_{d}(Γ) depends on d and does not stabilise as d → ∞.

The system { P_{d}(Γ) }_{d≥0} is universal for spaces where Γ is a net. Regard the Baum-Connes map

μ: K_{∗}(P_{d}(Γ)) ⟶ K_{∗}(C^{∗} |Γ|) .

3.1. Conjecture (Coarse Baum-Connes conjecture). The map

μ: lim_{d→∞} K_{∗}(P_{d}(Γ)) ⟶ K_{∗}(C^{∗} |Γ|)

is an isomorphism.

The limit process on the left hand side makes it coarse invariant.

3.2. Conjecture (Gromov-Lawson). If M is a compact aspherical manifold (M̃ is contractible), then M cannot carry a Riemannian metric of positive scalar curvature.

The coarse Baum-Connes conjecture for Γ = π_{1}M (as a metric space with a word metric) implies the Gromov-Lawson conjecture. In fact, rational injectivity of the map μ would be enough. We have

lim_{d→∞} K_{∗}(P_{d}(Γ)) ≃ K_{∗}(M̃) .

If μ: K_{∗}(M̃) → K_{∗}(C^{∗} |Γ|) is an isomorphism, then the index of the Dirac operator [D] ∈ K_{∗}(M̃) would be nonzero, so M cannot have positive scalar curvature.

4. The Novikov conjecture

Let M be a smooth closed manifold.

4.1. Definition. We say that M is rigid if whenever another smooth closed manifold M′ is homotopy equivalent to M, then M is homeomorphic to M′.

For example, lense spaces are not rigid. The Poincaré conjecture asks for the rigidity of spheres.

4.2. Conjecture (Borel). All aspherical manifolds are rigid.

4.3. Theorem (Novikov). The rational Pontrijagin classes are topological invariants (invariant under homeomorphisms).

The Borel conjecture and Novikov’s theorem together imply the Novikov conjecture for aspherical manifolds.

4.4. Conjecture (Novikov conjecture for aspherical manifolds). The rational Pontrijagin classes of aspherical manifolds are homotopy invariant.

In general, one would have to deal with higher signatures.

4.5. Theorem (Descent principle, Pedersen-Carlsson, Higson-Roe). The coarse Baum-Connes conjecture for Γ (as a metric space with the word metric) implies the Novikov conjecture.

4.6. Conjecture (Strong Novikov conjecture for aspherical manifolds). The map

μ: K_{∗}(M) = K_{∗}^{Γ}(\underline{E}Γ) = K_{∗}(BΓ) ⟶ K_{∗}(C_{r}^{∗}Γ)

is rationally injective.

4.7. Proposition The coarse Baum-Connes conjecture for Γ as a metric space implies the strong Novikov conjecture for Γ as a group.

Let ℂΓ be the group ring of Γ. Recall that ℂΓ acts on ℓ^{2}Γ by

∑ c_{γ}u_{γ} ⟼ ∑ c_{γ}λ_{γ} ,

where (λ_{γ}ξ)(g) = ξ(γ^{−1}g). Define an inclusion homomorphism

i: ℂΓ ⟶ ℂ |Γ|

by

∑ c_{γ}u_{γ} ⟼ ∑ c_{γ}λ_{γ} ⟼ (∑ c_{λ}λ_{γ} ⊗ P_{0} ,

where P_{0} is a rank one projectioon in H. Recall that d(g_{1}g, g_{2}g) = g(g_{1}, g_{2}). This implies that the propagation of λ_{γ} is l(γ).

4.8. Lemma. ℂΓ⊗K(H) ≅ (ℂ |Γ|)^{Γ}, where the Γ-action on ℂ |Γ| is induced by the Gamma action on ℓ^{2}(Γ) ⊗ H from the right.

Proof. Exercise

In order to proof Proposition 4.7, we have to show that

μ: K_{∗}(M) ⟶ K_{∗}(C_{r}^{∗}Γ)

is injective. Because C_{r}^{∗}Γ is too difficult, regard instead the map

K_{∗}(M) ⟶^{μ} K_{∗}(C_{r}^{∗}Γ) = K_{∗}(C_{r}^{∗}Γ ⊗ K) ⟶^{≅} K_{∗}(C^{∗} |Γ|^{Γ}) ⟶ K_{∗}((C_{b}(M̃) ⊗ C^{∗} |Γ|)^{Γ}) ,

where C_{b} denotes bounded continuous functions. The algebra on the very right is a family of coarse algebras over M (Borel construction, homotopy quotient). We can use the coarse Baum-Connes conjecture fibre-wise to compute the right hand side and obtain

K_{∗}((C_{b}(M̃) ⊗ C^{∗} |Γ|)^{Γ}) ≅ K_{∗}(M̃ ×_{Γ} M̃) ≅ K_{∗}(M) ,

and the composition of all the maps above is the identity. In particular, the very first arrow must be injective. □

Let Γ be a finite metric space. The coarse Baum-Connes conjecture says that

μ: lim_{d→∞} K_{∗}(P_{d}(Γ)) ⟶ K_{∗}(C^{∗} |Γ|)

is an isomorphism. If one can prove surjectivity in a canonical way, then will injectivity follow.

Let M be a Riemannian spin manifold (for convenience), and let Γ ⊂ M be a net. Let D be the Dirac operator (or any other K-homology element), and set F = \frac{D}{\sqrt{1+D^{2}}} . Let \{ U_{γ} \}_{γ∈Γ} be an open cover of M of bounded diameter with γ ∈ U_{γ}, and let { ψ_{γ} }_{γ} be a subordinate partition of unity. Set

F′ = ∑_{γ∈Γ} ψγ^{\frac{1}{2}} F ψγ^{\frac{1}{2}} ,

then F′ has finite propagation. If Γ is sufficiently ε-dense, we can acchieve that the diameter of the U_{γ} become arbitraily small, and so the propagation of F′ becomes very small.

If T is a bounded operator on L^{2}(M), then

(1) the propagation of T is the smallest number r ≥ 0 such that fT g = 0 for all f, g ∈ C_{0}(M) with d(supp(f), supp(g)) > r;

(2) T is locally compact if fT and T f are compact operators for all f ∈ C_{0}(M). We define a C^{∗}-algebra C^{∗}(M, L^{2}(M)) to be the C^{∗}-algebra closure of the space of locally compact operators T of finite propagation. Then as a C^{∗}-algebra

C^{∗}(M, L^{2}(M)) ≅ C^{∗} |Γ|

for any net Γ ⊂ M.

We may thus forget about the net Γ. Given ε > 0, let { U_{i} }_{i} be any open cover with diam U_{i} < ε for all i. Then F′ has propagation at most ε But, one has to be very careful with bundles, e.g., one can identify L^{2}(M, S) ⊂ ⨁_{N} L^{2}(M).

Let C_{L}^{∗}(M) be the localisation algebra, generated by all f: [0, ∞) → C^{∗}(M, L^{2}(M)) such that (1) f is uniformly bounded and continuous,

(2) the propagation of f(t) goes to 0 as t → ∞. Define

F (t) = (t − n)F_{\frac{1}{n}} + (1 − (t − n)) F_{\frac{1}{n+1}} .

if t ∈ [n, n + 1], n ∈ ℤ_{+}.

Define the local index ind_{L} F = ind F (t) ∈ K_{∗}(C_{L}(M, L^{2}(M))). We have a local index map ind_{L},

(4.9)  lim_{d→∞} K_{∗}(P_{d}(Γ)) ⟶_{≅}^{ind_{L}} lim_{d→∞} K_{∗}(C_{L}^{∗}(P_{d}(Γ))) ⟶^{e} K_{∗}(C^{∗} |Γ|) ,

where e is induced by f ↦ f(0). To prove that ind_{L} is an isomorphism, you need a Mayer-Vietoris map. We want to understand which elements of K_{∗}(C^{∗} |Γ|) come from lim_{d→∞} K_{∗}(P_{d}(Γ)). But these elements have to admit “localisation” in lim_{d→∞} K_{∗}(C_{L}^{∗}(P_{d}(Γ))).

4.10. Example. Let M = ℝ^{2}, Γ = ℤ^{2}. A general operator has finite, but probably very large propagation. Consider the map g: x ↦ \frac{x}{2}. For [P] ∈ K_{0}(C^{∗}(ℝ^{2})), consider [Ad(V_{g})(P)] = [P], but the propagation of Ad(V_{g})(P) is half of the propagation of P. If P is a projection, it has to be approximated by a quasi-projection, which has finite propagation, and thus is in the image of the index map ind = e_{∗} ∘ ind_{L}. This process in fact works for all Hadamard manifolds.

4.11. Example. Let M = ℝ^{2}, Γ = ℤ^{2}. Cover M by fundamental domains for a rescaled ℤ^{2} actions, and declare that neighbouring vertices has distance 1. If in the original metric, d(x, y) ≤ R, then now, we have d(x, y) < ε. In order for this process to work, the complexity of the covering must not increase with this rescaling.

4.12. Definition. (Gromov). The large scale dimension of any metric space X is defined to be the smallest integer n such that for all r > 0, there exists a uniformly bounded cover { U_{i} }_{i} of X of uniformly bounded diameter, such that every ball with radius B(x, r) intersects at most n + 1 of the U_{i} for all x ∈ X.

4.13. Theorem. The coarse Baum-Connes conjecture holds for Γ if Γ has finite large scale dimension.

But, one has to be careful. The localisation process of (4.9) does not work. One needs a quantitative version of localisation.

4.14. Theorem. The coarse Baum-Connes conjecture holds for Γ if Γ is uniformly embeddable into a Hilbert space and has bounded geometry.

4.15. Definition. (Gromov). Let H be a Hilbert space. A map f: Γ → H is called a uniform embedding if there exist non-decreasing functions ρ_{1}, ρ_{2} on [0, ∞), such that

ρ_{1}(d(x, y)) ≤ ∥f(x) − f(y)∥_{H} ≤ ρ_{2}(d(x, y)) , (1)

lim_{r→∞} ρ_{1}(r) = ∞ . (2)

4.16. Definition. Γ has bounded geometry if for all r there exists an N such that #B(x, r) ≤ N for all x ∈ Γ.

4.17. Example. Let X be the disjoint union of S^{2n} over all n ∈ ℕ. Let d|_{S^{2n}} be n times the standard metric, and assume that d(S^{2n}, S^{2m}) ≥ n + M . There exists a Dirac operator D = ⨁_{n=1}^{∞} D_{n}, and a net Γ in X. Then

0 ≠ [D] ∈ lim_{d→∞} K_{∗}(P_{d}(Γ)) ≅ ∏_{n∈ℕ} ℤ / ⨁_{n∈ℕ} ℤ ,

but

0 = ind D ∈ K_{∗}(C^{∗} |Γ|) .

The scalar curvature is

k_{n} = \frac{2n(2n − 1)}{n^{2}} ≥ 2 ,

so D is invertible.