Higher rho-invariants for the signature operator A survey and - - PowerPoint PPT Presentation

Higher rho-invariants for the signature operator A survey and perspectives

Charlotte Wahl

Hannover

Copenhagen, 11-15/6/2018

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 1 / 20

Basics in noncommutative index theory: A noncommutative Chern character

Given A C ∗-algebra

• p. e. A = C(B), B closed manifold

A∞ ⊂ A “smooth” subalgebra (=closed under holomorphic functional calculus, dense, etc.) A∞ = C ∞(B)

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 2 / 20

Basics in noncommutative index theory: A noncommutative Chern character

Given A C ∗-algebra

• p. e. A = C(B), B closed manifold

A∞ ⊂ A “smooth” subalgebra (=closed under holomorphic functional calculus, dense, etc.) A∞ = C ∞(B)

• ne gets

a Z Z-graded Fr´ echet algebra ˆ Ω∗A∞ of noncommutative differential forms with differential d : ˆ ΩkA∞ → ˆ Ωk+1A∞ a Chern character ch : K∗(A) → HdR

∗ (A∞)

HdR

∗ (A∞) pairs with continuous reduced cyclic cocycles on A∞

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 2 / 20

Basics in noncommutative index theory: A noncommutative Chern character

Given A C ∗-algebra

• p. e. A = C(B), B closed manifold

A∞ ⊂ A “smooth” subalgebra (=closed under holomorphic functional calculus, dense, etc.) A∞ = C ∞(B)

• ne gets

a Z Z-graded Fr´ echet algebra ˆ Ω∗A∞ of noncommutative differential forms with differential d : ˆ ΩkA∞ → ˆ Ωk+1A∞ a Chern character ch : K∗(A) → HdR

∗ (A∞)

HdR

∗ (A∞) pairs with continuous reduced cyclic cocycles on A∞

Motivating example used in higher index theory: Γ finitely generated group with length function, A = C ∗Γ, A∞ the Connes-Moscovici algebra

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 2 / 20

Dirac operators over C ∗-algebras

Given (M, g) closed oriented Riemannian manifold E → M hermitian bundle with Clifford action and compatible connection (Z Z/2-graded, if dim M even) P ∈ C ∞(M, Mn(A∞)) projection we get an A-vector bundle F := P(An × M) → M and a (odd) Dirac operator DF : C ∞(M, E ⊗ F) → C ∞(M, E ⊗ F).

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 3 / 20

Dirac operators over C ∗-algebras

Given (M, g) closed oriented Riemannian manifold E → M hermitian bundle with Clifford action and compatible connection (Z Z/2-graded, if dim M even) P ∈ C ∞(M, Mn(A∞)) projection we get an A-vector bundle F := P(An × M) → M and a (odd) Dirac operator DF : C ∞(M, E ⊗ F) → C ∞(M, E ⊗ F). Important example for higher index theory: the Mishchenko C ∗Γ-vector bundle: F = ˜ M ×Γ C ∗Γ with Γ = π1(M). E = S the spinor bundle (gives twisted spin Dirac operator) E = Λ∗(T ∗M) (gives twisted de Rham or signature operator).

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 3 / 20

Index theory

The Dirac operator DF is Fredholm on the Hilbert A-module L2(M, E ⊗ F) with ind(DF) ∈ K∗(A).

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 4 / 20

Index theory

The Dirac operator DF is Fredholm on the Hilbert A-module L2(M, E ⊗ F) with ind(DF) ∈ K∗(A).

Proposition (Atiyah-Singer index theorem)

ch(ind(DF)) =

• M

ˆ A(M) ch(E/S) ch(F) ∈ HdR

∗ (A∞) .

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 4 / 20

Index theory

The Dirac operator DF is Fredholm on the Hilbert A-module L2(M, E ⊗ F) with ind(DF) ∈ K∗(A).

Proposition (Atiyah-Singer index theorem)

ch(ind(DF)) =

• M

ˆ A(M) ch(E/S) ch(F) ∈ HdR

∗ (A∞) .

Application in higher index theory: If DF is the signature operator twisted by F = ˜ M ×Γ C ∗Γ , then ind(DF) is homotopy invariant. The proposition implies: By pairing ch(ind(DF)) with cyclic cocycles one gets higher signatures. This can be used to prove the Novikov conjecture for Gromov hyperbolic groups (Connes–Moscovici 1990).

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 4 / 20

Secondary invariants

Let A be a smoothing symmetric operator on L2(M, E ⊗ F) such that DF + A is invertible. (A should be odd if dim M is even.) Then one can define η(DF, A) ∈ ˆ Ω∗A∞/[ˆ Ω∗A∞, ˆ Ω∗A∞] + d ˆ Ω∗A∞ generalizing the classical η-invariant (with A = C) η(DF, A) = 1 √π ∞ t− 1

2 Tr(DFe−t(DF+A)2)dt . Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 5 / 20

Secondary invariants

Let A be a smoothing symmetric operator on L2(M, E ⊗ F) such that DF + A is invertible. (A should be odd if dim M is even.) Then one can define η(DF, A) ∈ ˆ Ω∗A∞/[ˆ Ω∗A∞, ˆ Ω∗A∞] + d ˆ Ω∗A∞ generalizing the classical η-invariant (with A = C) η(DF, A) = 1 √π ∞ t− 1

2 Tr(DFe−t(DF+A)2)dt .

Higher η-invariants were introduced by Lott (1992). The general definition is implicit in work of Lott (1999).

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 5 / 20

Atiyah–Patodi–Singer index theorem

Let M be an oriented Riemannian manifold with cylindric end Z = I R+ × ∂M. On Z all structures are assumed of product type. If dim M is even, then on Z D+

F = c(dx)(∂x − D∂ F) .

Let A be a symmetric, smoothing operator on L2(∂M, E + ⊗ F) such that D∂

F + A is invertible.

Let χ: M → I R be smooth, supp χ ⊂ Z; supp(χ − 1) compact.

Proposition (W., 2009)

ch ind(D+

P − c(dx)χ(x)A) =

• M

ˆ A(M) ch(E/S) ch(F) − η(D∂

F, A) .

The proposition generalizes the higher APS index theorem proven by Leichtnam–Piazza (1997–2000).

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 6 / 20

Higher ρ-invariants for the signature operator (Leichtnam–Piazza, Wahl)

M oriented Riemannian manifold with dim M = 2m − 1. Assume that the m-th Novikov Shubin invariant of M is ∞+. Let DF be the signature operator twisted by the Mishchenko bundle F. Let I be the involution which is 1 on forms of degree < m and −1 on the

• complement. Then for t small DF + tI is invertible.

We define ρ(M) := [η(DF, tI)] ∈ ˆ Ω∗A∞/[ˆ Ω∗A∞, ˆ Ω∗A∞] + d ˆ Ω∗A∞ + ˆ Ωe

∗ A∞.

Product formula: If both sides are defined, then ρ(M × N) = ρ(M) ch(sign(N)) .

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 7 / 20

Generalization to an almost flat setting I (Azzali–W., work in progress)

Given c ∈ H2(Γ) one may construct a Mishchenko bundle Fσs associated to a twisted group C ∗-algebra C ∗(Γ, σs) with [σs] = eisc ∈ H2(Γ, U(1)). The bundle is almost flat. The C ∗-algebras assemble to an upper-semicontinuous field. Crucial property: If a differential operator associated to that field is invertible at s = 0, then it is invertible for s near 0. In this case we can guarantee that the signature operator twisted by Fσs can be perturbed to an invertible operator and define the higher ρ-invariants. They have the usual properties: Metric independence, product formula. Some properties follow from the flat case using the Hanke–Schick method

• f comparison.

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 8 / 20

Generalization to an almost flat setting II

(Fn, ∇n) sequence of Hermitian bundles with connection such that (∇n)2 n→∞ → 0. Can we construct an invertible perturbation for the signature operator such that it leads to a well-defined ρ-invariant?

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 9 / 20

Generalization to an almost flat setting III

For geometric applications, one wants to assign a ρ-invariant to a map f : M → BΓ, thus better start with quasirepresentations of Γ instead of flat bundles: Fix a finite set F ⊂ Γ and ε > 0. A map φ: Γ → A is an (F, ε)-unitary representation if φ(e) = 1 and it holds that φ(g) ≤ 1, ∀g ∈ Γ , φ(g−1) − φ(g)∗ ≤ ε ∀g ∈ F , φ(gh) − φ(g)φ(h) ≤ ε ∀g, h ∈ F .

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 10 / 20

Generalization to an almost flat setting III

For geometric applications, one wants to assign a ρ-invariant to a map f : M → BΓ, thus better start with quasirepresentations of Γ instead of flat bundles: Fix a finite set F ⊂ Γ and ε > 0. A map φ: Γ → A is an (F, ε)-unitary representation if φ(e) = 1 and it holds that φ(g) ≤ 1, ∀g ∈ Γ , φ(g−1) − φ(g)∗ ≤ ε ∀g ∈ F , φ(gh) − φ(g)φ(h) ≤ ε ∀g, h ∈ F . For manifolds with psc metrics we define ρ-invariants for the spin Dirac

• perators twisted by almost flat bundles associated to a sequence of

(Fn, 1

n)-unitary representations (see work by Dardalat).

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 10 / 20

Generalization to an almost flat setting III

For geometric applications, one wants to assign a ρ-invariant to a map f : M → BΓ, thus better start with quasirepresentations of Γ instead of flat bundles: Fix a finite set F ⊂ Γ and ε > 0. A map φ: Γ → A is an (F, ε)-unitary representation if φ(e) = 1 and it holds that φ(g) ≤ 1, ∀g ∈ Γ , φ(g−1) − φ(g)∗ ≤ ε ∀g ∈ F , φ(gh) − φ(g)φ(h) ≤ ε ∀g, h ∈ F . For manifolds with psc metrics we define ρ-invariants for the spin Dirac

• perators twisted by almost flat bundles associated to a sequence of

(Fn, 1

n)-unitary representations (see work by Dardalat).

For the signature operator additional conditions may be necessary, e. g. completely positive asymptotic representations of C ∗Γ.

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 10 / 20

Higher ρ-invariants for homotopy equivalences

Let M, N be odd-dimensional oriented closed Riemannian manifolds, f : M → N a smooth orientation preserving homotopy equivalence. A = C ∗Γ, Γ = π1(N). FN = ˜ N ×Γ C ∗Γ Mishchenko bundle, FM = f ∗FN. DF signature operator on N ∪ Mop twisted by FN ∪ FM.

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 11 / 20

Higher ρ-invariants for homotopy equivalences

Let M, N be odd-dimensional oriented closed Riemannian manifolds, f : M → N a smooth orientation preserving homotopy equivalence. A = C ∗Γ, Γ = π1(N). FN = ˜ N ×Γ C ∗Γ Mishchenko bundle, FM = f ∗FN. DF signature operator on N ∪ Mop twisted by FN ∪ FM. There is a smoothing operator A(f ) such that DF + A(f ) is invertible (Hilsum–Skandalis 1992, Piazza–Schick 2007).

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 11 / 20

Higher ρ-invariants for homotopy equivalences

Let M, N be odd-dimensional oriented closed Riemannian manifolds, f : M → N a smooth orientation preserving homotopy equivalence. A = C ∗Γ, Γ = π1(N). FN = ˜ N ×Γ C ∗Γ Mishchenko bundle, FM = f ∗FN. DF signature operator on N ∪ Mop twisted by FN ∪ FM. There is a smoothing operator A(f ) such that DF + A(f ) is invertible (Hilsum–Skandalis 1992, Piazza–Schick 2007). Let ˆ Ωe

∗ (A∞) = Cg0 d g1 . . . d gm | g0g1 . . . gm = e ⊂ ˆ

Ω∗A∞.

Definition

ρ(f ) := [η(DF, A(f ))] ∈ ˆ Ω∗A∞/[ˆ Ω∗A∞, ˆ Ω∗A∞] + d ˆ Ω∗A∞ + ˆ Ωe

∗ A∞ .

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 11 / 20

Analytic gluing along homotopic boundaries

Assume that M, N are even-dimensional oriented Riemannian manifolds with boundary and a homotopy equivalence f : ∂M ≃ ∂N. Then we can define an analytic signature signan(M ∪∂ Nop) by using A(f ) for the definition of the boundary conditions for the signature operator.

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 12 / 20

Analytic gluing along homotopic boundaries

Assume that M, N are even-dimensional oriented Riemannian manifolds with boundary and a homotopy equivalence f : ∂M ≃ ∂N. Then we can define an analytic signature signan(M ∪∂ Nop) by using A(f ) for the definition of the boundary conditions for the signature operator. The L-groups Ln(Z ZΓ) can be represented by such pairs: More specifically by a normal map F : W → [0, 1] × M such that the restriction to ∂0W is a diffeomorphism and the restriction to ∂1W a homotopy equivalence. Thus we get an induced map signan : Ln(Z ZΓ) → K0(C ∗Γ).

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 12 / 20

Analytic gluing along homotopic boundaries

Assume that M, N are even-dimensional oriented Riemannian manifolds with boundary and a homotopy equivalence f : ∂M ≃ ∂N. Then we can define an analytic signature signan(M ∪∂ Nop) by using A(f ) for the definition of the boundary conditions for the signature operator. The L-groups Ln(Z ZΓ) can be represented by such pairs: More specifically by a normal map F : W → [0, 1] × M such that the restriction to ∂0W is a diffeomorphism and the restriction to ∂1W a homotopy equivalence. Thus we get an induced map signan : Ln(Z ZΓ) → K0(C ∗Γ). Open question: Does the map for n = 4k agree with the standard map using quadratic forms and periodicity?

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 12 / 20

Properties

1 By pairing with suitable traces one recovers ρAPS(N) − ρAPS(M)

• resp. ρL2(N) − ρL2(M) (Piazza–Schick 2007).

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 13 / 20

Properties

1 By pairing with suitable traces one recovers ρAPS(N) − ρAPS(M)

• resp. ρL2(N) − ρL2(M) (Piazza–Schick 2007).

2 If the higher ρ-invariants are defined, then ρ(f ) = ρ(M) − ρ(N). Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 13 / 20

Properties

1 By pairing with suitable traces one recovers ρAPS(N) − ρAPS(M)

• resp. ρL2(N) − ρL2(M) (Piazza–Schick 2007).

2 If the higher ρ-invariants are defined, then ρ(f ) = ρ(M) − ρ(N). 3 (Product formula) If N = N1 × X, M = M1 × X, f = f1 × idX, then

ρ(f ) = ρ(f1) ch(ind(DFX )).

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 13 / 20

Properties

1 By pairing with suitable traces one recovers ρAPS(N) − ρAPS(M)

• resp. ρL2(N) − ρL2(M) (Piazza–Schick 2007).

2 If the higher ρ-invariants are defined, then ρ(f ) = ρ(M) − ρ(N). 3 (Product formula) If N = N1 × X, M = M1 × X, f = f1 × idX, then

ρ(f ) = ρ(f1) ch(ind(DFX )).

4 The following diagram is well-defined and commutes

Ln+1(Z ZΓ)

• signan
• S(N)

ρ

• Kn+1(C ∗Γ)

ch

ˆ

Ω∗A∞/[ˆ Ω∗A∞, ˆ Ω∗A∞] + d ˆ Ω∗A∞ + ˆ Ωe

∗ A∞ .

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 13 / 20

Applications

dim N = 4k − 1, k ≥ 2, Γ = π1(N) with torsion

Proposition (Chang–Weinberger 2003)

There are homotopy equivalences fi : Mi → N, i ∈ I N such that ρL2(Mi) = ρL2(Mj), i = j. Thus [(Mi, fi)] are distinct in S(N).

Corollary

Let X be a closed manifold with a non-zero higher signature. Assume that π1(X), Γ are Gromov hyperbolic. Then [(Mi × X, fi × idX)] are distinct in S(N × X) and distinguished by ρ(fi × idX).

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 14 / 20

Proposition

Assume that N is a closed oriented connected odd-dimensional manifold whose fundamental group is a product Γ = Γ1 × Γ2. We assume that the following conditions hold:

1 Γ1 contains a nontrivial torsion element, 2 there are k, m ∈ I

N with k ≥ 2 and dim N + 1 − m = 4k such that Hm(BΓ2, Q) = 0,

3 Γ2 = Z

Zm or Γ1, Γ2 have property (RD) and Γ2 has in addition property (PC). Then S(N) is infinite. Proof adapts methods by Leichtnam–Piazza for psc manifolds.

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 15 / 20

Heuristic application

Heuristic Prop.

Assume that Γ has torsion. Then Z Z{g | g = e, g has finite order and g of polynomial growth} acts freely on S(N) if dim N = 4k − 1, k ≥ 2.

Heuristic proof.

The trace τg is continuous on the Connes–Moscovici algebra A∞ ⊂ C ∗Γ. Let n be the order of g. The projections Pg = 1

n

• k=1 gk define

elements in L0(QΓ) ⊗ Q ∼ = L0(Z ZΓ) ⊗ Q ∼ = L4k(Z ZΓ) ⊗ Q. We get lifts pg ∈ L4k(Z ZΓ), and τg(ph) = 0 if and only if g = h. Thus τg can be used to build a dual basis to the images of the pg in K0(C ∗Γ) ⊗ C.

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 16 / 20

Heuristic proof cont.

Now the assertion follows from the diagram L4k(Z ZΓ)

• S(N)

ρ

• K0(C ∗Γ) ⊗ C ch
• τg
• ˆ

Ω∗A∞/[ˆ Ω∗A∞, ˆ Ω∗A∞] + d ˆ Ω∗A∞ + ˆ Ω<e>

∗

A∞

τg

• C

=

C .

Problem: Need that signan agrees with the standard map L4k(Z ZΓ) → K0(C ∗Γ). Remark: This application is inspired by arguments of Weinberger–Yu and also follows in stronger form from their work.

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 17 / 20

Comparing signan to the standard map: A Kaminker–Miller type strategy

Let M, N be manifolds with homotopic boundary.

1 Associate to them simplicial complexes Ms, Ns with a Poincar´

e duality structure (and further appropriate properties). There is a “homotopy equivalence” FM : Ms → M at the level of complexes (using the Whitney map).

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 18 / 20

Comparing signan to the standard map: A Kaminker–Miller type strategy

Let M, N be manifolds with homotopic boundary.

1 Associate to them simplicial complexes Ms, Ns with a Poincar´

e duality structure (and further appropriate properties). There is a “homotopy equivalence” FM : Ms → M at the level of complexes (using the Whitney map).

2 We should be able to define signan(Ms ∪∂ Ns,opp),

signan(Ms ∪FM Mopp), signan(Ns ∪FN Nopp), signan(M ∪∂ Nopp) using the Hilsum–Skandalis formalism.

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 18 / 20

Comparing signan to the standard map: A Kaminker–Miller type strategy

Let M, N be manifolds with homotopic boundary.

1 Associate to them simplicial complexes Ms, Ns with a Poincar´

e duality structure (and further appropriate properties). There is a “homotopy equivalence” FM : Ms → M at the level of complexes (using the Whitney map).

2 We should be able to define signan(Ms ∪∂ Ns,opp),

signan(Ms ∪FM Mopp), signan(Ns ∪FN Nopp), signan(M ∪∂ Nopp) using the Hilsum–Skandalis formalism.

3 Then the following additivity property should hold:

signan(Ms∪FMMopp)+signan(M∪∂Nopp) = signan(Ms∪∂Ns,opp)+signan(N

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 18 / 20

Comparing signan to the standard map: A Kaminker–Miller type strategy

Let M, N be manifolds with homotopic boundary.

1 Associate to them simplicial complexes Ms, Ns with a Poincar´

e duality structure (and further appropriate properties). There is a “homotopy equivalence” FM : Ms → M at the level of complexes (using the Whitney map).

2 We should be able to define signan(Ms ∪∂ Ns,opp),

signan(Ms ∪FM Mopp), signan(Ns ∪FN Nopp), signan(M ∪∂ Nopp) using the Hilsum–Skandalis formalism.

3 Then the following additivity property should hold:

signan(Ms∪FMMopp)+signan(M∪∂Nopp) = signan(Ms∪∂Ns,opp)+signan(N

4 It follows that signan(M ∪∂ Nopp) = signan(Ms ∪∂ Ns,opp). Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 18 / 20

Comparing signan to the standard map: A Kaminker–Miller type strategy

Let M, N be manifolds with homotopic boundary.

1 Associate to them simplicial complexes Ms, Ns with a Poincar´

e duality structure (and further appropriate properties). There is a “homotopy equivalence” FM : Ms → M at the level of complexes (using the Whitney map).

2 We should be able to define signan(Ms ∪∂ Ns,opp),

signan(Ms ∪FM Mopp), signan(Ns ∪FN Nopp), signan(M ∪∂ Nopp) using the Hilsum–Skandalis formalism.

3 Then the following additivity property should hold:

signan(Ms∪FMMopp)+signan(M∪∂Nopp) = signan(Ms∪∂Ns,opp)+signan(N

4 It follows that signan(M ∪∂ Nopp) = signan(Ms ∪∂ Ns,opp). 5 Identify signan(Ms ∪∂ Ns,opp) ∈ K0(C ∗Γ) with the image of the

Ranicki L-theoretic signature in K0(C ∗Γ).

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 18 / 20

Generalization to the almost flat setting

should be possible for twisted group C ∗-algebras as well as almost flat bundles by using the Hilsum–Skandalis framework

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 19 / 20

Generalization to the almost flat setting

should be possible for twisted group C ∗-algebras as well as almost flat bundles by using the Hilsum–Skandalis framework In the almost flat setting the relation to the ordinary resp. higher ρ-invariants cannot be proven by directly adapting the methods from the flat case: Limit process by Piazza–Schick uses flatness of the connection.

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 19 / 20

Further open questions

What is the connection with the Higson–Roe map “from surgery to analysis” (2004)? Ln+1(Z ZΓ)

• S(N)
• N(N)
• Ln(Z

ZΓ)

• Kn+1(C ∗

r Γ)

Kn+1(D∗

ΓN)

Kn(BΓ) Kn(C ∗

r Γ)

the analytic version by Piazza–Schick?

• ther interpretations of ρ-invariants (Higson–Roe, Deeley–Goffeng,

Weinberger–Xie–Yu)?

Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 20 / 20