Higher localised <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mover accent="true"><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:math>-genera for proper actions and applications

Proper actions and proper invariant metrics

We show that if a locally compact group G acts properly on a locally compact σ-compact space X, then there is a family of G-invariant proper continuous finite-valued pseudometrics which induces the topology of X. If X is, furthermore, metrizable, then G acts properly on X if and only if there exists a G-invariant proper compatible metric on X.

Singular Reduction for Proper Actions

Geometric Quantization for Proper Actions

We first introduce an invariant index for G-equivariant elliptic differential operators on a locally compact manifold M admitting a proper cocompact action of a locally compact group G. It generalizes the Kawasaki index for orbifolds to the case of proper cocompact actions. Our invariant index is used to show that an analog of the Guillemin-Sternberg geometric quantization conjecture holds if M...

Kk-theoretic Duality for Proper Twisted Actions

Let the discrete group G act properly and isometrically on the Riemannian manifold X. Let C0(X, δ) be the section algebra of a smooth locally trivial G-equivariant bundle of elementary C∗-algebras representing an element δ of the Brauer group BrG(X). Then C0(X, δ ) ⋊ G is KK-theoretically Poincaré dual to ( C0(X, δ)⊗̂C0(X)Cτ (X) ) ⋊ G, where δ is the inverse of δ in the Brauer group. We deduce t...

Higher-order Genera of Knots

For certain classes of knots we define geometric invariants called higher-order genera. Each of these invariants is a refinement of the slice genus of a knot. We find lower bounds for the higherorder genera in terms of certain von Neumann ρ-invariants, which we call higher-order signatures. The higher-order genera offer a refinement of the Grope filtration of the knot concordance group.