Shriek map

Abstract: In this paper we define K-theoretic secondary invariants attached to a Lie groupoid $G$. The K-theory of $C^*_r(G_{ad}^0)$ (where $G_{ad}^0$ is the adiabatic deformation $G$ restricted to the interval $[0,1)$) is the receptacle for K-theoretic secondary invariants. We give a Lie groupoid version of construction given by Piazza and Schick in the setting of the Coarse Geometry. Our construction directly generalises to more involved geometrical situation, such as foliations, well encoded by a Lie groupoid. Along the way we tackle the problem of producing a wrong-way functoriality between adiabatic deformation groupoid K-groups with respect to transverse maps. This extends the construction of the lower shriek map given by Connes and Skandalis. Moreover we attach a secondary invariant to the two following operators: the signature operator on a pair of homotopically equivalent Lie groupoids; the Dirac operator on a Lie groupoid equipped with a metric that has positive scalar curvature $s$-fiber-wise. Furthermore we prove a Lie groupoid version of the Delocalized APS Index Theorem of Piazza and Schick. Finally we give a product formula for the secondary invariants and we state stability results about cobordism classes of Lie groupoid structures and bordism classes of Lie groupoid metric with positive scalar curvature along the $s$-fibers. This is the revised version accepted by Advances in Mathematics.

Abstract: We construct an unbounded representative for the shriek class associated to the embeddings of spheres into Euclidean space. We equip this unbounded Kasparov cycle with a connection and compute the unbounded Kasparov product with the Dirac operator on $\mathbb R^{n+1}$. We find that the resulting spectral triple for the algebra $C(\mathbb S^n)$ differs from the Dirac operator on the round sphere by a so-called index cycle, whose class in $KK_0(\mathbb C, \mathbb C)$ represents the multiplicative unit. At all points we check that our construction involving the unbounded Kasparov product is compatible with the bounded Kasparov product using Kucerovsky's criterion and we thus capture the composition law for the shriek map for these immersions at the unbounded KK-theoretical level.

Abstract: We establish equalities between cochain and chain type levels of maps by making use of exact functors which connect appropriate derived and coderived categories. Relevant conditions for levels of maps to be finite are extracted from the equalities which we call duality on the levels. Moreover, we give a lower bound of the cochain type level of the diagonal map on the classifying space of a Lie group by considering the ghostness of a shriek map which appears in derived string topology. A variant of Koszul duality for a differential graded algebra is also discussed.