2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/74947When the index bundle of a longitudinal Dirac type operator is transversely smooth, we define its Chern character in Haefliger cohomology and relate it to the Chern character of the $K-$theory index. This result gives a concrete connection between the topology of the foliation and the longitudinal index formula. Moreover, the usual spectral assumption on the Novikov-Shubin invariants of the operator is improved.Geometric TopologyK-Theory and HomologyOperator Algebras57R30text