2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62693We compare the higher analytic torsion of Bismut and Lott of a fibre bundle p: M -> B equipped with a flat vector bundle F -> M and a fibre-wise Morse function h on M with a higher torsion T that is constructed in terms of a families Thom-Smale complex associated to h and F, thereby extending previous joint work with Bismut. Under additional conditions on F, the torsion T is related to Igusa's higher Franz-Reidemeister torsion. As an application, we use the higher analytic torsion to detect infinite families of smooth bundles p_i: M -> B with diffeomorphic fibres that are homeomorphic but not diffeomorphic as bundles.80 pages, AmSTeX, uses pstricks. v2: new abstract, more examples and applications. v3: 94 pages, major revision, submitted; Thm 0.2 and Sections 3.c, 5.d-5.f addedDifferential GeometryK-Theory and Homology58J52; 57R22, 57Q10text