Homotopy invariance of twisted higher signatures on manifolds with boundary
摘要
Let M be an oriented compact manifold with boundary. We assume that TTi (M) is the product of a non-trivial finite group F and of a group F which is either of polynomial growth or Gromov hyperbolic. We fix a non-trivial representation p: F -> U{) and let Ep be the associated unitary flat bundle on M^We denote by M the universal cover of M and we consider the F-Galois covering TT-.M/F -^ M, the lifted flat bundle Ep = 7r*(-Ep) and the associated twisted signature operator. Under the additional assumption that the induced twisted signature operator on the boundary of M /F is -Z^-invertible, Lott has introduced in [L2] the twisted higher signatures of M; our main result, a positive answer to a form of Novikov conjecture on manifolds with boundary, is that these are homotopy invariants of the pair {M^QM). The proof depends heavily on the 6-^?-pseudodifferential calculus developed in [LP1], on the higher APS index theorem of [LP1] (here extended so as to cover Gromov-hyperbolic groups) and on a classical result of Kaminker-Miller, stating the equality of the index classes associated to two homotopy-equivalent hermitian Fredholm complexes. RESUME. -INVARIANCE PAR HOMOTOPIE DES HAUTES SIGNATURES "TWISTEES" SUR DES VARIETES A BORD. -Soit M une variete compacte orientee a bord. On suppose que 71-1 (M) est Ie produit d'un groupe fini non trivial F et d'un groupe F qui est soit a croissance polynomiale, soit hyperbolique au sens de Gromov. On se donne une representation non triviale p:F -^ U() et on considere Ie fibre plat unitaire associe Ep. On designe par M Ie revetement universel de M et on considere Ie revetement F-galoisien TT'.M/F -> M, Ie fibre plat releve Ep = Tr*{Ep) et Poperateur de signature twiste associe. Sous Phypothese supplementaire que Poperateur de signature twiste induit sur Ie bord de M/F est Z^-inversible, Lott a introduit dans [L2] les hautes signatures twistees de M. Notre resultat principal -une reponse positive a une conjecture de type Novikov pour le