Indices of Toeplitz tuples on pseudoregular domains

This paper proves an index theorem of Toeplitz tuples on pseudoregular domains in Cⁿ. Geometrically, the index of Toeplitz tupleT _n is (-1) ⁿ time wrapping number of φ _n around the origin. As one of the applications of the index theorem, we completely characterize the automorphism groups of Toeplitz algebras on Poincaré domain. As another application, it is shown thatC*(Ω)≅C*(Bn) for every Poincare domain Ω inC ⁿ (n≠2). It is also noticed thatC*(Ω)≅C*(B ₂) if and only if the Poincaré conjecture is true for ∂Ω.