Surgery and index formulas on stratified manifolds

The elliptic theory of pseudodifferential operators (PDOs) on stratified manifolds with any stratification length, is proved. Any elliptic operator is Fredholm. A set of symbols on strata is the symbol of some elliptic operator if and only if the given symbols satisfy a certain condition, called the compatibility condition. In this situation, the principal symbol of a pseudodifferential operator is the set of symbols on strata. Each of these symbols is an operator-valued function defined on the cotangent bundle of the corresponding open stratum. The ellipticity condition on an operator means that the principal symbol must be invertible.