**Abstract. **We study mathematical programs with vanishing constraints (MPVCs) from a topological point of view. We introduce the new concept of a T-stationary point for MPVC. Under the Linear Independence Constraint Qualification we derive an equivariant Morse Lemma at nondegenerate T-stationary points.

Then, two basic theorems from Morse Theory (deformation theorem and cell-attachment theorem) are proved. Outside the T-stationary point set, continuous deformation of lower level sets can be performed. As a consequence, the topological data (such as the number of connected components) then remain invariant. However, when passing a T-stationary level, the topology of the lower level set changes via the attachment of a *q*-dimensional cell. The dimension *q* equals the stationary T-index of the (nondegenerate) T-stationary point. The stationary T-index depends on both the restricted Hessian of the Lagrangian and the number of bi-active vanishing constraints.

Further, we prove that all T-stationary points are generically nondegenerate. The latter property is shown to be stable under C^{2}-perturbations of the defining functions. Finally, some relations with other stationarity concepts, such as strong, weak, and M-stationarity, are discussed.

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