The study of topological superconductivity is largely based on the analysis of mean-field Hamiltonians that violate particle number conservation and have only short-range interactions. Although this approach has been very successful, it is not clear that it captures the topological properties of real superconductors, which are described by number-conserving Hamiltonians with long-range interactions. To address this issue, we study topological superconductivity directly in the number-conserving setting. We focus on a diagnostic for topological superconductivity that compares the fermion parity $mathcal{P}$ of the ground state of a system in a ring geometry and in the presence of zero vs. $Phi_{text{sc}}=frac{h}{2e} equiv pi$ flux of an external magnetic field. A version of this diagnostic exists in any dimension and provides a $mathbb{Z}_2$ invariant $ u=mathcal{P}_0mathcal{P}_{pi}$ for topological superconductivity. In this paper we prove that the mean-field approximation correctly predicts the value of $ u$ for a large family of number-conserving models of spinless superconductors. Our result applies directly to the cases of greatest physical interest, including $p$-wave and $p_x+ip_y$ superconductors in one and two dimensions, and gives strong evidence for the validity of the mean-field approximation in the study of (at least some aspects of) topological superconductivity.
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