  We complete the classification (started by Bray and the second author) of all
closed 3-manifolds with Yamabe invariant greater than that of $\RP^3$, by
showing that such manifolds are either $S^3$ or finite connected sums $# m(S^2
\times S^1) # n(S^2 \tilde{\times} S^1)$ for $m + n \geq 1$, where $S^2
\tilde{\times} S^1$ is the nonorientable $S^2$-bundle over $S^1$.
  A key ingredient is Aubin's Lemma, which says that if the Yamabe constant is
positive, then it is strictly less than the Yamabe constant of any of its
non-trivial finite conformal coverings. This lemma, combined with inverse mean
curvature flow and with analysis of the Green's functions for the conformal
Laplacians on specific finite and normal infinite Riemannian coverings, will
allow us to construct a family of nice test functions on the finite coverings
and thus prove the desired result.
