In this paper we compute the σ-invariants (sometimes also called the smooth Yamabe invariants) of RP3 and RP2 × S 1 (which are equal) and show that the only prime 3-manifolds with larger σ-invariants are S 3 , S 2 × S 1 , and ˜ S 2 ×S 1 (the nonorientable S 2 bundle over S 1 ). More generally, we show that any 3-manifold with σ-invariant greater than RP3 is either S 3 , a connect sum with an S 2 bundle over S 1 , or has more than one nonorientable prime component. A corollary.
