  We shall prove a new non-vanishing theorem for the stable cohomotopy
Seiberg-Witten invariant of connected sums of 4-manifolds with positive first
Betti number. The non-vanishing theorem enables us to find many new examples of
4-manifolds with non-trivial stable cohomotopy Seiberg-Witten invariants and it
also gives a partial, but strong affirmative answer to a conjecture concerning
non-vanishing of the invariant. Various new applications of the non-vanishing
theorem are also given. For example, we shall introduce variants
$\bar{\lambda}_k$ of Perelman's $\bar{\lambda}$ invariants for real numbers $k$
and compute the values for a large class of 4-manifolds including connected
sums of certain K{\"{a}}hler surfaces. The non-vanishing theorem is also used
to construct the first examples of 4-manifolds with non-zero simplicial volume
and satisfying the strict Gromov-Hitchin-Thorpe inequality, but admitting
infinitely many distinct smooth structures for which no compatible Einstein
metric exists. Moreover, we are able to prove a new result on the existence of
exotic smooth structures.
