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We shall prove a new non-vanishing theorem for the stable cohomotopySeiberg-Witten invariant of connected sums of 4-manifolds with positive firstBetti number. The non-vanishing theorem enables us to find many new examples of4-manifolds with non-trivial stable cohomotopy Seiberg-Witten invariants and italso gives a partial, but strong affirmative answer to a conjecture concerningnon-vanishing of the invariant. Various new applications of the non-vanishingtheorem 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 connectedsums of certain K{\"{a}}hler surfaces. The non-vanishing theorem is also usedto construct the first examples of 4-manifolds with non-zero simplicial volumeand satisfying the strict Gromov-Hitchin-Thorpe inequality, but admittinginfinitely many distinct smooth structures for which no compatible Einsteinmetric exists. Moreover, we are able to prove a new result on the existence ofexotic smooth structures.

Stable Cohomotopy Seiberg-Witten Invariants of Connected Sums of Four-Manifolds with Positive First Betti Number