2008/04/22 by Ishida, Masashi, Sasahira, Hirofumi
#53C25 #57R57 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.0804.3452
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 λk of Perelman's λ invariants for real numbers k and compute the values for a large class of 4-manifolds including connected sums of certain Kähler 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.