Stable cohomotopy Seiberg-Witten invariants of connected sums of four-manifolds with positive first Betti number. II: Applications (Q1688579)

This paper is the continuation of [Int. J. Math. 26, No. 6, Article ID 1541004, 23 p. (2015; Zbl 1322.57024)], where a generalization of a nonvanishing theorem for stable cohomotopy Seiberg-Witten invariants has been proved. In the current article the authors give various applications of their non-vanishing theorem to the differential geometry and topology of 4-manifolds. The four theorems proved in the paper refer to the existence of exotic smooth structures, smooth connected sum decomposition of 4-manifolds and computations of Perelman's \(\overline{\lambda}\) invariant. For \(X\) a closed smooth Riemannian 4-manifold, \(b^+(X)\) (resp. \(b^-(X)\)) denotes the dimension of the maximal positive (resp.negative) definite linear subspace in the second cohomology of \(X\). Then \(\chi(X)\) and \(\tau(X)\) denote respectively the Euler characteristic and the signature of \(X\), \(c_1^2(X)=2\chi(X)+3\tau(X)\), and \(SW_X\), \(BF_X\) are respectively the Seiberg-Witten invariant and the stable cohomotopy Seiberg-Witten invariant of \(X\). The notion of BF-admissible 4-manifold is the one defined by the authors in the first part of the article [op. cit.]. Theorem A. Let \(X\) be a closed, simply connected, non-spin, symplectic 4-manifold with \(b^+(X)\equiv 3\) (mod 4), homeomorphic but not diffeomorphic to \(Y=p\mathbb{C}\mathbb{P}^2\#q\overline{\mathbb{C}\mathbb{P}}^2\). Let \(X_m\) be BF-admissible 4-manifolds with \(b^+(X_m)-b_1(X_m)>1\) for \( m=1,2\). Then, after connected summing both \(X\) and \(Y\) with either \(X_1\) or \(X_1\#X_2\), we still get homeomorphic but not diffeomorphic 4-manifolds. Theorem B. For \(m=1,2,3\), let \(X_m\) be BF-admissible 4-manifiolds with \(b^+(X_m)-b_1(X_m)>1\) and set \(X:=\#_{m=1}^nX_m\), where \(n=2,3\). Then \(X\) is never diffeomorphic to \(\#_{m=1}^NY_m\) with \(b^+(Y_m)>0\) and \(N>n\). Theorem D. Let \(N\) be a closed oriented smooth 4-manifold with \(b^+(N)=0\). For \(m=1,2,3\), let \(X_m\) be BF-admissible 4-manifolds with \(b^+(X_m)-b_1(X_m)>11\). For \(n=2,3\), \(M:=(\#_{m=1}^nX_m)\#N\) cannot admit any Einstein metric if \(4n-c^2_1(N)\geq\frac{1}{3}\sum_{m=1}^nc^2_1(X_m)\). Theorem C is an inequality which refers to an invariant \(\overline{\lambda}_k(X)\), that generalizes Perelman's \(\overline{\lambda}\) invariant (\(\overline{\lambda}_1=\overline{\lambda}\) ), in the case \(X=(\#_{m=1}^nX_m)\#N\), with the \(X_m\) being BF-admissible 4-manifolds satisfying \(b^+(X_m)-b_1(X_m)>1\) and \(N\) being a closed oriented manifold with \(b^+(N)=0\).