The Yamabe invariants of Inoue surfaces, Kodaira surfaces, and their blowups

On the spectral sets of Inoue surfaces

“…Remark 1.3. Inoue surfaces do not admit metrics of positive scalar metric, as was proved by Albanese [4,Theorem 4.5]. It also follows from Cecchini and Schick [10], making use of Hasegawa's result [15] that Inoue surfaces are solvmanifolds and hence are enlargeable in the sense of Gromov and Lawson [14].…”

“…32J15 53C55 57R57 58J50. The first author was partially supported by NSF Grants DMS-1811111 and DMS-1952790, and the second author was partially supported by NSF Grant DMS-1952762. of the matrices M to construct a fake RP 4 and interesting fibered 2-spheres in a homotopy 4-sphere. From this point of view, the manifolds S M are given by surgery on this homotopy sphere along those knots.…”

Kodaira Dimension and the Yamabe Problem, II

“…However, surfaces of class VII must also be excluded from Theorem A, because their Yamabe invariants are not always of the same sign. Indeed, while the Hopf surfaces discussed above have positive Yamabe invariant, a class of minimal class-VII surfaces discovered by Inoue [22] were shown by the first author [2] to have Yamabe invariant zero. We will call these examples Inoue-Bombieri surfaces, both because Inoue's paper credited Bombieri with their independent discovery, and to distinguish them from various other class-VII surfaces that also bear Inoue's name.…”

“…Our second proof, laid out in §3 below, instead deduces the lemma from a curvature estimate proved by Kronheimer [25], and is closer in spirit to [29,31] because of the leading role played by the Seiberg-Witten equations. We then go on, in §4, to explain why Theorems A and B require the exclusion of the Kod = −∞ case, while in the process giving a simplied proof of the main result of [2]. We then conclude, in §5, with a discussion of various open problems and pertinent related results.…”

On the Scalar Curvature of 4-Manifolds

“…Fortunately, combining the Schoen-Yau and Gromov-Lawson methods yields the following useful generalization of a theorem of Albanese [4,5]: Proposition 3.1. Let N 3 be a smooth compact oriented connected enlargeable 3-manifold, and let X be a smooth compact oriented 4-manifold that admits a smooth submersion ̟ : X → S 1 with fiber N. Let P be any smooth compact oriented 4-manifold, and let M = X#P .…”