Spin Manifolds, Einstein Metrics, and Differential Topology

Abstract

Abstract. We show that there exist smooth, simply connected, four-dimensional spin manifolds which do not admit Einstein metrics, but nevertheless satisfy the strict Hitchin-Thorpe inequality. Our construction makes use of the Bauer/Furuta cohomotopy refinement of the Seiberg-Witten invariant [4,3], in conjunction with curvature estimates previously proved by the second author [17]. These methods also allow one to easily construct many examples of topological 4-manifolds which admit an Einstein metric for one smooth structure, but which have infinitely many other smooth structures for which no Einstein metric can exist.

Families of 4-Manifolds with Nontrivial Stable Cohomotopy Seiberg–Witten Invariants, and Normalized Ricci Flow

“…Namely, suppose that there exists a positive integer m 0 such that Z ℓ 1 ,ℓ 2 g,h (m 0 ) is diffeomorphic to Z ℓ 1 ,ℓ 2 g,h (m) for any integer m ≥ m 0 . Then, by taking m → ∞, we see that the set of monopole classes of 4-manifold Z ℓ 1 ,ℓ 2 g,h (m 0 ) is unbounded by (33). However, this is a contradiction because the set of monopole classes of any given smooth 4-manifold with b + > 1 must be finite by Proposition 14.…”

Homotopy theoretical considerations of the Bauer–Furuta stable homotopy Seiberg–Witten invariants

“…(2) (Ishida-LeBrun [15,13,12]) Non-existence of Einstein metrics and computations of the Yamabe invariants under some circumstances.…”

“…(3) Constructions of spin 4-manifolds without Einstein metric and computations of their Yamabe invariants (see the work of Ishida and LeBrun [12,13,15]).…”

An essential relation between Einstein metrics, volume entropy, and exotic smooth structures

“…One can use the bandwidth argument (cf. [15,16]) to see this. Alternatively, one can also see this by using only the finiteness property of the set of special monopole classes (cf.…”