Lefschetz fibrations and exotic symplectic structures on cotangent bundles of spheres: Figure 1.

Abstract

(J. Topol. 3 (2010) 157–180) In the proof of [4, Lemma 1.1], we appealed to an explicit isotopy of totally real spheres, constructed in [3, Section 5]. That construction works in the lowest dimension ( n = 2 ), but is wrong in general (one of the endpoints is not the desired sphere). Here, we explain a different approach, leading to a corrected version of [4, Lemma 1.1], which requires an additional assumption. Independently, while [4, Lemma 1.2] makes a statement about homotopy classes of almost complex structures, its proof only determines the isomorphism class of the tangent bundle as an abstract complex vector bundle, which is a priori a weaker statement. The argument here also fills that gap. The rest of the original paper is unaffected. Consider M = M m as in [4], in complex dimension n > 2 . The construction of E depends on a choice of Lagrangian sphere S = S δ m + 1 ⊂ M . By [1], the smooth isotopy class of S depends only on [ S ] ∈ H n ( M ) . Since S is Lagrangian, it comes with a canonical formal Legendrian structure (more precisely, a formal Legendrian structure for { 0 } × S ⊂ R × M , as defined in [5]). Given two homologous Lagrangian spheres, we can use a smooth isotopy between them to compare their canonical formal Legendrian structures. If these coincide, then the resulting manifolds E are diffeomorphic, compatibly with the homotopy classes of their almost complex structures. In general, the difference between two formal Legendrian structures for a given n-sphere is described by an element of π n + 1 ( V n , 2 n + 1 , U n ) , where V n , 2 n + 1 is the Stiefel manifold. That homotopy group was analyzed in [5, Lemmas A.5–A.7], with the following implications for our situation (cf. [5, Theorem A.4]). Crossing over one marked point. Lemma 1.1 (Corrected).Suppose that n = 2 . Then, any choice of δ m + 1 leads to a manifold E which is diffeomorphic to T * S n + 1 , and this diffeomorphism is compatible with the homotopy classes of almost complex structures. For higher even n , the same holds under the following additional assumption: ( *) δ m + 1 can be connected to a ‘standard path’ by an isotopy (rel endpoints ) which crosses over an even number of marked points in the plane (here, the ‘standard paths’ are the δ k , l from [4, Section 7]). It remains to consider the case n = 6 . Then, for any choice of δ m + 1 , the resulting E will be diffeomorphic to T * S 7 ≅ S 7 × R 7 (one shows this using the h-cobordism theorem, and the fact that any seven-dimensional vector bundle over S 7 is trivial). However, there are two possible homotopy classes of almost complex structures ( π 7 ( O 14 / U 7 ) ≅ π 7 ( O ∞ / U ∞ ) ≅ Z / 2 ), and it is not clear which one will arise if ( *) is dropped. In particular, we still do not know what element of (2) appears there. We thank Emmy Murphy for helpful suggestions.