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**Noncomplex smooth 4-manifolds with Lefschetz fibrations.**
*(English)*
Zbl 0977.57020

A smooth 4-dimensional manifold is said to admit a Lefschetz fibration if it is the total space of a fibration by Riemann surfaces, where all but a finite number of the fibers are smooth, and where the singular fibers are immersed with a single transverse self-intersection. Originally defined by B. Moishezon in the 1970’s, these fibrations have been increasingly studied of late, in large part because of work by S. Donaldson showing that any symplectic 4-manifold (after perhaps being blown up) admits a Lefschetz fibration. Existing explicit descriptions of Lefschetz fibrations have tended to be on complex manifolds, hence Donaldson’s theorem raises the question of finding explicit descriptions of Lefschetz fibrations on symplectic 4-manifolds which are noncomplex.

This paper fills that void by giving, for any fixed fiber genus \(g\geq 2\), a construction of an infinite family of Lefschetz fibrations whose total spaces are not complex (with either orientation). Moreover, as the examples in each family are readily shown to have distinct fundamental groups, each family consists of mutually nonhomeomorphic members. The author does this by generalizing a construction of B. Ozbagci and A. I. Stipsicz [Proc. Am. Math. Soc. 128, No. 10, 3125-3128 (2000; Zbl 0951.57015)], who gave a family of genus \(2\) Lefschetz fibrations by forming “twisted” fiber sums of two copies of a known complex fibration of genus \(2\) discovered by Y. Matsumoto, and demonstrating by a straightforward fundamental group calculation that the resulting manifolds cannot be complex. In the paper under review, the author generalizes Matsumoto’s example to get analogous examples for any arbitrary fiber genus \(g\geq 2\), from which noncomplex examples can again be constructed via twisted fiber sums. The bulk of the paper is devoted to the generalization of Matsumoto’s example, a result which is of interest in its own right. The author generalizes Matsumoto’s example by exploiting a well-known correspondence (via global monodromies) between Lefschetz fibrations and relations in mapping class groups given by right-handed Dehn twists; Matsumoto’s example is generalized by an ingenious lifting of a relation in the braid group \(B_{2g+2}\) on \(2g+2\) strings to a relation in the mapping class group of a genus \(g\) surface. (Matsumoto’s example has also been generalized by C. Cadavid [A remarkable set of words in the mapping class group, thesis, University of Texas, 1998]).

This paper fills that void by giving, for any fixed fiber genus \(g\geq 2\), a construction of an infinite family of Lefschetz fibrations whose total spaces are not complex (with either orientation). Moreover, as the examples in each family are readily shown to have distinct fundamental groups, each family consists of mutually nonhomeomorphic members. The author does this by generalizing a construction of B. Ozbagci and A. I. Stipsicz [Proc. Am. Math. Soc. 128, No. 10, 3125-3128 (2000; Zbl 0951.57015)], who gave a family of genus \(2\) Lefschetz fibrations by forming “twisted” fiber sums of two copies of a known complex fibration of genus \(2\) discovered by Y. Matsumoto, and demonstrating by a straightforward fundamental group calculation that the resulting manifolds cannot be complex. In the paper under review, the author generalizes Matsumoto’s example to get analogous examples for any arbitrary fiber genus \(g\geq 2\), from which noncomplex examples can again be constructed via twisted fiber sums. The bulk of the paper is devoted to the generalization of Matsumoto’s example, a result which is of interest in its own right. The author generalizes Matsumoto’s example by exploiting a well-known correspondence (via global monodromies) between Lefschetz fibrations and relations in mapping class groups given by right-handed Dehn twists; Matsumoto’s example is generalized by an ingenious lifting of a relation in the braid group \(B_{2g+2}\) on \(2g+2\) strings to a relation in the mapping class group of a genus \(g\) surface. (Matsumoto’s example has also been generalized by C. Cadavid [A remarkable set of words in the mapping class group, thesis, University of Texas, 1998]).

Reviewer: Terry Fuller (Northridge)

### MSC:

57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |