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Diffeomorphisms of fake Enriques surfaces. (English) Zbl 0827.14024
Catanese, F. (ed.) et al., Problems in the theory of surfaces and their classification. Papers from the meeting held at the Scuola Normale Superiore, Cortona, Italy, October 10-15, 1988. London: Academic Press. Symp. Math. 32, 237-245 (1991).
Let $$X$$ be a smooth oriented closed 4-manifold, $$\text{Diff} (X)$$ its group of orientation preserving diffeomorphisms. $$\text{Diff} (X)$$ has a natural representation $$\rho : \text{Diff} (X) \to O (H(X))$$ in the orthogonal group of the lattice $$H(X) : = H^2 (X, \mathbb{Z}) /{\text{Tors}}$$ whose symmetric bilinear form is induced by the intersection form. The purpose of this note is to study the image $$\text{Diff}^* (X) : = \text{Im} (\rho)$$ of this representation for a certain class of algebraic surfaces. Recall that an Enriques surface is a (minimal) algebraic surface $$X$$ of Kodaira dimension $$\text{kod} (X) = 0$$ with fundamental group $$\pi_1 (X) \cong \mathbb{Z}/2$$. – A fake Enriques surface is by definition an algebraic surface which is not an ordinary Enriques surface but has the same homotopy type. These surfaces have been investigated by the author [Topology 27, No. 4, 415-427 (1988; Zbl 0682.14027)] where it was shown that fake Enriques surfaces are always homeomorphic, but never diffeomorphic to ordinary Enriques surfaces. They can be characterized as those regular honestly elliptic surfaces with geometric genus $$p_g = 0$$ which have two multiple fibres $$2pF_{2p}$$, $$2qF_{2q}$$, where $$p,q$$ are relatively prime odd positive integers, not both equal to 1. We denote a surface of this type by $$X_{2p, 2q}$$. These surfaces induce infinitely many distinct $$C^\infty$$-structures on the topological manifold underlying an Enriques surface [loc. cit.]. They have fewer symmetries than ordinary Enriques surfaces. Let $$E_8$$ be the root lattice of the corresponding Lie group. We show:
Theorem. Let $$X_{2p, 2q}$$ be a fake Enriques surface. The group $$\text{Diff}^* (X_{2p, 2q})$$ contains an extended Weyl group isomorphic to $$O(E_8) \propto E_8$$ as subgroup of finite index. The index divides $$2(2pq)^8$$.
This result gives a “lower bound” on $$\text{Diff}^* (X_{2p, 2q})$$. The techniques of R. Friedman and F. W. Morgan [J. Differ. Geom. 27, No. 2, 297-369 (1988; Zbl 0669.57016)] can be adapted to show that $$\text{Diff}^* (X_{2p, 2q})$$ has infinite index in $$O(H(X_{2p, 2q}))$$. This gives an “upper bound” for the size of $$\text{Diff}^* (X_{2p, 2q})$$.
For the entire collection see [Zbl 0824.00026].
##### MSC:
 14J28 $$K3$$ surfaces and Enriques surfaces 57S05 Topological properties of groups of homeomorphisms or diffeomorphisms 58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
##### Citations:
Zbl 0682.14027; Zbl 0669.57016