Gluck surgery along a 2-sphere in a 4-manifold
by
Osamu Saeki
Department of Mathematics, Faculty of Science, Hiroshima University, Higashi-Hiroshima 739-8526, Japan
Coauthors: Yuichi Yamada (Tokyo, Japan), Masakazu Teragaito (Hiroshima, Japan), Atsuko Katanaga (Tsukuba, Japan)

One of the well-known methods to construct a new 4-manifold from an old one is the Gluck surgery along an embedded 2-sphere with trivial normal bundle, which is defined as follows (see ). Let M be a smooth 4-manifold and K a smoothly embedded 2-sphere in M. We suppose that the tubular neighborhood N(K) of K in M is diffeomorphic to S² ×D². Let \tau be the self-diffeomorphism of S² ×S¹ = \partial(S² ×D²) defined by \tau(z, \alpha) = (\alphaz, \alpha), where we identify S¹ with the unit circle of C and S² with the Riemann sphere [^(C)] = C \cup {\infty}. Then consider the 4-manifold obtained from M - IntN(K) by regluing S² ×D² along the boundary using \tau. We say that the resulting 4-manifold, denoted by \Sigma(K), is obtained from M by the Gluck surgery along K (see ). When the ambient 4-manifold is the 4-sphere S⁴, we call a smoothly embedded 2-sphere K in S⁴ a 2-knot. In this case, the resulting 4-manifold \Sigma(K) is always a homotopy 4-sphere. It has been known that for certain 2-knots K, \Sigma(K) is again diffeomorphic to S⁴ (see, for example, , ). It has not been known if the Gluck surgery along a 2-knot K in S⁴ produces a 4-manifold \Sigma(K) not diffeomorphic to S⁴ for some K (see 4.11, 4.24, 4.45 and ). On the other hand, for 2-spheres embedded in 4-manifolds M not necessarily diffeomorphic to S⁴, Akbulut , has constructed an example of an embedded 2-sphere K in such an M such that \Sigma(K) is homeomorphic, but not diffeomorphic to M. Price has considered a similar construction using embedded projective planes in S⁴. Let P be a smoothly embedded projective plane in S⁴. In the following, we fix an orientation for S⁴. Then it is known that the tubular neighborhood N(P) of P is always diffeomorphic to the nonorientable D²-bundle over RP² with Euler number +/- 2 (see , ), which we denote by N_e, where e = +/- 2 is the Euler number. Note that \partialN_e is diffeomorphic to the quaternion space Q, whose fundamental group is isomorphic to the quaternion group of order 8. Then consider the closed orientable 4-manifold \Pi(P)_j obtained from S⁴ - IntN(P) by regluing N_e along the boundary using a self-diffeomorphism j of Q. Then we say that \Pi(P)_j is obtained from S⁴ by a Price surgery along P with respect to j. In fact, Price has shown that there are exactly six isotopy classes of orientation preserving self-diffeomorphisms of Q - thus we have essentially six choices for j - and that exactly four of them produce homotopy 4-spheres by a Price surgery. Furthermore, he has also shown that there are at most two diffeomorphism types among the four homotopy 4-spheres thus constructed, one of which is the standard 4-sphere. In the following, \Pi(P) will denote the unique homotopy 4-sphere obtained by the Price surgery along P with respect to a nontrivial self-diffeomorphism of Q, which may not be diffeomorphic to the 4-sphere. Obviously we can generalize the above definition of Price surgeries to those along projective planes embedded in arbitrary 4-manifolds with normal Euler number +/- 2. Let P₀ be a standardly embedded projective plane in S⁴, whose normal Euler number is equal to either 2 or -2 (for example, see , , , ). One of our main results of this talk is the following theorem concerning the relationship between Gluck surgeries and Price surgeries. Let K be a 2-knot in S⁴. Then the homotopy 4-sphere \Sigma(K) obtained by the Gluck surgery along K is diffeomorphic to the homotopy 4-sphere \Pi(P₀ \sharp K) obtained by the Price surgery along the projective plane P₀ \sharp K, where \sharp denotes the connected sum. In fact, the above theorem is a direct consequence of a more general result as follows. In the following, for a projective plane P smoothly embedded in S⁴, we denote by N(P) and E(P) its tubular neighborhood in S⁴ and S⁴ - IntN(P) respectively, and we call E(P) the exterior of P. Let K and K' be an arbitrary pair of 2-knots in S⁴. Then there exist four self-diffeomorphisms j_j (j = 1, 2, 3, 4) of Q such that the closed oriented 4-manifold E(P₀ \sharp K) \cup _jj -E(P₀ \sharp K') obtained by gluing E(P₀ \sharp K) and -E(P₀ \sharp K') along their boundaries using j_j is orientation preservingly diffeomorphic to S⁴, \Sigma(K), \Sigma(K'!) and \Sigma(K) \sharp \Sigma(K'!) for j = 1, 2, 3 and 4 respectively, where -E(P₀ \sharp K') denotes E(P₀ \sharp K') with the reversed orientation and K'! the mirror image of K'. Furthermore, using our Theorem , we will show that the Gluck surgery along a smoothly embedded 2-sphere K in an arbitrary 4-manifold M is always realized by a Price surgery along the connected sum P₀ \sharp K of K and a standardly embedded projective plane P₀ contained in a 4-disk in M.

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Date received: May 26, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabo-02.