Topology Atlas || Conferences
 XI Brazilian Meeting of Topology August 3-7, 1998 IGCE - Universidade Estadual Paulista - UNESP Rio Claro, S.P., Brazil

Organizers
Joao Peres Vieira, Marcelo Jose Saia, Oziride Manzoli Neto, Suely Druck, Alice Kimie Miwa Libardi, Izabel Cristina Rossini

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 Ne, where e = +/- 2 is the Euler number. Note that \partialNe 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 Ne 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 jj (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 jj 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.

[]
S. Akbulut, On fake S3 [×\tilde] S1\sharp S2 ×S2, in ^^ ^^ Combinatorial methods in topology and algebraic geometry", Contemporary Math. 44, Amer. Math. Soc., 1985, pp.281-286.
[]
S. Akbulut, Constructing a fake 4-manifold by Gluck construction to a standard 4-manifold, Topology 27 (1988), 239-243.
[]
H. Gluck, The embedding of two-spheres in the four-sphere, Trans. Amer. Math. Soc. 104 (1962), 308-333.
[]
R. E. Gompf, On Cappell-Shaneson 4-spheres, Topology Appl. 38 (1991), 123-136.
[]
C. McA. Gordon, Knots in the 4-sphere, Comment. Math. Helv. 51 (1976), 585-596.
[]
A. Katanaga and O. Saeki, Embeddings of quaternion space in S4, preprint, 1997.
[]
R. C. Kirby, 4-Manifold problems, in ^^ ^^ Four-manifold theory", Contemporary Math. 35, Amer. Math. Soc., 1984, pp.513-528.
[]
T. Lawson, Splitting S4 on RP2 via the branched cover of CP2 over S4, Proc. Amer. Math. Soc. 86 (1982), 328-330.
[]
T. Lawson, Detecting the standard embedding of RP2 in S4, Math. Ann. 267 (1984), 439-448.
[]
W. S. Massey, Proof of a conjecture of Whitney, Pacific J. Math. 31 (1969), 143-156.
[]
W. S. Massey, Imbeddings of projective planes and related manifolds in spheres, Indiana Univ. Math. J. 23 (1973), 791-812.
[]
J. M. Montesinos, On twins in the four-sphere I, Quart. J. Math. Oxford 34 (1983), 171-199.
[]
T. Price, Homeomorphisms of quaternion space and projective planes in four space, J. Austral. Math. Soc. 23 (1977), 112-128.
[]
T. Price and D. Roseman, Embeddings of the projective plane in four space, preprint, 1975.
[]
Y. Yamada, Decomposition of S4 as a twisted double of a certain manifold, Tokyo J. Math. 20 (1997), 23-33.