Algebraic surfaces and irrational connected sums of four manifolds

This chapter examines various ways to construct symplectic manifolds and submanifolds. It begins by studying blowing up and down in both the complex and the symplectic contexts. The next section is devoted to a discussion of fibre connected sums and describes Gompf’s construction of symplectic four-manifolds with arbitrary fundamental group. The chapter also contains an exposition of Gromov’s telescope construction, which shows that for open manifolds the h-principle rules and the inclusion of the space of symplectic forms into the space of nondegenerate 2-forms is a homotopy equivalence. The final section outlines Donaldson’s construction of codimension two symplectic submanifolds and explains the associated decompositions of the ambient manifold.

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Stable cohomotopy Seiberg–Witten invariants of connected sums of four-manifolds with positive first Betti number, I: Non-vanishing theorem

International Journal of Mathematics ◽

10.1142/s0129167x15410049 ◽

2015 ◽

Vol 26 (06) ◽

pp. 1541004 ◽

Cited By ~ 1

Author(s):

Masashi Ishida ◽

Hirofumi Sasahira

Keyword(s):

Betti Number ◽

Witten Invariant ◽

Vanishing Theorem ◽

Connected Sums ◽

Four Manifolds ◽

Stable Cohomotopy ◽

Invent Math

We shall prove a new non-vanishing theorem for the stable cohomotopy Seiberg–Witten invariant [S. Bauer and M. Furuta, Stable cohomotopy refinement of Seiberg–Witten invariants: I, Invent. Math.155 (2004) 1–19; S. Bauer, Stable cohomotopy refinement of Seiberg–Witten invariants: II, Invent. Math.155 (2004) 21–40.] of connected sums of 4-manifolds with positive first Betti number.

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Critical metrics on connected sums of Einstein four-manifolds

Advances in Mathematics ◽

10.1016/j.aim.2015.11.054 ◽

2016 ◽

Vol 292 ◽

pp. 210-315 ◽

Cited By ~ 4

Author(s):

Matthew J. Gursky ◽

Jeff A. Viaclovsky

Keyword(s):

Critical Metrics ◽

Connected Sums ◽

Four Manifolds

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Symmetries of simply-connected four-manifolds, especially algebraic surfaces

Transformation Groups - Lecture Notes in Mathematics ◽

10.1007/bfb0085623 ◽

1989 ◽

pp. 347-367

Author(s):

Steven H. Weintraub

Keyword(s):

Algebraic Surfaces ◽

Four Manifolds ◽

Simply Connected

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Embedded surfaces in four-manifolds, branched covers, and SO(3)-invariants

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100073114 ◽

1995 ◽

Vol 117 (2) ◽

pp. 275-286 ◽

Cited By ~ 1

Author(s):

D. Kotschick ◽

G. Matić

Keyword(s):

Algebraic Curves ◽

Homology Class ◽

Algebraic Surfaces ◽

Branched Covers ◽

Minimal Genus ◽

Adjunction Formula ◽

Embedded Surfaces ◽

Four Manifolds ◽

Image Position ◽

Simply Connected

One of the outstanding problems in four-dimensional topology is to find the minimal genus of an oriented smoothly embedded surface representing a given homology class in a smooth four-manifold. For an arbitrary homology class in an arbitrary smooth manifold not even a conjectural lower bound is known. However, for the classes represented by smooth algebraic curves in (simply connected) algebraic surfaces, it is possible that the genus of the algebraic curve, given by the adjunction formulais the minimal genus. This is usually called the (generalized) Thom conjecture. It is mentioned in Kirby's problem list [11] as Problem 4·36.

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