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40
A decomposition of smooth simplyconnected hcobordant 4manifolds
 J. Differential Geom
, 1996
"... Introduction and the Statement. In [A] S. Akbulut obtained an example of the exotic manifold cutting off the contractible submanifold from the standard manifold and regluing it via nontrivial involution of the boundary. In these notes we give a proof of a decomposition theorem stated below. It gener ..."
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Cited by 14 (2 self)
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Introduction and the Statement. In [A] S. Akbulut obtained an example of the exotic manifold cutting off the contractible submanifold from the standard manifold and regluing it via nontrivial involution of the boundary. In these notes we give a proof of a decomposition theorem stated below. It generalizes the
Akbulut's Corks and HCobordisms of Smooth, Simply Connected 4Manifolds
"... F35> 0 = @A 1 , is an involution [9]. Corollary: Any homotopy 4sphere, \Sigma 4 , can be constructed by cutting out a contractible 4manifold, A 0 from S 4 and gluing it back in by an involution of @A 0 . Remark: Since there are many examples of nontrivial hcobordisms (the first ones were di ..."
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Cited by 11 (1 self)
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F35> 0 = @A 1 , is an involution [9]. Corollary: Any homotopy 4sphere, \Sigma 4 , can be constructed by cutting out a contractible 4manifold, A 0 from S 4 and gluing it back in by an involution of @A 0 . Remark: Since there are many examples of nontrivial hcobordisms (the first ones were discovered by Donaldson [4]), there are as many examples of nontrivial, rel boundary, hcobordisms A. However these A are delicate objects; their nontriviality vanishes when a trivial hcobordism is added. That is, if we add A 0 \Theta I to A along @A 0 \Theta I, then it follows from the Addenda that we have an hcobordism between S 4 on the bottom as well as S 4 on the top; thus the hcobordism is the trivial<F54.1
Symplectic genus, minimal genus and diffeomorphisms
"... Abstract. In this paper, the symplectic genus for any 2−dimensional class in a 4−manifold admitting a symplectic structure is introduced, and its relation with the minimal genus is studied. It is used to describe which classes in rational and irrational ruled manifolds are realized by connected symp ..."
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Cited by 10 (4 self)
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Abstract. In this paper, the symplectic genus for any 2−dimensional class in a 4−manifold admitting a symplectic structure is introduced, and its relation with the minimal genus is studied. It is used to describe which classes in rational and irrational ruled manifolds are realized by connected symplectic surfaces. In particular, we completely determine which classes with square at least −1 in such manifolds can be represented by embedded spheres. Moreover, based on a new characterization of the action of the diffeomorphism group on the intersection forms of a rational manifold, we are able to determine the orbits of the diffeomorphism group on the set of classes represented by embedded spheres of square at least −1 in any 4−manifold admitting a symplectic structure. Let M be a smooth, closed oriented 4−manifold. An orientationcompatible symplectic form on M is a closed two−form ω such that ω ∧ω is nowhere vanishing and agrees with the orientation. For any oriented 4−manifold M, its symplectic cone CM is defined as the set of cohomology classes which are represented by orientationcompatible
The stable mapping class group of simply connected 4manifolds
, 2006
"... We consider mapping class groups Γ(M) = π0Diff(M fix ∂M) of smooth compact simply connected oriented 4–manifolds M bounded by a collection of 3–spheres. We show that if M contains CP 2 or CP 2 as a connected summand then Γ(M) is independent of the number of boundary components. By repackaging cla ..."
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Cited by 6 (1 self)
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We consider mapping class groups Γ(M) = π0Diff(M fix ∂M) of smooth compact simply connected oriented 4–manifolds M bounded by a collection of 3–spheres. We show that if M contains CP 2 or CP 2 as a connected summand then Γ(M) is independent of the number of boundary components. By repackaging classical results of Wall, Kreck and Quinn, we show that the natural homomorphism from the mapping class group to the group of automorphisms of the intersection form becomes an isomorphism after stabilization with respect to connected sum with CP 2 #CP 2. We next consider the 3+1 dimensional cobordism 2–category C of 3–spheres, 4–manifolds (as above) and enriched with isotopy classes of diffeomorphisms as 2–morphisms. We identify the homotopy type of the classifying space of this category as the Hermitian algebraic Ktheory of the integers. We also comment on versions of these results for simply connected spin 4–manifolds.
EMBEDDED SURFACES AND ALMOST COMPLEX STRUCTURES
, 1998
"... Abstract. In this paper, we prove necessary and sufficient conditions for a smooth surface in a smooth 4manifold X to be pseudoholomorphic with respect to an almost complex structure on X. In particular, this provides a systematic approach to the construction of pseudoholomorphic curves that do not ..."
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Abstract. In this paper, we prove necessary and sufficient conditions for a smooth surface in a smooth 4manifold X to be pseudoholomorphic with respect to an almost complex structure on X. In particular, this provides a systematic approach to the construction of pseudoholomorphic curves that do not minimize the genus in their homology class.
The Borel/Novikov conjectures and stable diffeomorphisms of 4manifolds
"... this paper we make the following two conjectures, relate them to standard conjectures in manifold theory, and thereby prove the following two conjectures for large classes of fundamental groups. Conjecture 0.1 If M and N are closed, orientable, smooth 4manifolds which are homotopy equivalent and ha ..."
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this paper we make the following two conjectures, relate them to standard conjectures in manifold theory, and thereby prove the following two conjectures for large classes of fundamental groups. Conjecture 0.1 If M and N are closed, orientable, smooth 4manifolds which are homotopy equivalent and have torsionfree fundamental group, then they are stably di#eomorphic. Conjecture 0.2 If M and N are closed, orientable, topological 4manifolds which are homotopy equivalent, have torsionfree fundamental group, and have the same KirbySiebenmann invariant, then they are stably homeomorphic. In the simplyconnected case the validity of Conjecture 0.1 is wellknown by the work of Wall [W1], who showed that homotopy equivalent, smooth, simplyconnected 4manifolds are hcobordant, and that hcobordant, smooth, simplyconnected manifolds are stably di#eomorphic. Using gauge theory, Donaldson [D] showed that they need not be di#eomorphic. In the simplyconnected case, Conjecture 0.2 follows from the work of Freedman [F], with the stronger conclusion that the manifolds are actually homeomorphic. P. Teichner in his thesis [T1] constructed an example of two closed, orientable, # Supported by the Alexander von HumboldtStiftung and the National Science Foundation. The author wishes to thank the Johannes GutenbergUniversitat in Mainz for its hospitality while this work was carried out. 1 homotopy equivalent, smooth 4manifolds with finite fundamental group which are not stably di#eomorphic. There is a map # 2 : H 2 (#; Z 2 ) # L 4 (Z#), which appears in the surgery classification of highdimensional manifolds. (Here L = L h , and refers to the Witt group of quadratic forms on free Z#modules.) As we shall see, this map is conjectured to be injective for all torsionfree groups...
Einstein metrics and the number of smooth structures on a fourmanifold
, 2003
"... We prove that for every natural number k there are simply connected topological four–manifolds which have at least k distinct smooth structures supporting Einstein metrics, and also have infinitely many distinct smooth structures not supporting Einstein metrics. Moreover, all these smooth structur ..."
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Cited by 5 (2 self)
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We prove that for every natural number k there are simply connected topological four–manifolds which have at least k distinct smooth structures supporting Einstein metrics, and also have infinitely many distinct smooth structures not supporting Einstein metrics. Moreover, all these smooth structures become diffeomorphic to each other after connected sum with only one copy of the complex projective plane. We prove that manifolds with these properties cover a large geographical area.
Monopole classes and Perelman’s invariant of fourmanifolds
, 2006
"... We calculate Perelman’s invariant for compact complex surfaces and a few other smooth fourmanifolds. We also prove some results concerning the dependence of Perelman’s invariant on the smooth structure. ..."
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Cited by 5 (0 self)
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We calculate Perelman’s invariant for compact complex surfaces and a few other smooth fourmanifolds. We also prove some results concerning the dependence of Perelman’s invariant on the smooth structure.
An obstruction to smooth isotopy in dimension 4
, 1998
"... It is certainly wellknown that a suitable count of solutions to the YangMills or SeibergWitten equations gives rise to invariants of a smooth 4manifold. In recent years, these invariants have become reasonably computable, and have led to many advances in 4manifold theory. In this paper, we use ..."
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Cited by 5 (1 self)
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It is certainly wellknown that a suitable count of solutions to the YangMills or SeibergWitten equations gives rise to invariants of a smooth 4manifold. In recent years, these invariants have become reasonably computable, and have led to many advances in 4manifold theory. In this paper, we use the tools of gauge theory to describe invariants of a diffeomorphism