Simply-connected Manifold Research Articles

Simply-connected irreducible 4-manifolds with no symplectic structures

Simply-connected irreducible 4-manifolds with no symplectic structures

A non-extended hermitian form over ℤ[ℤ

We describe a nonsingular hermitian form of rank 4 over the group ring Z[Z] which is not extended from the integers. Moreover, we show that under certain indefiniteness asumptions, every nonsingular hermitian form on a free Z[Z]module is extended from the integers. As a corollary, there exists a closed oriented 4-dimensional manifold with fundamental group Z which is not the connected sum of S × S with a simply-connected 4-manifold.

Four-Manifolds With Surface Fundamental Groups

We study the homotopy type of closed connected topological 4 4 -manifolds whose fundamental group is that of an aspherical surface F F . Then we use surgery theory to show that these manifolds are s s -cobordant to connected sums of simply-connected manifolds with an S 2 \mathbb {S}^{2} -bundle over F F .

A decomposition theorem forh-cobordant smooth simply-connected compact 4-manifolds

Theorem. Let M and N be smooth ;h -cobordant compact 1-connected 4-manifolds. There exist decompositions M = M0 [ M1 and N = N0 [ N1 where M0 and N0 are smooth compact contractible 4-manifolds with boundary ; so that (M1;);and (N1;)are dieomorphic. If M and N are closed; then we may further arrange that M1 and N1 are 1-connected. In fact; if W is an h-cobordism connecting M and N; then W can be written W = W0 [I W1 where (W0;M0;N0)is an (often) nontrivial h-cobordism and (W1;M1;N1)is smoothly the product h-cobordism (M1 I;M1 f0 g;M1 f1 g). This theorem lies in the mainstream of combinatorial 4-manifold theory circa 1975, but, oddly, the result was overlooked. It now ts into a rich context (Donaldson theory) and raises new questions when compared with the known examples. In 1988 Akbulut [A] showed that a manifold X homotopy equivalent to a K3-surface could, after blowing up a point, be transformed into a rational surface Q by cutting out and regluing a certain contractible \Mazur" manifold by an involution of its boundary . At the time a rather delicate argument of Fintushel and Stern (relying on the observed presence of the homology sphere 2;3;7 X) showed that certain Donaldson invariants of X (and its blow up X + ) were nontrivial. This distinguished X + from the h-cobordant manifold Q. In retrospect, this was the rst nontrivial example of our theorem. The result was a surprise: the Donaldson invariant, which derives from the second homology, had been localized on a contractible piece. Recently Bizaca and Gompf [BG] have shown that X is, in fact, dieomorphic to K3, allowing a more direct analysis of its Donaldson invariant. (See e.g. [QK] or [KM].) They also nd an innite family of examples

A hypercomplex stiefel manifold

We construct a hypercomplex structure on a compact simply-connected manifold derived from the hyperkähler structure of flat space.

A decomposition theorem for h -cobordant smooth simply-connected compact 4-manifolds

A decomposition theorem for h -cobordant smooth simply-connected compact 4-manifolds

A decomposition of smooth simply-connected $h$-cobordant 4-manifolds

We prove that any two smooth h-cobordant simply-connected 4-manifolds can be obtained by taking two manifolds with boundary, one of which is contractible, and gluing them along the boundary via two different attaching maps.

Involutions on spin $4$-manifolds

We show that a simply-connected spin 4-manifold which admits a locally linear involution must have vanishing signature. We also show that the codimensions of all components of the fixed point set of an involution on a spin 4-manifold are the same modulo 4. There is no assumption of local linearity in this result, which extends a lemma of Atiyah and Bott.

A structure theorem and a splitting theorem for simply-connected smooth 4-manifolds

A structure theorem and a splitting theorem for simply-connected smooth 4-manifolds

Uniqueness of connected sum decompositions in dimension 4

Suppose W is a 4-manifold with good fundamental group and M is a closed simply-connected 4-manifold. Suppose we are given two decompositions h 1: W ⋍ M# W 1 and h 2: W ⋍ M# W 2 inducing the same decomposition of π 2 W. In this paper we study when we can conclude that W 1 and W 2 are homeomorphic. As a consequence we conclude that the ∗ operation for changing the Kirby-Siebenmann invariant of a 4-manifold is well defined. We will also use this discussion to relate the ambient approach to classification to the surgery approach.

Realization of simply-connected 4-manifolds with a given boundary

Realization of simply-connected 4-manifolds with a given boundary

Simply-connected 4-manifolds with a given boundary

Let M be a closed, oriented, connected 3-manifold and let ( Z n , L) be a symmetric bilinear form which presents H ∗( M; Z ). Lt V L ( M) be the set of all oriented homeomorphism types of compact, 1-connected, oriented 4-manifolds with boundary M and intersection pairing isomorphic to ( Z n , L). We will give a complete description of the sets V L ( M).

Fields of 2-planes on compact simply-connected smooth 4-manifolds

Fields of 2-planes on compact simply-connected smooth 4-manifolds

Simply-Connected 4-Manifolds with a given Boundary

Simply-Connected 4-Manifolds with a given Boundary

Homologically Trivial Group Actions I: Simply Connected Manifolds

Homologically Trivial Group Actions I: Simply Connected Manifolds

Stochastic Differential Geometry, a Coupling Property, and Harmonic Maps

A Riemannian manifold has the Brownian coupling property if two Brownian motions can be constructed on it, with arbitrary initial points, and such that they are sure to meet at some time. While Euclidean space has this property, simply-connected manifolds with negative curvature bounded away from zero do not. Such a coupling is then not possible even if Γ-martingales of bounded dilatation are used rather than Brownian motions. A generalised little Picard theorem for harmonic maps is proved using these results.

Simply-connected 4-manifolds with a given boundary

Let M M be a closed, oriented, connected 3 3 -manifold. For each bilinear, symmetric pairing ( Z n , L ) ({{\mathbf {Z}}^n},\,L) , our goal is to calculate the set V L ( M ) {\mathcal {V}_L}(M) of all oriented homeomorphism types of compact, 1 1 -connected, oriented 4 4 -manifolds with boundary M M and intersection pairing isomorphic to ( Z n , L ) ({{\mathbf {Z}}^n},\,L) . For each pair ( Z n , L ) ({{\mathbf {Z}}^n},\,L) which presents H ∗ ( M ) {H_ \ast }(M) , we construct a double coset space B L t ( M ) B_L^t(M) and a function c L t : V L ( M ) → B L t ( M ) c_L^t:{\mathcal {V}_L}(M) \to B_L^t(M) . The set B L t ( M ) B_L^t(M) is the quotient of the group of all link-pairing preserving isomorphisms of the torsion subgroup of H 1 ( M ) {H_1}(M) by two naturally occuring subgroups. When ( Z n , L ) ({{\mathbf {Z}}^n},\,L) is an even pairing, we construct another double coset space B ^ L ( M ) {\hat B_L}(M) , a function c ^ L : V L ( M ) → B ^ L ( M ) {\hat c_L}:{\mathcal {V}_L}(M) \to {\hat B_L}(M) and a projection p 2 : B ^ L ( M ) → B L t ( M ) {p_2}:{\hat B_L}(M) \to B_L^t(M) such that p 2 ⋅ c ^ L = c L t {p_2} \cdot {\hat c_L} = c_L^t . Our main result states that when ( Z n , L ) ({{\mathbf {Z}}^n},\,L) is even the function c ^ L {\hat c_L} is injective, as is the function c L t × Δ : V L ( M ) → B L t ( M ) × Z / 2 c_L^t \times \Delta :{\mathcal {V}_L}(M) \to B_L^t(M) \times {\mathbf {Z}}/2 when ( Z n , L ) ({{\mathbf {Z}}^n},\,L) is odd. Here Δ \Delta is a Kirby-Siebenmann obstruction to smoothing. It follows that the sets V L ( M ) {\mathcal {V}_L}(M) are finite and of an order bounded above by a constant depending only on H 1 ( M ) {H_1}(M) . We also show that when H 1 ( M ; Q ) ≅ 0 {H_1}(M;{\mathbf {Q}}) \cong 0 and ( Z n , L ) ({{\mathbf {Z}}^n},\,L) is even, c L t = p 2 ⋅ c ^ L c_L^t = {p_2} \cdot {\hat c_L} is injective. It seems likely that via the functions c L t × Δ c_L^t \times \Delta and c ^ L {\hat c_L} , the sets B L t ( M ) × Z / 2 B_L^t(M) \times {\mathbf {Z}}/2 and B ^ L ( M ) {\hat B_L}(M) calculate V L ( M ) {\mathcal {V}_L}(M) when ( Z n , L ) ({{\mathbf {Z}}^n},\,L) is respectively odd and even. We verify this in several cases, most notably when H 1 ( M ) {H_1}(M) is free abelian. The results above are based on a theorem which gives necessary and sufficient conditions for the existence of a homeomorphism between two 1 1 -connected 4 4 -manifolds extending a given homeomorphism of their boundaries. The theory developed is then applied to show that there is an m > 0 m > 0 , depending only on H 1 ( M ) {H_1}(M) , such that for any self-homeomorphism f f of M M , f m {f^m} extends to a self-homeomorphism of any 1 1 -connected, compact 4 4 -manifold with boundary M M .

Morse theory for codimension-one foliations

It is shown that a smooth codimension-one foliation on a compact simply-connected manifold has a compact leaf if and only if every smooth real-valued function on the manifold has a cusp singularity.

Toral actions on 5-manifolds

We are mainly concerned with closed orientable manifolds of dimension 5 5 supporting effective three-dimensional torus actions. We obtain a complete classification of simply-connected manifolds of this type and a partial classification for the nonsimply-connected case.

Tori in the diffeomorphism groups of simply-connected 4-manifolds

Let M be a closed simply-connected 4-manifold. All manifolds will be assumed smooth and oriented. The purpose of this paper is to classify up to conjugacy the topological subgroups of Diff(M) isomorphic to the 2-dimensional torus T2 (Theorem 1), and to give an explicit formula for the number of such conjugacy classes (Theorem 2). Such a conjugacy class corresponds uniquely to a weak equivalence class of effective T2-actions on M. Thus the classification problem is trivial unless M supports an effective T2-action. Orlik and Raymond showed that this happens if and only if M is a connected sum of copies of ± ;P2 and S2 × S2 (2), and so this paper is really a study of the different T2-actions on these manifolds.