4-manifolds: 1-connected

Revision as of 11:20, 8 June 2010

1 Introduction

Any finitely presentable group may occur as the fundamental group of a smooth closed 4-manifold. On the other hand, the class of simply connected (topological or smooth) 4-manifolds still appears to be quite rich, so it appears reasonable to consider the classification of simply connected 4-manifolds in particular.

It appears that the intersection form is the main algebro-topological invariant of simply-connected 4-manifolds.

Technical remark: When we mention the term 4-manifold without the explicit mention of topological or smooth we shall mean the larger class of topological 4-manifolds.

2 Construction and examples

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2.1 First examples

The first examples that come to one's mind are the 4-sphere $<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> {{Stub}} == Introduction == <wikitex>; Any finitely presentable group may occur as the fundamental group of a smooth closed 4-manifold. On the other hand, the class of simply connected (topological or smooth) 4-manifolds still appears to be quite rich, so it appears reasonable to consider the classification of simply connected 4-manifolds in particular. It appears that the [[Intersection_form|intersection form]] is the main algebro-topological invariant of simply-connected 4-manifolds. Technical remark: When we mention the term ''4-manifold'' without the explicit mention of topological or smooth we shall mean the larger class of topological 4-manifolds. </wikitex> == Construction and examples == <wikitex>; ... </wikitex> === First examples === <wikitex>; The first examples that come to one's mind are the 4-sphere $S^4$, the complex projective space $\mathbb{CP}^2$, the complex projective space with its opposite (non-complex) orientation $\overline{\mathbb{CP}^2}$, the product $S^2 \times S^2$, various connected sums of these, and in particular $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$. </wikitex> == Invariants == <wikitex>; ... </wikitex> == Classification/Characterization == <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]]S^4$, the complex projective space $\mathbb{CP}^2$, the complex projective space with its opposite (non-complex) orientation $\overline{\mathbb{CP}^2}$, the product $S^2 \times S^2$, various connected sums of these, and in particular $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$.

3 Invariants

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4 Classification/Characterization

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5 Further discussion

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6 References