4-manifolds: 1-connected
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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 $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$. | 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 $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$. | ||
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| + | The intersection form of the 4-sphere is the "empty form" of rank 0. The intersection forms of the others are given by | ||
| + | $$ | ||
| + | q_{\mathbb{CP}^2} & = ( \ 1 \ ) \\ | ||
| + | q_{\overline{\mathbb{CP}^2}} = ( \, -1 \ ) \\ | ||
| + | \\ | ||
| + | q_{S^2 \times S^2} = | ||
| + | $$ | ||
| + | |||
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| + | === Hypersurfaces in $\mathbb{CP}^3$ === | ||
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| + | === Elliptic surfaces === | ||
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| + | === Branched coverings === | ||
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| + | === The $E_8$ manifold === | ||
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== Invariants == | == Invariants == | ||
Revision as of 11:49, 8 June 2010
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This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
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, their intersection forms
...
2.1 First examples
The first examples that come to one's mind are the 4-sphere 
, the complex projective space 
, the complex projective space with its opposite (non-complex) orientation 
, the product 
, various connected sums of these, and in particular 
.
The intersection form of the 4-sphere is the "empty form" of rank 0. The intersection forms of the others are given by
Tex syntax error
2.2 Hypersurfaces in $\mathbb{CP}^3$
2.3 Elliptic surfaces
2.4 Branched coverings
2.5 The $E_8$ manifold
3 Invariants
...
4 Classification/Characterization
...
5 Further discussion
...
