4-manifolds: 1-connected

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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

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2.1 First examples

The first examples that come to one's mind are the 4-sphere $S^4$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> {{Stub}} == Introduction == <wikitex>; Any finitely presentable group may occur as the fundamental group of a smooth closed 4-manifold. On the other hand, the class of simply connected (topological or smooth) 4-manifolds still appears to be quite rich, so it appears reasonable to consider the classification of simply connected 4-manifolds in particular. It appears that the [[Intersection_forms|intersection form]] is the main algebro-topological invariant of simply-connected 4-manifolds. Technical remark: When we mention the term ''4-manifold'' without the explicit mention of topological or smooth we shall mean the larger class of topological 4-manifolds. </wikitex> == Construction and examples, their intersection forms == <wikitex>; ... </wikitex> === First examples === <wikitex>; The first examples that come to one's mind are the 4-sphere $S^4$, the complex projective space $\mathbb{CP}^2$, the complex projective space with its opposite (non-complex) orientation $\overline{\mathbb{CP}^2}$, the product $S^2 \times S^2$, various connected sums of these, and in particular $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$. The intersection form of the 4-sphere is the "empty form" of rank 0. The intersection forms of the others are given by $$ q_{\mathbb{CP}^2} = ( \ 1 \ ) , $$ $$ q_{\overline{\mathbb{CP}^2}} = ( \, -1 \ ) , $$ $$ q_{S^2 \times S^2} = \begin{pmatrix} \ 0 \ & \ 1 \ \ 1 & 0 \end{pmatrix} , $$ $$ q_{\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}} = \begin{pmatrix} \ 1 \ & \ 0 \ \ 0 & -1 \end{pmatrix} . $$ The manifolds $S^2 \times S^2$ and $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$ both have indefinite intersection forms of same rank and signature, but of different type. Therefore they are not homotopy-equvialent. </wikitex> === Hypersurfaces in CP³ === <wikitex>; For an integer $d \geq 1$ we define a subset $X_d$ of $\mathbb{CP}^3$ by the formula $$ S_d = \{ X_0^d + X_1^d + X_2^d + X_3^d = 0 | [X_0:X_1:X_2:X_3] \in \mathbb{CP}^3 \} . $$ It is easy to check that in each chart of $mathbb{CP}^3$ the $S_d$ is cut out transversally by the homogeneous polynomial of degree $d$. This is a special case of a [[Complete_intersections|complete intersection]]. </wikitex> === Elliptic surfaces === <wikitex>; </wikitex> === Branched coverings === <wikitex>; </wikitex> === The E₈ manifold === <wikitex>; </wikitex> == Invariants == <wikitex>; ... </wikitex> == Topological classification == <wikitex>; ... </wikitex> == Non-existence results for smooth 4-manifolds == <wikitex>; ... </wikitex> == The Seiberg-Witten invariants == <wikitex>; ... </wikitex> == Failure of the h-cobordism theorem == <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]]S^4$, the complex projective space $\mathbb{CP}^2$$\mathbb{CP}^2$, the complex projective space with its opposite (non-complex) orientation $\overline{\mathbb{CP}^2}$$\overline{\mathbb{CP}^2}$, the product $S^2 \times S^2$$S^2 \times S^2$, various connected sums of these, and in particular $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$$\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$.

The intersection form of the 4-sphere is the "empty form" of rank 0. The intersection forms of the others are given by

$\displaystyle q_{\mathbb{CP}^2} = ( \ 1 \ ) ,$ $$\displaystyle q_{\mathbb{CP}^2} = ( \ 1 \ ) ,$$

$\displaystyle q_{\overline{\mathbb{CP}^2}} = ( \, -1 \ ) ,$ $$\displaystyle q_{\overline{\mathbb{CP}^2}} = ( \, -1 \ ) ,$$

$\displaystyle q_{S^2 \times S^2} = \begin{pmatrix} \ 0 \ & \ 1 \ \\ 1 & 0 \end{pmatrix} ,$ $$\displaystyle q_{S^2 \times S^2} = \begin{pmatrix} \ 0 \ & \ 1 \ \\ 1 & 0 \end{pmatrix} ,$$

$\displaystyle q_{\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}} = \begin{pmatrix} \ 1 \ & \ 0 \ \\ 0 & -1 \end{pmatrix} .$ $$\displaystyle q_{\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}} = \begin{pmatrix} \ 1 \ & \ 0 \ \\ 0 & -1 \end{pmatrix} .$$

The manifolds $S^2 \times S^2$$S^2 \times S^2$ and $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$$\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$ both have indefinite intersection forms of same rank and signature, but of different type. Therefore they are not homotopy-equvialent.

2.2 Hypersurfaces in CP³

For an integer $d \geq 1$$d \geq 1$ we define a subset $X_d$$X_d$ of $\mathbb{CP}^3$$\mathbb{CP}^3$ by the formula

$\displaystyle S_d = \{ X_0^d + X_1^d + X_2^d + X_3^d = 0 | [X_0:X_1:X_2:X_3] \in \mathbb{CP}^3 \} .$ $$\displaystyle S_d = \{ X_0^d + X_1^d + X_2^d + X_3^d = 0 | [X_0:X_1:X_2:X_3] \in \mathbb{CP}^3 \} .$$

It is easy to check that in each chart of $mathbb{CP}^3$$mathbb{CP}^3$ the $S_d$$S_d$ is cut out transversally by the homogeneous polynomial of degree $d$$d$. This is a special case of a complete intersection.

3 Invariants

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4 Topological classification

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5 Non-existence results for smooth 4-manifolds

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6 The Seiberg-Witten invariants

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7 Failure of the h-cobordism theorem

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8 Further discussion

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