Stable classification of 4-manifolds

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1 Introduction

In this page we report about the stable classification of $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 --> <!-- DON'T FORGET TO ENTER YOUR USER NAME BELOW: E.G. {{Authors|Uncle Otto}} .--> {{Authors|Matthias Kreck}}{{Stub}} == Introduction == <wikitex>; In this page we report about the [[stable classification of manifolds|stable classification]] of $-manifolds. </wikitex> == Construction and examples == <wikitex>; We begin with the construction of manifolds which give many stable diffeomorphism types of $-manifolds: * $S^4$ * $S^2 \times S^2$ * $\CP^2$ * $K:= \{x \in \CP^3 | \sum x_i^4 =0\}$, the Kummer surface. Let $P=<g_1, \dots , g_n| r_1,\dots,r_m>$ be the presentation of a group $\pi$. Then we build a $-dimensional complex $X(P)$ by taking a wedge of $n$ circles and attaching a $-cell via each relation $r_i$. Then we thicken $X(P)$ to a smooth compact manifold with boundary $W(P)$ in $\mathbb R^5$ and consider its boundary denoted by $M(P)$. For details and why this is well defined see [[Thickenings]]. $M(P)$ is a smooth $-manifold with fundamental group $\pi$ and we add it to our list * $M(P)$ </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]]4$-manifolds.

2 Construction and examples

We begin with the construction of manifolds which give many stable diffeomorphism types of $4$$4$-manifolds:

• $S^4$$S^4$
• $S^2 \times S^2$$S^2 \times S^2$
• $\CP^2$$\CP^2$
• $K:= \{x \in \CP^3 | \sum x_i^4 =0\}$$K:= \{x \in \CP^3 | \sum x_i^4 =0\}$, the Kummer surface.

Let $P=<g_1, \dots , g_n| r_1,\dots,r_m>$$P=<g_1, \dots , g_n| r_1,\dots,r_m>$ be the presentation of a group $\pi$$\pi$. Then we build a $2$$2$-dimensional complex $X(P)$$X(P)$ by taking a wedge of $n$$n$ circles and attaching a $2$$2$-cell via each relation $r_i$$r_i$. Then we thicken $X(P)$$X(P)$ to a smooth compact manifold with boundary $W(P)$$W(P)$ in $\mathbb R^5$$\mathbb R^5$ and consider its boundary denoted by $M(P)$$M(P)$. For details and why this is well defined see Thickenings. $M(P)$$M(P)$ is a smooth $4$$4$-manifold with fundamental group $\pi$$\pi$ and we add it to our list

• $M(P)$$M(P)$

3 Invariants

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4 Classification/Characterization

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5 Further discussion

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