# Stable classification of 4-manifolds

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## 1 Introduction

In this page we report about the stable classification of $4$${{Author|Matthias Kreck}}{{Stub}} == Introduction == ; In this page we report about the [[stable classification of manifolds|stable classification]] of -manifolds. == Construction and examples == ; 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= 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) == Invariants == ; ... == Classification/Characterization == ; ... == Further discussion == ; ... == References == {{#RefList:}} [[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=$$P=$ be the presentation of a group $\pi$$\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)$

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