Stable classification of 4-manifolds

Revision as of 12:36, 27 November 2010 (view source) Matthias Kreck (Talk | contribs) ← Older edit		Revision as of 14:25, 27 November 2010 (view source) Diarmuid Crowley (Talk | contribs) m Newer edit →
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	Let $P=<g_1, \dots , g_n| r_1,\dots,r_m>$ be the presentation of a group $\pi$. Then we build a $2$-dimensional complex $X(P)$ by taking a wedge of $n$ circles and attaching a $2$-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 $4$-manifold with fundamental group $\pi$ and we add it to our list:		Let $P=<g_1, \dots , g_n| r_1,\dots,r_m>$ be the presentation of a group $\pi$. Then we build a $2$-dimensional complex $X(P)$ by taking a wedge of $n$ circles and attaching a $2$-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 $4$-manifold with fundamental group $\pi$ and we add it to our list:
	* $M(P)$		* $M(P)$
−	Let $\pi$ be a finitely presentable group. Then for each element $\alpha$ in $H_4(K(\pi_1,1)$ there is a smooth, closed, oriented manifold $M(\alpha)$ with signature zero and a map $f: M(\alpha) \to K(\pi_1,1)$ mapping the fundamental class to $\alpha$. The existence follows (for example using the Atiyah-Hirzebruch spectral sequence [[B-Bordism#Spectral sequences]] from the fact that the oriented bordism group is zero in degree $1$, $2$ and $3$ [[Oriented bordism]].	+	Let $\pi$ be a finitely presentable group. Then for each element $\alpha$ in $H_4(K(\pi_1,1)$ there is a smooth, closed, oriented manifold $M(\alpha)$ with signature zero and a map $f: M(\alpha) \to K(\pi_1,1)$ mapping the fundamental class to $\alpha$. The existence follows (for example using the [[B-Bordism#Spectral sequences|Atiyah-Hirzebruch spectral sequence]]) from the fact that the oriented bordism groups are zero in degree $1$, $2$ and $3$: [[Oriented bordism]].
−	This manifold is - of course - not unique. But we will see that its stable diffeomorphism class is unique, if we require that the universal covering is non spinnable. We add it to our list:	+	This manifold is, of course, not unique. But we will see that its stable diffeomorphism class is unique, if we require that the universal covering is non spinnable. We add it to our list:
	* $M(\alpha)$		* $M(\alpha)$
	</wikitex>		</wikitex>
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	</wikitex>		</wikitex>
−	== Classification/Characterization ==	+	== Classification ==
	<wikitex>;		<wikitex>;
	{{beginthm|Theorem}} Let $M$ and $N$ be $4$-dimensional compact smooth manifolds with non spinnable universal covering. Then $M$ and $N$ are stably diffeomorphic if and only if the invariants above agree.		{{beginthm|Theorem}} Let $M$ and $N$ be $4$-dimensional compact smooth manifolds with non spinnable universal covering. Then $M$ and $N$ are stably diffeomorphic if and only if the invariants above agree.
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	{{endthm}}		{{endthm}}
−	The proof of this result is an easy consequence of the general stable classification theorem (\cite{Kreck1999}, [[Stable classification of manifolds]]). Namely, the normal $1$-type is $K(\pi,1) \times BSO \to BO$ [[Stable classification of manifolds#The normal k-type]]. Thus the $B$-bordism group is $\Omega ^{SO}(K(\pi_1,1)$, which by the Atiyah-Hirzebruch spectral sequence is ismorphic to $\mathbb Z \oplus H_4(K(\pi,1);\mathbb Z)$ under the signature and the image of the fundamental class. Now the statement follows from [[Stable classification of manifolds#Theorem 3.1.]]	+	The proof of this result is an easy consequence of the general stable classification theorem (\cite{Kreck1999}, [[Stable classification of manifolds]]). Namely, the normal $1$-type is $K(\pi,1) \times BSO \to BO$, see [[Stable classification of manifolds#The normal k-type|Stable classification of manfifolds]]. Thus the $B$-bordism group is $\Omega ^{SO}(K(\pi_1,1)$, which by the Atiyah-Hirzebruch spectral sequence is ismorphic to $\mathbb Z \oplus H_4(K(\pi,1);\mathbb Z)$ under the signature and the image of the fundamental class. Now the statement follows from Theorem 3.1 of [[Stable classification of manifolds#The stable classification of normal (k-1)-smoothings on 2k-dimensional manifolds|Stable classification of manifolds]].
		+
		+	$$
		+	\xymatrix{
		+	B \ar[r]^{} \ar[d]^{} & K(\pi, 1) \ar[d]^{\hat w_2} \\
		+	BSO \ar[r]^{w_2} & K(\Zz/2, 2)
		+	}
		+	$$
		+
	</wikitex>		</wikitex>

---

Revision as of 14:25, 27 November 2010

The user responsible for this page is Matthias Kreck. No other user may edit this page at present.

This page has not been refereed. The information given here might be incomplete or provisional.

Contents 1 Introduction 2 Construction and examples 3 Invariants 4 Classification 5 Further discussion 6 References

1 Introduction

In this page we report about the stable classification of clsoed oriented ${{Author|Matthias Kreck}}{{Stub}} == Introduction == <wikitex>; In this page we report about the [[stable classification of manifolds|stable classification]] of clsoed oriented $-manifolds. We will begin with a special class of closed oriented $-manifolds, namely those, where the universal covering is not spinnable. </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: * $M(P)$ Let $\pi$ be a finitely presentable group. Then for each element $\alpha$ in $H_4(K(\pi_1,1)$ there is a smooth, closed, oriented manifold $M(\alpha)$ with signature zero and a map $f: M(\alpha) \to K(\pi_1,1)$ mapping the fundamental class to $\alpha$. The existence follows (for example using the Atiyah-Hirzebruch spectral sequence [[B-Bordism#Spectral sequences]] from the fact that the oriented bordism group is zero in degree 4$-manifolds. We will begin with a special class of closed oriented $4$-manifolds, namely those, where the universal covering is not spinnable.

2 Construction and examples

We begin with the construction of manifolds which give many stable diffeomorphism types of $4$-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 $2$-dimensional complex $X(P)$ by taking a wedge of $n$ circles and attaching a $2$-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 $4$-manifold with fundamental group $\pi$ and we add it to our list:

• $M(P)$

Let $\pi$ be a finitely presentable group. Then for each element $\alpha$ in $H_4(K(\pi_1,1)$ there is a smooth, closed, oriented manifold $M(\alpha)$ with signature zero and a map $f: M(\alpha) \to K(\pi_1,1)$ mapping the fundamental class to $\alpha$. The existence follows (for example using the Atiyah-Hirzebruch spectral sequence) from the fact that the oriented bordism groups are zero in degree $1$, $2$ and $3$: Oriented bordism. This manifold is, of course, not unique. But we will see that its stable diffeomorphism class is unique, if we require that the universal covering is non spinnable. We add it to our list:

• $M(\alpha)$

3 Invariants

The following is a complete list of invariants for the stable classification of closed, smooth oriented $4$-manifolds whose universal covering is not spinnable:

• The Euler characteristic $\chi (M)$
• The signature $\sigma (M)$
• The fundamanetal group $\pi_1(M)$
• The image of the fundamental class $[u_*([M])]\in H_4(K(\pi_1(M),1)/Out(\pi_1(M))$of $M$.

Here $u:M \to K(\pi_1(M),1)$ is a classifying map of the universal covering and $Out(\pi_1(M))$ is the outer automorphism group which acts on the homology of $K(\pi_1(M),1)$.

4 Classification

The proof of this result is an easy consequence of the general stable classification theorem ([Kreck1999], Stable classification of manifolds). Namely, the normal $1$-type is $K(\pi,1) \times BSO \to BO$, see Stable classification of manfifolds. Thus the $B$-bordism group is $\Omega ^{SO}(K(\pi_1,1)$, which by the Atiyah-Hirzebruch spectral sequence is ismorphic to $\mathbb Z \oplus H_4(K(\pi,1);\mathbb Z)$ under the signature and the image of the fundamental class. Now the statement follows from Theorem 3.1 of Stable classification of manifolds.

5 Further discussion

...

6 References

• [Kreck1999] M. Kreck, Surgery and duality, Ann. of Math. (2) 149 (1999), no.3, 707–754. MR1709301 (2001a:57051) Zbl 0935.57039

$, $ and $ [[Oriented bordism]]. This manifold is - of course - not unique. But we will see that its stable diffeomorphism class is unique, if we require that the universal covering is non spinnable. We add it to our list: * $M(\alpha)$ == Invariants == ; The following is a complete list of invariants for the stable classification of closed, smooth oriented $-manifolds whose universal covering is not spinnable: * The Euler characteristic $\chi (M)$ * The signature $\sigma (M)$ * The fundamanetal group $\pi_1(M)$ * The image of the fundamental class $[u_*([M])]\in H_4(K(\pi_1(M),1)/Out(\pi_1(M))$of $M$. Here $u:M \to K(\pi_1(M),1)$ is a classifying map of the universal covering and $Out(\pi_1(M))$ is the outer automorphism group which acts on the homology of $K(\pi_1(M),1)$. == Classification/Characterization == ; {{beginthm|Theorem}} Let $M$ and $N$ be $-dimensional compact smooth manifolds with non spinnable universal covering. Then $M$ and $N$ are stably diffeomorphic if and only if the invariants above agree. The different stable diffeomorphism classes of manifolds with fundamental group $\pi$ are given by $M(\alpha ) \sharp_k \CP^2 \sharp_s (-\CP^2)$. {{endthm}} The proof of this result is an easy consequence of the general stable classification theorem (\cite{Kreck1999}, [[Stable classification of manifolds]]). Namely, the normal 4-manifolds. We will begin with a special class of closed oriented $4$-manifolds, namely those, where the universal covering is not spinnable.

2 Construction and examples

We begin with the construction of manifolds which give many stable diffeomorphism types of $4$-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 $2$-dimensional complex $X(P)$ by taking a wedge of $n$ circles and attaching a $2$-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 $4$-manifold with fundamental group $\pi$ and we add it to our list:

• $M(P)$

Let $\pi$ be a finitely presentable group. Then for each element $\alpha$ in $H_4(K(\pi_1,1)$ there is a smooth, closed, oriented manifold $M(\alpha)$ with signature zero and a map $f: M(\alpha) \to K(\pi_1,1)$ mapping the fundamental class to $\alpha$. The existence follows (for example using the Atiyah-Hirzebruch spectral sequence) from the fact that the oriented bordism groups are zero in degree $1$, $2$ and $3$: Oriented bordism. This manifold is, of course, not unique. But we will see that its stable diffeomorphism class is unique, if we require that the universal covering is non spinnable. We add it to our list:

• $M(\alpha)$

3 Invariants

The following is a complete list of invariants for the stable classification of closed, smooth oriented $4$-manifolds whose universal covering is not spinnable:

• The Euler characteristic $\chi (M)$
• The signature $\sigma (M)$
• The fundamanetal group $\pi_1(M)$
• The image of the fundamental class $[u_*([M])]\in H_4(K(\pi_1(M),1)/Out(\pi_1(M))$of $M$.

Here $u:M \to K(\pi_1(M),1)$ is a classifying map of the universal covering and $Out(\pi_1(M))$ is the outer automorphism group which acts on the homology of $K(\pi_1(M),1)$.

4 Classification

Theorem 4.1. Let $M$ and $N$ be $4$-dimensional compact smooth manifolds with non spinnable universal covering. Then $M$ and $N$ are stably diffeomorphic if and only if the invariants above agree.

The different stable diffeomorphism classes of manifolds with fundamental group $\pi$ are given by $M(\alpha ) \sharp_k \CP^2 \sharp_s (-\CP^2)$.

The proof of this result is an easy consequence of the general stable classification theorem ([Kreck1999], Stable classification of manifolds). Namely, the normal $1$-type is $K(\pi,1) \times BSO \to BO$, see Stable classification of manfifolds. Thus the $B$-bordism group is $\Omega ^{SO}(K(\pi_1,1)$, which by the Atiyah-Hirzebruch spectral sequence is ismorphic to $\mathbb Z \oplus H_4(K(\pi,1);\mathbb Z)$ under the signature and the image of the fundamental class. Now the statement follows from Theorem 3.1 of Stable classification of manifolds.

5 Further discussion

...

6 References

• [Kreck1999] M. Kreck, Surgery and duality, Ann. of Math. (2) 149 (1999), no.3, 707–754. MR1709301 (2001a:57051) Zbl 0935.57039

$-type is $K(\pi,1) \times BSO \to BO$ [[Stable classification of manifolds#The normal k-type]]. Thus the $B$-bordism group is $\Omega ^{SO}(K(\pi_1,1)$, which by the Atiyah-Hirzebruch spectral sequence is ismorphic to $\mathbb Z \oplus H_4(K(\pi,1);\mathbb Z)$ under the signature and the image of the fundamental class. Now the statement follows from [[Stable classification of manifolds#Theorem 3.1.]] == Further discussion == ; ... == References == {{#RefList:}} [[Category:Manifolds]]4-manifolds. We will begin with a special class of closed oriented $4$-manifolds, namely those, where the universal covering is not spinnable.

2 Construction and examples

We begin with the construction of manifolds which give many stable diffeomorphism types of $4$-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 $2$-dimensional complex $X(P)$ by taking a wedge of $n$ circles and attaching a $2$-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 $4$-manifold with fundamental group $\pi$ and we add it to our list:

• $M(P)$

Let $\pi$ be a finitely presentable group. Then for each element $\alpha$ in $H_4(K(\pi_1,1)$ there is a smooth, closed, oriented manifold $M(\alpha)$ with signature zero and a map $f: M(\alpha) \to K(\pi_1,1)$ mapping the fundamental class to $\alpha$. The existence follows (for example using the Atiyah-Hirzebruch spectral sequence) from the fact that the oriented bordism groups are zero in degree $1$, $2$ and $3$: Oriented bordism. This manifold is, of course, not unique. But we will see that its stable diffeomorphism class is unique, if we require that the universal covering is non spinnable. We add it to our list:

• $M(\alpha)$

3 Invariants

The following is a complete list of invariants for the stable classification of closed, smooth oriented $4$-manifolds whose universal covering is not spinnable:

• The Euler characteristic $\chi (M)$
• The signature $\sigma (M)$
• The fundamanetal group $\pi_1(M)$
• The image of the fundamental class $[u_*([M])]\in H_4(K(\pi_1(M),1)/Out(\pi_1(M))$of $M$.

Here $u:M \to K(\pi_1(M),1)$ is a classifying map of the universal covering and $Out(\pi_1(M))$ is the outer automorphism group which acts on the homology of $K(\pi_1(M),1)$.

4 Classification

The proof of this result is an easy consequence of the general stable classification theorem ([Kreck1999], Stable classification of manifolds). Namely, the normal $1$-type is $K(\pi,1) \times BSO \to BO$, see Stable classification of manfifolds. Thus the $B$-bordism group is $\Omega ^{SO}(K(\pi_1,1)$, which by the Atiyah-Hirzebruch spectral sequence is ismorphic to $\mathbb Z \oplus H_4(K(\pi,1);\mathbb Z)$ under the signature and the image of the fundamental class. Now the statement follows from Theorem 3.1 of Stable classification of manifolds.

5 Further discussion

...

6 References

• [Kreck1999] M. Kreck, Surgery and duality, Ann. of Math. (2) 149 (1999), no.3, 707–754. MR1709301 (2001a:57051) Zbl 0935.57039