Stable classification of 4-manifolds

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1 Introduction

In this page we report about the stable classification of closed oriented $4$${{Author|Matthias Kreck}}{{Stub}} == Introduction == <wikitex>; In this page we report about the [[stable classification of manifolds|stable classification]] of closed oriented $-manifolds. We will begin with a special class of closed oriented $-manifolds, namely those, where the universal covering is not spinnable. </wikitex> == Construction and examples == <wikitex>; We begin with the construction of manifolds which give many stable diffeomorphism types of $-manifolds: * $S^4$ * $S^2 \times S^2$ * $\CP^2$ * $K:= \{x \in \CP^3 | \sum x_i^4 =0\}$, the Kummer surface. Let $P=<g_1, \dots , g_n| r_1,\dots,r_m>$ be the presentation of a group $\pi$. Then we build a $-dimensional complex $X(P)$ by taking a wedge of $n$ circles and attaching a $-cell via each relation $r_i$. Then we thicken $X(P)$ to a smooth compact manifold with boundary $W(P)$ in $\mathbb R^5$ and consider its boundary denoted by $M(P)$. For details and why this is well defined see [[Thickenings]]. $M(P)$ is a smooth $-manifold with fundamental group $\pi$ and we add it to our list: * $M(P)$ Let $\pi$ be a finitely presentable group. Then for each element $\alpha$ in $H_4(K(\pi_1,1)$ there is a smooth, closed, oriented manifold $M(\alpha)$ with signature zero and a map $f: M(\alpha) \to K(\pi_1,1)$ mapping the fundamental class to $\alpha$. The existence follows (for example using the [[B-Bordism#Spectral sequences|Atiyah-Hirzebruch spectral sequence]]) from the fact that the oriented bordism groups are zero in degree 4$-manifolds. We will begin with a special class of closed oriented $4$$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$$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)$

Let $\pi$$\pi$ be a finitely presentable group. Then for each element $\alpha$$\alpha$ in $H_4(K(\pi_1,1)$$H_4(K(\pi_1,1)$ there is a smooth, closed, oriented manifold $M(\alpha)$$M(\alpha)$ with signature zero and a map $f: M(\alpha) \to K(\pi_1,1)$$f: M(\alpha) \to K(\pi_1,1)$ mapping the fundamental class to $\alpha$$\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$$1$, $2$$2$ and $3$$3$: see 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)$$M(\alpha)$

3 Invariants

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

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

  Tex syntax error

  $M$.

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

4 Classification

Theorem 4.1. Let

Tex syntax error

$M$ and $N$$N$ be $4$$4$-dimensional compact smooth manifolds with non spinnable universal covering. Then

Tex syntax error

$M$ and $N$$N$ are stably diffeomorphic if and only if the invariants above agree.

The different stable diffeomorphism classes of manifolds with fundamental group $\pi$$\pi$ are given by $M(\alpha ) \sharp_k \CP^2 \sharp_s (-\CP^2)$$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$$1$-type is $K(\pi,1) \times BSO \to BO$$K(\pi,1) \times BSO \to BO$, see Stable classification of manfifolds. Thus the $B$$B$-bordism group is $\Omega ^{SO}(K(\pi_1,1)$$\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)$$\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.

$\displaystyle \xymatrix{ B \ar[r]^{} \ar[d]^{} & K(\pi, 1) \ar[d]^{\hat w_2} \\ BSO \ar[r]^{w_2} & K(\Zz/2, 2) }$

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 $: see [[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 == ; {{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$$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$$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)$

Let $\pi$$\pi$ be a finitely presentable group. Then for each element $\alpha$$\alpha$ in $H_4(K(\pi_1,1)$$H_4(K(\pi_1,1)$ there is a smooth, closed, oriented manifold $M(\alpha)$$M(\alpha)$ with signature zero and a map $f: M(\alpha) \to K(\pi_1,1)$$f: M(\alpha) \to K(\pi_1,1)$ mapping the fundamental class to $\alpha$$\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$$1$, $2$$2$ and $3$$3$: see 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)$$M(\alpha)$

3 Invariants

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

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

  Tex syntax error

  $M$.

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

4 Classification

Theorem 4.1. Let

Tex syntax error

$M$ and $N$$N$ be $4$$4$-dimensional compact smooth manifolds with non spinnable universal covering. Then

Tex syntax error

$M$ and $N$$N$ are stably diffeomorphic if and only if the invariants above agree.

The different stable diffeomorphism classes of manifolds with fundamental group $\pi$$\pi$ are given by $M(\alpha ) \sharp_k \CP^2 \sharp_s (-\CP^2)$$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$$1$-type is $K(\pi,1) \times BSO \to BO$$K(\pi,1) \times BSO \to BO$, see Stable classification of manfifolds. Thus the $B$$B$-bordism group is $\Omega ^{SO}(K(\pi_1,1)$$\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)$$\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.

$\displaystyle \xymatrix{ B \ar[r]^{} \ar[d]^{} & K(\pi, 1) \ar[d]^{\hat w_2} \\ BSO \ar[r]^{w_2} & K(\Zz/2, 2) }$

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 $K(\pi,1) \times BSO \to BO$, 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]]. $$ \xymatrix{ B \ar[r]^{} \ar[d]^{} & K(\pi, 1) \ar[d]^{\hat w_2} \ BSO \ar[r]^{w_2} & K(\Zz/2, 2) } $$ == Further discussion == ; ... == References == {{#RefList:}} [[Category:Manifolds]]4-manifolds. We will begin with a special class of closed oriented $4$$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$$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)$

Let $\pi$$\pi$ be a finitely presentable group. Then for each element $\alpha$$\alpha$ in $H_4(K(\pi_1,1)$$H_4(K(\pi_1,1)$ there is a smooth, closed, oriented manifold $M(\alpha)$$M(\alpha)$ with signature zero and a map $f: M(\alpha) \to K(\pi_1,1)$$f: M(\alpha) \to K(\pi_1,1)$ mapping the fundamental class to $\alpha$$\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$$1$, $2$$2$ and $3$$3$: see 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)$$M(\alpha)$

3 Invariants

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

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

  Tex syntax error

  $M$.

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

4 Classification

Theorem 4.1. Let

Tex syntax error

$M$ and $N$$N$ be $4$$4$-dimensional compact smooth manifolds with non spinnable universal covering. Then

Tex syntax error

$M$ and $N$$N$ are stably diffeomorphic if and only if the invariants above agree.

The different stable diffeomorphism classes of manifolds with fundamental group $\pi$$\pi$ are given by $M(\alpha ) \sharp_k \CP^2 \sharp_s (-\CP^2)$$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$$1$-type is $K(\pi,1) \times BSO \to BO$$K(\pi,1) \times BSO \to BO$, see Stable classification of manfifolds. Thus the $B$$B$-bordism group is $\Omega ^{SO}(K(\pi_1,1)$$\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)$$\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.

$\displaystyle \xymatrix{ B \ar[r]^{} \ar[d]^{} & K(\pi, 1) \ar[d]^{\hat w_2} \\ BSO \ar[r]^{w_2} & K(\Zz/2, 2) }$

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