Stable classification of 4-manifolds

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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 $H_4(K(\pi,1)$ there is a smooth, closed, connected, oriented, non-spinnable manifold $M(\alpha)$ with signature zero, fundamental group $\pi$ and $u_*([M]) = \alpha$. This is proved in several steps by first using the [[B-Bordism#Spectral sequences|Atiyah-Hirzebruch spectral sequence]]) and the fact that the oriented bordism groups are zero in degree $1$, $2$ and $3$: see [[Oriented bordism]] to show that there is a closed, smooth, oriented manifold $M$ together with a map $f: N \to (K(\pi,1)$ with $f_*([M]) = \alpha$ and signature zero. Then by surgeries on $0$- and $1$-dimensional spheres one changes $M$ and $f$ in such a way, that $M$ is connected and $f_*$ is an isomorphism on $\pi_1$. Finally we form the connected sum $\mathbb{CP}^2 \oplus (-\mathbb {CP}^2)$ to make $M$ non-spinnable. This manifold is of course not unique but we will se that it is unique up to stable diffeomorphism and we abbreviate it by
+
Let $\pi$ be a finitely presentable group. Then for each element $\alpha$ in $H_4(K(\pi,1))$ there is a smooth, closed, connected, oriented, non-spinnable manifold $M(\alpha)$ with signature zero, fundamental group $\pi$ and $u_*([M]) = \alpha$. This is proved in several steps by first using the [[B-Bordism#Spectral sequences|Atiyah-Hirzebruch spectral sequence]]) and the fact that the oriented bordism groups are zero in degree $1$, $2$ and $3$: see [[Oriented bordism]] to show that there is a closed, smooth, oriented manifold $M$ together with a map $f: N \to K(\pi,1)$ with $f_*([M]) = \alpha$ and signature zero. Then by surgeries on $0$- and $1$-dimensional spheres one changes $M$ and $f$ in such a way, that $M$ is connected and $f_*$ is an isomorphism on $\pi_1$ (reference). Finally we form the connected sum with $\mathbb{CP}^2 \oplus (-\mathbb {CP}^2)$ to make $M$ non-spinnable. This manifold is of course not unique but we will see that it is unique up to stable diffeomorphisms and we abbreviate it by
* $M(\alpha)$
* $M(\alpha)$
</wikitex>
</wikitex>

Revision as of 15:45, 31 March 2011

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

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Contents

1 Introduction

In this page we report about the stable classification of closed oriented 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)) there is a smooth, closed, connected, oriented, non-spinnable manifold M(\alpha) with signature zero, fundamental group \pi and u_*([M]) = \alpha. This is proved in several steps by first using the Atiyah-Hirzebruch spectral sequence) and the fact that the oriented bordism groups are zero in degree 1, 2 and 3: see Oriented bordism to show that there is a closed, smooth, oriented manifold
Tex syntax error
together with a map f: N \to K(\pi,1) with f_*([M]) = \alpha and signature zero. Then by surgeries on 0- and 1-dimensional spheres one changes
Tex syntax error
and f in such a way, that
Tex syntax error
is connected and f_* is an isomorphism on \pi_1 (reference). Finally we form the connected sum with \mathbb{CP}^2 \oplus (-\mathbb {CP}^2) to make
Tex syntax error
non-spinnable. This manifold is of course not unique but we will see that it is unique up to stable diffeomorphisms and we abbreviate it by
  • 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
    Tex syntax error
    .

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
Tex syntax error
and N be 4-dimensional compact smooth manifolds with non spinnable universal covering. Then
Tex syntax error
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.

\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

$, $ and $: see [[Oriented bordism]] to show that there is a closed, smooth, oriented manifold $M$ together with a map $f: N \to K(\pi,1)$ with $f_*([M]) = \alpha$ and signature zero. Then by surgeries on -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)) there is a smooth, closed, connected, oriented, non-spinnable manifold M(\alpha) with signature zero, fundamental group \pi and u_*([M]) = \alpha. This is proved in several steps by first using the Atiyah-Hirzebruch spectral sequence) and the fact that the oriented bordism groups are zero in degree 1, 2 and 3: see Oriented bordism to show that there is a closed, smooth, oriented manifold
Tex syntax error
together with a map f: N \to K(\pi,1) with f_*([M]) = \alpha and signature zero. Then by surgeries on 0- and 1-dimensional spheres one changes
Tex syntax error
and f in such a way, that
Tex syntax error
is connected and f_* is an isomorphism on \pi_1 (reference). Finally we form the connected sum with \mathbb{CP}^2 \oplus (-\mathbb {CP}^2) to make
Tex syntax error
non-spinnable. This manifold is of course not unique but we will see that it is unique up to stable diffeomorphisms and we abbreviate it by
  • 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
    Tex syntax error
    .

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
Tex syntax error
and N be 4-dimensional compact smooth manifolds with non spinnable universal covering. Then
Tex syntax error
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.

\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

$- and 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)) there is a smooth, closed, connected, oriented, non-spinnable manifold M(\alpha) with signature zero, fundamental group \pi and u_*([M]) = \alpha. This is proved in several steps by first using the Atiyah-Hirzebruch spectral sequence) and the fact that the oriented bordism groups are zero in degree 1, 2 and 3: see Oriented bordism to show that there is a closed, smooth, oriented manifold
Tex syntax error
together with a map f: N \to K(\pi,1) with f_*([M]) = \alpha and signature zero. Then by surgeries on 0- and 1-dimensional spheres one changes
Tex syntax error
and f in such a way, that
Tex syntax error
is connected and f_* is an isomorphism on \pi_1 (reference). Finally we form the connected sum with \mathbb{CP}^2 \oplus (-\mathbb {CP}^2) to make
Tex syntax error
non-spinnable. This manifold is of course not unique but we will see that it is unique up to stable diffeomorphisms and we abbreviate it by
  • 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
    Tex syntax error
    .

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
Tex syntax error
and N be 4-dimensional compact smooth manifolds with non spinnable universal covering. Then
Tex syntax error
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.

\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

$-dimensional spheres one changes $M$ and $f$ in such a way, that $M$ is connected and $f_*$ is an isomorphism on $\pi_1$ (reference). Finally we form the connected sum with $\mathbb{CP}^2 \oplus (-\mathbb {CP}^2)$ to make $M$ non-spinnable. This manifold is of course not unique but we will see that it is unique up to stable diffeomorphisms and we abbreviate it by * $M(\alpha)$ == 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-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)) there is a smooth, closed, connected, oriented, non-spinnable manifold M(\alpha) with signature zero, fundamental group \pi and u_*([M]) = \alpha. This is proved in several steps by first using the Atiyah-Hirzebruch spectral sequence) and the fact that the oriented bordism groups are zero in degree 1, 2 and 3: see Oriented bordism to show that there is a closed, smooth, oriented manifold
Tex syntax error
together with a map f: N \to K(\pi,1) with f_*([M]) = \alpha and signature zero. Then by surgeries on 0- and 1-dimensional spheres one changes
Tex syntax error
and f in such a way, that
Tex syntax error
is connected and f_* is an isomorphism on \pi_1 (reference). Finally we form the connected sum with \mathbb{CP}^2 \oplus (-\mathbb {CP}^2) to make
Tex syntax error
non-spinnable. This manifold is of course not unique but we will see that it is unique up to stable diffeomorphisms and we abbreviate it by
  • 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
    Tex syntax error
    .

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
Tex syntax error
and N be 4-dimensional compact smooth manifolds with non spinnable universal covering. Then
Tex syntax error
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.

\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

$-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-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)) there is a smooth, closed, connected, oriented, non-spinnable manifold M(\alpha) with signature zero, fundamental group \pi and u_*([M]) = \alpha. This is proved in several steps by first using the Atiyah-Hirzebruch spectral sequence) and the fact that the oriented bordism groups are zero in degree 1, 2 and 3: see Oriented bordism to show that there is a closed, smooth, oriented manifold
Tex syntax error
together with a map f: N \to K(\pi,1) with f_*([M]) = \alpha and signature zero. Then by surgeries on 0- and 1-dimensional spheres one changes
Tex syntax error
and f in such a way, that
Tex syntax error
is connected and f_* is an isomorphism on \pi_1 (reference). Finally we form the connected sum with \mathbb{CP}^2 \oplus (-\mathbb {CP}^2) to make
Tex syntax error
non-spinnable. This manifold is of course not unique but we will see that it is unique up to stable diffeomorphisms and we abbreviate it by
  • 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
    Tex syntax error
    .

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
Tex syntax error
and N be 4-dimensional compact smooth manifolds with non spinnable universal covering. Then
Tex syntax error
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.

\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

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