Stable classification of 4-manifolds
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− | == Construction and examples == | + | == Construction and examples I== |
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− | We begin with the construction of manifolds which give many stable diffeomorphism types of $4$-manifolds: | + | We begin with the construction of two classes of manifolds which can be used to give many stable diffeomorphism types of non-spinnable $4$-manifolds. The first is: |
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* $\CP^2$ | * $\CP^2$ | ||
− | + | The second is a large class of manifolds associated to certain algebraic data. | |
− | + | 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 sure that $M$ is 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 | |
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− | + | ||
− | 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> | ||
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== Classification == | == Classification == | ||
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− | {{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 $K(\pi,1) \times BSO \to BO$, 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]]. | 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]]. | ||
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+ | The different stable diffeomorphism classes of manifolds with fundamental group $\pi$ are given by $M(\alpha ) \sharp_k \CP^2 \sharp_s (-\CP^2)$. Here $k+s + \chi | ||
$$ | $$ |
Revision as of 15:55, 31 March 2011
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
In this page we report about the 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.
2 Construction and examples I
We begin with the construction of two classes of manifolds which can be used to give many stable diffeomorphism types of non-spinnable -manifolds. The first is:
The second is a large class of manifolds associated to certain algebraic data. Let be a finitely presentable group. Then for each element in there is a smooth, closed, connected, oriented, non-spinnable manifold with signature zero, fundamental group and . 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 , and : see Oriented bordism to show that there is a closed, smooth, oriented manifold together with a map with and signature zero. Then by surgeries on - and -dimensional spheres one changes and in such a way, that is connected and is an isomorphism on (reference). Finally we form the connected sum with to make sure that is 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
3 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
- The signature
- The fundamanetal group
- The image of the fundamental class of .
Here is a classifying map of the universal covering and is the outer automorphism group which acts on the homology of .
4 Classification
Theorem 4.1. Let and be -dimensional compact smooth manifolds with non-spinnable universal covering. Then and are stably diffeomorphic if and only if the invariants above agree.
The proof of this result is an easy consequence of the general stable classification theorem ([Kreck1999], Stable classification of manifolds). Namely, the normal -type is , see Stable classification of manfifolds. Thus the -bordism group is , which by the Atiyah-Hirzebruch spectral sequence is ismorphic to under the signature and the image of the fundamental class. Now the statement follows from Theorem 3.1 of Stable classification of manifolds.
The different stable diffeomorphism classes of manifolds with fundamental group are given by . Here $k+s + \chi
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
2 Construction and examples I
We begin with the construction of two classes of manifolds which can be used to give many stable diffeomorphism types of non-spinnable -manifolds. The first is:
The second is a large class of manifolds associated to certain algebraic data. Let be a finitely presentable group. Then for each element in there is a smooth, closed, connected, oriented, non-spinnable manifold with signature zero, fundamental group and . 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 , and : see Oriented bordism to show that there is a closed, smooth, oriented manifold together with a map with and signature zero. Then by surgeries on - and -dimensional spheres one changes and in such a way, that is connected and is an isomorphism on (reference). Finally we form the connected sum with to make sure that is 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
3 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
- The signature
- The fundamanetal group
- The image of the fundamental class of .
Here is a classifying map of the universal covering and is the outer automorphism group which acts on the homology of .
4 Classification
Theorem 4.1. Let and be -dimensional compact smooth manifolds with non-spinnable universal covering. Then and are stably diffeomorphic if and only if the invariants above agree.
The proof of this result is an easy consequence of the general stable classification theorem ([Kreck1999], Stable classification of manifolds). Namely, the normal -type is , see Stable classification of manfifolds. Thus the -bordism group is , which by the Atiyah-Hirzebruch spectral sequence is ismorphic to under the signature and the image of the fundamental class. Now the statement follows from Theorem 3.1 of Stable classification of manifolds.
The different stable diffeomorphism classes of manifolds with fundamental group are given by . Here $k+s + \chi
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
2 Construction and examples I
We begin with the construction of two classes of manifolds which can be used to give many stable diffeomorphism types of non-spinnable -manifolds. The first is:
The second is a large class of manifolds associated to certain algebraic data. Let be a finitely presentable group. Then for each element in there is a smooth, closed, connected, oriented, non-spinnable manifold with signature zero, fundamental group and . 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 , and : see Oriented bordism to show that there is a closed, smooth, oriented manifold together with a map with and signature zero. Then by surgeries on - and -dimensional spheres one changes and in such a way, that is connected and is an isomorphism on (reference). Finally we form the connected sum with to make sure that is 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
3 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
- The signature
- The fundamanetal group
- The image of the fundamental class of .
Here is a classifying map of the universal covering and is the outer automorphism group which acts on the homology of .
4 Classification
Theorem 4.1. Let and be -dimensional compact smooth manifolds with non-spinnable universal covering. Then and are stably diffeomorphic if and only if the invariants above agree.
The proof of this result is an easy consequence of the general stable classification theorem ([Kreck1999], Stable classification of manifolds). Namely, the normal -type is , see Stable classification of manfifolds. Thus the -bordism group is , which by the Atiyah-Hirzebruch spectral sequence is ismorphic to under the signature and the image of the fundamental class. Now the statement follows from Theorem 3.1 of Stable classification of manifolds.
The different stable diffeomorphism classes of manifolds with fundamental group are given by . Here $k+s + \chi
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
2 Construction and examples I
We begin with the construction of two classes of manifolds which can be used to give many stable diffeomorphism types of non-spinnable -manifolds. The first is:
The second is a large class of manifolds associated to certain algebraic data. Let be a finitely presentable group. Then for each element in there is a smooth, closed, connected, oriented, non-spinnable manifold with signature zero, fundamental group and . 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 , and : see Oriented bordism to show that there is a closed, smooth, oriented manifold together with a map with and signature zero. Then by surgeries on - and -dimensional spheres one changes and in such a way, that is connected and is an isomorphism on (reference). Finally we form the connected sum with to make sure that is 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
3 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
- The signature
- The fundamanetal group
- The image of the fundamental class of .
Here is a classifying map of the universal covering and is the outer automorphism group which acts on the homology of .
4 Classification
Theorem 4.1. Let and be -dimensional compact smooth manifolds with non-spinnable universal covering. Then and are stably diffeomorphic if and only if the invariants above agree.
The proof of this result is an easy consequence of the general stable classification theorem ([Kreck1999], Stable classification of manifolds). Namely, the normal -type is , see Stable classification of manfifolds. Thus the -bordism group is , which by the Atiyah-Hirzebruch spectral sequence is ismorphic to under the signature and the image of the fundamental class. Now the statement follows from Theorem 3.1 of Stable classification of manifolds.
The different stable diffeomorphism classes of manifolds with fundamental group are given by . Here $k+s + \chi
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
2 Construction and examples I
We begin with the construction of two classes of manifolds which can be used to give many stable diffeomorphism types of non-spinnable -manifolds. The first is:
The second is a large class of manifolds associated to certain algebraic data. Let be a finitely presentable group. Then for each element in there is a smooth, closed, connected, oriented, non-spinnable manifold with signature zero, fundamental group and . 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 , and : see Oriented bordism to show that there is a closed, smooth, oriented manifold together with a map with and signature zero. Then by surgeries on - and -dimensional spheres one changes and in such a way, that is connected and is an isomorphism on (reference). Finally we form the connected sum with to make sure that is 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
3 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
- The signature
- The fundamanetal group
- The image of the fundamental class of .
Here is a classifying map of the universal covering and is the outer automorphism group which acts on the homology of .
4 Classification
Theorem 4.1. Let and be -dimensional compact smooth manifolds with non-spinnable universal covering. Then and are stably diffeomorphic if and only if the invariants above agree.
The proof of this result is an easy consequence of the general stable classification theorem ([Kreck1999], Stable classification of manifolds). Namely, the normal -type is , see Stable classification of manfifolds. Thus the -bordism group is , which by the Atiyah-Hirzebruch spectral sequence is ismorphic to under the signature and the image of the fundamental class. Now the statement follows from Theorem 3.1 of Stable classification of manifolds.
The different stable diffeomorphism classes of manifolds with fundamental group are given by . Here $k+s + \chi
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