Stable classification of 4-manifolds

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== Introduction ==
== Introduction ==
<wikitex>;
<wikitex>;
In this page we report about the [[stable classification of manifolds|stable classification]] of $4$-manifolds.
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In this page we report about the [[stable classification of manifolds|stable classification]] of closed oriented $4$-manifolds. For the general concept and results about stable classification see the page on stable classification. We will begin with a special class of closed oriented $4$-manifolds, namely those, where the universal covering is not spinnable.
</wikitex>
</wikitex>
== Construction and examples ==
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== Construction and examples I==
<wikitex>;
<wikitex>;
We begin with the construction of manifolds which give many stable diffeomorphism types of $4$-manifolds:
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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:
* $S^4$
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* $S^2 \times S^2$
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* $\CP^2$
* $\CP^2$
* $K:= \{x \in \CP^3 | \sum x_i^4 =0\}$, the Kummer surface.
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The second is a large class of manifolds associated to certain algebraic data.
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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 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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* $M(\alpha)$
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</wikitex>
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
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== Invariants ==
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<wikitex>;
* $M(P)$
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The following is a complete list of invariants for the stable classification of closed, smooth oriented $4$-manifolds whose universal covering is not spinnable:
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* The Euler characteristic $\chi (M)$
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* The signature $\sigma (M)$
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* The fundamanetal group $\pi_1(M)$
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* The image of the fundamental class $[u_*([M])]\in H_4(K(\pi_1(M),1)/Out(\pi_1(M))$of $M$.
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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 $K(\pi_1(M),1)$.
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</wikitex>
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== Classification ==
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<wikitex>;
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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.
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{{endthm}}
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The proof of this result is an easy consequence of the general stable classification theorem (\cite{Kreck1999}, [[Stable classification of manifolds]]), see the page on stable classification. 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]].
</wikitex>
</wikitex>
== Invariants ==
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== Realization of the invariants ==
<wikitex>;
<wikitex>;
...
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Given the Theorem above one wonders about the realization of the invariants. Partial answers are easy, but in general this is a complicated open question, where the answer is only known for special fundamental groups. In the simply connected case there are only two invariants, the Euler characteristic and the signature. The Euler characteristic of of a simply connected $4$-manifold is $\ge 2$ and it is $2$ if and only if $M$ is a homotopy sphere. Since a homotopy sphere is spinnable it cannot occur in our context. Thus the Euler characteristic is at least $3$, the Euler characteristic of $\mathbb {CP}^2$. By connected sum with copies of $\mathbb {CP}^2$ we can achieve all values $\ge 3$.
</wikitex>
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If the Euler characteristic of a simply connected closed $4$-manifold is $k$, the second Betti number is $b=k-2$ and so the possible values of the signature are $-b \le s \le b$ with $s = k mod 2$, since the Euler characteristic and the signature agree mod $2$. For given $b$ these values $s$ are realized by $\sharp _{(s+b )/2 }\mathbb {CP}^2 \sharp_ {(s+b)2-s} (-\mathbb { CP}^2)$. Thus we see that all possible values are realized and so we obtain:
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{{beginthm|Theorem}} The stable diffeomorphism types of simply connected, closed, non-spinnable $4$-manifolds are given by connected sums of copies $\mathbb {CP}^2$ and $-\mathbb {CP}^2$.
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{{endthm}}
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What are the problems for realizing the invariants in the non-simply-connected case? If we look at the simply connected case we started with the question, what is the minimal Euler characteristic of a smooth closed non-spinnable $4$-manifold? In the non-simply-connected case the corresponding question for manifolds with fundamental group $\pi$ is the following:
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{{beginthm|Problem}} Given a finitely presentable group $\pi$ and a class $\alpha \in H_4(K(\pi_1,1))$, what is the minimal Euler characacteristic of a closed, oriented $4$-manifold with universal covering non-spinnable and image of the fundamental class under the classifying map of the universal covering $\alpha$? This minimal Euler characteristic is a function $H_4(K(\pi_1,1)) \to \mathbb Z$. It might be an interesting invariant of the group $\pi$ but probably has no good properties.
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{{endthm}}
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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
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$$
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\xymatrix{
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B \ar[r]^{} \ar[d]^{} & K(\pi, 1) \ar[d]^{\hat w_2} \\
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BSO \ar[r]^{w_2} & K(\Zz/2, 2)
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}
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$$ -->
== Classification/Characterization ==
<wikitex>;
...
</wikitex>
</wikitex>
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[[Category:Manifolds]]
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[[Category:Theory]]

Latest revision as of 16:24, 4 January 2013

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 4-manifolds. For the general concept and results about stable classification see the page on stable classification. 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 I

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:

  • \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 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

  • 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 proof of this result is an easy consequence of the general stable classification theorem ([Kreck1999], Stable classification of manifolds), see the page on stable classification. 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 Realization of the invariants

Given the Theorem above one wonders about the realization of the invariants. Partial answers are easy, but in general this is a complicated open question, where the answer is only known for special fundamental groups. In the simply connected case there are only two invariants, the Euler characteristic and the signature. The Euler characteristic of of a simply connected 4-manifold is \ge 2 and it is 2 if and only if M is a homotopy sphere. Since a homotopy sphere is spinnable it cannot occur in our context. Thus the Euler characteristic is at least 3, the Euler characteristic of \mathbb {CP}^2. By connected sum with copies of \mathbb {CP}^2 we can achieve all values \ge 3.

If the Euler characteristic of a simply connected closed 4-manifold is
Tex syntax error
, the second Betti number is b=k-2 and so the possible values of the signature are -b \le s \le b with s = k mod 2, since the Euler characteristic and the signature agree mod 2. For given b these values s are realized by \sharp _{(s+b )/2 }\mathbb {CP}^2 \sharp_ {(s+b)2-s} (-\mathbb { CP}^2). Thus we see that all possible values are realized and so we obtain:


Theorem 5.1. The stable diffeomorphism types of simply connected, closed, non-spinnable 4-manifolds are given by connected sums of copies \mathbb {CP}^2 and -\mathbb {CP}^2.

What are the problems for realizing the invariants in the non-simply-connected case? If we look at the simply connected case we started with the question, what is the minimal Euler characteristic of a smooth closed non-spinnable 4-manifold? In the non-simply-connected case the corresponding question for manifolds with fundamental group \pi is the following:

Problem 5.2. Given a finitely presentable group \pi and a class \alpha \in H_4(K(\pi_1,1)), what is the minimal Euler characacteristic of a closed, oriented 4-manifold with universal covering non-spinnable and image of the fundamental class under the classifying map of the universal covering \alpha? This minimal Euler characteristic is a function H_4(K(\pi_1,1)) \to \mathbb Z. It might be an interesting invariant of the group \pi but probably has no good properties.





6 Further discussion

...

7 References

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