On the signature of four-manifolds with universal covering spin
On the signature of four-manifolds with universal covering spin
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DOI：
10.1007/bf01444915
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发表时间：
1993
影响因子：
1.4
通讯作者：
P. Teichner
中科院分区：
文献类型：
--
作者：
P. Teichner
本文研究了泛覆盖为自旋的闭定向4-流形，并提出了对签名整除性是否有限制的问题。由于任何自然数都是r 2，s的连通和的签名，如果没有关于泛覆盖的假设，就不可能存在任何限制。当然，最著名的这种限制是由Rohlin在[10]中证明的，在那里他证明了光滑的4维自旋流形的签名a可以被16整除（比较我们的主要定理的第（2）部分以获得新的证明）。库默曲面K表明这是可能的最佳一般结果。除以K上的某个自由全纯对合，得到Enriques曲面（比较[1]），它的构造特征为8，基本群为7//2。此外，Hitchin在[5]中证明了Enriques曲面上存在一个反全纯自由对合.我们将商称为希钦流形，它的签名为4，基本群为7//2 x 7//2。Rohlin定理是对非旋四维流形的一个很好的推广，比较[4，定理6.3]：a（M）= FoF-2-fl（F）mod 16.这里F是M中与W2 M对偶的（不一定可定向的）曲面.这意味着在M\F上存在一个Pin-结构，它导致F上的Pin-结构，从而导致F上的7//2-交形式的二次加细。因此，可以定义77/8值Brown-Aft不变量fl（F）。从曲面F开始，上面的公式表明可以构造具有小签名和泛覆盖自旋的4-流形的进一步例子。但到目前为止，所有使用这种方法的尝试都失败了，因此我们试图找到一种不同的方法来解决这个问题。这是出于1990年Oberwolfach拓扑会议上几个人的问题，他们想知道是否所有闭定向4-流形的签名，其泛
In this note we study closed oriented 4-manifolds whose universal covering is spin and ask whether there are restrictions on the divisibility of the signature. Since any natural number appears as the signature of a connected sum of r 2, s, without the assumption on the universal covering there cannot exist any restrictions. Certainly, the most famous such restriction was proved by Rohlin in [10], where he showed that the signature a of a smooth 4-dimensional spin manifold is divisible by 16 (compare part (2) of our Main Theorem for a new proof). The Kummer surface K shows that this is the best possible general result. Dividing by a certain free holomorphic involution on K, one obtains the Enriques surface (compare [1]) which by construction has signature 8 and fundamental group 7//2. Furthermore, Hitchin showed in [5] that there exists an antiholomorphic free involution on the Enriques surface. We will refer to the quotient as the Hitchin manifold which then has signature 4 and fundamental group 7//2 x 7//2. Rohlin's theorem admits a nice generalization to nonspin 4-manifolds, compare [4, Theorem 6.3]: a (M)= FoF-2-fl (F) mod 16.Here F is a (not necessarily orientable) surface in M which is dual to W2 M. This implies that there exists a Pin--structure on M\F which induces a Pin--structure on F and hereby a quadratic refinement of the 7//2-intersection form on F. Thus the 77/8-valued Brown-Aft invariant fl (F) can be defined. Starting with a surface F, the above formula suggests that one can construct further examples of 4-manifolds with small signature and universal covering spin. But so far, all attempts using this method failed and therefore we tried to find a different way of attacking the problem. This was motivated by the questions of several people at the Oberwolfach Topology Conference in 1990 who wanted to know whether the signature of all closed oriented 4-manifolds whose universal