Quadratic Functions of Cocycles and Pin Structures
Quadratic Functions of Cocycles and Pin Structures
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共循环和销结构的二次函数
DOI：
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发表时间：
2018
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通讯作者：
J. Morgan
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作者：
G. Brumfiel;J. Morgan
我们构造了紧致$n$-流形$M^n$上的Pin$^-$结构的等价类与定义在$n-1$相对$\mathbb {Z}/2$余圈上的$\mathbb {Z}/4$值“二次函数”$Q$之间的自然双射对应，$Q \colon Z^{n-1}（M^n，\partial M^n ; \mathbb{Z} /2）\to \mathbb{Z}/4$. $Q（p+q）$的“二次”性质和上边界上的值$Q（dc）$用Steenrod的更高的$\cup_i$乘积表示。对于$n = 2$的结果扩展了旧的结果有关Pin$^-$结构的封闭曲面上的二次加细杯产品配对$H ^1（M^n ; \mathbb{Z} /2）$。在定向的情况下，也就是说，对于自旋流形，结果扩展了Kapustin的结果，参见arXiv：1505.05856v2，以及我们以前关于Pontrjagin对偶四维自旋边数的论文中的结果，参见arXiv：1803.08147。在本文中，将这些结果推广到Pin$^-$流形需要一种不同的方法，涉及到Postnikov塔的稳定同伦理论。
We construct a natural bijective correspondence between equivalence classes of Pin$^-$ structures on a compact simplicial $n$-manifold $M^n$, possibly with boundary, and $\mathbb{Z}/4$-valued 'quadratic functions' $Q$ defined on degree $n-1$ relative $\mathbb{Z}/2$ cocycles, $Q \colon Z^{n-1}(M^n, \partial M^n ; \mathbb{Z} /2) \to \mathbb{Z}/4$. The 'quadratic' property of $Q(p+q)$ and the values $Q(dc)$ on coboundaries are expressed in terms of higher $\cup_i$ products of Steenrod. For $n = 2$ the results extend old results relating Pin$^-$ structures on closed surfaces to quadratic refinements of the cup product pairing on $H^1(M^n ; \mathbb{Z} /2)$. In the oriented case, that is, for Spin manifolds, the results extend results of Kapustin, see arXiv:1505.05856v2, and results in our previous paper on the Pontrjagin dual 4-dimensional Spin bordism, see arXiv:1803.08147. The extension of those results to Pin$^-$ manifolds in this paper required a different approach, involving some stable homotopy theory of Postnikov towers.