Gysin图和Atiyah-Hirzebruch谱序列
Gysin map and Atiyah-Hirzebruch spectral sequence
摘要：
在具有有限CW-复形同伦类型的空间上，讨论了乘法上同调理论的Atiyah-Hirzebruch谱序列与Gysin映射之间的关系。特别地，让我们固定这样一个乘法上同调理论h*，让我们考虑一个n维光滑流形X和一个p维紧致子流形Y，满足适当的可定向性假设。我们证明，从Atiyah-Hirzebruch谱序列开始，Y在X中的Poincarè对偶，在我们的设置中，它是一个单上同调类，系数在h^{n-p}（一点）中，如果这样的类生存到最后一步，它是通过Y的单位上同调类的Gysin映射的图像表示的。然后我们证明了Y上一般上同调类的类似陈述。
英文摘要：
We discuss the relations between the Atiyah-Hirzebruch spectral sequence and the Gysin map for a multiplicative cohomology theory, on spaces having the homotopy type of a finite CW-complex. In particular, let us fix such a multiplicative cohomology theory h* and let us consider a smooth manifold X of dimension n and a compact submanifold Y of dimension p, satisfying suitable hypotheses about orientability. We prove that, starting the Atiyah-Hirzebruch spectral sequence with the Poincarè dual of Y in X, which, in our setting, is a simplicial cohomology class with coefficients in h^{n-p}(one-point), if such a class survives until the last step, it is represented by the image via the Gysin map of the unit cohomology class of Y. We then prove the analogous statement for a generic cohomology class on Y.
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