A Double Points Formula in Complex K-theory and an Application
A Double Points Formula in Complex K-theory and an Application
复制标题
复K理论中的双点公式及其应用
DOI：
10.1112/blms/25.2.184
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发表时间：
1993
期刊：
影响因子：
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通讯作者：
András Szücs
中科院分区：
文献类型：
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作者：
András Szücs
1959年，Lashof和Smale发表了一个公式，表达了对任意自然数r由光滑浸没的r元点的流形实现的同调类。他们关于r^ 3的一些论点是不正确的。F. Ronga和RJ Herbert给出了完整的证明。M. Audin将结果推广到r= 2的有向协阵和复协阵。这里我们想证明Lashof和Smale对r= 2的证明（这是正确的）可以推广到任何由乘谱产生的超然上同调理论，特别是ku -理论。利用Grothendieck-Riemann-Roch定理，我们将这个公式从ku理论转化为通常的上同调。通过这种方法，我们得到了一系列公式，首先是Lashof和Smale的原始公式。最后，我们应用这些公式证明了某些共乘类的流形不包含浸没的双点流形。
In 1959, Lashof and Smale published a formula expressing for any natural number r the homology class realized by the manifold of the r-tuple points of a smooth immersion. Some of their arguments concerning the case r^ 3 were incorrect. F. Ronga and RJ Herbert gave complete proofs. M. Audin generalized the result to the oriented and complex cobordisms for r= 2. Here we want to show that the proof of Lashof and Smale for r= 2 (which is correct) can be generalized to any extraordinary cohomology theory arising from a multiplicative spectrum, in particular to the KU-theory. Using the Grothendieck-Riemann-Roch theorem, we translate this formula from the KU-theory into the usual cohomologies. In this way we obtain a series of formulas, the first being the original formula of Lashof and Smale. Finally, we apply these formulas to show that certain cobordism classes of manifolds do not contain double point manifolds of immersions.