THE COHOMOLOGY OF QUOTIENTS OF CLASSICAL GROUPS
THE COHOMOLOGY OF QUOTIENTS OF CLASSICAL GROUPS
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经典群商的上同调
DOI：
10.1016/0040-9383(65)90001-7
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发表时间：
1965
期刊：
影响因子：
--
通讯作者：
W. Browder
中科院分区：
文献类型：
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作者：
P. Baum;W. Browder
经典紧李群so（n），SU（n），Sp（n）的同调和上同调环是众所周知的（例如参见Bore 1 [3]）。在[4]中，Bore 1利用中心子群研究了这类群的同调环，特别是计算了具有Zp系数p素的上同调环。在本文中，我们进一步研究这些商，扩展了博雷尔的结果。我们获得额外的信息，积分上同调，我们完全确定对角映射上同调2，系数p素，和行动的Steenrod代数在任何商的这些群体之一的中心子群。这些信息导致了一些应用，如同伦等价的紧致连通单群是同构的，以及证明关于真实的射影空间上的向量场的事实的技术（见9美元）。这些结果可用于证明真实的射影空间的非浸入定理。特别地，证明了当n= 2 ′ +3，r2 3时，P*-1和P ″不浸入R2 ″-7中。Mahowald [9]和Sanderson [12]已经证明P”确实浸入RZn-'中，因此这是P”和P”-'的最佳可能浸入。
THE HOMOLOGY and cohomology rings of the classical compact Lie groups so (n), SU (n), Sp (n) are well known (for example see Bore1 [3]). Most of these groups have non-trivial centers, and in [4], Bore1 investigated the quotients of these groups by central subgroups, in particular, calculating cohomology rings with Zp coefficients, p prime. In this paper we pursue the investigation of these quotients further, extending Borel’s results. We obtain extra information on the integral cohomology and we completely determine the diagonal maps in cohomology with 2, coefficients p prime, and the action of the Steenrod algebra in any quotient of one of these groups by a central subgroup. This information leads to some applications such as the fact that homotopy equivalent compact connected simple groups are isomorphic, and a technique to prove facts about vector fields on real projective spaces,(see $9). These results on vector fields may be applied to prove non-immersion theorems for real projective spaces. In particular, it is shown that if n= 2’+ 3, r 2 3, then P*-l and P” do not immerse in R2”-7. Mahowald [9] and Sanderson [12] have shown that P” does immerse in RZn-‘, so that this is the best possible immersion for P” and P”-‘.