Davis-Januszkiewicz空间上具有预定特征类的向量丛
Vector bundles over Davis-Januszkiewicz spaces with prescribed characteristic classes
摘要：
对于任意n-1维单纯复形，我们在Davis-Januszkiewicz空间上构造了一个特殊的n维复向量丛，其Chern类由Stanley Reisner代数生成元中的初等对称多项式给出.我们证明了这个复向量丛的同构类型以及它的实化完全由它的特征类决定。这使我们能够表明，着色性能的单纯复形反映了分裂性质的这个丛，反之亦然。对于具有特定特征的Pontrjagin类和Euler类的2n维实向量丛，也讨论了类似的问题.我们还分析了这些丛承认一个复结构。结果表明，这些丛与拟素流形和矩角复形的切丛密切相关。
英文摘要：
For any (n-1)-dimensional simplicial complex, we construct a particular n-dimensional complex vector bundle over the associated Davis-Januszkiewicz space whose Chern classes are given by the elementary symmetric polynomials in the generators of the Stanley Reisner algebra. We show that the isomorphism type of this complex vector bundle as well as of its realification are completely determined by its characteristic classes. This allows us to show that coloring properties of the simplicial complex are reflected by splitting properties of this bundle and vice versa. Similar question are also discussed for 2n-dimensional real vector bundles with particular prescribed characteristic Pontrjagin and Euler classes. We also analyze which of these bundles admit a complex structure. It turns out that all these bundles are closely related to the tangent bundles of quasitoric manifolds and moment angle complexes.
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