重心拉直与有界上同调
Barycentric straightening and bounded cohomology
摘要：
研究了非紧型不可约对称空间中单形的重心直化问题。本文证明了对于秩r>1的n维对称空间，只要p至少为n-r+2，p-Jacobian就有一致有界范数。因此，对于一个非紧的，连通的，半单的实李群G，在n-r+2次及以上，每个上同调类都有一个有界的代表。这在小余维中回答了Dupont问题。我们还给出了对称空间的例子，其中的重心拉直的n-r维的单形有无界的体积，显示的范围内，我们得到的有界的p-Jacobian是非常接近最佳的。
英文摘要：
We study the barycentric straightening of simplices in irreducible symmetric spaces of non-compact type. We show that, for an n-dimensional symmetric space of rank r>1, the p-Jacobian has uniformly bounded norm, as soon as p is at least n-r+2. As a consequence, for a non-compact, connected, semisimple real Lie group G, in degrees n-r+2 and higher, every cohomology class has a bounded representative. This answers Dupont's problem in small codimension. We also give examples of symmetric spaces where the barycentrically straightened simplices of dimension n-r have unbounded volume, showing that the range in which we obtain boundedness of the p-Jacobian is very close to optimal.
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