（有界）连续上同调与Gromov比例原理
(Bounded) continuous cohomology and Gromov proportionality principle
摘要：
设X是拓扑空间，C（X）是X上的实系数奇异上链复形.我们用Cc（X）表示由连续上链给出的子复形，即由这样的上链，其对单形空间的限制（赋予紧开拓扑）定义了连续实函数。我们证明了至少对于“合理”空间，C_c（X）在C（X）中的包含诱导了上同调同构，从而回答了Mostow提出的一个问题。我们还证明了这样的同构是等距的关于L^infty范数的上同调定义的Gromov。作为应用，讨论了黎曼流形的单形体积的Gromov比例原理的上同调证明。
英文摘要：
Let X be a topological space, and let C(X) be the complex of singular cochains on X with real coefficients. We denote by Cc(X) the subcomplex given by continuous cochains, i.e. by such cochains whose restriction to the space of simplices (endowed with the compact-open topology) defines a continuous real function. We prove that at least for "reasonable" spaces the inclusion of Cc(X) in C(X) induces an isomorphism in cohomology, thus answering a question posed by Mostow. We also prove that such isomorphism is isometric with respect to the L^infty-norm on cohomology defined by Gromov. As an application, we discuss a cohomological proof of Gromov's proportionality principle for the simplicial volume of Riemannian manifolds.
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