We collect three observations on the homology for Smale spaces defined by Putnam. The definition of such homology groups involves four complexes. It is shown here that a simple convergence theorem for spectral sequences can be used to prove that all complexes yield the same homology. Furthermore, we introduce a simplicial framework by which the various complexes can be understood as suitable "symmetric" Moore complexes associated to the simplicial structure. The last section discusses projective resolutions in the context of dynamical systems. It is shown that the projective cover of a Smale space is realized by the system of shift spaces and factor maps onto it.

中文翻译：

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关于Smale空间的同源性的注记

我们收集关于Putnam定义的Smale空间的同源性的三个观察结果。这种同源基团的定义涉及四个复合物。在此表明，可以使用一个简单的频谱序列收敛定理来证明所有复合物都具有相同的同源性。此外，我们介绍了一个简单框架，通过该框架，可以将各种复合物理解为与该简单结构相关的合适的“对称”摩尔复合物。最后一部分讨论了动力学系统中的投影分辨率。结果表明，Smale空间的投影覆盖是通过移位空间和因子映射系统实现的。