Summary

Abstract

We can now compute the lower algebraic K-theory of the 73 split crystallographic groups. Recall that Theorem 5.1 tells us that, for all such groups Γ, we have an isomorphism

$$\displaystyle{K_{n}(\mathbb{Z}\varGamma )\mathop{\cong}H_{{\ast}}^{\varGamma }(E_{ \mathcal{F}\mathcal{I}\mathcal{N}}(\varGamma ); \mathbb{K}\mathbb{Z}^{-\infty }) \oplus \bigoplus _{\hat{\ell} \in \mathcal{T}^{{\prime\prime}}}H_{n}^{\varGamma _{\hat{\ell}}}(E_{ \mathcal{F}\mathcal{I}\mathcal{N}}(\varGamma _{\hat{\ell}}) \rightarrow {\ast};\; \mathbb{K}\mathbb{Z}^{-\infty }).}$$

For all 73 of our groups, we have:

• Explicitly computed in Chap. 7 the homology groups

  $$\displaystyle{H_{{\ast}}^{\varGamma }(E_{ \mathcal{F}\mathcal{I}\mathcal{N}}(\varGamma ); \mathbb{K}\mathbb{Z}^{-\infty }),}$$

  and summarized the results in Table 7.8.