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Summary

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2113)

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.

Keywords

  • Group Theory
  • Point Group
  • Explicit Calculation
  • Homology Group
  • Cell Complex

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. F.T. Farrell, L. Jones, The lower algebraic K-theory of virtually infinite cyclic groups. K Theory 9, 13–30 (1995)

    Google Scholar 

  2. C. Weibel, NK 0 and NK 1 of the groups C 4 and D 4. Comment. Math. Helv. 84, 339–349 (2009)

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© 2014 Springer International Publishing Switzerland

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Farley, D.S., Ortiz, I.J. (2014). Summary. In: Algebraic K-theory of Crystallographic Groups. Lecture Notes in Mathematics, vol 2113. Springer, Cham. https://doi.org/10.1007/978-3-319-08153-3_10

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