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I'm looking for a reference or an idea about how to calculate the homology groups of dihedral groups $D_{2n}$ with integer coefficients or any abelian group when the degree $n$ is even. Here https://groupprops.subwiki.org/wiki/Group_cohomology_of_dihedral_groups is the result that I need but there is no reference nor idea of how to calculate the homology groups.
In the case when $n$ is odd I have made the computations via spectral sequences and the fact that $n$ is odd helps to have $E^2=E^\infty$ what doesn't happen when $n$ is even.

Martin
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  • This paper provides a computation for $D_8$ in characteristic $2$ : https://bchetard.files.wordpress.com/2017/10/group_cohomology.pdf you can presumably generalize the argument to get $D_{2n}$, but with $\mathbb F_2$ coefficients, not $\mathbb Z$ – Maxime Ramzi Aug 06 '19 at 11:32

1 Answers1

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There is a nice explicit free resolution of $D_{2n}$ constructed by Suguru Hamada and C. T. C. Wall in the early 1960's. See the following paper for a simplified description of the Hamada-Wall resolution:

Handel, David. "On products in the cohomology of the dihedral groups." Tohoku Mathematical Journal, Second Series 45, no. 1 (1993): 13-42.

Handel uses this free resolution to compute the integral cohomology of $D_{2m}$ (Theorems 5.2 and 5.3 of his paper). See this answer for a summary of Handel's result. Once we know the cohomology we can compute the homology using the universal coefficient theorem.

For the record, when $m\geq 3$ is odd and $n\geq 1$, the integral homology of $D_{2m}$ is $$ H_n(D_{2m};\mathbb{Z}) \;=\; \begin{cases} 0 & \text{if }n\text{ is even,} \\ \mathbb{Z}/2 & \text{if }n\equiv 1\pmod 4, \\ \mathbb{Z}/2m & \text{if }n\equiv 3\pmod 4. \end{cases} $$ When $m\geq 2$ is even and $n\geq 1$, the integral homology of $D_{2m}$ is $$ H_n(D_{2m};\mathbb{Z}) \;=\; \begin{cases} (\mathbb{Z}/2)^{n/2} & \text{if }n\text{ is even}, \\ (\mathbb{Z}/2)^{(n+3)/2} & \text{if }n\equiv 1\pmod 4, \\ (\mathbb{Z}/2)^{(n+1)/2} \oplus (\mathbb{Z}/m) & \text{if }n\equiv 3\pmod 4. \end{cases} $$ (Of course $H_0(D_{2m};\mathbb{Z})\cong\mathbb{Z}$ in both cases.)

Note that we can now compute the homology with respect to any abelian group by using the universal coefficient theorem again.

Jim Belk
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