I'm looking for some references for the computation of the homology in integer coefficients of the dihedral group $D_n$. Most precisely, I am interested in explicit pairing relations with the well-known integer cohomology ring.

I've found something here, but I need some results in terms of pairing with the cohomology (which seems more understood) and not only the abstract isomorphism (as given there).

Some examples of the computations I'm looking for For example, I was told that $H_2(D_n;\mathbb{Z})\cong \mathbb{Z}\langle d \rangle$ when $n=2 \pmod{4}$, where $d$ is the dual of $x_1\cdot x_2$, the two generators of the cohomology ring in $\mathbb{Z}_2$-coefficients.

Via UCT is easy to see that for $n$ odd, $$H_5(D_{2n},\mathbb{Z})\cong \text{Ext}(H^6(D_{2n},\mathbb{Z})\cong \mathbb{Z}_2$$ and by a quick application of the Bockestein e.s. one sees that such $\mathbb{Z}_2$ is generated by an integral lift of $r_5$, where $r_5$ is the generator of $H_5(D_{2n};\mathbb{Z}_2)$ dual to the generator of $H^5(D_{2n};\mathbb{Z}_2)\cong \mathbb{Z}_2\langle r^5\rangle$ (recall that $H^*(D_{2n};\mathbb{Z}_2)\cong \mathbb{Z}_2[r]$

in general, when $n$ is even, the Bockstein e.s. doesn't provide much help, since there are $n$-torsion groups coming up.

Since the integer cohomology ring is known in terms of explicit generators (I'm especially interested in the case $n=2 \pmod{4}$) I was wondering if homology had a nice description in terms of pairings with the cohomology generators.


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