The set of isomorphism classes of Z/nZ-equivariant line bundles over a 2 dimensional Z/nZ-CW complex

Question

Suppose I wish to find the set of isomorphism classes of $\mathbb{Z}/n\mathbb{Z}$-equivariant line bundles over a 2-dimensional, compact $\mathbb{Z}/n\mathbb{Z}$-CW-complex $X$, i.e. $\mathrm{Vect}^{1}_{\mathbb{Z}/n\mathbb{Z}}(X)$, I can write it using equivariant homotopy theory as

$\mathrm{Vect}^{1}_{\mathbb{Z}/n\mathbb{Z}}(X)\approx[X,B_{\mathbb{Z}/n\mathbb{Z}}GL_{1}(\mathbb{C})]_{\mathbb{Z}/n\mathbb{Z}}$.

Nonequivariantly, $B_{\mathbb{Z}/n\mathbb{Z}}GL_{1}(\mathbb{C})$ is $K(\mathbb{Z},2)$ so the above would seem like a Bredon cohomology group $H^2_{\mathbb{Z}/n\mathbb{Z}}(X;M)$ with $M$ some coefficient system and I could try to use the universal coefficient spectral sequence to attempt at computing it, however it seems like an awful lot of machinery for one of the simpler cases.

If the action is relatively nice, though it has fixed points, is there a simpler way of finding $\mathrm{Vect}^{1}_{\mathbb{Z}/n\mathbb{Z}}(X)$?

If not, which coefficient system $M$ is the right one?

Thanks for your time

Answer 1

You can replace $GL_1(\mathbb C)$ with its maximal compact subgroup, which is $S^1$. Since $S^1$ is an abelian compact Lie group, there is a natural $\mathbb Z/n$-equivariant equivalence $$B_{\mathbb Z/n}S^1\xrightarrow{\simeq} \mbox{Map}(E\mathbb Z/n, BS^1).$$

See, for example, this MO question for a discussion of this equivalence, with references and some generalizations.

It follows that for a compact $\mathbb Z/n$-complex $X$, $\mbox{Vect}^1_{\mathbb Z/n}(X)$ is $\pi_0$ of the space $$\mbox{Map}(X\times E\mathbb Z/n, K(\mathbb Z, 2))^{\mathbb Z/n}\simeq \mbox{Map}(X\times_{\mathbb Z/n} E\mathbb Z/n, K(\mathbb Z, 2))$$

So the answer is the second cohomology group of the homotopy orbit space $X\times_{\mathbb Z/n} E\mathbb Z/n$, rather than a Bredon cohomology group of $X$ with coefficients in a Mackey functor.

The Leray-Serre spectral sequence for computing the (co)homology of the homotopy orbit space is often tractable.