Let us assume that $X=\mathbb{R}\times S^1$ is given with a $G=\mathbb{Z}_2$ action that corresponds to the symmetry $(x,e^{i\theta})\mapsto(-x,e^{-i\theta})$. I want to compute the equivariant cohomology of $X$ relative to the fixed point set $X^G=\{(0,-1),(0,1)\}$. Since the subspace $X^G$ is $G$-invariant, there is a natural generalization of Borel's construction to the relative equivariant cohomology $$ H^*_G(X, X^G) = H^*(X\times_G EG,X^G \times_G EG). $$ And we know that this has an associated long exact sequence $$ \cdots \rightarrow H^n_G(X, X^G) \rightarrow H^n_G(X) \rightarrow H^n_G(X^G) \rightarrow H^{n+1}_G(X, X^G)\rightarrow \cdots. $$ How do we really calculate $H^n_G(X)$? When we take a field like $F_2$ with two elements, one can at least take advantage of Smith theory to conclude that since the cochain $C^*(X, X^G)$ is $G$-free, the relative equivariant cohomology can be identified with the cohomology of the subcomplex of invariants, which vanishes above the dimension of $X$ [1]. Can anybody help me with the lower relative cohomology groups?

[1] R.B. Sher, R.J. Daverman, Handbook of Geometric Topology, North Holland, 1st ed., p. 13

Best,

AB