Geodesic
Substitutional lemma for G-spaces of 1-dimensional groups
Glasgow mathematical journal, Tome 38 (1996) no. 2, pp. 215-220

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Let G be a compact Lie group and X a G-CW complex. We are interested in the calculation of the Borel cohomology of Xwhere EG is a universal free G-space and we use on the right hand side cellular cohomology. For an introduction to G-CW complexes see Matumoto [4] and for a good exposition on Borel cohomology see for instance torn Dieck [2], We want to replace X with an ordinary CW complex Y in order to find an ordinary CW structure on the Borel construction EG ΧGY so we can use cellular chains to compute the Borel cohomology of X. For every compact Lie group one has an extensionwhere G0 is the identity component, so for our case G0 is isomorphic to the circle group . We are dealing with the case in which π0(G) is isomorphic to C2, the cyclic group of order 
Pérez, Juan Antonio. Substitutional lemma for G-spaces of 1-dimensional groups. Glasgow mathematical journal, Tome 38 (1996) no. 2, pp. 215-220. doi: 10.1017/S0017089500031463
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[1] 1.Brown, K. S., Cohomology of groups (Springer Verlag, 1982). Google Scholar | DOI

[2] 2.Dieck, T. torn, Transformation groups. (Walter de Gruyter, 1987). Google Scholar | DOI

[3] 3.Greenlees, J. P. C. and May, J. P., Generalized Tate cohomology, Mem. Amer. Math. Soc. (to appear). Google Scholar

[4] 4.Matumoto, T., On G-CW complexes and a theorem of J. H. C. Whitehead, J. Fac. Sci. Univ. Tokio 18 (1971) 363–374. Google Scholar

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