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The Borel localization theorem in (Borel) equivariant cohomology states that if $T$ is a torus and $M$ a smooth $T$-manifold, with fixed point set $M^T$, then upon localizing the coefficient ring $H^*_T := H^*(BT)$ (that is, tensoring with its field of fractions), the restriction map $$H^*_T(M) \to H^*_T(M^T)$$ becomes an isomorphism.

The earliest reference to this result I know of is in Hsiang Wu-Yi's classic book on transformation groups, where he refers to the result as a "localization theorem of Borel–Atiyah–Segal type." It seems from Hsiang's presentation that Borel only proved this result for $T = S^1$; in any event, I've never been able to find the more general result in the Seminar on Transformation Groups. But I'd like to attribute it to someone.

Who originated this result? Failing that, what is the earliest citation you know?

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Here is the reference trail, according to this source:

Borel made the key observation [1] that the cohomology of the fixed point set was closely related to a torsion-free quotient. In the 1960’s, this was formalized as the “localization theorem” of Borel-Atiyah-Segal-Quillen [2,3].

  1. A. Borel, Seminar on transformation groups, Annals of Math. Studies 46, Princeton (1960) [Sect. XII Theorem 3.4]

  2. M.F. Atiyah and G. Segal, Equivariant cohomology and localization, lecture notes, 1965, Warwick.

  3. D. Quillen, The Spectrum of an Equivariant Cohomology Ring I, Annals of Mathematics 94, 549–572 (1971). [Theorem 4.4].

A more extensive overview of the literature leading up to, and following after the localization theorem can be found here (section 1.7). In addition to the unpublished 1965 lecture notes of Atiyah and Segal, there is a 1968 publication [4], crediting the theorem to Segal [5] (who writes that "The theory was invented by Professor Atiyah, and most of the results are due to him.").

  1. M.F. Atiyah and G.B Segal: Index of elliptic operators II, Ann. Math. 87, 531–545 (1968).

  2. G.B. Segal, Equivariant K-theory, Publ. Math. Inst. Hautes Etudes (Paris) 34, 129-151 (1968).

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  • $\begingroup$ Thanks! This may be the wrong forum to ask, but do you have an idea how I could obtain a copy of the cited lecture notes? $\endgroup$
    – jdc
    Commented Feb 3, 2015 at 2:50
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    $\begingroup$ no, but I added links to more easily accessible publications by Atiyah and Segal. $\endgroup$
    – Carlo Beenakker
    Commented Feb 3, 2015 at 7:41
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    $\begingroup$ FWIW, I emailed Atiyah about this a few weeks after reading your answer, and he believes that the result, in this form, is probably due to himself. $\endgroup$
    – jdc
    Commented Jul 9, 2015 at 4:00

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