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?**



Citation: earliest incidence of the Borel localization theorem
Here is the reference trail, according to this [source](http://www.jstor.org/stable/1971419):



> 
> 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,](http://www.indiana.edu/~jfdavis/seminar/Borel,Seminar_on_Transformation_Groups.pdf) 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](http://www.jstor.org/stable/1970770), 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](http://www.math.ias.edu/~goresky/pdf/equivariant.jour.pdf) (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 In addition to the unpublished 1965 lecture notes of Atiyah and Segal, there is a 1968 publication [4], crediting the theorem to Segal [5] (who writes that *"The theory was invented by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."*).


4. M.F. Atiyah and G.B Segal: [Index of elliptic operators II,](http://folk.uio.no/rognes/suprema/trace/index-II.pdf) Ann. Math. **87**, 531–545 (1968).
5. G.B. Segal, [Equivariant K-theory,](http://link.springer.com/article/10.1007%2FBF02684593?LI=true) Publ. Math. Inst. Hautes Etudes (Paris) **34**, 129-151 (1968).


