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Fix a group $G$. For an abelian coefficient group $A$ let $H^*_G(-;A):=H^*(EG\times_G-;A)$ be the Borel cohomology with $A$ coefficents. This is a functor from $G$-spaces to graded abelian groups which is $G$-homotopy invariant and has long exact Mayer-Vietoris sequences. If $A$ is a commutative ring, then $H^*_G(-;A)$ is a graded commutative ring.

Definition: A Borel equivariant cohomology operation of type $(A,B,m,n)$ is a natural transformation $$ \Theta:H^m_G(-;A)\to H^n_G(-;B) $$ of functors from $G$-spaces to abelian groups.

Definition: A stable Borel equivariant cohomology operation of type $(A,B,k)$ is a sequence of natural transformations $$ \Theta_m:H^m_G(-;A)\to H^{m+k}_G(-;B)\quad (m\ge0) $$ from $G$ spaces to abelian groups which commute with boundary maps in Mayer-Vietoris sequences.

Has there been a study of such operations in Borel equivariant cohomology?

I'm aware that there is a body of work on operations in $RO(G)$-graded Bredon cohomology, but I'm hoping computations might be simpler in Borel cohomology. I'm also sort of hopeful that there may be more Borel equivariant cohomology operations than just those coming from ordinary (non-equivariant) cohomology operations, because here we only ask for naturality with respect to maps of Borel constructions induced by $G$-maps.

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    $\begingroup$ Try Greenlees $\S2$. $\endgroup$
    – Tyrone
    Commented May 10, 2021 at 16:38
  • $\begingroup$ @Tyrone: This looks like an answer to me. $\endgroup$
    – Mark Grant
    Commented May 10, 2021 at 18:55
  • $\begingroup$ @Mark Grant do you want the index $G$ on the right side of your definition of Borel cohomology? $\endgroup$
    – user51223
    Commented May 11, 2021 at 11:10
  • $\begingroup$ @user51223: oops no, I'll correct that. $\endgroup$
    – Mark Grant
    Commented May 11, 2021 at 12:29

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