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Edit. After thinking about this problem a bit longer, I am not so sure anymore that the Bredon cohomology proposed by Adem and Ruan gives me the invariants I am looking for. I have therefore moved that part to the very bottom and changed some of the questions a bit.

Is there a definition of equivariant or orbifold Chern classes that restrict to $H^\bullet(-;\mathbb Z)$-valued Chern classes if the group $G$ is trivial, or if the orbifold is a smooth manifold in the sense that all isotropy groups are trivial?

Q1 How does one define a suitable orbifold cohomology theory $H^\bullet_{\mathrm{orb}}$ that becomes integral cohomology if all isotropy groups are trivial? The answer probably depends on compatible choices of $H^\bullet_{\mathrm{orb}}(\mathbb R^n/\Gamma)$ for all linear actions of finite groups.

Q2 Are there (natural etc.) Chern classes $c_k(V)\in H^\bullet_{\mathrm{orb}}(B)$?

For simplicity, assume we are dealing with an effective orbifold $B$. Then there exists a Lie group $G$ and a smooth manifold such that the orbifold is isomorphic to $M/G$. Note that $M$ and $G$ are not unique. A canonical choice for $M$ would be the orthogonal frame orbibundle of $TB$ (if $TB$ is given a Riemannian metric) and $G=O(n)$.

If $B=M/G$ then an orbifold vector bundle $V\to B$ can be identified by pullback along $p\colon M\to B$ with a $G$-equivariant vector bundle $p^*V$ on $M$.

Q3 Are there equivariant Chern classes $c_k(p^*V)\in H^\bullet_G(M)$ (where one should probably consider one of the definitions proposed in this question)?

Q4 Is there a (natural etc.) pullback map $p^*\colon H^\bullet_{\mathrm{orb}}(B)\to H^\bullet_G(M)$ such that $p^*c_k(V)=c_k(p^*V)$?

Note. I am particularly interested in torsion phenomena, so a $\mathbb Q$-version would not really help. On the other hand, I probably only need the non-equivariant part of $c_k(p^*V)$ (but vector bundles on $|B|$ instead of orbibundles will not suffice for my purpose).


To give some background, I've been reading Adem-Ruan's article on twisted orbifold $K$-theory. In Section 3, a cohomology theory is introduced, whose integral version would be defined as follows. Assume that a compact Lie group $G$ acts with finite stabilisers on a $C^\infty$-manifold $M$, so $B=G\backslash M$ is an orbifold. Assume that a $G$-CW structure on $M$ has been fixed. To each orbit type $G/H$, associate a cellular chain group $$C_\bullet^G(M)(G/H)=C_\bullet(M^H/WH_0)\;,$$ where $WH_0$ is the maximal connected subgroup of $N_G(H)/H$. Now, let $R(-)$ denote the complex representation ring and consider $$C_G^\bullet(M;R)=\mathrm{Hom}_{Or(G)}(C_\bullet^G(M),R)\subset\prod_H\mathrm{Hom}(C_\bullet(M^H/WH_0),R(H))\;,$$ where the subscript $Or(G)$ means that for each inclusion $K\subset H$, the homomorphisms should intertwine the pullback induced by $M^H/WH_0\to M^K/WK_0$ with the restriction $R(H)\to R(K)$. Then consider $$H_{\mathrm{orb}}(B)=H_G^\bullet(M;R)=H^\bullet(C_G^\bullet(M;R))\;.$$ Note that Adem and Ruan only consider everything $\otimes\mathbb Q$, which they show is independent of the choice of $M$ and $G$.

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  • $\begingroup$ I am readding this paper webusers.imj-prg.fr/~michele.vergne/publications/… . $\endgroup$
    – user21574
    Apr 19, 2016 at 14:24
  • $\begingroup$ Say we declare Q4 to be an isomorphism, and thereby give an answer to Q1. Then Q2 and Q3 are the same question, and the answer is yes, there exist equivariant characteristic classes (they're the ordinary characteristic classes of Borel-construction(the bundle map)). $\endgroup$ Apr 26, 2016 at 23:03
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    $\begingroup$ Also, your "canonical choice" only works if your orbifold is generically a manifold, not e.g. $pt/\Gamma$. $\endgroup$ Apr 26, 2016 at 23:04
  • $\begingroup$ @AllenKnutson Thanks for your comments. Do you have a good reference for these constructions? Moreover, I once heard people claim that the Borel construction would not give the desired information in all cases, so I guess there might be other underlying equivariant cohomology theories? And regarding the second comment, I was referring to effective orbifolds at that point, that is, all isotropy groups act effectively (there might be other terms around). $\endgroup$ Apr 27, 2016 at 9:34
  • $\begingroup$ Are $\mathbb{Q}$-valued Chern classes of an orbibundle defined in Adem-Ruan or any other reference? $\endgroup$ Mar 24, 2021 at 13:51

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