It is known that for $n \not\equiv 0 \mod 4$, the oriented cobordism ring $MSO\_n$ is finite. That is, for oriented n-dimensional manifold $Y$, there exists $m\in \mathbb{N}$, such that $mY$ bounds. 


Does it hold for equivariant oriented cobordism with compact Lie group action?


Addition: @Oscar Randal-Williams shows that for almost all even n, equivariant oriented cobordism is not finite. 


An additional question is: **Is it finite for n odd?** (especially for circle action) 



Is equivariant oriented cobordism finite?
No. For a $G$-manifold $M$, taking the signature of the fixed points $M^G$ defines a homomorphism $\phi : \Omega\_n^G \to \mathbb{Z}$, as if $W : M\_0 \leadsto M\_1$ is a cobordism then so is $W^G : M\_0^G \leadsto M\_1^G$.


Now let $G=S^1$, $X=\mathbb{CP}^k$ with the $G$-action having $(k+1)$ fixed points, $Y=\mathbb{CP}^2$ with the trivial action, then we have 
$$\phi(X \times Y) = (k+1)\cdot\mathrm{sign}(Y) = k+1,$$
so $\Omega\_{2k+4}^{S^1}$ is not finite.


