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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)

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    $\begingroup$ Your claim about $MSO_n$ is only true for $n \not\equiv 0 \mod 4$. $\endgroup$
    – Oscar Randal-Williams
    Commented Dec 21, 2016 at 15:15
  • $\begingroup$ @Oscar Randal-Williams thank you for the correction! $\endgroup$
    – Bo Liu
    Commented Dec 21, 2016 at 15:19

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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.

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    $\begingroup$ Thank you very much! Does it finite when n odd and the fixed point set is also odd dimensional? $\endgroup$
    – Bo Liu
    Commented Dec 21, 2016 at 15:45

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