Assume a finite group acts smoothly on a manifold $M$ of dimension $n$. Suppose $a\in H_i(M)$, where $i=1,\ldots,[n/2]$.
Is there a way to kill $a$ with equivariant surgery and keep the same fixed point sets?
In the sense that after surgery we obtain a smooth action of $G$ on a new manifold $M'$ with $H_i(M')$ isomorphic to $H_i(M)$ with $a$ killed and $M^G\cong M'^G$.
If there is not an exact solution, then at least some results in that direction will be very nice to me.