$Z_2^k$-actions with connected fixed point set

In this paper we describe the equivariant cobordism classification of smooth actions $(M^m,\phi)$ of the group $G=\mathbb{Z}_2^k$ on closed smooth $m$-dimensional manifolds $M^m$, for which the fixed point set of the action is a connected manifold of dimension n and $2^k n - 2^{k-1} \leq m < 2^k n$. Here, $\mathbb{Z}_2^k$ is considered as the group generated by $k$ commuting smooth involutions defined on $M^m$. This generalizes a previous result of 2008 of the second author, who obtained this type of classification for $k=2$ and $m=4n-1$ or $m=4n-2$.