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We describe the equivariant cobordism classification of smooth actions $(M^m,\Phi )$ of the group $G=Z{₂}^k$ on closed smooth m-dimensional manifolds $M^m$ for which the fixed point set of the action is the union F = p ∪ Vⁿ, where p is a point and Vⁿ is a connected manifold of dimension n with n > 0. The description is given in terms of the set of equivariant cobordism classes of involutions fixing p ∪ Vⁿ. This generalizes a lot of previously obtained particular cases of the above question; additionally, the result yields...

Let Fⁿ be a connected, smooth and closed n-dimensional manifold. We call Fⁿ a manifold with property when it has the following property: if $N^m$ is any smooth closed m-dimensional manifold with m > n and $T:N^m\to N^m$ is a smooth involution whose fixed point set is Fⁿ, then m = 2n. Examples of manifolds with this property are: the real, complex and quaternionic even-dimensional projective spaces $R{P}^{2n}$, $C{P}^{2n}$ and $H{P}^{2n}$, and the connected sum of $R{P}^{2n}$ and any number of copies of Sⁿ × Sⁿ, where Sⁿ is the n-sphere and n is not...

Let Fⁿ be a connected, smooth and closed n-dimensional manifold satisfying the following property: if $N^m$ is any smooth and closed m-dimensional manifold with m > n and $T:N^m\to N^m$ is a smooth involution whose fixed point set is Fⁿ, then m = 2n. We describe the equivariant cobordism classification of smooth actions $(M^m;\Phi )$ of the group $G=Z{₂}^k$ on closed smooth m-dimensional manifolds $M^m$ for which the fixed point set of the action is a submanifold Fⁿ with the above property. This generalizes a result of F. L. Capobianco,...

Let E be an oriented, smooth and closed m-dimensional manifold with m ≥ 2 and V ⊂ E an oriented, connected, smooth and closed (m-2)-dimensional submanifold which is homologous to zero in E. Let $S^{n-2}\subset Sⁿ$ be the standard inclusion, where Sⁿ is the n-sphere and n ≥ 3. We prove the following extension result: if $h:V\to S^{n-2}$ is a smooth map, then h extends to a smooth map g: E → Sⁿ transverse to $S^{n-2}$ and with $g^{-1}\left(S^{n-2}\right)=V$. Using this result, we give a new and simpler proof of a theorem of Carlos Biasi related to the ambiental bordism...