Some Computations in Equivariant cobordism in relation to Milnor manifolds

Abstract

Let ${\mathcal{N}}_{\ast }$ be the unoriented cobordism algebra, let $G=({\mathbb{Z}}_2{)}^n$ and let $Z_{\ast }(G)$ denote the equivariant cobordism algebra of $G$-manifolds with finite stationary point sets. Let ${\epsilon }_{\ast }:Z_{\ast }(G)\to {\mathcal{N}}_{\ast }$ be the homomorphism which forgets the $G$-action. We use Milnor manifolds (degree 1 hypersurfaces in $\mathbb{R}{P}^m\times \mathbb{R}{P}^n$) to construct non-trivial elements in $Z_{\ast }(G)$. We prove that these elements give rise to indecomposable elements in $Z_{\ast }(G)$ in degrees up to $2^n-5$. Moreover, in most cases these elements can be arranged to be in $Ker({\epsilon }_{\ast })$.