$\textbf {Z}_{(2)}$-knot cobordism in codimension two, and involutions on homotopy spheres

Abstract:

Let ${Z_{(2)}}$ denote the ring of rational 2-adic integers. In this paper, we consider the group ${\Psi _k}$ of ${Z_{(2)}}$-cobordism classes of ${Z_{(2)}} - \operatorname {knot} (\Sigma ^{k + 2}, {K^k})$, where $\Sigma$ is a 1-connected ${Z_{(2)}}$-sphere ${Z_{(2)}}$-cobordant to ${S^{k + 2}}$, and K is a 1-connected ${Z_{(2)}}$-sphere embedded in $\Sigma$ with trivial normal bundle. For $n \geqslant 3$, we will prove that ${\Psi _{2n}} = 0$ and ${\Psi _{2n - 1}} = {C_\varepsilon }({Z_{(2)}})$, $\varepsilon = {( - 1)^n}$. Also, we will show that the group $\Theta _{4m - 1}^{4m + 1}$ of L-equivalence classes of differentiable involutions on $(4m + 1)$-homotopy spheres with codimension two fixed point sets defined by Bredon contains infinitely many copies of Z.