Wall manifolds with involution

Consider smooth manifolds $W$ with involution $t$ and a Wall structure described by a map $f:W \to {S^1}$ such that $ft = f$. For such objects we define cobordism theories ${\text{W}}_\ast ^I$ (in case $W$ is closed, $t$ unrestricted), ${\text{W}}_ \ast ^F$ (for $W$ closed, $t$ fixed-point free), and ${\text{W}}_ \ast ^{{\text{rel}}}$ ($W$ with boundary, $t$ free on $W$). We prove that there is an exact sequence

$$0 \to {\text{W}}_ \ast ^I \to {\text{W}}_ \ast ^{{\text{rel}}} \to {\text{W}}_ \ast ^F \to 0.$$

As a corollary, ${\text{W}}_ \ast ^I$ imbeds in the cobordism of unoriented manifolds with involution. We also describe how ${\text{W}}_ \ast ^I$ determines the $2$-torsion in the cobordism of oriented manifolds with involution.