Let $M$ be a manifold with fundamental group $Z_2$. $S^{diff}(M)$ and $S^{diff}_t(M)$ respectively denotes smooth structure set of $M$ and Tangential smooth structure set of $M$. then, we have 2 maps fitting into respective surgery exact sequences. Let them be,
$S^{diff}_t(M) \xrightarrow{\eta^t} [M,SG]$; where $SG=$ all degree 1 stable map between spheres. $\eta^t$ is tangential normal invariant.
$S^{diff}(M) \xrightarrow{\eta} [M,G/O]$; where $\eta$ is normal invariant.
1) Here is what I am trying to do: Both our cases the map is onto. And, RHS as a group is known. I want to find one inverse for each element on the RHS. The reason is I want to see what manifolds will come up in the structure set upto addition with homotopy sphere.
2) Here is what I can think of: Normal invariant of pinch maps may generate the RHS. But, they are zero. So, that will not help. Maybe the self homotopy equivalence group of $M$ and their normal invariants? Other than this I have no idea.
3) Special structures on $M$: $M$ is a sphere bundle with section. $M = X \vee Y$ stably, where $X$ is a manifold and $Y$ is the thom space of a vector bundle over a manifold. $M$ is spin.
I have seen similar investigations only for $PL$ and Topological categories. For smooth and Tangential smooth category I could not find any reference. So, if any references is out there that may give me any idea about how to proceed please tell me. Thank You.
Edit: It is not necessay the references and ideas match my specific conditions.
