In classical surgery theory, there is a map
$$L_{n+1}(\pi_1M)\to S(M^n)$$

Element in $L_{n+1}(\pi_1M)$ is realized as surgery obstruction of a surgery problem to $M\times I$ with one boundary piece the identity map and the other a homotopy equivalence (Wall's realization). The map is defined by sending the element in $L$ to the boundary piece which is a homotopy equivalence (a structure on $M$).It is NOT clear to me if the domain manifold of the structure is homeomorphic to $M$.

In   https://mathoverflow.net/questions/211160/homology-spheres-and-fundamental-group/211170#211170  

(when $n+1$ is even) Danny Ruberman commented that "The effect of the action is to change the multisignature, and hence it changes the homeomorphism type of M"

What's the correct reference if i want to understand some details about the effect of Wall's realization on multisignature?