Recall that two 4-manifolds $M$ and $N$ are *stably diffeomorphic* if there exist $m,n$ such that
$$M \#_n (S^2 \times S^2) \cong N \#_n (S^2 \times S^2).$$
That is, they become diffeomorphic after taking sufficiently many connected sums with $S^2 \times S^2$.

I am interested to find examples $M$ and $N$ which are homotopy equivalent $M \simeq N$, but where $M$ and $N$ *fail* to be stably diffeomorphic.

I know of two sources of examples of such manifolds. In Example 5.2.4 of

Topological 4-manifolds with finite fundamental group P. Teichner, PhD Thesis, University of Mainz, Germany, Shaker Verlag 1992, ISBN 3-86111-182-9.

Teichner constructs a pair of $M$ and $N$ where the fundamental group $\pi$ is any finite group with Sylow 2-subgroup a generalized Quaterion group $Q_{8n}$ with $n \geq 2$.

Another pair of $M$ and $N$ with fundamental group the infinite dihedral group was constructed in:

On the star-construction for topological 4-manifolds. P. Teichner, Proc. of the Georgia International Topology Conference 1993. Geom. top. AMS/IP Stud. Adv. Math. 2 300-312 A.M.S. (1997)

Are there any other known examples of this phenomenon? I have been unsuccessful in finding any others in the literature, but this is not my area of expertise. Are there any general results about when this can occur?

simply connectedclosed 4-manifolds are stably diffeomorphic. $\endgroup$ – Mike Miller Eismeier Dec 10 '20 at 18:25