TITLE: Homotopy equivalent smooth 4-manifolds which are not stably diffeomorphic?
QUESTION [10 upvotes]: 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?

REPLY [3 votes]: For orientable 4-manifolds, I believe you gave a complete list of the known examples. For non-orientable, the phenomenon does generalise. Kreck showed that for every 1-type $(\pi,w)$ with $\pi$ a finitely presented group and $w \colon \pi \to C_2$ a surjective homomorphism, there is a 4-manifold $M$ with $\pi_1(M) \cong \pi$ and orientation character $w$, such that $M\# K3$ and $M \#^{11} S^2 \times S^2$ are homeomorphic (so in particular homotopy equivalent) but not stably diffeomorphic.
@incollection {Kreck-84,
AUTHOR = {Kreck, M.},
TITLE = {Some closed {$4$}-manifolds with exotic differentiable
structure},
BOOKTITLE = {Algebraic topology, {A}arhus 1982 ({A}arhus, 1982)},
SERIES = {Lecture Notes in Math.},
VOLUME = {1051},
PAGES = {246--262},
PUBLISHER = {Springer, Berlin},
YEAR = {1984}
}