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](http://people.mpim-bonn.mpg.de/teichner/Math/ewExternalFiles/phd.pdf) 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](http://people.mpim-bonn.mpg.de/teichner/Math/ewExternalFiles/star.pdf). 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?



Homotopy equivalent smooth 4-manifolds which are not stably diffeomorphic?
$\newcommand{\Z}{\mathbb Z}\newcommand{\RP}{\mathbb{RP}}$ $\RP^4$ and Capell-Shaneson's fake $\RP^4$, which I'll
denote $Q$, are an example with fundamental group $\Z/2$. I don't know if this generalizes, but I like this example
for TFT reasons: [David Reutter proved that](https://arxiv.org/abs/2001.02288) semisimple 4d TFTs cannot
distinguish oriented, stably diffeomorphic $4$-manifolds, but there is a semisimple TFT which distinguishes $\RP^4$
from $Q$.


Kreck's modified surgery theory determines whether two closed $4$-manifolds $X$ and $Y$ are $(S^2\times
S^2)$-stably diffeomorphic using bordism. Specifically, $X$ and $Y$ must have the same stable normal $1$-type
$\xi\colon B\to BO$. (See [Kreck](https://arxiv.org/abs/math/9905211) for the definition of a stable normal
$1$-type.) Then, one computes the set $S(\xi) := \Omega\_4^\xi/\mathrm{Aut}(\xi)$, where $\mathrm{Aut}(\xi)$ denotes
the fiber homotopy equivalences of $\xi\colon B\to BO$. $X$ and $Y$ determine classes in $S(\xi)$; they are
stably diffeomorphic iff these classes are equal.


In the case of $\RP^4$ and $Q$, the stable normal type is $\xi\colon B\mathit{SO}\times B\Z/2\to BO$, where the map
is classified by the rank-zero virtual vector bundle $V\_{\mathit{SO}}\oplus (\sigma - 1)$; here $V\_{\mathit{SO}}\to
B\mathit{SO}$ and $\sigma\to B\Z/2$ are the tautological bundles. A lift of the classifying map across $\xi$ is
equivalent to a pin$^+$ structure on the tangent bundle, so we look at $\Omega\_4^{\mathit{Pin}^+}\cong\Z/16$. The
$\mathrm{Aut}(\xi)$-action on $\Z/16$ sends $x\mapsto \pm x$.


[Kirby-Taylor](https://www.maths.ed.ac.uk/%7Ev1ranick/papers/kirbytaylorpin.pdf) choose an isomorphism
$\Omega\_4^{\mathit{Pin}^+}\to\Z/16$ and show that under this isomorphism, the two pin$^+$ structures on $\RP^4$ are
sent to $\pm 1$, and the two pin$^+$ structures on $Q$ are sent to $\pm 9$. Thus when we send $x\mapsto -x$, these
two remain distinct.




---


TFT digression: to construct a 4d unoriented TFT that distinguishes $\RP^4$ from $Q$, begin with the pin$^+$
invertible TFT whose partition function is the $\eta$-invariant defining the isomorphism
$\Omega\_4^{\mathit{Pin}^+}\to\mu\_{16}$ (here $\mu\_{16}$ denotes the 16th roots of unity in $\mathbb C$). Then
perform the finite path integral over pin$^+$ structures. Both of these operations are mathematically understood
for once-extended TFT, so the result is a once-extended (hence semisimple) unoriented TFT which distinguishes
$\RP^4$ from $Q$. I wrote about this in little more detail [in another MO
answer](https://mathoverflow.net/a/362517/97265).


