Homotopy equivalent smooth 4-manifolds which are not stably diffeomorphic?

Question

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?

Answer 1

$\newcommand{\Z}{\mathbb Z}\newcommand{\RP}{\mathbb{RP}}$ $\RP^4$ 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 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 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 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.

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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.

Answer 2

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} }