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?

  • $\begingroup$ Is it possible that exotic 4-spheres could give examples? Or perhaps it is known they are all stably diffeomorphic? $\endgroup$ Dec 10, 2020 at 17:36
  • 3
    $\begingroup$ @ConnorMalin Wall proved many moons ago that homotopy equivalent simply connected closed 4-manifolds are stably diffeomorphic. $\endgroup$
    – mme
    Dec 10, 2020 at 18:25
  • 6
    $\begingroup$ If you find an exotic 4-sphere, let me know. $\endgroup$ Dec 10, 2020 at 18:25

2 Answers 2


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

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.

  • 1
    $\begingroup$ This is a good example. I was thinking about oriented manifolds and should have specified that in the OP. So this is one more example of what I was looking for. Is this plus what I listed the complete list of known examples? $\endgroup$ Dec 10, 2020 at 22:04
  • $\begingroup$ @ChrisSchommer-Pries I don't know whether this is a complete list of what's known, unfortunately. I personally suspect this example generalizes to unorientable 4-manifolds with some other fundamental groups, but unfortunately I am not familiar with the state of the literature. Sorry about that. $\endgroup$ Dec 10, 2020 at 22:27

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} }


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.