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

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?

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


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.

• 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? Dec 10, 2020 at 22:04
• @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. 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} }