# Every _______ $d$-manifold has an $S$-structure

I am looking for some analogous nontrivial but known statements and references about statements of the form:

Every _______ $$d$$-manifold has an $$S$$-structure.

Here _______ is a placeholder for certain conditions, like orientability, complex, triangulable, PL, topological, etc.

This is very vague. What do I really mean?

For example:

The precise statement on J. W. Morgan's "The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds (MN-44)" that a $$4$$-manifold $$X$$ admits a $$\mathit{Spin}^c$$ structure (Lemma 3.1.2) seems to be that every orientable $$4$$-manifold $$X$$ admits a $$\mathit{Spin}^c$$ structure. This means that we impose the $$w_1(X)=0$$ for the orientable $$4$$-manifold $$X$$. Note that

$$Spin^c=\frac{(Spin \times U(1))}{\mathbb{Z}/2\mathbb{Z}}.$$

Note that $$\mathit{Pin^c}=\frac{(\mathit{Pin}^+ \times U(1))}{\mathbb{Z}/2\mathbb{Z}}=\frac{(Pin^- \times U(1))}{\mathbb{Z}/2\mathbb{Z}},$$ $$\mathit{Pin}^{\tilde c+}=\frac{(\mathit{Pin}^+ \ltimes U(1))}{\mathbb{Z}/2\mathbb{Z}},$$ $$\mathit{Pin}^{\tilde c-}=\frac{(\mathit{Pin}^- \ltimes U(1))}{\mathbb{Z}/2\mathbb{Z}}.$$

$$\mathit{Spin}^H (n)=\mathit{Spin}(n) \times \mathrm{SU}(2)/{\mathbb{Z}/2\mathbb{Z}}=\mathit{Spin}(n) \times \mathrm{SU}(2)/\{ 1,-1\}$$

Do we have similar statements analogous to every _______ $$d$$-manifold has a $$\mathit{Spin}^H$$ structure? Such as every orientable $$d$$-manifold has a $$\mathit{Spin}^H$$ structure for some $$d$$? (like $$d=5$$?)

1. Are there other similar statements?
• All the examples you gave are examples of $G$-structures, so maybe using '$G$-structure' instead of '$S$-structure' would be a more informative title. Unless you're interested in a broader question. – Michael Albanese Mar 17 '20 at 22:15
• @YCor: I just replaced the Latex command with ______. Do you feel it is still an issue? I was just trying to clarify the existence of a placeholder. – Michael Albanese Mar 17 '20 at 22:36
• Actually in the title I'd even suggest to even replace it with words, but maybe it's better if the OP does it. The long underscore catches the eye a bit too much in the front list (to my taste). – YCor Mar 17 '20 at 22:54
• @Sebastian: I don't know how Gauss-Bonnet can be used to prove that. More generally, any orientable three-manifold admits a spin structure. Similarly, any closed three-manifold admits a pin$^-$ structure. – Michael Albanese Mar 18 '20 at 22:45
• Use the Euler characteristic: it's even. – Tyrone Mar 19 '20 at 9:33

In this paper, Aleksandar Milivojevic and I prove that every orientable manifold of dimension $$\leq 7$$ is spin$$^h$$. We also construct, for every $$d \geq 8$$, infinitely many homotopy types of closed, simply connected $$d$$-manifolds which are not spin$$^h$$.
• Every manifold of dimension $$d \leq 2$$ is pin$$^-$$. In addition, every compact $$3$$-manifold is pin$$^-$$. I don't know if there is a non-compact 3-manifold which is not pin$$^-$$.
• Every orientable manifold of dimension $$d \leq 3$$ is spin, and the orientable four-dimensional manifold $$\mathbb{CP}^2$$ is not spin. Note that for orientable manifolds, the spin, pin$$^+$$, and pin$$^-$$ conditions are equivalent, so one can make the same statements for orientable pin$$^+$$ manifolds or orientable pin$$^-$$ manifolds.
• Every orientable manifold of dimension $$d \leq 4$$ is spin$$^c$$, and the Wu manifold $$SU(3)/SO(3)$$ is an orientable five-manifold which is not spin$$^c$$. The proof you refer to only works for the closed case, for the general case, see this note by Teichner and Vogt.
• Every orientable 2-manifold is complex, while $$S^4$$ is an orientable 4-manifold which is not complex.
• Every orientable manifold of dimension $$d \leq 4$$ is stably almost complex, and the Wu manifold $$SU(3)/SO(3)$$ is an orientable five-manifold which is not stably almost complex.
The third and fifth statements are very similar. This is due to the fact that every stably almost complex manifold is spin$$^c$$.