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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:

  1. Every orientable $4$-manifold has a $\mathit{Spin}^c$ Structure is true.

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

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

  1. Every nonorientable $4$-manifold has a $\mathit{Pin^c}$, $\mathit{Pin^{\tilde c+}}$ or $\mathit{Pin^{\tilde c-}}$ structure is false.

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

  1. Spin-H structures

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

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?
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    $\begingroup$ 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. $\endgroup$ – Michael Albanese Mar 17 '20 at 22:15
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    $\begingroup$ @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. $\endgroup$ – Michael Albanese Mar 17 '20 at 22:36
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    $\begingroup$ 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). $\endgroup$ – YCor Mar 17 '20 at 22:54
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    $\begingroup$ @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. $\endgroup$ – Michael Albanese Mar 18 '20 at 22:45
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    $\begingroup$ Use the Euler characteristic: it's even. $\endgroup$ – Tyrone Mar 19 '20 at 9:33
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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$^h$.

There are several statements of the form you seek:

  • 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$^c$.

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