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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 at 22:15
  • $\begingroup$ @MichaelAlbanese isn't your replacement for "..." overly emphatic? I hesitated to change into simply "(...)"; I agree I was puzzled by "..." but I'm skeptical about forged math mode. $\endgroup$ – YCor Mar 17 at 22:32
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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 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 at 22:45
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    $\begingroup$ Use the Euler characteristic: it's even. $\endgroup$ – Tyrone Mar 19 at 9:33
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In a paper in preparation, Aleksandar Milivojevic and I show that every orientable five-manifold admits a spin${}^h$ structure, and we provide examples of (higher-dimensional) orientable manifolds which do not admit a spin${}^h$ structure.

The proof that every orientable five-manifold admits a spin${}^h$ structure is relatively simple. First, a spin${}^h$ structure on an orientable $n$-manifold is equivalent to the existence of an immersion into a spin $(n + 3)$-manifold. Now, by Cohen's immersion theorem, every five-manifold immerses into $\mathbb{R}^8$, which is spin.

The construction of orientable manifolds which do not admit a spin${}^h$ structure is harder. We construct examples in dimension $8$ and above. We do not know whether there are such manifolds in dimensions $6$ or $7$.

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