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 $d=5$?)

- Are there other similar statements?