# 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 at 22:15
• @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. – YCor Mar 17 at 22:32
• 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 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 at 22:45
• Use the Euler characteristic: it's even. – Tyrone Mar 19 at 9:33

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 $$7$$.