Understanding terminology of, fibers, clutchings and Hopf.

Question

I have some questions regarding the terminology of fiber bundles as used in section 3 of this paper; http://www.sciencedirect.com/science/article/pii/S0723086907000151

The section starts off by recalling the Hopf Fibration $S^7\hookrightarrow S^{15}\rightarrow S^8$. It says that because the fibers of this bundle are (diffeo to) $S^7$, it intersects with an 8-dimensional vector space of $\mathbb{R}^{16}$. I'm not sure why the 7 dimensional $S^7$ it intersects an 8-dimensional space though, is it because the way it is parameterized with 8 variables/embedded in $\mathbb{R}^8$?
It goes on to say that these fibers form an "8-plane vector bundle $\zeta$ over $S^8$". In the texts I've been consulting I have not come across this term "8-plane" bundle, and google is no help. Does this just literally mean 8 planes? I think that would be sensible since we're considering the 8 dimensional spaces that intersect $S^7$ over $S^8$, and if I'm understanding $\zeta$ correctly it's fibers are those 8 dimensional sub-spaces of $\mathbb{R}^{16}$ that intersect $S^7$.

We then get that the clutching function for $\zeta$ is the map $a\rightarrow A(x)=ax$. Now I'm pretty new to clutching functions (A la' Cohen and Hatcher texts on fiber bundles), and I'm not sure how to realize this as the clutching function. My guess is that since we're dealing with a vector bundle we need the "linearity", and this map should be similar to the clutching function for the $S^7$-bundle which I believe is just rotation by octonions. Also since our map is just repeated multiplication, $x^n$ I think that has a role to play, but that's just a heuristic guess.

I've been pouring through texts and papers for days trying to get my head around all of this, so any clarity you can bring would be greatly appreciated!

Answer 1

Grab a copy of Milnor and Stasheff's "Characteristic Classes" and read the first 4 or 5 sections, and you'll know all sorts of stuff about vector bundles, explained by the masters.

The authors are placing $S^{15}$ inside $R^{16}$ as the standard unit sphere. So everything here is taking place inside $R^{16}$. I assume that the Hopf map here is something like $(t, s) \mapsto t s^{-1}$, where $t$ and $s$ are Cayley numbers --- that's how I recall it, anyhow. For a fixed value in the codomain-- say the identity, to make things easy, we get, for the fiber over that point, all pairs $(t, s)$ with $t s^{-1} = 1$, i.e., with $t = s$ and with $$\| (t, s) \|^2 = \| (t, t) \|^2 = \|t\|^2 + \|t\|^2 = 1.$$

The set of such $(t, t)$ pairs lies in an 8-plane in 16-space (just the way a great circle in 3-space lies in some plane in 3-space). I think that gets is past your first sentence.

The next claim --- that this set of planes forms a bundle of $S^8$ --- is a little trickier. I think it's pretty plausible that if instead of $st^{-1} = 1$, we'd looked at pairs $(s, t)$ with $ts^{-1}= u$, where $u$ is very close to $1$, we'd have gotten a nearby 8-plane in 16-space. In fact, if we'd taken a whole neighborhood $U$ of the target value $1$ (by which I mean the Cayley number $(1, 0, 0, 0, 0, 0, 0, 0)$) we'd have gotten, as a preimage something that looks like $U \times R^8$. That's a typical "coordinate chart" for an 8-dimensional "vector bundle" over $S^7$.

By the way, if you have a $k$-dimensional vector bundle over some manifold $M$ (i.e., roughly, a choice of a $k$-plane for each point of $M$), we sometimes call this a $k$-plane bundle over $M$. So while you were looking for "8-plane bundle", you should ahve been looking for "vector bundle" and then saying "with a fiber dimension of 8".

The clutching functions? Without reading the paper, I don't know what they're talking about. But I'm pretty sure you need to get the basics down a little better, and Milnor and Stasheff is a treasure for that.

Answer 2

The way I prefer to think of the Hopf bundle is as the sphere bundle of the tautological bundle of projective space. With this view, it is quite explicit that the fiber of the Hopf fibration is an $n$-sphere living in an $(n+1)$-plane.

And yes, the phrase $n$-plane just means an $n$-dimensional linear subspace.

For example in the complex case, we have the projective complex line $\mathbb{P}^1(\mathbb{C})$ which is a 2-sphere. Projective space is a quotient of the vector space $\mathbb{C}^2$, namely $\mathbb{P}^1(\mathbb{C})\cong \mathbb{C}^2/(v\sim\lambda v)$ and the tautological bundle $\mathcal{O}_{\mathbb{C}}(-1)$ is a subbundle of the trivial bundle $\mathcal{O}_{\mathbb{C}}(-1)\times\mathbb{C}^2$ with that fiber:

$$ \mathcal{O}_{\mathbb{C}}(-1)=\{(\ell,v)\in\mathbb{P}^1(\mathbb{C})\times\mathbb{C}^2|v\in\ell\}. $$

In words, $\mathcal{O}_{\mathbb{C}}(-1)$ is the set of ordered pairs of lines and vectors contained in those lines.

Then the sphere bundle of $\mathcal{O}_{\mathbb{C}}(-1)$ is the set of ordered pairs of unit vectors in each line. Each fiber of $\mathcal{O}_{\mathbb{C}}(-1)$ is a linear subspace of $\mathbb{C}^2$; a complex line. The unit vectors in a complex line form an $S^1$.

Similarly we may consider $\mathcal{O}_{\mathbb{K}}(-1)$ for $\mathbb{K}$ the reals, the quaternions, and octonions. The tautological bundle is a subbundle of $\mathcal{O}_{\mathbb{K}}(-1)\times\mathbb{K}^2,$ each fiber a linear subspace.

So for $\mathcal{O}_{\mathbb{R}}(-1)$, we have a bundle with whose fibers are $S^0$ living in 1-dimensional real vector spaces. For $\mathcal{O}_{\mathbb{H}}(-1)$, we have a bundle with whose fibers are $S^3$ living in 4-planes. For $\mathcal{O}_{\mathbb{O}}(-1)$, we have a bundle with whose fibers are $S^7$ living in 8-planes.