The 7-sphere is the highest dimensional sphere that is parallelisable, and the next highest is the 3-sphere. An inherent property of each of these spheres is that they're embedded in a space of dimensionality at least one greater than their own: 4 and 8.

We know that quaternions and octonions are important algebras in mathematics in general and physics which have properties not found in other dimensions.

Is this property of parallelisability derived from the spheres themselves, or from the 4- and 8- dimensional spaces in which they're embedded?

If it's derived from the spaces, are there other special properties these spaces share as a result and are there other examples of mathematical objects: shapes, knots, networks, algebras, combinatorics, vectors, polynomials etc. which are parallelisable or have some other analogous property only in dimensions 4 and 8?

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    $\begingroup$ There are other subsets of theor $4$- and $8$-dimensioal spaces which are not parallelizable... $\endgroup$ – Mariano Suárez-Álvarez Aug 7 '16 at 18:47
  • $\begingroup$ @MarianoSuárez-Alvarez thanks, but what property of the space makes it possible to construct a parallelisable, spherical subset in 4 and 8 dimensions? A sphere is essentially just the set of points equidistant from one point. Is it uniquely this rule in combination with the dimensions 4 and 8 which leads to parallelisability only in these dimensions or do dimensions 4 and 8 grant this property (or similar) to other objects too? $\endgroup$ – user334732 Aug 7 '16 at 18:58
  • $\begingroup$ The parallelizable spheres $S^0,S^1,S^3,S^7$ are the unit elements of the normed division algebras $\Bbb R,\Bbb C, \Bbb H, \Bbb O$. There are many exceptional structures related to these algebras, such as Hopf fibrations, projective spaces, simple formally real Jordan algebras (historically intended to be an algebra of quantum observables), and exceptional simple lie algebras. $\endgroup$ – arctic tern Aug 7 '16 at 19:05
  • $\begingroup$ @arctictern thanks, I'm aware of that. Those are algebras of 1,2,4 and 8 dimensions (although 1 and 2 are regarded as trivially parallelisable). I'm seeking more insight into what makes these special. $\endgroup$ – user334732 Aug 8 '16 at 6:25
  • $\begingroup$ The dimensions are special because they are the dimensions of normed division algebras. The division algebras have those dimensions because of the Cayley-Dixson multiplication laws. $\endgroup$ – arctic tern Aug 8 '16 at 6:32

An $n$-dimensional manifold is parallelisable if and only if there exists $n$ vectors fields $X_1,...,X_n$ defined on $M$ such that for every $x\in M$, $X_1(x),...,X_n(x)$ is a base of $T_xM$, so to be parallelisable by definition does not depend of the manifold where $M$ can be embedded. Remark that if $M$ is compact, it can be embedded in $R^{2n+1}$ by using Withney theorem.

Concerning the sphere $S^3$ and $S^7$, the structure of $R^4$ and $R^8$ can be used to show that they are parallelisable. You can define $S^3$ as the set of quaternions of module $1$, the multiplication of the set of quaternion induces on $S^3$ a structure of a Lie group, this shows that $S^3$ is parallelisable.

  • $\begingroup$ I'm taking this answer to say that the property of parallelisability does belong primarily to the 3- and 7-spheres, but since each sphere is the unit element of the normed division algebra in 4- and 8-dimensions, this property is also inherently linked to the 4- and 8-dimensional spaces too. Would that be fair? $\endgroup$ – user334732 Aug 8 '16 at 6:56

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