A222010 Dimensions of spheres that admit continuous multiplications with unit element.

0, 1, 3, 7

Offset

0,3

Comments

Adams's (1960) Hopf invariant one theorem states that S^0, S^1, S^3, S^7 are the only spheres that are H-spaces, i.e., that admit continuous multiplications with unit element.

This is related to the fact that nontrivial cross products only exist in vector spaces of 3 or 7 dimensions. [Jonathan Vos Post, Feb 09 2013]

Links

Table of n, a(n) for n=0..3.

J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. Math., 72 (1960), 20-104.

Peter F. McLoughlin, When does a cross product on R^{n} exist?, arXiv:1212.3515 [math.HO], 2012-2013.

Wikipedia, H-space

Wikipedia, Hopf invariant one

Formula

a(n) = 2^n - 1 for n = 0, 1, 2, 3.

a(n) = A222011(n) - 1.

Example

0, 1, 3, 7 are members because multiplications on S^0, S^1, S^3, S^7 are defined by regarding them as the unit spheres in the real, complex, quaternion, and Cayley numbers, respectively.

Crossrefs

Cf. A222011.

Sequence in context: A157596 A111316 A297530 * A152590 A261873 A293525

Adjacent sequences: A222007 A222008 A222009 * A222011 A222012 A222013

Keyword

nonn,fini,full,nice

Author

Jonathan Sondow, Feb 06 2013

Status

approved