

A222010


Dimensions of spheres that admit continuous multiplications with unit element.


1




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 Hspaces, 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 nonexistence of elements of Hopf invariant one, Ann. Math., 72 (1960), 20104.
Peter F. McLoughlin, When does a cross product on R^{n} exist?, arXiv:1212.3515 [math.HO], 20122013.
Wikipedia, Hspace
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



