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A222010 Dimensions of spheres that admit continuous multiplications with unit element. 1


%S 0,1,3,7

%N Dimensions of spheres that admit continuous multiplications with unit element.

%C 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.

%C 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]

%H J. F. Adams, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/adams1.pdf">On the non-existence of elements of Hopf invariant one</a>, Ann. Math., 72 (1960), 20-104.

%H Peter F. McLoughlin, <a href="http://arxiv.org/abs/1212.3515">When does a cross product on R^{n} exist?</a>, arXiv:1212.3515 [math.HO], 2012-2013.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/H-space">H-space</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hopf_invariant_one">Hopf invariant one</a>

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

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

%e 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.

%Y Cf. A222011.

%K nonn,fini,full,nice

%O 0,3

%A _Jonathan Sondow_, Feb 06 2013

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