most recent 30 from http://mathoverflow.net 2013-06-20T05:32:48Z http://mathoverflow.net/feeds/question/3107 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/3107/construction-of-maps-s3-s2-with-arbitrary-hopf-invariant jc 2009-10-28T18:32:41Z 2009-10-28T20:03:57Z

<p>The well-known Hopf fibration S^1 -> S^3 -> S^2 has explicit constructions involving the geometry of C^2 and intersections of complex lines with the 3-sphere. They don't seem to generalize easily to "higher" Hopf maps from S^3 -> S^2 with Hopf invariant not equal to one. Are there any simple expressions for those maps?</p>

http://mathoverflow.net/questions/3107/construction-of-maps-s3-s2-with-arbitrary-hopf-invariant/3126#3126 Eric Wofsey 2009-10-28T19:49:02Z 2009-10-28T19:49:02Z

<p>You can get them by precomposing with a degree n map from S^3 to itself. In particular, this gives an interpretation in terms of the group structure: if h:S^3 \to S^2 is the Hopf map (which is just modding out by the subgroup S^1=U(1) of S^3=Sp(1)), then a map of Hopf invariant n is given by x \mapsto h(x^n), where x^n is using the group multiplication on S^3.</p>

http://mathoverflow.net/questions/3107/construction-of-maps-s3-s2-with-arbitrary-hopf-invariant/3128#3128 Ilya Nikokoshev 2009-10-28T19:57:50Z 2009-10-28T19:57:50Z

<p>Actually, yes, there is a construction involving complex projective line. </p> <p>Consider all points (x₁, x₂, x₃, x₄) on a 3-sphere in the 4-dimensional space. Our goal is to map them to <code>S</code>² which is the same as <code>CP</code>¹. </p> <p>To do this, take a quaternion </p> <blockquote> <p><code>x</code>₁ + <code>x</code>₂<code>i</code> + <code>x</code>₃<code>j</code> + <code>x</code>₄<code>k</code></p> </blockquote> <p>raise it to the <code>n</code>-th power (this is that group law on a 3-sphere) and decompose back into two complex numbers z₁ + z₂<code>j</code>. Now <code>z</code>_i<code>:z</code>_i is a point of a complex projective line, that's it!</p>